Figure 2-6: Schematic drawing of the slotted beam system . ...... The concrete part, steel reinforcement parts, steel tube part, steel plate parts, and steel anchors ...
UNIVERSITY OF CALGARY
Development of Innovative Self-Centering Concrete Beam-Column Connections Reinforced
using Shape Memory Alloys
by
Fadi Oudah
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF CIVIL ENGINEERING
CALGARY, ALBERTA
DECEMBER, 2014
© Fadi Oudah 2014
Abstract The use of smart materials in the design of structures is emerging in the research community due to their unique ability to sense the environment and react upon thermal and/or mechanical stimulus. The main objective of this research study is to design innovative self-centering concrete beamcolumn connections reinforced using Shape Memory Alloy (SMA) bars with relocated plastic hinges. The proposed systems utilize the Psuedoelastic (PE) response of SMA in re-centering the connection after being subjected to earthquake-like motion. The behaviours of the systems are investigated experimentally, analytically, and using finite element simulation. In the first stage of this project, the design of steel-reinforced Single Slotted Beam (SSB) and Double Slotted Beam (DSB) connections with relocated plastic hinges is conducted and validated using experimental testing of large-scale connections. The SSB system includes a vertical slot made into the bottom fibre of the beam member while the DSB system includes vertical slots on both the top and the bottom fibres of the beam member. The location of the plastic hinge, is therefore, relocated by moving the vertical slots away from the face of the column. Test results indicated the efficiency of the newly developed systems in relocating the plastic hinge away from the face of the column, while a relocating distance equivalent to the shear depth of the beam member was found to provide the best seismic performance. The design of SSB and DSB connections reinforced using PE SMA bars at the locations of the plastic hinges was conducted in the second stage of this research. The transfer of forces in the concrete joint is different than in conventional design, and thus, experimental testing of joint-like specimens was conducted to examine the transfer of forces. The test results were also used to validate an analytical strut-and-tie model that was developed to verify the effect of the PE SMA anchorage on the behaviour of joints in beam-column connections. Furthermore, mechanical steel
ii
anchors and couplers were modified in the context of this research in order to provide adequate anchorage of the PE SMA bars into the concrete joint and beam members. Experimental testing of large-scale SSB and DSB connections reinforced using PE SMA indicated their superior performance as compared with conventional concrete connections in terms of low permanent deformation and high drift capabilities. Analytical models were developed to predict the load-displacement relationships of the connections in the third stage of this project. The sliding shear behaviour at the location of the vertical slots was also included in the models via a newly developed Two Distinct Element (TDE) shear deformation theory. A parametric study of the PE SMA-reinforced connections was also conducted. It was concluded that changing the concrete hinge depth and the relocation distance have significant effect on the stiffness, ultimate load, and ultimate displacement of the connections. Three-dimensional Finite Element Method (FEM) models of the tested connections were also developed using the ABAQUS software in the third stage. The models were validated with the experimental results in terms of the load-displacement envelopes, beam rotation, strain in bottom reinforcement, and cracking pattern. The models were found to predict the response of the connections reasonably. However, the accuracy is reduced as the vertical slot is moved away from the face of the column by 1.7 times the effective shear depth.
iii
Preface Structural engineering is a creative research area in which experimental and theoretical research play major roles in its development. The topic of robustness and resilience of structures is particularly highlighted in the context of earthquake engineering. The threats coming from future earthquakes represent real challenges to our modern seismic design approaches as well as our infrastructure. Therefore, we aim in the context of this research project to develop smart structural systems capable of re-centering themselves after being subjected to earthquake motion while maintaining an insignificant level of damage. This can be achieved by using a smart structural material called Shape Memory Alloy (SMA). The SMA material was originally developed by mechanical engineering division for aerospace applications. However, it was used lately in the seismic design of structures due to its excellent performance in terms of stable hysteretic damping and high fatigue resistance properties. The material can dissipate large amount of energy through inelastic deformation and return to its pre-deformed shape when heated and/or mechanically unloaded. The primary research outcome of the present study is to reduce the seismic risk in Canada and to minimize the impact of future earthquakes on the country’s lives and economy. In this research, innovative concrete structural systems reinforced using SMA bars at the plastic hinge locations are developed and investigated experimentally, analytically, and using finite element simulation. The experimental part consisted of testing nine large-scale beam-column connections reinforced using steel and SMA bars and subjected to quasi-static loading, while the analytical modeling consisted of predicting and validating the response of the tested connections. The models were also used to conduct a parametric study to investigate the effect of different design variables on the behaviour of the connections. Finally, finite element simulation models were developed to
iv
examine the behaviour of the connections locally by studying the effect of loading on the beam rotation, strain in the reinforcement, and cracking pattern. My personal motivation of conducting this research stems from my eagerness to make a change in this world. I choose this area of research (earthquake engineering) because it affects our societies and mankind directly. I hope that my efforts in this research will save human lives in the future. At the end, I believe that there is nothing hard as along as WE ALL work hard.
Calgary, December 2014
Fadi Oudah
v
Acknowledgements Foremost, my sincere thanks goes to my supervisor Dr. Raafat El-Hacha for his guidance and continuous support toward completing this thesis. I am thankful for his aspiring guidance, motivation, and friendly advice. Besides my advisor, I would like to thank the rest of the supervisory and examining committee members: Dr. Thomas Brown, Dr. Nigel Shrive, Dr. Neil Duncan, Dr. Les Sudak, and Dr. Maged Youssef, for their insightful comments and constructive criticism. I also would like to express my gratitude to Lafarge Canada for proving the concrete, the technical staff at the University of Calgary (Terry Quinn, Daniel Tilleman, Mirsad Berbic, Donald Anson, and Daniel Larson) for their help in conducing the experiments, and to the University of Calgary for the financial support through the funds awarded to Dr. Raafat El-Hacha. A big thank you goes to my friends and colleagues Khaled Abdelrahman and Donna Chen for their input in resolving the technical difficulties encountered during this course of research.
vi
Dedication
To my dear mother (Afaf Al-Taher) and dear father (Samir Oudah) To my brothers (Youssef and Abdellatif) and sister (Mona) To those who struggle for their freedom To those who pursue a meaning for their lives To mankind
vii
Table of Contents Abstract ............................................................................................................................... ii Preface................................................................................................................................ iv Acknowledgements ............................................................................................................ vi Dedication ......................................................................................................................... vii Table of Contents ............................................................................................................. viii List of Tables ................................................................................................................... xiii List of Figures and Illustrations ....................................................................................... xvi List of Symbols, Abbreviations and Nomenclature ..................................................... xxviii Epigraph ............................................................................................................................. xl CHAPTER ONE: INTRODUCTION ..................................................................................1 1.1 General .......................................................................................................................1 1.2 Idea and Motivation ...................................................................................................2 1.2.1 Single-Slotted Beam (SSB) System ..................................................................5 1.2.2 Double-Slotted Beam (DSB) System ................................................................7 1.3 Research Significance ................................................................................................8 1.4 Objectives ..................................................................................................................8 1.5 Scope of Work .........................................................................................................10 1.6 Thesis Content .........................................................................................................11 CHAPTER TWO: LITERATURE REVIEW ....................................................................13 2.1 Introduction ..............................................................................................................13 2.2 Capacity Design Philosophy ....................................................................................16 2.3 Relocation of Plastic Hinges ....................................................................................19 2.4 Research Progress in the Behaviour of SSB System ...............................................22 2.5 Shape Memory Alloy (SMA) ..................................................................................29 2.5.1 Overview .........................................................................................................29 2.5.2 Factors Affecting the Behaviour of PE SMA ..................................................34 2.5.2.1 Effect of Temperature ............................................................................34 2.5.2.2 Effect of Loading Rate ...........................................................................38 2.5.2.3 Effect of Cyclic Loading........................................................................43 2.5.2.4 Fatigue Life ............................................................................................45 2.5.2.5 Size Effect ..............................................................................................47 2.5.3 Applications .....................................................................................................50 2.5.3.1 General Applications .............................................................................50 2.5.3.2 Self-Centering Concrete Structures Reinforced using PE SMA ...........52 2.6 Summary ..................................................................................................................65 CHAPTER THREE: DESIGN AND BEHAVIOUR OF STEEL-REINFORCED SSB AND DSB SYSTEMS WITH RELOCATED VERTICAL SLOTS ………..………….. 68 3.1 Introduction ..............................................................................................................68 3.2 Design of Connections .............................................................................................68 3.2.1 Analysis of Forces Applied onto the Connections ..........................................68 3.2.2 Connection Types and Geometries ..................................................................71 3.2.3 Beam Design ...................................................................................................73 viii
3.2.3.1 Flexural Design ......................................................................................73 3.2.3.2 Shear Design ..........................................................................................76 3.2.4 Column Design ................................................................................................84 3.2.5 Joint Design .....................................................................................................85 3.2.6 Anchorage and Buckling of the Longitudinal Reinforcement ........................86 3.3 Prediction of Rotational Capacity ............................................................................87 3.4 Geometry, Detailing, and Construction of the Connections ....................................92 3.5 Test Setup and Loading Regime ..............................................................................95 3.6 Instrumentations.......................................................................................................98 3.7 Experimental Behaviour of the SSB System .........................................................102 3.7.1 Hysteretic Response ......................................................................................102 3.7.2 Cracking Pattern ............................................................................................109 3.7.3 Beam Rotation ...............................................................................................116 3.7.4 Beam Elongation ...........................................................................................120 3.7.5 Strain Profile in Longitudinal Reinforcement ...............................................122 3.7.6 Shear Mechanism ..........................................................................................128 3.7.6.1 Shear in the Beam ................................................................................128 3.7.6.2 Shear in the Joint..................................................................................132 3.7.7 Components of Deformation .........................................................................135 3.8 Experimental Behaviour of the DSB System ........................................................139 3.8.1 Hysteretic Response ......................................................................................139 3.8.2 Cracking Pattern ............................................................................................146 3.8.3 Beam Rotation ...............................................................................................150 3.8.4 Beam Elongation ...........................................................................................153 3.8.5 Strain Profile in Longitudinal Reinforcement ...............................................155 3.8.6 Shear Mechanism ..........................................................................................160 3.8.6.1 Shear in the beam .................................................................................160 3.8.6.2 Shear in the Joint..................................................................................163 3.8.7 Components of Deformation .........................................................................166 3.9 Comparison and Damage Assessment ...................................................................170 3.9.1 Ductility-Based Damage Assessment............................................................171 3.9.2 Energy-Based Damage Assessment ..............................................................174 3.9.3 Other Damage Assessment Indices ...............................................................179 3.10 Summary ..............................................................................................................183 CHAPTER FOUR: DESIGN AND PERFORMANCE OF SELF-CENTERING CONCRETE BEAM-COLUMN CONNECTIONS REINFORCED USING PE SMA ….….... 185 4.1 Introduction ............................................................................................................185 4.2 Testing Matrix........................................................................................................186 4.3 Design Considerations ...........................................................................................186 4.4 Material Properties of the PE SMA Material.........................................................188 4.4.1 Test Setup, Instrumentations, and Loading Regime ......................................188 4.4.2 Monotonic Behaviour ....................................................................................191 4.4.3 Cyclic Behaviour ...........................................................................................193 4.5 Anchorage Performance ........................................................................................199 4.5.1 Anchorage Performance in Joint-Like Specimens ........................................200 4.5.1.1 Design Considerations .........................................................................202 ix
4.5.1.2 Testing Matrix......................................................................................204 4.5.1.3 Test Regime and Set-up .......................................................................206 4.5.1.4 Instrumentations...................................................................................207 4.5.1.5 Test Results ..........................................................................................208 4.5.1.6 Discussion and Analytical Modeling ...................................................213 4.5.2 Anchorage Modification ................................................................................224 4.6 Coupler Performance .............................................................................................231 4.7 Geometry, Detailing, and Construction of the Connections ..................................234 4.8 Test Setup and Loading Regime ............................................................................236 4.9 Instrumentations.....................................................................................................236 4.10 Experimental Behaviour of the Self-Centering Connections ..............................239 4.10.1 Hysteretic Response ....................................................................................239 4.10.2 Cracking Pattern ..........................................................................................245 4.10.3 Beam Rotation .............................................................................................249 4.10.4 Beam Elongation .........................................................................................252 4.10.5 Strain Profile in the PE SMA Bars ..............................................................254 4.10.6 Shear Mechanism ........................................................................................257 4.10.6.1 Shear in the Beam ..............................................................................257 4.10.6.2 Shear in the Joint................................................................................261 4.10.7 Components of Deformation .......................................................................263 4.11 Comparison and Damage Assessment .................................................................267 4.11.1 Ductility-Based Damage Assessment..........................................................267 4.11.2 Energy-Based Damage Assessment ............................................................268 4.11.3 Other Damage Assessment Indices .............................................................272 4.12 Summary ..............................................................................................................275 CHAPTER FIVE: ANALYTICAL PREDICTION MODEL ………..……………….. 277 5.1 Introduction ............................................................................................................277 5.2 Development of the Load-Deflection Prediction Model .......................................277 5.2.1 Beam Bending ...............................................................................................278 5.2.1.1 Constitutive Model for Concrete .........................................................281 5.2.1.2 Constitutive Model for Steel ................................................................283 5.2.2 Stub Bending .................................................................................................285 5.2.3 Column Bending ............................................................................................287 5.2.4 Joint Shear .....................................................................................................289 5.2.5 Rigid-body Rotation ......................................................................................291 5.2.5.1 Steel-Reinforced Connections .............................................................292 5.2.5.2 PE SMA-Reinforced Connections .......................................................295 5.2.6 Sliding Shear .................................................................................................296 5.2.6.1 Critical Review ....................................................................................297 5.2.6.2 Proposed TDE model ...........................................................................300 5.2.6.3 TDE Model Formulation .....................................................................302 5.2.6.4 Solution Scheme ..................................................................................319 5.2.6.5 TDE Model Validation ........................................................................321 5.3 Application and Sensitivity of the Load-Displacement model ..............................322 5.4 Model Validation ...................................................................................................323 5.5 Parametric Study ....................................................................................................329 x
5.5.1 Effect of the Concrete Strength .....................................................................331 5.5.2 Effect of the Concrete Hinge Depth ..............................................................332 5.5.3 Effect of the Yield Strength ...........................................................................333 5.5.4 Effect of the Bottom-to-Top Reinforcement Ratio and the Amount of Bottom Reinforcement in the SSB System .................................................................334 5.5.5 Effect of the Amount of the Diagonal Reinforcement ..................................335 5.5.6 Effect of the Relocation Distance ..................................................................336 5.6 Summary ................................................................................................................338 CHAPTER SIX: FINITE ELEMENT SIMULATION OF THE TESTED CONNECTIONS .................................................................................................................................340 6.1 Introduction ............................................................................................................340 6.2 Model Development ..............................................................................................340 6.2.1 Element Type.................................................................................................340 6.2.2 Material Constitutive Models ........................................................................341 6.2.2.1 Concrete ...............................................................................................341 6.2.2.2 Steel Reinforcement and Steel Tube ....................................................344 6.2.2.3 PE SMA ...............................................................................................344 6.2.3 Geometry .......................................................................................................345 6.2.4 Boundary Conditions and Loading ................................................................349 6.2.5 Mesh Sensitivity Analysis .............................................................................351 6.3 Model Validation ...................................................................................................354 6.4 Summary ................................................................................................................364 CHAPTER SEVEN: CONCLUSIONS AND RECOMMENDATIONS ........................366 7.1 General ...................................................................................................................366 7.2 Conclusions ............................................................................................................368 7.2.1 Design of Steel-Reinforced SSB and DSB Connections ...............................368 7.2.2 Behaviour of Steel-Reinforced SSB and DSB Connections .........................368 7.2.3 Behaviour of Joint-Like Specimens, Anchor, and Coupler ...........................371 7.2.4 Behaviour of PE SAM-Reinforced SSB and DSB Connections ...................371 7.2.5 Analytical Prediction Model and Parametric Study ......................................373 7.2.6 Finite Element Simulation .............................................................................373 7.3 Recommendations ..................................................................................................374 REFERNCES ………………………………………………………………………….. 376 APPENDIX A: SPECIMEN DESIGN, CONSTRUCTION, AND MATERIAL PROPERTIES .........................................................................................................396 A.1 Introduction ...........................................................................................................396 A.2 Analysis of Forces Applied onto the Connections ................................................396 A.3 Design of Steel-Reinforced SSB and DSB Connections .....................................399 A.3.1 Beam Design.................................................................................................399 A.3.1.1 Flexural Design ...................................................................................399 A.3.1.2 Shear Design .......................................................................................402 A.3.1.3 Anchorage ...........................................................................................405 A.3.2 Column Design .............................................................................................406 xi
A.3.2.1 Strong-Column Weak-Beam Requirement .........................................407 A.3.2.2 Shear Design .......................................................................................409 A.3.3 Joint Design ..................................................................................................411 A.4 Design of SSB and DSB Connections Reinforced using SMA and Control Connection ..............................................................................................................................412 A.5 Fabrication of Connections ...................................................................................413 A.5.1 Steel-Reinforced SSB and DSB Connections ..............................................413 A.5.1.1 Formwork ............................................................................................413 A.5.1.2 Steel Cage ...........................................................................................414 A.5.1.3 Concrete Casting .................................................................................418 A.5.2 Control, SSB, and DSB Connections Reinforced using PE SMA................419 A.6 Fabrication of Joint-Like Specimens ....................................................................421 A.7 Material Properties ................................................................................................423 A.7.1 Concrete ........................................................................................................423 A.7.2 Steel ..............................................................................................................424 APPENDIX B: DEFORMATION ANALYSIS METHODS ..........................................426 B.1 Beam Rotation .......................................................................................................426 B.2 Beam Elongation ...................................................................................................426 B.3 Beam Shear Distortion ..........................................................................................427 B.4 Joint Shear Distortion ............................................................................................428 B.5 Components of Deformation .................................................................................430 APPENDIX C: SUPPLEMENTARY TEST RESULTS .................................................434 C.1 Joint-Like Specimens ............................................................................................434 C.2 Positive Bending Test of the DSB-P-1.0 Connection ...........................................437 APPENDIX D: VERIFICATION OF THE TDE MODEL .............................................440 D.1 Introduction ...........................................................................................................440 D.2 Description of the Examined Shear Walls ............................................................440 D.3 Flexure-Axial-Shear Interaction ...........................................................................441 D.4 Analysis Results ....................................................................................................443 D.5 Conclusions ...........................................................................................................445
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List of Tables Table 3-1: Testing matrix of the SSB and the DSB systems ........................................................ 72
Table 3-2: Predicted steel strain and rotational capacity of the steel-reinforced connections ...... 92
Table 3-3: Load comparison at the yielding and the ultimate conditions of the SSB
connections ......................................................................................................................... 107
Table 3-4: Evaluation of plastic hinge relocation of the SSB connections................................. 107
Table 3-5: Flexural overstrength factors, φo, for the SSB connections ...................................... 108
Table 3-6: Member contributions of the SSB connections at the ultimate cycle........................ 138
Table 3-7: Load comparison at the yielding and the ultimate conditions of the DSB
connections ......................................................................................................................... 142
Table 3-8: Evaluation of plastic hinge relocation of the DSB connections ................................ 143
Table 3-9: Flexural overstrength factors, φo, for the DSB connections...................................... 144
Table 3-10: Member contributions of the DSB connections at the ultimate cycle ..................... 169
Table 3-11: The average values of θy, Δy, and Py in the SSB and DSB connections.................. 173
Table 3-12: Ductility-based assessment of the steel-reinforced SSB and DSB connections ..... 174
Table 3-13: Energy-based assessment of the steel-reinforced SSB and DSB connections ........ 176
Table 3-14: Other damage assessment indices of the steel-reinforced SSB and DSB
connections ......................................................................................................................... 181
Table 4-1: Chemical composition of the PE SMA material* ..................................................... 188
Table 4-2: Material properties of the PE SMA subjected to monotonic loading........................ 191
Table 4-3: Material properties of the PE SMA subjected to cyclic loading ............................... 196
Table 4-4: Testing matrix of the joint-like specimens ................................................................ 204
Table 4-5: Test results of the joint-like specimens ..................................................................... 208
Table 4-6: Strain and stress test results at failure of the modified anchors ................................ 229
Table 4-7: Load comparison at the yielding and the ultimate conditions in the Control, SSB P-1.0, and DSB-P-1.0 connections ..................................................................................... 244
Table 4-8: Flexural overstrength factors, φo, for the DSB connections...................................... 245
xiii
Table 4-9: Ultimate strain values at the critical sections in the Control, SSB-P-1.0, and DSB P-1.0 connections ................................................................................................................ 256
Table 4-10: Member contributions of the Control, SSB-P-1.0, and DSB-P-1.0 connections at
the ultimate cycle ................................................................................................................ 265
Table 4-11: The average values of θy, Δy, Py in the Control, SSB-P-1.0, and DSB-P-1.0
connections ......................................................................................................................... 268
Table 4-12: Ductility-based damage assessment of the Control, SSB-P-1.0, and DSB-P-1.0
connections ......................................................................................................................... 268
Table 4-13: Energy-based assessment of the Control, SSB-P-1.0, and DSB-P-1.0 connections 269
Table 4-14: Other damage assessment indices of the Control, SSB-P-1.0, and DSB-P-1.0
connections ......................................................................................................................... 272
Table 5-1: Parameters of the proposed rotation ratio (Rθ) versus the stub rotation (θb1)
relationships ........................................................................................................................ 287
Table 5-2: Comparison of the Analytical and Experimental loads at ultimate ........................... 324
Table 5-3: Parametric Study matrix for the PE SMA-reinforced SSB connections ................... 330
Table 5-4: Parametric Study matrix for the PE SMA-reinforced DSB connections .................. 331
Table 5-5: Optimum relocation distances of the PE SMA-reinforced connections.................... 338
Table 6-1: Comparison of the FEM results with the experimental results under positive
bending................................................................................................................................ 355
Table 6-2: Comparison of the FEM results with the experimental results under negative
bending................................................................................................................................ 355
Table A-1: End-actions obtained from the analysis at the designated connection ..................... 398
Table A-2: Scaled end-actions obtained from the analysis at the designated connection .......... 398
Table A-3: Factored axial load, shear, and moments at the designated column and beam ........ 398
Table A-4: Design moments of the SSB system ......................................................................... 402
Table A-5: Design moments of the DSB system ........................................................................ 402
Table A-6: Detailing for standard 90o hooks for deformed bars ................................................ 405
Table A-7: Design moments of the SSB-P-1.0 connection ........................................................ 413
Table A-8: Design moments of the DSB-P-1.0 connection ........................................................ 413
xiv
Table A-9: Design moments of the Control connection ............................................................. 413
Table A-10: Concrete compression test results .......................................................................... 424
Table A-11: Steel tension coupon test results ............................................................................. 425
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List of Figures and Illustrations Figure 1-1: Typical stress-strain relationship of SMA (Youssef et al., 2008) ................................ 3
Figure 1-2: Schematic drawings of the main features in the proposed SSB and DSB systems ..... 5
Figure 2-1: The fundamental procedure of the DDBD method (Priestley et al., 2007) ............... 14
Figure 2-2: Capacity design analogy (Elnashai and Di Sarno, 2008) ........................................... 17
Figure 2-3: Comparison of mechanism in SCWB and WCSB configurations (Paulay and
Priestley, 1992) ..................................................................................................................... 18
Figure 2-4: Joint shear failure in RC structures subjected to Kocaeli, Turkey earthquake
(Sezen et al., 2003)................................................................................................................ 19
Figure 2-5: Plastic hinge relocation techniques ............................................................................ 21
Figure 2-6: Schematic drawing of the slotted beam system ......................................................... 24
Figure 2-7: Hysteretic response of the connections tested by Ohkubo and Hamamoto (2004) ... 25
Figure 2-8: Cracking pattern of the connections tested by Ohkubo and Hamamoto (2004) ........ 26
Figure 2-9: Hysteretic response of connections tested by Au (2010) ........................................... 27
Figure 2-10: Hysteretic response of two interior SSB connections tested by Byrne and Bull
(2012) .................................................................................................................................... 27
Figure 2-11: Cracking pattern in the RC SSB superassembly tested by Muir et al. (2012b) ....... 28
Figure 2-12: A typical phase diagram of SMA material............................................................... 30
Figure 2-13: Stress-strain relationships of SMA wires at T > Af and T < Af (reproduced from
Matsui et al. (2006)).............................................................................................................. 32
Figure 2-14: Variation of the surface temperature of a PE SMA wire under uniaxial tension
test under cyclic temperature (Tamai and Kitagawa, 2002) ................................................. 35
Figure 2-15: Temperature change versus the transformation strain (Tobushi et al., 1999) .......... 36
Figure 2-16: The effect of temperature on the stress-strain responses of SMA wires
(Pieczyska et al., 2005) ......................................................................................................... 37
Figure 2-17: Energy definitions and temperature effect (Pieczyska et al., 2005)......................... 37
Figure 2-18: The effect of the loading rate on the hysteretic response of PE SMA wires
(Tobushi et al., 1998) ............................................................................................................ 39
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Figure 2-19: The effect of the loading rate on the transformation stresses (Tobushi et al., 1998) ..................................................................................................................................... 40
Figure 2-20: Behaviour of PE SMA under uncontrolled conditions (Pieczyska et al., 2005) ...... 42
Figure 2-21: The effect of the testing condition on the energy behaviour of PE SMA
(Pieczyska et al., 2005) ......................................................................................................... 42
Figure 2-22: Mechanical training of PE SMA wires for 100 cycles (Soul et al., 2010) ............... 44
Figure 2-23: Effect of cyclic loading on the stress-strain relationship of PE SMA (Tobushi et
al., 1998) ............................................................................................................................... 44
Figure 2-24: Fatigue life prediction of PE SMA obtained from the literature.............................. 47
Figure 2-25: Stress-strain relationships of PE SMA material with different diameters
(DesRoches et al., 2004) ....................................................................................................... 48
Figure 2-26: Effect of bar size on the stress and strain characteristics of PE SMA material
(DesRoches et al., 2004) ....................................................................................................... 49
Figure 2-27: Hysteretic response of concrete columns reinforced using PE SMA tested by
Saiidi and Wang (2006) ........................................................................................................ 53
Figure 2-28: Hysteretic response of beam-connections tested by Youssef et al. (2008) .............. 55
Figure 2-29: Comparison of the cumulative energy versus story drift relationships of
connections tested by Youssef et al. (2008).......................................................................... 55
Figure 2-30: Load-displacement modeling of beam tested by Youssef et al. (2008) (Elbahy et
al., 2010) ............................................................................................................................... 56
Figure 2-31: FE modeling of concrete structures reinforced using PE SMA (Alam et al.,
2008) ..................................................................................................................................... 57
Figure 2-32: Base shear-top storey drift relationships of concrete buildings (Alam et al.,
2009) ..................................................................................................................................... 58
Figure 2-33: Inter-storey drift relationships of concrete buildings (Alam et al., 2009)................ 58
Figure 2-34: Hysteretic response of beam-column connections tested by Nehdi et al. (2010) .... 61
Figure 2-35: Base shear-displacement relationships of the corrosion free concrete columns
(Billah and Alam, 2012) ....................................................................................................... 62
Figure 2-36: Hysteretic response of one-storey one-bay frame reinforced using PE SMA
(Khaloo and Mobini, 2011)................................................................................................... 63
xvii
Figure 2-37: Base shear-drift relationships of the shear walls with different amount of SMA reinforcement (Ghassemieh et al., 2012) .............................................................................. 64
Figure 2-38: Tip deflection of the coupled shear wall reinforced using steel or SMA
subjected to Koyna earthquake (Ghassemieh et al., 2013) ................................................... 65
Figure 3-1: Elevation view of the building with the designated connection ................................ 69
Figure 3-2: Plan view of the building with the column identification .......................................... 70
Figure 3-3: Definition of parameters in the SSB and the DSB systems ....................................... 72
Figure 3-4: Force distribution in the SSB and the DSB systems under positive and negative
moments ................................................................................................................................ 74
Figure 3-5: Visualization of the compressive strains in the SSB connections under positive
bending.................................................................................................................................. 78
Figure 3-6: Visualization of the compressive strains in the SSB connections under negative
bending.................................................................................................................................. 79
Figure 3-7: Visualization of the compressive strains in the DSB connections under positive
bending.................................................................................................................................. 82
Figure 3-8: Shear design regions and details in the SSB and the DSB systems........................... 84
Figure 3-9: Joint shear mechanism in the SSB and DSB connections ......................................... 86
Figure 3-10: Schematic drawing of the rotational capacity prediction model .............................. 89
Figure 3-11: Details of the steel-reinforced SSB connections (all dimensions are in mm) .......... 93
Figure 3-12: Details of the steel-reinforced DSB connections (all dimensions are in mm) ......... 94
Figure 3-13: Selection of connection test setup: (a) drift behaviour in a single storey, (b) drift
behaviour in exterior connection, and (c) actual drift behaviour of the tested connections (Hanson and Conner, 1967) .................................................................................................. 95
Figure 3-14: Test setup of the beam-column connections (all dimensions are in mm) ................ 96
Figure 3-15: Quasi-static loading history ..................................................................................... 97
Figure 3-16: Positions and labels of the LSC and laser devices (all dimensions are in mm) ....... 99
Figure 3-17: Locations of the SGs in the SSB connections (all dimensions are in mm) ............ 100
Figure 3-18: Locations of the SGs in the DSB connections (all dimensions are in mm) ........... 101
Figure 3-19: DICT instrumentation of the SSB and DSB connections ...................................... 102
xviii
Figure 3-20: Hysteretic behaviour responses of the SSB connections ....................................... 104
Figure 3-21: Alternative definitions of yielding condition (reproduced from Park (1989))....... 105
Figure 3-22: Cracking patterns at the ultimate condition in the SSB connections ..................... 111
Figure 3-23: Schematic representation of the diagonal cracks in the SSB connections ............. 112
Figure 3-24: Cracking pattern at the top side of the SSB connections ....................................... 113
Figure 3-25: Cracking pattern at the bottom side of the SSB connections ................................. 114
Figure 3-26: Bar fractures in the SSB connections..................................................................... 114
Figure 3-27: Secondary failure in the SSB-S-1.7 connection ..................................................... 115
Figure 3-28: Cracking pattern at the joint in the SSB connections............................................. 116
Figure 3-29: Rotation profiles of the beams in the SSB connections ......................................... 119
Figure 3-30: Change in length in the SSB connections .............................................................. 121
Figure 3-31: Beam elongation in the SSB connections .............................................................. 121
Figure 3-32: Strain profiles of the bottom reinforcement in the SSB connections ..................... 125
Figure 3-33: Strain profiles of the top reinforcement in the SSB connections ........................... 128
Figure 3-34: Locations of the LSCs and strain gauges used to examine the shear behaviour in
the beams of the SSB connections ...................................................................................... 129
Figure 3-35: Beam shear distortion in the SSB connections ...................................................... 130
Figure 3-36: Moment versus strain in the beam stirrup of the SSB connections ....................... 131
Figure 3-37: Moment versus strain in the diagonal reinforcement of the SSB connections ...... 131
Figure 3-38: Locations of the LSCs and strain gauges used to examine the joint behaviour of
the SSB connections ........................................................................................................... 132
Figure 3-39: Joint shear distortion in the SSB connections ........................................................ 133
Figure 3-40: Envelopes of the moment-joint strain relationships in the SSB connections ........ 135
Figure 3-41: Member contributions to the total applied displacement in the SSB connections. 137
Figure 3-42: Comparison of the rotation ratios (θ2/θ1) of the SSB-S-1.0 and SSB-S-1.7
connections ......................................................................................................................... 139
xix
Figure 3-43: Hysteretic behaviour of the DSB connections ....................................................... 141
Figure 3-44: Variation of the M+/M- ratio with the increase in the displacement amplitude ..... 144
Figure 3-45: Cracking patterns at the ultimate condition in the DSB connections .................... 146
Figure 3-46: Cracking pattern at the top side in the DSB connections ....................................... 149
Figure 3-47: Cracking pattern at the joint in the DSB connections ............................................ 150
Figure 3-48: Rotation profiles of the beams in the DSB connections ........................................ 152
Figure 3-49: Change in length in the DSB connections ............................................................. 154
Figure 3-50: Beam elongation in the DSB connections.............................................................. 154
Figure 3-51: Strain profiles of the bottom reinforcement in the DSB connections .................... 157
Figure 3-52: Strain profiles of the top reinforcement in the DSB connections .......................... 160
Figure 3-53: Locations of the LSCs and strain gauges used to examine the shear behaviour in
the beams of the DSB connections ..................................................................................... 161
Figure 3-54: Beam shear distortion in the DSB connections ...................................................... 162
Figure 3-55: Moment versus strain in the beam stirrup of the DSB connections ....................... 162
Figure 3-56: Moment versus strain in the diagonal reinforcement of the DSB connections ...... 163
Figure 3-57: Joint shear distortion in the DSB connections ....................................................... 164
Figure 3-58: Envelopes of the moment-joint strain relationships in the DSB connections ........ 166
Figure 3-59: Member contributions to the total applied displacement in the DSB connections 168
Figure 3-60: Comparison of the θ2/θ1 ratios of the DSB-S-1.0 and DSB-S-1.7 connections..... 170
Figure 3-61: Cumulative energy versus displacement ductility relationships of the SSB and
DSB connections................................................................................................................. 177
Figure 3-62: Energy index versus displacement ductility relationships of the SSB and DSB
connections ......................................................................................................................... 178
Figure 3-63: Equivalent viscous damping versus displacement ductility relationships of the
SSB and DSB connections.................................................................................................. 179
Figure 3-64: Residual displacement index versus displacement ductility relationships of the
SSB and DSB connections.................................................................................................. 181
xx
Figure 3-65: Work index versus displacement ductility relationships of the SSB and DSB connections ......................................................................................................................... 182
Figure 3-66: Stiffness index versus displacement ductility relationships of the SSB and DSB
connections ......................................................................................................................... 183
Figure 4-1: Design considerations in the PE SMA-reinforced SSB and DSB connections ....... 187
Figure 4-2: Test setup and locations of the LSCs and strain gauges in the PE SMA monotonic
and cyclic tests .................................................................................................................... 189
Figure 4-3: Loading regime of the tension-tension cyclic coupon test of the PE SMA material 190
Figure 4-4: Stress-strain relationship of the PE SMA material subjected to monotonic loading 191
Figure 4-5: Strain distribution along the gauge length of the PE SMA material subjected to
monotonic loading .............................................................................................................. 192
Figure 4-6: Stress-strain relationship of the PE SMA material subjected to cyclic loading....... 194
Figure 4-7: Definition of the parameters used to evaluate the cyclic response of the PE SMA . 195
Figure 4-8: Envelopes of the stress-strain relationships of the PE SMA material subjected to
cyclic loading ...................................................................................................................... 195
Figure 4-9: Evaluation of the effect of cyclic loading on the behaviour of PE SMA................. 197
Figure 4-10: Strain distribution along the gauge length of the PE SMA material subjected to
cyclic loading ...................................................................................................................... 198
Figure 4-11: Comparison of the stress-strain relationships between steel and PE SMA bars .... 199
Figure 4-12: Mechanical anchors for connecting PE SMA bar to steel developed by Alam et
al. (2010) ............................................................................................................................. 200
Figure 4-13: Mechanical anchor and coupler used in this research ............................................ 201
Figure 4-14: Column behaviour subjected to shear and bending moment ................................. 203
Figure 4-15: Details of the joint-like specimens ......................................................................... 204
Figure 4-16: Test setup of the joint-like specimens .................................................................... 206
Figure 4-17: Positions and labels of the LSC devices in the joint-like specimens ..................... 207
Figure 4-18: Locations of the strain gauges in the joint-like specimens .................................... 207
Figure 4-19: Typical failure of the Steel-reinforced joint-like specimens .................................. 209
xxi
Figure 4-20: Typical failures of the PE SMA-reinforced joint-like specimens .......................... 209
Figure 4-21: Stress-strain relationships of the vertical reinforcement in the joint-like
specimens ............................................................................................................................ 211
Figure 4-22: Moment-curvature relationships of the joint-like specimens ................................. 212
Figure 4-23: Moment-strain relationships of the stirrup in the joint-like specimens ................. 212
Figure 4-24: Force resisting mechanisms in the anchored PE SMA and deformed steel
reinforcement ...................................................................................................................... 214
Figure 4-25: Bond and splitting components of the bearing stresses (Thompson, 2002)........... 215
Figure 4-26: Strut-and-tie models of the joint-like specimens ................................................... 220
Figure 4-27: Moment-curvature prediction of the steel-reinforced joint-like specimens ........... 223
Figure 4-28: Moment-curvature prediction of the PE SMA-reinforced joint-like specimens.... 223
Figure 4-29: Test setup and locations of the LSC devices in the modified anchor testing......... 224
Figure 4-30: Modified steel anchors ........................................................................................... 227
Figure 4-31: Stress-strain relationships of the PE SMA bars in the modified anchors .............. 227
Figure 4-32: Failure modes of the modified steel anchors ......................................................... 228
Figure 4-33: Stress-slip relationships of the PE SMA bars in the modified anchors ................. 230
Figure 4-34: Modified steel coupler ........................................................................................... 231
Figure 4-35: Stress-strain relationships of the PE SMA bars in the modified couplers ............. 233
Figure 4-36: Failure modes of the modified steel couplers ........................................................ 233
Figure 4-37: Stress-slip relationships of the PE SMA bars in the modified coupler .................. 234
Figure 4-38: Details of the Control, SSB-P-1.0, and DSB-P-1.0 connections ........................... 236
Figure 4-39: Positions and labels of the LSC and laser devices ................................................. 237
Figure 4-40: Locations of the strain gauges in the Control, SSB-P-1.0, and DSB-P-1.0
connections ......................................................................................................................... 238
Figure 4-41: Hysteretic response of the Control, SSB-P-1.0, and DSB-P-1.0 connections ....... 242
Figure 4-42: Definition of the yielding point in the Control, SSB-P-1.0, and DSB-P-1.0
connections ......................................................................................................................... 243
xxii
Figure 4-43: Cracking patterns at the ultimate condition in the Control, SSB-P-1.0, and DSB P-1.0 connections ................................................................................................................ 246
Figure 4-44: Cracking pattern at the top side of the Control, SSB-P-1.0, and DSB-P-1.0
connections ......................................................................................................................... 248
Figure 4-45: Cracking pattern at the bottom side of the Control, SSB-P-1.0, and DSB-P-1.0
connections ......................................................................................................................... 249
Figure 4-46: Rotation profiles of the beams in the Control, SSB-P-1.0, and DSB-P-1.0
connections ......................................................................................................................... 252
Figure 4-47: Change in length in the Control, SSB-P-1.0, and DSB-P-1.0 connections............ 253
Figure 4-48: Beam elongation in the Control, SSB-P-1.0, and DSB-P-1.0 connections ............ 254
Figure 4-49: Moment-strain relationships of the bottom reinforcement at the critical section
in the Control, SSB-P-1.0, and DSB-P-1.0 connections..................................................... 255
Figure 4-50: Moment-strain relationships of the top reinforcement at the critical section in
the Control, SSB-P-1.0, and DSB-P-1.0 connections ......................................................... 255
Figure 4-51: Strain profiles in the top reinforcement in the SSB-P-1.0 connection ................... 257
Figure 4-52: Locations of the LSCs and strain gauges used to examine the shear behaviour in
the beams of the Control, SSB-P-1.0, and DSB-P-1.0 connections.................................... 258
Figure 4-53: Beam shear distortion at location A in the Control, SSB-P-1.0, and DSB-P-1.0
connections ......................................................................................................................... 259
Figure 4-54: Beam shear distortion at location B in the Control, SSB-P-1.0, and DSB-P-1.0
connections ......................................................................................................................... 259
Figure 4-55: Moment versus strain in the beam stirrup of the Control, SSB-P-1.0, and DSB-
P-1.0 connections ................................................................................................................ 260
Figure 4-56: Moment versus strain in the diagonal reinforcement of the SSB-P-1.0 and DSB-
P-1.0 connections ................................................................................................................ 261
Figure 4-57: Joint shear distortion in the SSB connections ........................................................ 262
Figure 4-58: Envelopes of the moment-joint strain relationships of the middle stirrup in the
Control, SSB-P-1.0, and DSB-P-1.0 connections............................................................... 263
Figure 4-59: Member contributions to the total applied displacement in the Control, SSB-P 1.0, and DSB-P-1.0 connections ......................................................................................... 265
Figure 4-60: Comparison of the θ2/θ1 ratios of the SSB-P-1.0 and DSB-P-1.0 connections ..... 266
xxiii
Figure 4-61: Comparison of the cumulative energy versus displacement ductility relationships of the Control, SSB-P-1.0, and DSB-P-1.0 connections ............................... 270
Figure 4-62: Comparison of the energy index versus displacement ductility relationships of
the Control, SSB-P-1.0, and DSB-P-1.0 connections ......................................................... 270
Figure 4-63: Comparison of the equivalent viscous damping versus displacement ductility
relationships of the Control, SSB-P-1.0, and DSB-P-1.0 connections ............................... 271
Figure 4-64: Comparison of the residual displacement index versus displacement ductility
relationships of the Control, SSB-P-1.0, and DSB-P-1.0 connections ............................... 273
Figure 4-65: Comparison of the work index versus displacement ductility relationships of the
Control, SSB-P-1.0, and DSB-P-1.0 connections............................................................... 274
Figure 4-66: Comparison of the stiffness index versus displacement ductility relationships of
the Control, SSB-P-1.0, and DSB-P-1.0 connections ......................................................... 275
Figure 5-1: Concrete stress-strain relationships used in the analytical model ............................ 282
Figure 5-2: Steel stress-strain relationships used in the analytical model .................................. 284
Figure 5-3: Rotation ratio (Rθ) versus the stub rotation (θb1) experimentally obtained
relationships ........................................................................................................................ 286
Figure 5-4: Parameter definition of the rotation ratio (Rθ) versus the stub rotation (θb1) ........... 287
Figure 5-5: The critical horizontal shear forces in the joints of the SSB and DSB systems ...... 291
Figure 5-6: Bar extension model by Alsiwat and Saatcioglu (1992); (a) reinforcing bar
embedded in concrete; (b) stress distribution; (c) strain distribution; (d) bond stress between concrete and steel.................................................................................................. 293
Figure 5-7: Comparison of experimental and analytical stress-slip relationships ...................... 296
Figure 5-8: Schematic drawing showing the difference between the MCFT and TDE models . 301
Figure 5-9: Applied forces (a) parallel to the crack surface in y-direction in the S-element, (b)
parallel to the crack surface in x-direction in the S-element, (c) parallel to the crack surface in y-direction in the C-element, and (d) parallel to the crack surface in x-
direction in the C-element ................................................................................................... 303
Figure 5-10: Stresses (a) parallel to the crack surface in the S-element, (b) at the crack
surface in the S-element, (c) parallel to the crack surface in the C-element, and (d) at the crack surface in C-element.................................................................................................. 307
Figure 5-11: Constitutive models used in the TDE model ......................................................... 311
xxiv
Figure 5-12: Solution scheme of the TDE model for cracked concrete subjected to shear loading................................................................................................................................. 321
Figure 5-13: Sensitivity of the analytical model to concrete tension stiffening ......................... 323
Figure 5-14: Validation of the load-displacement relationships of the analytical model with
the steel-reinforced SSB and DSB connections.................................................................. 325
Figure 5-15: Validation of the load-displacement relationships of the analytical model with
the PE SMA reinforced SSB and DSB connections ........................................................... 326
Figure 5-16: Member contributions obtained from the analytical model ................................... 329
Figure 5-17: Effect of concrete strength on the load-displacement behaviour of the
connections ......................................................................................................................... 332
Figure 5-18: Effect of concrete hinge depth on the load-displacement behaviour of the
connections ......................................................................................................................... 333
Figure 5-19: Effect of yield strength on the load-displacement behaviour of the connections .. 334
Figure 5-20: Effect of bottom-to-top reinforcement ratio and the amount of the bottom
reinforcement on the load-displacement behaviour of the SSB connection ....................... 335
Figure 5-21: Effect of the amount of the diagonal reinforcement on the load-displacement
behaviour............................................................................................................................. 336
Figure 5-22: Effect of the relocation distance on the load-displacement behaviour .................. 337
Figure 5-23: Evaluation of the optimum relocating distance in the PE SMA-reinforced
connections ......................................................................................................................... 338
Figure 6-1: Concrete constitutive model used in the FEM modeling......................................... 341
Figure 6-2: Definition of the parameters used in calculating the concrete compressive and
tensile damage ..................................................................................................................... 343
Figure 6-3: Concrete damage versus strain relationships ........................................................... 344
Figure 6-4: Modeled connection (SSB-S-1.0 connection) .......................................................... 346
Figure 6-5: Modeled mechanical steel anchor and coupler ........................................................ 346
Figure 6-6: Modeled steel plates ................................................................................................. 347
Figure 6-7: Assembled FEM models .......................................................................................... 348
Figure 6-8: Tube-to-tube contact elements in the steel-reinforced connections ......................... 349
xxv
Figure 6-9: Boundary connections applied onto the connections ............................................... 350
Figure 6-10: Loading applied onto the modeled connections ..................................................... 351
Figure 6-11: Mesh sensitivity results of the first stage ............................................................... 352
Figure 6-12: Mesh sensitivity results of the second stage .......................................................... 353
Figure 6-13: Optimum mesh density of the modeled connections ............................................. 354
Figure 6-14: Comparison of the experimental and FEM load-displacement response of the
steel-reinforced connections ............................................................................................... 357
Figure 6-15: Comparison of the experimental and FEM load-displacement response of the PE
SMA reinforced connections .............................................................................................. 357
Figure 6-16: Comparison of the experimental and FEM beam rotation profiles in the steel-
reinforced connections ........................................................................................................ 359
Figure 6-17: Comparison of the experimental and FEM beam rotation profiles in the PE
SMA reinforced connections .............................................................................................. 359
Figure 6-18: Comparison of the experimental and FEM strain profiles in the bottom
reinforcement of the steel-reinforced connections .............................................................. 361
Figure 6-19: Comparison of the experimental and FEM strain profiles in the bottom
reinforcement of the PE SMA reinforced connections ....................................................... 361
Figure 6-20: Concrete tensile damage visualization of the tested connections .......................... 364
Figure A-1: Design spectral response acceleration..................................................................... 397
Figure A-2: Geometry of the designated connection (all dimensions are in mm) ...................... 399
Figure A-3: Beam dimensions and details at the vertical slots (all dimensions are in mm)....... 401
Figure A-4: Definition of the design parameters in a standard 90o hook for deformed bars ...... 405
Figure A-5: Column reinforcement details (all dimensions are in mm)..................................... 406
Figure A-6: Interaction diagram for column section .................................................................. 407
Figure A-7: Capacity design of columns for three factored load combinations ......................... 408
Figure A-8: Fabricated wooden formworks for concrete connections ....................................... 414
Figure A-9: Fixing the steel tube and the steel hooks ................................................................. 416
Figure A-10: Fixing the Styrofoam in connections .................................................................... 416
xxvi
Figure A-11: Steel cages of SSB and DSB connections reinforced using steel ......................... 417
Figure A-12: Concrete casting of the connections...................................................................... 418
Figure A-13: Fixing the anchors to the steel cage ...................................................................... 419
Figure A-14: Fixing SMA bars to couplers and to the steel cage ............................................... 420
Figure A-15: Steel cages of SSB and DSB connections reinforced using PE SMA and
Control connection.............................................................................................................. 421
Figure A-16: Fabricated wooden formworks for anchorage specimens ..................................... 422
Figure A-17: Concrete casting of the anchorage specimens ....................................................... 423
Figure B-1: Gaping opening and closing in SSB system............................................................ 427
Figure B-2: Calculation of beam shear deformation................................................................... 428
Figure B-3: Calculation of joint shear deformations .................................................................. 429
Figure B-4: Deformation components in the beam-column connections ................................... 433
Figure C-1: Hysteretic response of the PE SMA bars in the joint-like specimens ..................... 434
Figure C-2: Strain distribution at the mid-span section of the joint-like specimens .................. 435
Figure C-3: Strain distribution along the normalized gauge length of the modified anchors..... 436
Figure C-4: Strain distribution along the normalized gauge length of the modified coupler ..... 437
Figure C-5: Quasi-static loading history of the positive bending test of the DSB-P-1.0
connection ........................................................................................................................... 438
Figure C-6: Hysteretic response of the DSB-P-1.0 connection subjected to positive bending .. 438
Figure C-7: Cracking and fracture of top PE SMA bar at failure in the DSB-P-1.0 connection
subjected to positive bending test ....................................................................................... 439
Figure D-1: Details of the tested shear walls (reproduced from Zhang and Wang (2000), all
dimensions are in mm) ........................................................................................................ 441
Figure D-2: Moment-curvature relationships comparison between the TDE results and the
experimental results of shear walls ..................................................................................... 444
xxvii
List of Symbols, Abbreviations and Nomenclature Symbol a as A Ab Ac Ael Af Ag Ahys Aj As As Asb As,max As,min Asp Ast Av b bd bs bw c cm c1 C Cb Cc,b Cc,co Cd Cp Cs,b Cs,co Cw d da dagg
Definition depth of the equivalent rectangular stress block slope of the stress versus slip relationships area of the membrane element area of the reinforcement area of the concrete area enclosed by the elastic energy Austenite finish temperature gross-sectional area area enclosed by the hysteresis loop area of the joint Austenite start temperature area of the steel reinforcement amount of the bottom reinforcement maximum amount of the longitudinal steel minimum amount of the longitudinal steel total cross sectional area of all the transverse reinforcement which cross the potential plane of splitting amount of the top reinforcement amount of the transverse reinforcement defines the strain-hardening slope depth of the beam intercept of the stress versus slip relationship with the y-axis width of the beam depth of the neutral axis minimum of either the concrete cover or the side cover coefficient accounts for the clear spacing of the bars Carbon compression force in the beam compression force in the beam concrete compression force in the column concrete depth of the column coefficient of plastic strain compression force in the beam steel compression force in the column steel width of the column effective depth distance from the top fibre to the bottom side of the anchor maximum aggregate size xxviii
db dbh dbl dbx dby dLSC ds dsb dsd dst dts,n dv D Dc DL Dbh Dbv Dt Djv DDBD DICT DSFM E EA EA,c Ec E’c Ecum Eel Eeff,i Eh Ei Ein
bar diameter diameters of the stirrup reinforcement diameters of the longitudinal reinforcement diameters of the steel reinforcement in the x-direction diameters of the steel reinforcement in the y-direction diameter of the LSC distance from the concrete fibre to the centroid of the reinforcement distance from the top concrete fibre to the centroid of the bottom reinforcement distance from the top concrete fibre to the centroid of the diagonal steel reinforcement distance from the top concrete fibre to the centroid of the top steel reinforcement distance from the top fibre to the centroid of the concrete fibre nc effective shear depth dead load concrete compressive damage linear damage horizontal projection of the diagonal distance between the two pins connecting the LSCs to the concrete surface vertical projection of the diagonal distance between the two pins connecting the LSCs to the concrete surface concrete tensile damage vertical projection of the diagonal distance between the two pins Direct Displacement Based Design Digital Image Correlation Technique Disturbed Stress Field Model earthquake load Austenite modulus of elasticity modulus of elasticity of the Austenite phase calculated from the envelope of the cyclic response modulus of elasticity of concrete unloading concrete stiffness cumulative energy elastic energy effective stiffness of the stress-strain relationship in the PE SMA in the ith cycle, calculated from the envelope of the cyclic response modulus of elasticity of the hoop reinforcement energy per cycle i energy index xxix
EM Er Es Esh ESMA,s Es,t f fbx fby fc f’c f’cc fcx fcy fc1 fc2 fd fp fs fs,ih fsx fsx,cr fsy fsy,cr ft f’t fu fy fy,h fy,sb Fa Fc Fcr Fd Fdu Fs Fsb Fsd Fst Fts,n
Martinsite modulus of elasticity recoverable energy modulus of elasticity of steel stiffness of the strain hardening elastic modulus of the stress-slip relationship of SMA tangent modulus of the stress-strain curve of the reinforcing steel factor that allows for the non-uniform distribution of the shear stresses bond stresses in the x-direction bond stresses in the y-direction concrete compressive stress concrete compressive strength confined concrete compressive stress average concrete stress in the x-direction average concrete stress in the y-direction concrete tensile principle stress concrete compressive principle stress dowel stress reduced peak compressive strength stress in the reinforcement tension stress due to isotropic hardening average steel stress in the x-direction steel stress at the crack in the x-direction average steel stress in the y-direction steel stress at the crack in the y-direction concrete tensile stress concrete tensile strength stress at ultimate yield stress yield stress of the hoops yield stress of the bottom reinforcement acceleration-based site coefficients force in the concrete cracking load dowel force at a shear displacement of Δ at the crack ultimate dowel force force in the reinforcing bar Force in the bottom reinforcement Force in the diagonal reinforcement force in the top reinforcement tension-stiffening force in the concrete fibre nc xxx
Fv FEM g GF hch hco hw HL Ic Ie Ig jd jw kd,i ke kf ki kp ktr kv,cr kv,unc ky k1 K l lcbw lc-e lco ld ld,sp ln lo lpe lre ls ls-e lst lub
velocity-based site coefficients Finite Element Method maximum distance from the reinforcement crossing the crack fracture energy depth of the concrete hinge depth of the concrete column height of the shear wall height of lugs on the bar cracked moments of inertia effective moment of inertia gross moment of inertia depth of the joint width of the joint initial stiffness of the dowel force-crack width relationship effective stiffness of the structure at the ultimate condition elastic foundation stiffness of the surrounding concrete stiffness of the load versus displacement relationship in cycle i ratio of the factored axial load to the axial strength of the column term representing the effect of the transverse reinforcement on bond strength cracked shear stiffness of the joint un-cracked shear stiffness of the joint stiffness of the load versus displacement relationship at yielding factor accounts for the surface texture of the reinforcing bar coefficient in takes into account the effect of confinement steel, concrete cover, and bar spacing in calculating l d relocation distance and half of the joint width crack band width length of the region that contains C-elements column span development length development length at the onset of concrete splitting cracking moment arm distance between the neutral axis and the bottom reinforcement strain penetration length relocation distance shear span length of the region that contains S-elements length of the steel tube length of the unbonded region xxxi
lx ly L Le Lg Lpc Lsh Lt Lyp LSC me M Mb Mb,n Mb,p Mb,r Mco Mco,n Mcr MDSB Mex Mf Mf Min M s Ms M+SSB M-SSB M/M0
M+/MMCFT n nax nc ni np nsp
maximum distance between in the reinforcement in the x-direction maximum distance between in the reinforcement in the y-direction live load lengths of the elastic region gauge length of the SMA bar length of the pullout cone region length of the strain hardening region length of the beam member and half of the joint length of the yield plateau region Linear Strain Conversion effective mass of the structure participating in the fundamental mode of vibration applied moment beam moment nominal moment of the beam probable moment of the beam factored moment of resistance of the beam column moment nominal moment of the column cracking moment moment capacity of the DSB system external moment Martensite finish temperature factored moment internal moment Martensite start temperature moment capacity of the SSB system under positive bending moment capacity of the SSB system under negative bending ratio of the moment in the connections with the relocated vertical slots to that of the connection in which the slot is located at the face of the column ratio of the positive to the negative moment Modified Compression Filed Theory modulus ratio (E s E c ) axial stress concrete fibre number of cycles at a certain strain amplitude an optimum order of proposed equation type in the concrete stress-strain relationship number of bars developed along the plane of splitting xxxii
nx ny N Nf Nfi N Ni O P Pa Pe Pf Pi Pu Py Po PE R RB Ro Rθ RDI RC s ssp ssp,max ssp,1 sy sθx sθy S Sa Sd Sd,l Sd,u SL Ss Sw Sw,l
axial stresses in the x-direction axial stresses in the y-direction number of cycle fatigue life the fatigue life at a certain strain amplitude N itrogen Nickel O xygen applied load axial compression force load at first yield or 0.75 Pu factored axial load load at cycle i load at ultimate load at yielding axial strength of the column Psuedoelasticity Effect angle between the diagonal LSCs and the horizontal axes factor rules the steepness of the Bauschinger effect in the stress-strain curve value of the parameter RB during first loading rotation ratio Residual Displacement Index R einforced Concrete slip stirrup spacing maximum stirrup spacing maximum centre-to-centre spacing of the transverse reinforcement slip at yielding average diagonal crack spacing in the x-direction average diagonal crack spacing in the y-direction stress ratio spectral acceleration depth of slot in the SSB system depth of the lower slot in the DSB system depth of the upper slot in the DSB system clear spacing of lugs on the bar spacing distance between the middle stirrups width of the slot in the SSB system width of the lower slot in the DSB system xxxiii
Sw,u Sε S1
S2
SI SG SMA SME T Tb Te Ts,b Ts,co TDE Ti vagg vb vc vxy vy V Vb Vc Vco Vf Vf,I Vf,II Vf,III Vjh Vjh,avg Vjh,f Vr Vr,max Vs w w/c wc
width of the upper slot in the DSB system strain ratio diagonal compression strut transferring a portion of the compression forces from the compression reinforcement to the upper opposite corner of the joint diagonal compression strut transferring a portion the compression forces in the concrete hinge to the upper opposite corner of the joint Stiffness Index Strain Gauge Shape Memory Alloy Shape Memory Effect temperature tension force in the beam effective period at the maximum displacement response tension force in the beam steel tension force in the column steel Two Distinct Element Titanium shear stress due to aggregate interlock shear stress factor cracking concrete shear stress shear stress in the x-y plane shear stress at yielding of the hoop reinforcement applied load beam shear force concrete shear resistance column shear force factored shear load maximum factored shear force in region I maximum factored shear force in regions II maximum factored shear force in regions III horizontal shear force in the joint average horizontal shear force in the joint factored horizontal shear forces in the joints factored shear resistance maximum factored shear resistance stirrup shear resistance loading frequency water-cement ratio crack width xxxiv
wjh,1 wn Wd WI yt zb αo α αc αj αθ α1 β βc βd βj βv βθ β1 γc γs γu γxy γy δan δb δch δch+ δch - δdh δds δds,j δdv δel δext
horizontal shear weights natural frequency dissipated/absorbed energy Work Index distance from the extreme top fibre to the centroid of the section distance between the centroids of Tb and Cb constant that takes into account the type of the aggregate factor to modify the strain penetration length Poisson’s ratio of concrete the rotation angle of the vertical component of the joint with respect to the x-axis crack angle ratio of average stress in rectangular compression block to the specified concrete strength centroid of the compression force in the concrete hinge measured from the centroid of the section scaling coefficient in the fatigue life prediction model strength reduction factor the rotation angle of the horizontal component of the joint with respect to the y-axis factor accounting for shear resistance of cracked concrete stirrup angle ratio of depth of rectangular compression block to depth to the neutral axis exponent coefficient in the fatigue life prediction model shear distortion ultimate shear strain shear strain shear strain at yielding of the hoop reinforcement cumulative slip of the anchors bottom deformation obtained using LSC devices average change in the length of the beam member change in beam length under positive bending change in beam length under negative bending average horizontal expansion average displacement of the diagonal LSCs modified average diagonal displacement in the joint average vertical expansion beam elongation bar extension xxxv
δt δtot,b δtot,t Δ Δa Δb Δb1 Δb1,l Δb2 Δch Δco Δco,l Δfs Δi Δi,l Δi,Lt Δjs Δres Δrig Δs,j Δs Δss Δsb Δsd Δst Δstub Δs,x Δs,y ΔT Δy Δ1 Δ2 Δ3 Δ4 ε* εa εax,avg
top deformation obtained using LSC devices summation of the LSC readings mounted on the bottom of the beam member summation of the LSC readings mounted on the top of the beam member component of the crack width applied displacement at the tip of the beam beam tip displacement extrapolated displacement of the beam member in the l region displacement of the beam member in the l region beam displacement in ls region axial deformations in the concrete hinge Displacement due to column bending displacement due to column deformation at l difference in the steel stress between the current load stage and the beginning of the strain-hardening stage beam-tip displacement at cycle i total displacement at l applied displacement at Lt displacement due to joint shear residual displacement upon unloading (at zero load) displacement due to rigid body deformation horizontal shear displacement extrapolated displacement due to joint distortion displacement due to sliding shear axial deformation in the bottom reinforcement axial deformation in the diagonal reinforcement axial deformation in the top reinforcement displacement due to beam bending in the stub region components of the shear displacement in the x-direction components of the shear displacement in the y-direction total displacement displacement at yielding beam displacement due to the shift of the joint beam displacement due to the rotation of the joint fictitious column displacement due to the shift of the joint fictitious column displacement due to the rotation of the joint average normalized strain strain amplitude in the steel reinforcement average longitudinal strain values
xxxvi
εax,n εc ε’c
axial strain obtained from the flexural analysis and induced in each element strain in the concrete '
concrete strains at f c
εc,p εc,tot εcu εc2 εM εMs εmax εmax,i εmin εmin,i εp εs εsb εsd εsh εSMA εst εsu ε’ t
concrete plastic strain total concrete strain crushing strain principle concrete compressive strain transformation strain strain at the Martensitic start temperature maximum strain in the cycle maximum strain attained at cycle i minimum strain in the cycle minimum strain attained at cycle i reduced peak compressive strain strain in the steel strain in the bottom reinforcement strain in the diagonal reinforcement steel strain corresponding to the end of the yielding plateau strain in the SMA bar strain in the top reinforcement ultimate strain of the hoops
εu εut εx-d εy-d εy ε1 ε2 θ θ’ θb θb,vs θb1 θb2 θco θi θlb
strain at the ultimate load ultimate tensile strain in tension strain in the x-direction strain in the y-direction yield strain principle tensile strain principle compressive strain rotation angle inclination of the principle tensile stress with respect to the x-axis beam section rotation rotation at the vertical slot section rotation due to beam deformation in the l region rotation due to beam deformation in the Lt region rotation due to the column bending rotation at cycle i lower bound inclination angle
concrete strains at f t
'
xxxvii
θrig,s θrig,SMA θs θub θy θ1 θ2 λ λ1 λ2 λo μb μhys μΔ μΔ,cum μθ μθ,cum ρv ρx ρy ξ ξeq σ σAf σAs σdf σds σmax,i σmin,i σMf σMs σMs,c σMs,i σu ue uf ϕc
rotation due to rigid-body deformation rigid-body rotation of the PE SMA-reinforced connections rotation due to joint shear deformation upper bound inclination angle rotation at yielding rotation due to total deformation in the l region total rotation of the beam in the ls region factor to account for low-density concrete ratio of the actual to the specified yield strength factor that accounts for the increase in the stress resulting from the strain hardening overstrength factor bond stress hysteretic ductility index displacement ductility cumulative displacement ductility index rotation ductility cumulative rotation ductility index volumetric ratio of the transverse reinforcement steel volumetric ratio of the steel reinforcement in the x-direction volumetric ratio of the steel reinforcement in the y-direction mean strain normalized to the yield strain updated following a strain reversal equivalent viscous damping stress stress at the Austenite finish temperature stress at the Austenite start temperature detwinning finish stress detwinning start stress minimum stress attained at cycle i maximum stress attained at cycle i stress at the Martensitic finish temperature stress at the Martensitic start temperature stress at the start of the Martensitic phase calculated from the envelope of the cyclic response stress at the start of the Martensitic phase at cycle i stress at ultimate load average elastic bond stress frictional bond stress concrete material resistance factor xxxviii
ϕs φo ψro ψu 2D 3D
steel material resistance factor flexural overstrength curvature at the rotational spring curvature at the ultimate Two-Dimensions Three-Dimensions
xxxix
Epigraph
‘The history of the world is none other than the progress of the consciousness of freedom’ Friedrich Hegel ‘I think; therefore I am’ Rene Descartes ‘Thoughts are the shadows of our feelings - always darker, emptier and simpler’ Friedrich Nietzsche ‘In order for the light to shine so brightly, the darkness must be present’ Francis Bacon
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Chapter One: Introduction 1.1 General The seismic capacity design philosophy is primarily based on ensuring a ductile failure through energy dissipation at particular locations in the structure. These locations are denoted as plastic hinges (i.e. centre of rotation) and are detailed to ensure a flexural failure and prevent undesired shear failure. The locations and the progression of plastic hinges are designed to prevent collapse through the formation of the so-called `beam mechanism`. This mechanism refers to the formation of the plastic hinges at the face of the columns in the beams at all levels of the structure followed by the formation of plastic hinges in the columns at the base level (Paulay and Priestley, 1992). The energy imposed on the structure is thus dissipated via the flexural deformation in the plastic hinges. The behaviour of plastic hinges has been an important research topic over the last few decades as new structural systems are being developed. One of the major research contributions in the area of seismic design of structures is the development of `Self-Centering` structural components; beam, column, and beam-column joints. Self-centering components have the ability to dissipate the induced energy via external or internal energy dissipation devices, while returning to their initial equilibrium position upon unloading. These systems usually involve the use of post-tensioned reinforcement, complex steel detailing, and precast concrete segments (the reader is referred to the work by Park (2003) for a comprehensive state-of-the-art review of complex self-centering structural components). Therefore, the complexity of using such systems hinders their use in the construction industry. Alternatively, several attempts were made to develop monolithic (i.e. cast as a single piece) selfcentering concrete connections reinforced using Fibre Reinforced Polymers (FRP). Even though connections reinforced using FRP material sustain large drift ratios while maintaining minimal 1
amount of permanent deformation, the linear elastic response of FRP leads to low amounts of energy dissipation as compared with conventional connections (Sharbatdar et al., 2011). In a recent study, Shape Memory Alloy (SMA) was used as a flexural reinforcement at the plastic hinge of concrete beam-column connection (Youssef et al., 2008). The SMA is a unique alloy with the ability to undergo large deformation and energy dissipation capacities while maintaining a superelastic response. The excellent properties of SMA makes it a good candidate for the use in the seismic design of structures. In this research, the use of SMA in monolithic concrete beamcolumn connections is optimized through the development of innovative detailing of plastic hinges in concrete beam-column connections. The description of the proposed design and the motivation behind this research are included in the following sections.
1.2 Idea and Motivation The number one priority in seismic design is to minimize human casualties, which may require sacrificing the usability of the structure after being subjected to earthquake. However, revolutionary design techniques associated with the use of new construction materials can be used in order to achieve a self-centering behaviour with considerable amount of energy dissipation in monolithic Reinforced Concrete (RC) structures. The SMA is a unique alloy with the ability to undergo large deformations and return to its original shape through stress removal (psuedoelasticity) or heating (shape memory effect) (Kumar and Lagoudas, 2007). The typical tension-tension cyclic response of SMA is shown in Figure 1-1. The exceptional property of SMA in recovering substantial inelastic deformation upon unloading makes it very beneficial in seismic design. Thus, if this material is used at the location of plastic
2
hinges with proper design limitations, the structure will dissipate the demand energy and return to its original shape when unloaded.
Figure 1-1: Typical stress-strain relationship of SMA (Youssef et al., 2008)
The detailing of the amount of flexural and shear reinforcements in the plastic hinge is of a paramount importance in the seismic design of structures. In order to utilize the Psuedoelastic (PE) response of SMA, careful attention should be paid towards proper detailing of the plastic hinge in the beam-column connection. The main detailing criteria that were considered in this research are: 1. Continuity of the steel reinforcement. The continuity of the steel reinforcement will prevent the total collapse upon reaching the failure point. In the proposed systems, the failure is defined as the fracture of the PE SMA bars while the total collapse is referred to the breakage of the beam member at the location of the plastic hinge. This procedure increases the margin of safety in the design and sustains the integrity of the system after failure. 2. Plastic hinge relocation. Preventing the collapse of the structure is one of the essential objectives of the capacity design philosophy, and therefore, the vertical load resisting system needs not to lose its integrity during all stages of seismic loading. However, the 3
development of plastic hinges in the beam right at the face of the column leads to yield penetration into the beam-column joint, which result in cracking and bond deterioration. The current seismic design provisions in the CSA-A23.3-04 (2004) code overcome the problems associated with the severe consequences of yield penetration by increasing the amount of the transverse reinforcement and limiting the dimensions of the joint. However, several research studies examined new detailing techniques to relocate the plastic hinge away from the face of the column in concrete beam-column connections in order to reduce the steel congestion in the joint and to yield a more cost-effective design. As discussed in detail in Section 2.3, the most significant drawback of these techniques is in the inaccuracy in predicting the position of the relocated plastic hinges. This will, consequently, lead to poor prediction of the drift rotation capacity since the location of the centre of rotation is conventionally assumed to be positioned at the centre of the plastic hinge.
Two types of beam-column connections that satisfy the aforementioned criteria are considered in this research; (a) Single-Slotted Beam (SSB) and (b) Double-Slotted Beam (DSB) as shown in Figure 1-2. The SSB system was originally developed by Ohkubo et al. (1999) to improve the performance of ductile RC frames when subjected to severe earthquake motion. It is utilized in this research to develop a self-centering connection via the use of SMA reinforcing bars. The DSB system is a new beam-column connection that is proposed and verified in this research. The SMA material is used as a flexural reinforcement in the beams in order to achieve a selfcentering behaviour. Discussions about the SSB and the DSB systems are provided in the following sections.
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Figure 1-2: Schematic drawings of the main features in the proposed SSB and DSB systems
1.2.1 Single-Slotted Beam (SSB) System The SSB connection is a connection made between a strong column and a weak beam. A vertical slot is made in concrete at the face of the column such that the depth of the slot equals the overall depth of the beam minus the floor thickness. The system is designed such that the bottom reinforcement yields in both tension and compression while the top reinforcement remains elastic during the rotation of the plastic hinge. In addition to the self-centering behaviour that is achieved with the use of SMA and the monolithic nature of the system (elimination of the connection), the advantages of the SSB system are listed as follows:
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1. Minimal beam elongation. The elastic response of the top reinforcement under positive and negative bending will ensure minimal elongation of the beam. In RC frame, the beam is restrained from elongating freely, which results in axial compressive forces induced in the member. Thus, the moment capacity of the beam will increase with increasing beam elongation in accordance to the load-moment interaction diagram (PM). This behaviour will yield uncertainties in the frame behaviour and lead to inaccuracies in predicting the response of the frame if not taken into account in the design (the effect of beam elongation is described in detail in Section 2.4). 2. Non-tearing action. It is common to assume rigid diaphragm behaviour of the concrete slabs in concrete structures. However, the cracking of the slab as a result of the beam elongation may jeopardize this assumption, and hence, inaccuracy in predicting the structural response of the system is anticipated. Due to the elastic response of the top reinforcement in the SSB system, the attached slab will remain intact (i.e. non-tearing action) during the earthquake motion. Consequently, the rigid diaphragm assumption will remain valid throughout the earthquake motion and the predicted response of the structure will yield high levels of accuracy. 3. Reduced SMA cost. The SMA bars are used only as a bottom reinforcement, and thus, half of the SMA quantity used in reinforcing conventional connections is required to achieve a self-centering behaviour under both positive and negative bending. 4. Plastic hinge relocation. The centre of rotation, i.e. the plastic hinge, is located at the centre of the vertical slot. Therefore, moving the vertical slot away from the face of the column is anticipated to relocate the plastic hinge effectively.
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5. Minimal repair cost. The concentrated plastic rotation at the slot will allow the concrete in the other regions to remain un-cracked, and hence, the repair costs after earthquake loading should be reduced given that the connection does not fail (i.e. fracture of the bottom reinforcement due to low cyclic fatigue does not occur). However, a cost effectiveness study is out of the scope of the research project.
1.2.2 Double-Slotted Beam (DSB) System The DSB system is made of a strong column and a weak beam slotted from both the top and the bottom sides as shown in Figure 1-2. A large amount of longitudinal reinforcement is placed inbetween the slots in order for the reinforcement to remain elastic during the plastic rotation of the beam. The top and the bottom reinforcements will act as energy dissipation devices. The energy imposed on the system is dissipated via the hysteresis response of both reinforcements. In addition to the self-centering ability, the advantages of the DSB system are as follows: 1. Minimal beam elongation. The elastic response of the middle reinforcement under positive and negative bending will ensure minimal elongation of the beam. 2. Non-tearing action. The top slot is anticipated to open and close during the seismic loading. However, if a construction joint is made in the slab at the location of the top slot, a nontearing action of the slab can be achieved. 3. High drift capacity. The neutral axis at the vertical slot is approximately equal to half of the beam depth. This will lead to higher drift capacity since the strain level in the tension reinforcement decreases with increasing depth of neutral axis (the drift capacity of DSB connection was found to be 2.0 times that of the Control connection. Both connections were designed for the same moment capacity – refer to Section 4.10). 7
4. Plastic hinge relocation. Moving the vertical slot away from the face of the column will relocate the plastic hinge effectively. 5. Minimal repair cost. The plastic rotation is concentrated at the slot. Therefore, minimal concrete cracking is anticipated when the connections is subjected to lateral loading. (However, the repair cost of the DSB system is out of the scope of the thesis).
1.3 Research Significance Even though moment-resisting RC buildings designed according to the capacity design philosophy are safe, that is, with small probability of collapse at design earthquake ground-motion intensities, they may sustain extensive damage with repair costs of about one-third of the building replacement value (Ramirez et al., 2012). The use of SSB and DSB self-centering beam-column connections proposed in this research study will not only maintain the functionality of the structure after being subjected to earthquake motion, but it will also contribute in reducing the repair costs of the structural and non-structural components of RC moment-resisting frames. Moreover, the simplified monolithic nature of the systems provides a simplified alternative for the more complex post-tensioned precast self-centering connections that use damping devices. Therefore, the design procedure and the special detailing investigated in this research can be used in the current design codes as an alternative way of plastic hinge design in moment-resisting RC structure.
1.4 Objectives The overall objective of this research study is to design self-centering monolithic concrete beamcolumn connections reinforced using smart materials. In order to achieve this objective, the project is subdivided into sub-objectives as follows: 8
• Review the up-to-date technologies of beam-column connection systems and the factors that affect the response of SMA material. • Design and experimentally validate the SSB and DSB systems reinforced using conventional steel with relocated plastic hinges. This includes the testing of large-scale beam-column connections in order to determine the effect of the relocation distance on the performance of the connections. • Characterize the monotonic and cyclic properties of SMA material for the use in seismic design applications. • Determine suitable anchors and couplers for the SMA bar and modify them if necessary. • Design and experimentally validate the response of joint-like specimens reinforced using SMA bars and compare it with joints reinforced using steel reinforcement. • Design and experimentally validate the response of large-scale beam-column connections reinforced using SMA bars designed according to the optimum plastic hinge relocation distance. • Develop an analytical model that predicts the response of the SSB and DSB systems reinforced using steel and SMA bars. This includes the development of a new deformationbased shear prediction model that is capable of predicting the sliding shear failure in concrete structures (the model is used in this research to examine the sliding shear capacity at the concrete hinge between the vertical slots). • Develop and validate 3D Finite Element (FE) models of the tested connections using the FE software ABAQUS to be used for conducting a parametric study in future research.
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1.5 Scope of Work The research project is subdivided into three main phases that are built toward achieving the overall objective of the study. The first phase includes the design and validation of the SSB and DSB connections reinforced using steel with relocated plastic hinges. Simplified elastic 2D FE models are used to determine the flow of forces within the systems and to design the flexural and shear reinforcements accordingly. Then, experimental testing of three large-scale connections in each system with the vertical slots placed at 0, 1.0 dv, and 1.7 dv from the face of the column, where dv is the effective shear depth (under negative bending in the SSB system and both positive and negative bending in the DSB system, the compression and tensile forces are concentrated at the centroid of the top and bottom reinforcements, and hence, the effective shear depth corresponds to the distance between the top and the bottom reinforcements). The responses of the connections are then analyzed and compared in terms of the hysteretic response, cracking pattern, beam rotation, beam elongation, strain profiles in the flexural reinforcement, shear mechanism in the beam and the joint, components of deformation, and damage assessment. In the second phase, the material response of SMA bars under monotonic and cyclic loading is investigated in order to determine its suitability for seismic applications. After that, the responses of six reduced-scale joint-like specimens reinforced using conventional steel and SMA bars anchored to the concrete using mechanical anchors are studied. The responses are investigated experimentally and analytically in order to determine the effect of the anchor orientation, position, and the level of confinement on the performance of the joints reinforced using SMA bars. Furthermore, the anchors and the couplers are modified in order to reduce the slippage and attain high strain and stress levels. Based on the results obtained from the first phase and the response of the SMA material in joint-like specimens, the final design of the large-scale self-centering concrete 10
beam-column connections is conducted. The design is validated experimentally by testing three connections; SSB, DSB, and conventional connections. The self-centering ability and the performance of the connections is studied by examining the energy dissipation ability and the level of permanent deformation in the systems. The third phase of the project involves the development of analytical and FE models of the SSB and DSB systems reinforced using steel and SMA bars. It also involves the development and validation of a new deformation-based shear prediction model named as the Two-DistinctElements (TDE) shear model. The TDE model is used to predict the shear failure of concrete structures in general and predict the response of the SSB and DSB systems in particular.
1.6 Thesis Content The thesis consists of seven chapters. The need for the development of self-centering monolithic concrete beam-column connections is previously described in this chapter along with the objectives and the scope of work of the present study. With reference to the main phases of the research project discussed in Section 1.5, the contents of the additional chapters are: • Chapter 2. It includes the necessary background information, the different systems used to relocate the plastic hinge, comprehensive overview of the pseudoelastic response of SMA, and a literature review about the applications of SMA material in the design of concrete structures. • Chapter 3. The design and the experimental validation of the first phase are included in this chapter. • Chapter 4. The design and the experimental validation of the second phase are included in this chapter. 11
• Chapter 5. The development and the validation of the analytical model in the third phase are included in this chapter. • Chapter 6. The development and the validation of the FE models in the third phase are included in this chapter • Chapter 7. The main conclusions and the proposed recommendations for future research are included in this chapter.
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Literature Review 2.1 Introduction Over the past few decades, there has been an increasing trend in developing structural systems that possess high ductility capacity and can withstand severe earthquake excitations with minor damage. The Direct Displacement Based Design (DDBD) method is emerging as a substitute for the current Force Based Design (FBD) method due to the shortcoming of the latter (Priestley, 2007; Priestley et al., 2007; Dwairi et al., 2007). Fundamental problems inherited in the FBD method can result in an unsatisfactory structural performance or in over-designed structural members. The interdependency of the strength and stiffness, inaccuracy in determining the fundamental period, inconsistency in defining the ductility, and the simplistic relationship between the ductility capacity and the force reduction factors are all problems associated with the use of the FBD method in the seismic design of structures (Priestley et al., 2007). Alternatively, the emerging DDBD philosophy provides a more rational base in which the design of the structural members is conducted by determining the displacement induced by the earthquake rather than the force associated with its acceleration. Therefore, a good understanding of the damping and the ductility properties is highly emphasized in the DDBD method. The procedure adopted in the DDBD method is summarized as follows: First, the displacement ductility is determined for the particular structure based on the importance of the structure and the region in which it is built. Secondly, the maximum anticipated displacement is calculated. Thirdly, the damping ratio is determined based on the type of the structure and the level of the displacement ductility (Figure 2-1(a)). Fourthly, the effective period of the structure (period based on the ultimate condition) is obtained from the maximum anticipated displacement and the damping ratio (Figure 2-1(b)). Finally, the effective
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stiffness of the structure (stiffness based on the ultimate condition) is calculated using the wellknown equation of a SDOF oscillator as follows; 4π 2 me ke = Te2
Equation 2-1
where ke is the effective stiffness of the structure at the ultimate condition, me the effective mass of the structure participating in the fundamental mode of vibration, and Te is the effective period at the maximum displacement response.
(a) Damping ratio-ductility relationships
(b) Design displacement spectra
Figure 2-1: The fundamental procedure of the DDBD method (Priestley et al., 2007)
Based on the aforementioned procedure, it is clearly shown that the damping ratio and the ductility capacity are the most important aspects in this design method. For the same displacement ductility and earthquake induced displacement, an increase in the damping ratio leads to an increase in the period of the structure. Knowing that the period of the structure is inversely related 14
to the effective stiffness yields the following conclusion. If the structural system can exhibit stable and reliable hysteretic response associated with high levels of energy dissipation, the stiffness of the structure can be relaxed. This conclusion defeats the basic principle associated with the current code design method (FBD) which states that members with higher stiffness are desired in order to achieve a maximum upper bound displacement limit based on the ‘equal displacement concept’ (i.e. the displacement ductility factor is equal to the force-reduction factor). Based on the DDBD approach, in other words, hybrid systems involving materials with relatively low modulus of elasticity, such as Fibre Reinforced Polymer (FRP) and Shape Memory Alloys (SMAs), can show adequate seismic performance as long as they possess a stable hysteretic response associated with high energy dissipation capacity. This important observation opens the door for using innovative materials in the seismic design of structures keeping in mind the importance of achieving high levels of damping. Moreover, the current trends in the design of earthquake resisting structures aim to design structural systems that experience little, if any, damage after being subjected to high magnitude earthquakes. This criterion led to the development of the so called ‘Self-Centering’ connections in which the connection is capable of re-centering itself once the earthquake motion damps. Selfcentering implies, in most of cases, nonlinear elastic response in which the energy induced is dissipated by means of attached dampers. This leads to more design complications and results in increased construction cost. Therefore, further research is needed in order to develop simple, yet reliable systems in which the connection is capable of re-centering itself while dissipating the induced energy. From the discussion presented so far about the DDBD procedure and the criteria of practical self-centering systems, SMA materials with psuedoelastic response seem to be a good 15
candidate for the design of Self-Centering concrete connections although they possess low modulus of elasticity. The performance of such connections can be further improved if the plastic hinge is relocated away from the face of the column. Plastic hinge relocation helps in maintaining the integrity of the joint due to the mitigation of the yield penetration. In this study, the Single Slotted Beam (SSB) system and the Double Slotted Beam (DSB) system are used to relocate the plastic hinge. In the remainder of this chapter, a comprehensive literature review is presented about the topic of developing Self-Centering concrete connections with relocated plastic hinges using SMA. The capacity design philosophy is first addressed followed by a discussion about the different state-of-the art techniques of plastic hinge relocation. The research progress of the slotted beam system is also presented. Afterwards, the behaviour of Psuedoelasticity (PE) SMA is thoroughly discussed with emphasis on its applications in the context of earthquake engineering.
2.2 Capacity Design Philosophy Provisions for the seismic design in the current design codes adopt the capacity design philosophy in which distinct elements of the primary lateral force resisting system are suitably designed and detailed for energy dissipation under severe deformation. Figure 2-2 describes the analogy of the capacity design method where a very ductile link (usually lower strength) is used to achieve adequate ductility for the entire chain, while the other links are assumed to be brittle. The ductile link represents the plastic hinge in real life structures.
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Figure 2-2: Capacity design analogy (Elnashai and Di Sarno, 2008)
The Capacity design of earthquake resisting Reinforced Concrete (RC) structures involves determining a kinematically admissible plastic mechanism (Paulay and Priestley, 1992). The mechanism should satisfy the ductility demand while maintaining small inelastic rotation in the plastic hinges in order to achieve minimum permanent deformation. Two different designs: (a) Strong-Column Weak-Beam (SCWB) and (b) Weak-Column Strong-Beam (WCSB) are shown in Figure 2-3. The capacity of the columns in the SCWB configuration is higher than that of the beams, thus, the plastic hinges form in the beams close to the column face. However, the capacity of the beams in the WCSB configuration is higher than that of the columns, and thus, the plastic hinges form in the column near the joint. The plastic hinge rotations in the SCWB configuration, θ1, are much smaller than those in the WCSB configuration, θ2, for the same maximum displacement, ∆, at roof level. Therefore, the overall ductility demand is more efficiently achieved when the plastic hinges are placed in the weak beams in the SCWB configuration. The goal of the capacity design is to ensure that the mechanism shown in Figure 2-3 (a) can only be developed.
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(a) SCWB configuration
(b) WCSB configuration
Figure 2-3: Comparison of mechanism in SCWB and WCSB configurations (Paulay and Priestley, 1992)
The plastic hinges are designed to dissipate the energy absorbed from the earthquake through the yielding of the flexural steel. Therefore, the shear strength of the plastic hinge is designed to be higher than the flexural strength, even when the plastic hinge reaches the overstrength moment, in order to avoid brittle failure. In this way, the components of the structure other than the plastic hinges, behave in a linear manner theoretically (Paulay and Priestley, 1992). The yielding of the flexural reinforcement results in the elongation of the members, and hence, results in a permanent damage to the structure. The level of damage depends on many aspects such as the earthquake magnitude, structural design, construction practice, etc. According to this philosophy, the structure may not be usable after being subjected to the earthquake motion even if the structure does not fully collapse. The significant amount of post-earthquake damage necessitates the need to develop alternative designs and innovative construction practices to minimize the permanent damage of the structures without sacrificing the ductility capacity at the plastic hinges. 18
2.3 Relocation of Plastic Hinges The high seismic shear forces applied to beam-column connections may trigger brittle shear failures of the joints before even developing the full flexural capacity in the beams (Figure 2-4). The conventional way of mitigating this action is to heavily reinforce the joint such that to ensure adequate elastic performance.
Figure 2-4: Joint shear failure in RC structures subjected to Kocaeli, Turkey earthquake (Sezen et al., 2003)
The development of the negative moment (and in many cases the positive moment as well) in the beams right at the face of the column leads to the so called ‘yield penetration’ in which the high strain magnitudes (multiples of the yield strain) that are developed in the longitudinal bars penetrate into the joint (Zhao and Sritharan, 2007). The penetration of the yield strain is associated with a degradation in the bond, and consequently, severe degradation in the joint shear mechanism takes place (strut and truss mechanisms). Therefore, research studies were conducted to relocate the plastic hinge away from the face of the column, and thus, mitigate the severe consequences associated with yield penetration and joint shear distortion (Park and Miburn, 1983; Abdel-Fattah and Wight, 1987; Al-Ayed et al., 1993; Al-Haddad, 1990; Paulay and Priestley, 1992; Mahini and Ronagh, 2011; Dalalbashi et al., 2012). The relocation of the plastic hinge will also reduce the 19
congestion and the amount of shear reinforcement in the joint since the inelastic shear demand is reduced, and hence, relocation of plastic hinges results in economic benefits. It will also help in distributing the plasticity in the longitudinal reinforcement for longer distances and avoid sudden failures due to stress concentrations (the stress concentration is avoided by debonding the steel reinforcement from the concrete near the location of vertical slots in the SSB and DSB systems – refer to Section 3.2.3.2). The relocation of the plastic hinge can be achieved by special detailing of the steel reinforcement in the connection, changing the geometry of the structure, or reinforcing using FRP material. Examples of relocating the plastic hinge using special detailing of the steel reinforcement can be seen in Figure 2-5 (a), (b), and (c). Cross diagonal steel reinforcing bars are placed such that the critical distance is one beam depth away from the face of the column (Figure 2-5 (a)). It is noted that cross diagonal reinforcement can be also used at mid-span between beam stubs that are heavily reinforced to prevent beam hinging at the column face (Buchanan, 1979). Other methods of plastic hinge relocation using different detailing schemes can be seen in Figure 2-5 (b) and (c). These methods are based on over reinforcing the beam section such that the flexural strength near the column face can only reach the nominal flexural strength while the critical section can develop the overstrength moment (around 1.24 times the nominal flexural strength (Derecho and Kianoush, 2001)). An example of relocating the plastic hinge by changing the geometry of the structure is shown in Figure 2-5 (d). The increased depth of the beam near the column in the haunch configuration results in higher moment capacity, and thus, it will resist higher applied moments and remain elastic while the plastic hinge forms at a distance of one beam depth away from the face of the column. 20
(a) Cross diagonal reinforcement (Paulay and Priestley, 1992)
(b) Bent reinforcement detailing (Paulay and Priestley, 1992)
(c) Straight reinforcement detailing (Al-Ayed et al., 1993)
(d) Haunch configuration (Paulay and Priestley, 1992)
(e) Web-bonded FRP wraps (Mahini and Ronagh, 2011) Figure 2-5: Plastic hinge relocation techniques
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Plastic hinge relocation can also be achieved by installing web-bonded FRP sheets on the concrete beam near the column face (Figure 2-5 (e)). The FRP sheets enhances the flexural capacity of the beam member near the column. The unwrapped concrete section, therefore, will reach the yield moment while the wrapped section remains elastic. In other words, the plastic hinge forms at the interface between the wrapped and the unwrapped sections (Mahini and Ronagh, 2011; Dalalbashi et al., 2012). The major drawback of the aforementioned plastic hinge relocation techniques is the high level of inaccuracy in determining the location of centre of rotation. The drift capacity of concrete structures is conventionally calculated by assuming that the structure behaves like a rigid element with all the rotation concentrated at the plastic hinge. Even though the inaccuracy in determining the location of centre of rotation does not disqualify the system in terms of its structural performance, it leads to inaccurate prediction of the drift capacity of the structure. In this research, the SSB and the DSB systems are used to relocate the plastic hinge away from the face of the column resulting in enhanced performance of the connection and improved accuracy in determining the location of the centre of rotation. The relocation is achieved by moving the vertical slots away from the face of the column.
2.4 Research Progress in the Behaviour of SSB System The cracking of the floor slabs and the significant elongation in the beam length when RC structures are subjected to earthquake excitation are among the main reasons that motivated Ohkubo et al. (1999) to develop an innovative beam-column design configuration that mitigates these problems. The system includes a vertical slot in the beam at the column face, and thus, it is referred to as a ‘slotted beam’ system as shown in Figure 2-6. The simple design configuration, 22
yet the spectacular seismic performance of the system, makes it a potential seismic design alternative. The bottom reinforcement yields in tension and compression under positive and negative bending, respectively. This action will lead to an elastic response in the top reinforcement, and hence, the non-tearing action of the attached slab is achieved. Consequently, the plasticity is concentrated in the bottom reinforcement while less damage in the top concrete fibres is expected. In conventional design, high strains are developed in the tension side compared to low strains at the compression side of the beams, and thus, the average strain at the mid-section will be tension resulting in net elongation. The elongation of the beam member can lead to two possible mechanisms or a combination of both as follows: the columns on both sides of the beam will restrain the elongation of the member resulting in compression forces in the beam, and hence, an increase in the moment capacity is expected based on the load-moment interaction diagram (overstrength the beam). This behaviour is referred to as the ‘membrane effect’ and was observed in experimental testing of scaled RC moment resisting frames (Vecchio and Balopoulou, 1990). It was found that a beam elongation of approximately 5 mm (a value several times larger than would be predicted using elastic analysis) yields an increase in the beam moment capacity up to 25%. The second mechanism would occur in the case of columns with low bending stiffness in which the elongated beam acts like a point load positioned at the middle of the column. In this case, the column may suffer from increased bending stresses. It is anticipated that in real life scenarios, both mechanisms take place at the same time in different proportions. The mechanisms may lead to premature failure of the system or inaccuracies in predicting the response, and thus, implementing the slotted beam system will mitigate them due to its minimal beam elongation. The nearly elastic response of the top reinforcement in the SSB system will, therefore, ensure small beam elongation 23
during the earthquake excitation and avoid the above-mentioned consequences of beam elongation. The bottom reinforcement is unbonded from the surrounding concrete for a certain length in order to enhance the drift capacity and decrease the likelihood of the low cyclic fatigue failure (refer to Section 3.3 for more discussion about the low cyclic fatigue failure in SSB connections). Additionally, diagonal reinforcement extending from the joint into the bottom side of the beam is installed in order to enhance the shear resisting mechanism during the negative bending.
Figure 2-6: Schematic drawing of the slotted beam system
In the context of this research, this system is referred to as ‘Single Slotted Beam’ (SSB) in order to distinguish it from the ‘Double Slotted Beam’ (DSB) system in which two vertical slots are introduced. The designs of the SSB and DSB systems encounter several concerns such as, buckling of the bottom reinforcement, low cyclic fatigue failure, anchorage failure, and sliding shear failures. Only the general behaviour of the experimentally tested specimens is presented with regard to the research progress in this field.
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Ohkubo and Hamamoto (2004) tested two SSB interior beam-column connections with floor slabs under cyclic loading and compared the performance with conventional interior beamcolumn connection designed for the same ultimate flexural strength. The hysteretic response of one of the SSB connections and the conventional one are shown in Figure 2-7. The response of the slotted beam connection is stable and did not show any signs of pinching shear compared with the response of the conventional connection which experienced pinching shear early in the cyclic loading. Consequently, for the same drift ratio, the energy dissipated in the former is higher than the latter.
(a) SSB connection
(b) Conventional connection
Figure 2-7: Hysteretic response of the connections tested by Ohkubo and Hamamoto (2004)
The cracking pattern of the floor slabs of one of the slotted beam connections is compared with the cracking developed in the conventional one as shown in Figure 2-8. A remarkable reduction in the crack number and size is observed which proves the non-tearing action of the system. Also, the elongation of the beam in the SSB connection was negligible compared to the conventional one.
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(a) SSB connection
(b) Conventional connection
Figure 2-8: Cracking pattern of the connections tested by Ohkubo and Hamamoto (2004)
Considerable progress in the design and behaviour of SSB connections was achieved by Au (2010). The design considerations, the flexural mechanism, and the shear mechanism were all discussed and studied thoroughly. Experimental testing along with analytical analysis were conducted to confirm the behaviour of the system. One exterior and one interior SSB connection were tested under cyclic loading. The hysteretic responses of the two connections are shown in Figure 2-9. A stable hysteretic response was observed for the exterior connection up to a drift of 3.5% after which the bottom reinforcement fractured. Minimal cracking and concrete spalling took place during the test. Unlike the behaviour of the exterior connection, the interior connection showed less stable hysteretic response and pinching shear at 2.5% drift due to the strain penetration into the joint and the slippage of the bottom reinforcement. It is noted that the ultimate drift ratios of both connections are low, and thus, improvements into achieving higher drifts are needed. A simple analytical model and a multi-spring model were also developed to predict the flexural response of the connections. The models showed satisfactory results when compared with the experimental behaviour and were used to perform sensitivity analyses.
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(a) Exterior SSB connection
(b) Interior SSB connection
Figure 2-9: Hysteretic response of connections tested by Au (2010)
Two interior SSB connections were tested under cyclic loading by Byrne and Bull (2012). The specimens were designed based on the recommendations of Au (2010) and the NZS 3101-1. (2006) code. The hysteretic response was observed to be stable and dissipated large amount of energy without signs of pinching shear (Figure 2-10). The specimens failed by the fracture of the bottom steel reinforcement due to the low cyclic fatigue effect at a drift ratio of 5.5%. The top reinforcement remained elastic up to 3.5% drift leading to a total beam elongation of 0.7%, which is lower than 2-5% elongation observed in conventional connections as reported in Lau (2001) and Fenwich and Megget (1993).
Figure 2-10: Hysteretic response of two interior SSB connections tested by Byrne and Bull (2012) 27
A recent research project was conducted at the University of Canterbury to investigate the performance of two-storey two-by-one bay RC SSB superassembly under earthquake-like motion (Muir et al., 2012a,b,c). The superassembly consisted of precast floor members attached to 3D SSB connections as shown in Figure 2-11. The system and loading configurations of the superassmbely allowed the interaction between shear, torsion, and flexure. Test results indicated the excellent performance of the system when subjected to seismic loading in terms of minimal residual drift and damage. It was also observed that the diagonal reinforcing bars were heavily loaded. Fewer and smaller cracks were formed in the slotted system as compared with conventional connections (Figure 2-11). It was also observed that the beam elongation was restrained by the floors which induced compressive forces in the beams and tension forces in the floors.
Figure 2-11: Cracking pattern in the RC SSB superassembly tested by Muir et al. (2012b)
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2.5 Shape Memory Alloy (SMA) 2.5.1 Overview Shape Memory Alloys (SMAs) are characterized by the Shape Memory Effect (SME) and the Psuedoelasticity (PE) behaviour. The SME is defined as the ability of the material to recover its original parent shape when heated while the PE is defined as the ability of the material to undergo hysteretic superelastic response when mechanically loaded at high temperatures (Shaw and Kyriakides, 1995; Kumar and Lagoudas, 2007). The thermal and mechanical behaviours in SMA are coupled, and thus, it is best described through examining the phase diagram. The phase diagram for a typical SMA material is illustrated in Figure 2-12. The abscissa and the ordinate axes represent the stress and the temperatures, respectively. The material can be found in two phases; the high temperature Austenitic phase (parent) and the low temperature Martensitic phase (product). Furthermore, the Martensitic phase is divided into twinned and detwinned phases. The material is transformed from the Austenitic phase to the Martensitic phase via the forward transformation (Martensitic transformation) while it is transformed from the Martensitic phase to the Austenite phase via the reverse transformation. The transformation between the two phases can be achieved by the application of heat or by inducing mechanical loads. There are four characteristic temperatures; Martensite start temperature (Ms), Martensite finish temperature (Mf), Austenite start temperature (As), and Austenite finish temperature (Af). These temperatures increase when stress is applied during the Martensitic and the reverse transformations. The four inclined lines (transformation lines) represent the slope at which the transformation temperatures increase with the application of mechanical loads. Also, the detwinning start stress (σds) and the detwinning finish stress (σdf) are shown in Figure 2-12. For
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temperatures below Mf, the material is present in the twinned Martensitic phase if the stress is below σds while it is present in the detwinned Martensitic phase if the stress is higher than σdf.
Figure 2-12: A typical phase diagram of SMA material
When the material is cooled from the parent Austenitic phase to temperatures below Mf following path 1, it transforms to the twinned Martensitic phase which does not encounter a macroscopic shape change in the material. Loading the SMA mechanically for stresses higher than σdf following path 2 transforms it to the detwinned Martensitic phase which encounters a macroscopic shape change in the material. Upon unloading the material following path 3 and heating it following path 4, the material retains its original parent Austenitic phase with no macroscopic shape change. Thus, cooling following path 1, then loading following path 2, then unloading following path 3, and finally heating the material following path 4 is described as the SME. This behaviour is often referred to as ‘one-way’ SME since the transformation from Martensitic phase to the Austenitic phase is achieved with one cycle of heating. However, the 30
SMA material can experience repeatable shape changes when subjected to thermo-mechanical cycling (material training). This behaviour is termed the two-way SME. If the material is cooled while subjected to high stresses (above σdf) following path 5, then it will transform directly to the detwinned phase (macroscopic shape change) and will return to the Austenitic phase upon heating following path 6. The PE is exhibited when the material is mechanically loaded at a temperature higher than Af (Austenite phase) following path 7. The material starts the transformation to the Martensitic phase when the stress reaches the stress at the Martensite start temperature, σMs (the stress at the intersection between the loading path 7 and the transformation line of Ms), and completes when the stress is increased beyond the stress at the Martensite finish temperature, σMf (the stress at the intersection of the loading path 7 and the transformation line of Mf). When the material is unloaded following path 8, it transforms from the Martensitic phase to the Austenitic phase when the stress reaches the stress at the Austenite start temperature, σAs (the stress at the intersection between the loading path 8 and the transformation line of As), and completes when the stress is increased beyond the stress at the Austenite finish temperature, σAf (the stress at the intersection between the loading path 8 and the transformation line of Af). The stress-strain hysteretic responses of SMA wires subjected to mechanical loading at temperatures, T, greater and less than Af are shown in Figure 2-13 (diameters of 0.5 mm and 0.75 mm for T > Af and T < Af, respectively). When mechanically loaded at T > Af, the material exhibits a PE response. The stress increases linearly with increasing strain in accordance to the modulus of elasticity of the Austenitic phase, EA, until it reaches the σMs (see Figure 2-12). During the Martensitic transformation from the Austenitic phase to the Martensitic phase, the material exhibits an increase in the strain that is not associated with an increase in the stress when the material is 31
stretched. Beyond the completion of the transition (σMf is reached), the material is in the Martensitic phase and the stress increases linearly with increasing strain. Upon unloading, the stress decreases linearly in relation to the modulus of elasticity of the Martensitic phase, EM. The reverse transformation from the Martensitic phase to the Austenitic phase takes place when the stress reaches σAs and completes when the stress reaches σAf. During the reverse transformation, a clear plateau is observed in the stress-strain relationship that is similar to that in the Martensitic transformation. The typical values of EA range from 0.15Es to 0.42Es (where Es is the modulus of elasticity of steel, 200 GPa) while the typical values of EM range from 0.11Es to 0.21Es. It also exhibits an elongation at failure in the range of 5% to 50% (DesRoches et al., 2004). Overshoot of the stresses occurs at the initiation of the Martensitic and reverse transformations as can be seen in Figure 2-13. Afterwards, the stresses necessary to create the nucleus of the Martensitic and the Austenitic phases are relaxed in the transformation regions since there is less resistance to the movement of the transition fronts (Tobushi et al., 1998).
Figure 2-13: Stress-strain relationships of SMA wires at T > Af and T < Af (reproduced from Matsui et al. (2006)) 32
The behaviour of the material when loaded at T < Af is described as follows. The stress increases linearly with increasing strain up to a stress of approximately 300 MPa (stress corresponding to σMs). After that, the behaviour is softened before the stress goes back to linearly increasing with increasing strain. With the increase in the stress, the material starts to transform from the twinned Martensitic phase at a stress of σds to the detwinned Martensitic phase at a stress of σdf (see Figure 2-12). During this transformation, the stress-strain relationship exhibits a plateau in which the increase in the strain is not associated with an increase in the stress. The loading and the unloading linear portions of the curve have the slope of EM. A residual strain is thus attained which can be recovered upon heating the material to T > Af (i.e. SME). It should be noted that the stress-strain relationships shown in Figure 2-13 were obtained under controlled conditions in which the temperature of the material was kept constant during the loading and unloading schemes (refer to Section 2.5.2.1 for more information about the effect of the controlled testing condition). The PE behaviour of SMA (mechanically loading at T > Af) was found to be very advantageous in the seismic design and retrofitting of structural members due to it psuedoelastic response, energy dissipation capabilities, and corrosion resistance capabilities (Soul et al., 2010). It has been utilized in designing structural elements with re-centering abilities and in the design of dampers (refer to Section 2.5.3.2). On the other hand, the use of SME is utilized primarily for confinement of columns for strengthening purposes (refer to Section 2.5.3.1). In this research, the PE property of SMA is used in the design of concrete beam-column connections with selfcentering abilities, and therefore, the different factors that influence the PE behaviour of SMA are examined in the following sections.
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2.5.2 Factors Affecting the Behaviour of PE SMA 2.5.2.1 Effect of Temperature The coupled nature of the thermal and the mechanical behaviour of SMA implies that cooling or heating the material will result in changing the mechanical response in terms of the transformation stresses of the PE behaviour; σMs, σMf, σAs, and σAf. Knowing the actual strength of the structural members is considered to be one of the important parameters in the seismic design in addition to the stiffness and the ductility properties (Paulay and Priestley, 1992; Elnashai and Di Sarno, 2008). Therefore, critical evaluation with respect to thermal effect on the strength properties of the material is essential before it can be reliably used in the seismic design of structures. The Martensitic and the reverse transformations take place through fronts moving along the material. The moving fronts act as heat sources that release heat (exothermic) or absorb heat (endothermic) during the Martensitic and the reverse transformations, respectively. This energy is either transferred to the surrounding environment by heat convection and heat conduction along the wire or stored in the material (Lin et al., 1994; Soul et al., 2010). Therefore, loading the SMA mechanically above Af (PE behaviour) will increase the temperature of the material while unloading it tends to reduce it depending on the strain rate of the applied loading. Thus, two identical pieces of SMA material may exhibit different hysteretic responses depending on the environmental condition during testing; controlled or un-controlled. In controlled testing, the temperature of the material is adjusted during stressing to keep it constant while in the un controlled testing only the ambient temperature is controlled (i.e. the temperature of the material changes in accordance to the Martensitic and the reverse transformations) (Tamai and Kitagawa, 2002; Pieczyska et al., 2005). The uncontrolled temperature condition corresponds to applications in which the SMA is used in air without airflow while the controlled temperature condition 34
corresponds to applications in which the SMA is exposed to high velocity airflow or used in water (Pieczyska et al., 2005). The thermal conductivity of concrete is around 150 times that of dry air (Morabito, 1989; Hilsenrath, 1955), therefore, the behaviour of SMA in concrete structures is inbetween the controlled and un-controlled conditions. The variation of the surface temperature with respect to time during a uniaxial test of an SMA wire is shown in Figure 2-14. The variation of the strains and stresses are also shown in the same figure. It can be seen that the rise and the fall in the temperature during the Martensitic and the reverse transformations follow the same pattern and that the net temperature change is almost zero (Tamai and Kitagawa, 2002). It should be noted that the temperature measurements in this study were obtained from readings of the air conditioner in the test room and not from a thermocouple or thermograph.
Figure 2-14: Variation of the surface temperature of a PE SMA wire under uniaxial tension test under cyclic temperature (Tamai and Kitagawa, 2002)
Tobushi et al. (1999) concluded that the actual temperature of the SMA measured using a thermograph can be an order of magnitude larger than that measured through a thermocouple. The 35
variation of net temperature change obtained from the study by Tobushi et al. (1999) is plotted against the transformation strain at different strain rates as shown in Figure 2-15. The transformation strain, εM, is defined as the strain that is induced once the Martensitic transformation starts (once the stress reaches σMs in Figure 2-12 and Figure 2-13).
Figure 2-15: Temperature change versus the transformation strain (Tobushi et al., 1999)
The effect of changing the temperature of the SMA on the stress-strain relationship in a controlled environment is shown in Figure 2-16. The Af of the wire is 323° K. It is clear that the transformation stresses (σMs, σMf, σAs, and σAf) increase with increasing temperature. This behaviour can be easily understood by examining the phase diagram shown in Figure 2-12. Due to the positive slope of the transformation lines, the increase in the applied temperature results in higher characteristic stresses in relation to the slopes of the transformation lines. The PE SMA material is commonly used in damper devices, and hence, analyzing the effect of temperature on the energy dissipation and storage properties is important (Pieczyska et al., 2005). The area inside the hysteresis loop of the stress-strain curve represents the dissipated/absorbed energy, Wd, while the area under the unloading curve represents the strain 36
energy (recoverable energy), Er, as illustrated in Figure 2-17 (a). The effect of varying the temperature on the Wd and Er is shown in Figure 2-17 (b). It is noted that the Af temperature of the material is 323° K. The Wd increases linearly with increasing temperature, while Er is almost constant for temperatures below Af and increases linearly with increasing temperature for temperatures above Af. The temperature at which Wd and Er are equal is approximately 367.5° K. This value can be used for optimization purposes in which the balance between Wd and Er is required.
Figure 2-16: The effect of temperature on the stress-strain responses of SMA wires (Pieczyska et al., 2005)
(a) Definition of energies; Wd and Er
(b) Energy relationships with temperature
Figure 2-17: Energy definitions and temperature effect (Pieczyska et al., 2005) 37
2.5.2.2 Effect of Loading Rate The rate of the applied mechanical loading is one of the most important factors to be considered in the seismic design of structures due to the high frequency dynamic vibration associated with the earthquake excitation. Thus, the effect of rate of loading on the stress-strain of PE SMA needs to be evaluated thoroughly. The working seismic loading results in strain rates in the range of 5% s 1
(300% min-1) to 8% s-1 (480% min-1) (Tamai and Kitagawa, 2002), and thus, the dynamic effect
in the response of PE SMA should be also considered in relation with the thermal effect (Tobushi et al., 1999). The PE SMA subjected to different strain rates is first evaluated for controlled temperature conditions and then compared with the behaviour of un-controlled temperature conditions (refer to Section 2.5.2.1 for the definitions of controlled and un-controlled temperature conditions). The effect of the strain rate on the PE behaviour of SMA wires before and after training under controlled temperature is shown in Figure 2-18. It is noted that the strain rates 0.1, 0.5, 1, 2, 10, 30, 50, 100% min-1 correspond to strain rates of 1.67×10-3, 8.33×10-3, 1.67×10-2, 3.33×10-2, 1.67×10-1, 5.00×10-1, 8.33×10-1, 1.67×100% s-1 , respectively. The values σMs and σMf increase while the values of σAs and σAf decrease with increasing strain rate for strain rates higher than 10% min-1 (1.67×10-2 s-1). The slopes of the stress-strain curves in the Martensitic and reverse transformations increase with increasing strain rate. It is observed that the overshoot and undershoot (peak responses) in the stress values at the start and the end of the Martensitic and the Austenitic phases vanish with increasing strain rates. Training the material does not influence the effect of loading rate; however, the values of σMs and σMf are increased while the values of σAs and σAf are decreased as compared with the un-trained material. 38
(a) Before training
(b) After training
Figure 2-18: The effect of the loading rate on the hysteretic response of PE SMA wires (Tobushi et al., 1998)
The effect of the strain rate on the transformation stresses of trained and un-trained PE SMA can be better understood by examining Figure 2-19. The values of the transformation stresses do not depend on the strain rate for strain rates equal or below 2% min-1 (3.33×10-2 s-1) while they change with increasing strain rate for strain rates higher than 10% min-1 (1.67×10-2 s-1). For strain rates higher than 2% min-1 (3.33×10-2 s-1), the σMs and σMf increase with increasing strain rate at different slopes. The σAs exhibits a constant trend with increasing strain rate with a drop at strain rate of 2% min-1 (3.33×10-2 s-1), while the σAf experiences a decreasing slope with increasing strain rate. Training the material does not alter the trends of the transformation stresses. The time necessary for the stress to relax after the overshoot at the initiation of the Martensitic and Austenitic phases is found to be 2 ks (1 ks = 1000 s) (Tanaka et al., 1988). Thus, there is enough time for the stress to relax under low rates of strain. Under dynamic loading, internal friction resistance against the movement of the transition fronts increases since there is not enough time for the stress to relax.
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(a) σMs stress
(b) σMf stress
(c) σAs stress
(d) σAf stress
Figure 2-19: The effect of the loading rate on the transformation stresses (Tobushi et al., 1998)
In un-controlled conditions, the deformation properties of the PE in SMA differ than those in the controlled conditions (Pieczyska et al., 2005). The stress-strain relationships of two PE SMA wires with similar chemical composition tested under uncontrolled temperature conditions for two strain rates (6×10-3 min-1 (5×10-4 s-1) and 6 min-1 (10-1 s-1)) are shown in Figure 2-20 (a). Under low strain rates, the hysteresis loop in Figure 2-20 (a) exhibits the well-known flag shape experienced in PE behaviour. However, the hysteresis behaviour changes dramatically when subjected to high strain rates as compared with the response under approximately the same strain 40
rates but tested under controlled conditions (Figure 2-18). The variation of the Wd and Er energies during the applied loading are plotted against the increase in the strain rate as shown in Figure 2-21 for controlled and uncontrolled conditions. Unlike the behaviour of the controlled condition, the Er increases with increasing strain rate under the un-controlled conditions while the Wd decreases with the application of high strain rates. In order to examine the effect of the loading rate thoroughly, it is very important to consider the dynamic effect as well as the thermal effect on the PE behaviour of SMA. The high rate of loading leads to the increase in the temperature (up to 10-50 k Tamai and Kitagawa 2002) and the increase in the applied stress due to the well-known dynamic effect. In SMAs, the temperature is coupled with the transformation stress, and thus, the dynamic effect is more pronounced in the SMA than many other metals such as mild steel. Under temperature controlled testing (representing the isothermal case), the thermal effect is isolated and the change in the behaviour is solely due to the effect of the dynamic loading in increasing the applied stress during the loading and decreasing it during the unloading stages. However, in the case of the un-controlled conditions, both the thermal and the dynamic effects exist and affect the response of the material. The change in the behaviour is more significant in the case of high rates of loading since the time is very short for the heat generated by the Martensitic transformation to be transferred into the air. This behaviour is clearly shown in Figure 2-20 (b) in which the temperature variation and the stressstrain behaviour are plotted in the same figure. Due to the fact that the reverse transformation stresses increase with increasing temperature, the slope of the stress-strain curve during the unloading is steeper as compared with that of the controlled condition. It should be noted that under very high strain rates, the dissipated energy per cycle did not experience significant variation despite the increase in the temperature due to the dynamic effect, and therefore, the damping 41
properties of the SMA may not deteriorate under the dynamic loading (Tamai and Kitagawa 2002). Even though the dynamic loading has a significant effect on the behaviour of SMA, the recentering ability (low permanent deformation) does not depend on the testing rate (DesRoches et al., 2004).
(a) Effect of loading rate
(b) Temperature variation
Figure 2-20: Behaviour of PE SMA under uncontrolled conditions (Pieczyska et al., 2005)
(a) Er energy
(b) Wd energy
Figure 2-21: The effect of the testing condition on the energy behaviour of PE SMA (Pieczyska et al., 2005) 42
The beam-column connections were subjected to quasi-static loading in this research, and thus, the PE SMA bars were subjected to low loading rates. Based on the aforementioned literature review, the temperature effect vanishes and the change in the material response is solely due to the dynamic effect which is very small in this case.
2.5.2.3 Effect of Cyclic Loading Subjecting the SMA material to thermal or mechanical cycles until the hysteretic response stabilizes is referred to as training. In the design of structures, it is important to be able to determine the hysteretic response of the constituent materials in order to be able to perform dynamic and seismic analyses. This can be achieved if the characteristics of the stabilized hysteretic response are available. Only the effect of mechanical cycling is considered in this section since it is related to the present research study while the reader is referred to Hartl and Lagoudas (2007) for more information about the effect of thermal cycling. Cycling (training) the material can have significant effects on the properties and the hysteretic response as can be seen in Figure 2-22 in which 2.46 mm and 0.5 mm diameter PE SMA wires were subjected to training of 100 cycles (Soul et al., 2010). The energy and the residual strain at the stabilized cycle were approximately 60% less and 75% more than those at the first cycle, respectively, for the 2.46 mm PE SMA wire. Several studies examined the number of cycles needed to achieve the stabilized behaviour. Experimental testing concluded that the number of cycles range between 20 to 60 (Kawaguchi et al., 1991; Tobushi et al., 1991; Lin et al., 1994; Pieczyska et al., 2005; Moumni et al., 2005; Soul et al., 2010), and that cycling the material for 100 cycles will ensure a stabilized behaviour (Maletta et al., 2012). Cyclic hysteretic response of PE SMA wires tested for 20 cycles under temperature controlled condition is shown in Figure 2-23. It is observed that the Martensitic and the reverse transformation stresses 43
decrease with increasing number of cycles, while the residual strain increases with increasing number of cycles.
(a) 2.46 mm diameter wire
(b) 0.5 mm diameter wire
Figure 2-22: Mechanical training of PE SMA wires for 100 cycles (Soul et al., 2010)
Figure 2-23: Effect of cyclic loading on the stress-strain relationship of PE SMA (Tobushi et al., 1998) 44
During the first mechanical cycle, dislocations are generated and accumulated due to the Martensitic and the reverse transformations (Tobushi et al., 1998). These dislocations result in the development of internal stresses within the material which result in residual strains when unloading. In the following cycle, the Martensitic transformation takes place at lower stresses since the material is already internally stressed. The same behaviour takes place in the reverse transformation. With the increase in the number of cycles, the dislocations accumulate resulting in the increase in the internal stresses until the material stabilizes. The relatively large decrease in the Martensitic transformation stresses, as compared with the behaviour at the reverse transformation, is due to the fact that the former occurs at higher stresses than the latter, consequently, higher number of dislocations and higher internal stresses are developed in the former than the latter.
2.5.2.4 Fatigue Life Determining the fatigue characteristics of structural materials is essential in the seismic design and analysis of structures. Reinforcing bars in structures (steel, SMA, FRP, etc) are subjected to high amplitude cyclic loading during the earthquake excitation, and thus, examining the low-cyclic fatigue behaviour of the reinforcement as well as the fatigue life is needed. Different factors affect the fatigue behaviour besides the stochastic nature of the fatigue phenomenon such as; the material composition, testing environment, and loading conditions. Thus, a large database of fatigue tests of PE SMA is needed before reliable conclusions are made with regard to its fatigue life predictions. Unfortunately, up to date, the fatigue testing of SMA in seismic applications is very limited, and thus, understanding the fatigue mechanism is not yet developed (Maletta et al., 2012). The fatigue testing of SMA is conducted using different testing techniques; alternating-plane bending (Furuichi et al., 2003; Matsui et al., 2006), rotating-bending (Tobushi et al., 2000; Matsui 45
et al., 2006; Figueiredo et al., 2009), pulsating-plane bending (Matsui et al., 2006), and uniaxial fatigue tests (Tabanli et al., 1999; Moumni et al., 2005; Maletta et al., 2012). Only the last two are useful in determining the fatigue life for structural purposes since the stress ratio S (minimum stress/maximum stress) can be varied unlike the test results obtained in the first two testing techniques in which the S ratio is always one and the specimens are subjected to both tension and compression at the same time. The fatigue lives of PE SMA wires subjected to pulsating-plane bending fatigue loading at a frequency of 500 cpm at room temperature for different S ratios, plotted with respect to the maximum strain, are shown in Figure 2-24 (a). It can be seen that the fatigue life is higher for higher S ratios. The endurance limit is approximately 0.65% strain which is close to the strain corresponding to the initiation of the Martensitic transformation (Matsui et al., 2006). Another study examined the fatigue life of PE SMA subjected to uniaxial fatigue loading in the transformation region with S equal to zero (Maletta et al., 2012). The strain amplitude versus the fatigue life curves are plotted in Figure 2-24 (b). The strain amplitude was divided into elastic and inelastic components. The elastic strain is calculated based on EM, while the inelastic component is the difference between the total strain and the elastic strain. It can be seen that the elastic and the inelastic strain amplitudes can be predicted accurately using power equations. The energy approach was also implemented in examining the fatigue life of PE SMA material subjected to uniaxial fatigue test (Moumni et al., 2005). The stress amplitude versus the fatigue life is plotted in Figure 2-24 (c) while the energy of the stabilized cycles (after training) versus the fatigue life is plotted in Figure 2-24 (d). For the same stress amplitude, the fatigue life is enhanced if the mean stress is zero (case of S = -1) which indicates the beneficial effect of the compressive loading on the fatigue behaviour. This is because the compressive stress tends to close 46
the fatigue micro cracks (Moumni et al., 2005). A linear trend of the dissipated energy versus the fatigue life is found for all the S values as can be seen in Figure 2-24 (d).
(a) Maximum strain (Matsui et al., 2006)
(b) Strain amplitude (Maletta et al., 2012)
(c) Stress amplitude (Moumni et al., 2005)
(d) Energy prediction (Moumni et al., 2005)
Figure 2-24: Fatigue life prediction of PE SMA obtained from the literature
2.5.2.5 Size Effect Most of the research conducted on PE SMA involved the use of small diameter wires ranging from 0.026 mm to 2.7 mm diameter (Soul et al., 2010; Chen and Schuh, 2011). This is due to two main reasons; its high cost and its applications. The high cost of the SMA hinders its ability to become a common construction material. Moreover, most of the early applications of SMA were in the 47
medical field and aerospace engineering, and thus, no motivation was driven to examine the behaviour of larger diameter bars (Kumar and Lagoudas, 2007). Recently, more research is being conducted in the structural field in which large diameter SMA reinforcement is used. In this research, large diameter SMA bars are to be used as internal reinforcements, and thus, understanding the size effect on the mechanical properties is necessary. Limited studies examined the effect of large diameter bars on the PE behaviour of SMA (DesRoches et al., 2004; Soul et al., 2010). The stress-strain relationships of four PE SMA bars tested under the same loading up to 6% strain are shown in Figure 2-25. It is noted that the samples were heat treated and tested under the same conditions.
Figure 2-25: Stress-strain relationships of PE SMA material with different diameters (DesRoches et al., 2004)
The hysteresis area of the 1.8 mm bar is the largest while it is the lowest for the 12.7 mm bar. Moreover, the strain hardening in the 12.7 mm bar is more pronounced than the other bars. The difference in the behaviour is further examined in Figure 2-26 by plotting the loading plateau stress (σMs), unloading plateau stress (σAs), residual strain, and the equivalent viscous damping 48
against the increase in the maximum cyclical strain. All bars had similar values of the loading stress except for the 1.8 mm while they encountered small variation of the unloading stress. The residual strains of all bars were low (less than 0.75% after been subjected to 6% cyclical strain) which means that the re-centering ability of the PE SMA is insensitive to the bar diameter. However, the equivalent viscous damping was found to depend on the size of the SMA bar.
Figure 2-26: Effect of bar size on the stress and strain characteristics of PE SMA material (DesRoches et al., 2004)
The difference in the response of PE SMA material due to the size effect can be understood by considering the surface-to-volume ratio (Soul et al., 2010). The surface-to-volume ratio is proportional to 4/db (where db is the bar diameter). For small values of db, the latent heat generated
49
during the Martensitic and the reverse transformations is transferred more effectively due to the high surface-to-volume ratio resulting in larger hysteresis loop (see Sections 2.5.1 and 2.5.2.1 regarding the discussion about the effect of latent heat on the properties of the material). On the contrary, large diameter PE SMA will have less surface-to-volume ratio, and therefore, the latent heat is not transferred as effectively as in the case of small wires. This results in the steep loading and un-loading transformation plateau since the increase in the heat leads to the increase in the transformation stresses.
2.5.3 Applications 2.5.3.1 General Applications The exceptional PE and SME properties of the SMA material makes it a potential candidate for the use in smart structures. There is a wide spectrum of research avenues in which the unique characteristics of the SMA was utilized (Song et al., 2006). In the context of the structural design of concrete members, the SME of SMA has been utilized in active confinement of columns (Choi et al., 2010), prestressing applications (Orvis, 2009), and self-healing structures (Song et al., 2006). The PE SMA material was used in damping applications, confinement of columns, reinforcing concrete members, and seismic design of self-centering connections and systems. In this section, the former three applications of PE SMA are discussed briefly followed by a detailed discussion of the fourth application in the following section since it is related to the work conducted in this research. The SMA material was used in damping application due to its superior PE response and energy dissipation ability. The excellent damping property of SMA was utilized in the design of structural dampers and base isolation systems (Witting and Cozzarelli, 1992; Zhang and Zhu, 50
2007). The ability of the PE SMA to absorb the energy induced by the motion and to dissipate it in the form of latent heat during the Martensitic and reverse transformations is responsible of the increasing use of PE SMA in damping applications. Most of the research conducted on the confinement of concrete columns using SMA wires utilized the SME rather than the PE behaviour. A recent research study by Choi et al. (2008), however, proposed a new confinement method and performed experimental testing on concrete cylinders confined using PE SME. It was found that the proposed confinement method results in a slight increase in the strength associated with a significant increase in the ductility Several research studies examined the use of PE SMA wires and bars in the design of concrete beams subjected to bending experimentally and analytically. Saiidi et al. (2007) performed pilot tests on concrete beams reinforced with PE SMA bars. Beams reinforced PE SMA rods exhibited considerably smaller residual deformations compared with the conventional ones. Composite FRP-PE SMA bars were fabricated and used to reinforce concrete beams (Wierschem, 2009; Wierschem and Andrawes, 2010; Zafar and Andrawes, 2012). Results indicated enhanced energy dissipation of the concrete structures as compared with those reinforced using conventional FRP. Khaloo et al. (2010) studied concrete cantilever beams reinforced with different quantities of steel and PE SMA rebars analytically. The inclusion of PE SMA was found to reduce the permanent deformation and enhance the crack closure upon unloading. Abdulridha et al. (2013) developed a preliminary constitutive model applicable to FE algorithms of PE SMA-reinforced concrete beams, which showed promising results. A newly developed PE SMA called Fe-Mn-SiCr-Ni-VC was used for prestressing applications in concrete beams (Lee et al., 2013). Test results showed the excellent performance of the prestressed beams and their ability to recover the prestress loss. Self-centering cement mortar members reinforced with PE SMA fibres were tested and 51
analyzed under cyclic loading (Shajil et al., 2013). The tested members were found to exhibit outstanding self-centering characteristics. Shrestha et al. (2013) examined a newly developed CuAl-Mn SMA exhibiting PE behaviour and enhanced recovery strain in reinforcing concrete beams. The concrete members demonstrated strong re-centering ability and enhancement in the crack recover capacity.
2.5.3.2 Self-Centering Concrete Structures Reinforced using PE SMA The PE property of SMA was also utilized in the design of self-centering structural systems in the seismic design of structures. Self-centering structural systems refer to the systems in which the induced energy is dissipated during the motion and results in minimal permanent deformation when the system unloads. Research on the applications of PE SMA material in limit-state design and the seismic design of concrete structures is still in the preliminary stages. A comprehensive literature review is presented in this section about the research progress in this field. The exploratory study by Saiidi and Wang (2006) was one of the pioneering studies in which the PE behaviour of SMA was utilized in the seismic design of self-centering RC structures. The behaviour of a concrete column reinforced using 15.9 mm diameter PE SMA bars at the plastic hinge and subjected to an axial load index of 0.25 (ratio of the axial stress to the concrete compressive strength) and earthquake-like motion was examined. The column was repaired by Engineered Cementitious Composites (ECCs) and subjected again to the same earthquake record. The load-displacement relationships of the column before and after repair are shown in Figure 2-27. In both tests, the load-displacement behaviour exhibited the well-known flag shape due to the PE property of SMA. The strain measurements of the PE SMA bars indicated maximum tensile and compressive strain values of approximately 2.5% and 0.6%, respectively. With reference to 52
Figure 2-13, it can be seen that these strain measurements yield relatively low amounts of energy dissipation. Overall, the column experienced outstanding performance in terms of the selfcentering ability (low permanent damage). A multi-segment hysteresis model, S-Hyst, consisting of linear segments that reproduce the essential features of the tested columns was developed and implemented in a computer program. The model predictions correlated well with the test results.
Figure 2-27: Hysteretic response of concrete columns reinforced using PE SMA tested by Saiidi and Wang (2006)
The concrete column reinforced using PE SMA tested by Saiidi and Wang (2006) formed the base for an extended analytical study by Saiidi et al. (2006) to compare the behaviour of the proposed connection with conventional connections. Three configurations were considered; conventional design, SMA in the plastic hinge, SMA and ECC concrete in the plastic hinge location. The connections were subjected to an axial load index of 0.1 and subjected to pushover lateral load and cyclic loading. Reinforcing using PE SMA was proven to enhance the selfcentering ability of the structures. However, the energy dissipated in the SMA reinforced structure and the SMA plus ECC reinforced structure decreased by 77% and 54%, respectively, as compared with the conventional structure at a drift ratio of 5%. It should be noted that the material models 53
and the modeling scheme adopted in the study (performed using the OpenSees software) involved many simplifying assumptions and did not represent the actual response of the PE SMA material. Thus, it is meant to examine the general behaviour of the system and not to obtained accurate results. Experimental evaluation of concrete beam-column connections reinforced using PE SMA at the location of the plastic hinge was conducted by Youssef et al. (2008). Two specimens were tested under cyclic loading while being subjected to an axial load index of 0.07. The SMA bars were placed at the location of the plastic hinge and coupled with the steel reinforcement from both ends using mechanical couplers. The Af temperature of the 20.6 mm SMA NiTi bar was in the range of -15 oC to -10 oC. The σMs and the EA were 401 MPa and 62.5 GPa, respectively. The hysteresis loops of RC joints reinforced at the plastic hinge location using conventional steel and SMA are shown in Figure 2-28 (a) and (b), respectively. The wider hysteretic loop of the steelreinforced connection indicated higher energy dissipation as compared with the SMA reinforced connection. The cumulative energy versus the story drift of the two specimens is shown in Figure 2-29. It is noted that significant slippage in the SMA anchors led to the increase in the permanent deformation and the energy dissipation. The difference between the cumulative energy trends of the two connections increased with increasing story drift. At the collapse limit 3% (defined by Elnashai and Broderick (1994)), the energy of the steel-reinforced connection was 29% higher than that of the SMA reinforced one. At the maximum drift (7.9%), the difference increased to 37%. It was also observed that the large cracks were formed at the location of the mechanical coupler in the beam which indicated the formation of a high stress concentration region at the coupler. The strain value in the steel and the SMA bar at the column face reached a maximum value of approximately 1% and 0.6%, respectively. The low strain value in the SMA means that 54
the PE elastic behaviour of SMA was not utilized efficiently since the Martensitic transformation strain is approximately 0.7% (Matsui et al., 2006). Consequently, the energy dissipated is low. Despite the excellent self-centering behaviour of the SMA reinforced connection, the low energy dissipation, the single wide crack at the coupler, and the low strains in the PE initiated the motivation to better design the connections, in this current research project, such that the unique properties of the PE SMA bars are utilized effectively (refer to Section 1.5).
(a) Steel-reinforced connection (JBC1)
(b) SMA-reinforced connection (JBC2)
Figure 2-28: Hysteretic response of beam-connections tested by Youssef et al. (2008)
Figure 2-29: Comparison of the cumulative energy versus story drift relationships of connections tested by Youssef et al. (2008)
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The behaviours of the beam-column joints tested by Youssef et al. (2008) were further examined by using the moment-area method in calculating the deflection of the hysteretic envelopes as shown in Figure 2-30 (JBC-1 and JBC-2 refer to the steel-reinforced and SMAreinforced connections, respectively) (Elbahy et al., 2010). It is noted that the stress-strain behaviour of the SMA reinforcement was idealized as bi-linear. This assumption is adequate given that only the envelope of the hysteretic response is modeled. The percentage error at the ultimate load and displacement of the SMA reinforced connection is 0 and 11%, respectively.
Figure 2-30: Load-displacement modeling of beam tested by Youssef et al. (2008) (Elbahy et al., 2010)
The column tested by Saiidi and Wang (2006) and the beam-column connection tested by Youssef et al. (2008) were modeled and validated using the Finite Element (FE) software SeismoStruct in a study conducted by Alam et al. (2008). The plastic hinge length, the crack width and spacing, and the bond-slip model of the SMA reinforcement inside the couplers (only in the case of the beam-column connection) were all incorporated in the models (Figure 2-31). However, the models do not account for the increase in the residual strain with cycling the SMA. The column model showed good agreement with the experimentally obtained result (difference of 5.6% and 56
6.1% for the base shear and tip displacement, respectively). Similarly, the model of the beamcolumn connection showed good agreement with the test results. The beam-column connection was modeled using two approaches; one with slip-bond effect inside the coupler and one without slip-bond effect. It was found that the predicted energy dissipation of the model including the slipbond effect and the model excluding the slip-bond effect was 16% lower and 17% higher than the experimental results. It is, therefore, concluded that only partial slippage took place inside the coupler in the test by Youssef et al. (2008).
(a) Connection tested by Youssef et al. (2008) (b) Column tested by Saiidi and Wang (2006) Figure 2-31: FE modeling of concrete structures reinforced using PE SMA (Alam et al., 2008)
Alam et al. (2009) performed an analytical pushover and dynamic analyses on eight-storey moment resisting frame reinforced with SMA at the location of the beam-column plastic hinges 57
and compared the behaviour with a concrete frame reinforced with conventional steel. The yield stress and the σMs for the steel and the SMA bars were 500 MPa and 480 MPa, respectively, while the modulus of elasticity of the steel and the SMA bar were 200 GPa and 68.2 GPa, respectively. The base shear versus the top storey drift curves of the two buildings are plotted in Figure 2-32 while the intra-storey drift of the floor levels are shown in Figure 2-33.
(a) Steel-reinforced
(b) SMA reinforced
Figure 2-32: Base shear-top storey drift relationships of concrete buildings (Alam et al., 2009)
(a) Steel-reinforced
(b) SMA reinforced
Figure 2-33: Inter-storey drift relationships of concrete buildings (Alam et al., 2009) 58
The strength and the stiffness of the SMA-reinforced building are lower than those of the steel-reinforced building while the inter-storey drift of the former is higher than the latter. The dynamic analysis consisted of subjecting the buildings to 10 ground motion earthquake records. In general, the SMA-reinforced building experienced higher maximum top and inter-storey drifts while it experienced lower residual drifts. Therefore, the SMA was proven to help in recentering the structure (minimum permanent damage) at the expense of the amount of the maximum drift. Another analytical study aimed at examining the performance of RC structures reinforced with SMA and subjected to pushover and dynamic analyses was conducted by Elfeki (2009). In this study, conventional RC moment resisting frames were designed and subjected to five scaled versions of the horizontal and vertical components of ground motion records. The critical sections at which damage is maximized were identified. Then, SMA reinforcement was used in these critical sections in order to optimize the design of the structure and reduce the material cost. Similar to the results obtained from the study performed by Alam et al. (2008), the optimized design was found to exhibit outstanding performance in terms of low damage, reasonable values of interstorey drift, and low values of residual inter-storey drift as compared with steel-reinforced buildings. The effect of using PE SMA reinforcement on the seismic performance of concrete buildings was investigated analytically by Alam et al. (2012) in which three building configurations were considered: steel reinforcement only (Steel), SMA in the plastic hinge and steel in other regions (Steel-SMA), and beams fully reinforced with SMA and steel in other regions (SMA). For each configuration, three different storeys were investigated; 3, 6, and 8. Pushover and dynamic time history analyses were conducted on each building in order to examine the effect of using SMA on the overstrength (ratio of the actual-to-designed base shear) and drift 59
performance. The pushover results indicate that the overstrength of the SMA frame is similar to that of the Steel frame which means that the procedure in the National Building Code of Canada (NBCC, 2005) can be used to calculate the base shear of the former. The ductility of the SMA frame was at least 16% less than the Steel frame while the ductility of the Steel-SMA frame ranged from 8% to 18% less than the Steel frame. Based on the capacity/demand ratios obtained from the time history dynamic analysis, the SMA frame is more effective up to 6 storey buildings, while the Steel frame proved to be more effective for buildings with 8 storey and higher. Nehdi et al. (2010) developed and tested a corrosion free beam-column connection. The connection had the same dimensions and material properties as the SMA reinforced connection tested by Youssef et al. (2008) with the exception that FRP reinforcement was used instead of the steel reinforcement. Therefore, a special mechanical anchor that connects the SMA to the FRP reinforcements was developed and validated. The hysteretic response of the connection and a conventional counterpart are shown in Figure 2-34. SMA-FRP reinforced connection did not exhibit the flag-shape that was encountered in the SMA-steel connection tested by Youssef et al. (2008). This is due to significant slippage that took place at the mechanical coupler in the former, which contributed in dissipating large amounts of energy and resulted in a significant permanent deflection. The energy dissipated and the residual drift at 4% drift of the SMA-FRP reinforced connection were 7.5% and 11.1%, respectively, more than those of the conventional counterpart. Unlike the SMA-reinforced connection tested by Youssef et al. (2008), the SMA-FRP tested in this research experienced inferior performance in terms of the ability of the system to re-centre itself.
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(a) Steel-reinforced
(b) SMA-FRP reinforced
Figure 2-34: Hysteretic response of beam-column connections tested by Nehdi et al. (2010)
An extensive analytical study using the SeismoStruct was conducted by Billah and Alam (2012) in which the seismic performance of corrosion-free concrete columns were examined. The proposed design consisted of SMA reinforcement at the location of the plastic hinge coupled with FRP reinforcement in the remaining portion of the columns and designed in accordance to CSA A23.3-04 (2004). This configuration (referred hereafter as SMA-FRP) was compared with another three configurations; SMA in the plastic hinge and stainless steel in the remaining of the column (SMA-SS), stainless steel in the plastic hinge and FRP in the remaining of the column (SS-FRP), and conventional steel-reinforced column (SS). The pushover analysis results of the four columns are shown in Figure 2-35.The difference in the response is due to the difference in the modulus of elasticity and the ultimate strength of the materials. As expected, the SMA-FRP configuration experienced the lowest ductility, then the SMA-SS, SS-FRP, and SS. However, the SMA-FRP configuration succeeded in minimizing the residual drift due to its PE behaviour. The results of the dynamic analysis (10 earthquake records were used) concluded that the SMA-FRP
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configuration resulted in the highest amount of drift, which means that the columns need to be designed to sustain large induced moments in order to avoid premature failure of the system.
Figure 2-35: Base shear-displacement relationships of the corrosion free concrete columns (Billah and Alam, 2012)
In an attempt to optimize the seismic design of SMA-reinforced structures, Khaloo and Mobini (2011) performed an analytical study in which a one storey one-span frame was designed with different ratios of SMA to steel reinforcement. Due to the low modulus of elasticity of the SMA, it was argued that it is not feasible and economical to replace all the longitudinal steel with SMA bars, and thus, four different ratios of SMA-to-steel bars were used in this study (0.13, 0.3, 0.43, and 0.58). The SMA reinforcements were placed at the plastic hinge locations. The load-drift relationships obtained from one cycle of induced displacement are shown in Figure 2-36. The dissipated energy decreased by 12.1% , 23.6%, 35.2%, and 45.7% while the residual displacement decreased by 2.6%, 9.3%, 24.8%, and 44.8% for the frames with SMA-to-steel ratios of 0.13, 0.30, 0.43, and 0.58, respectively, as compared with the conventional counterpart. Thus, it is observed that the increase in the amount of the SMA bars led to reductions in the energy dissipation and in 62
the permanent deflection of the frame. However, the reduction in the energy dissipated is almost linear, while the permanent deflection decreased in an increasing rate with increasing SMA-to steel ratio.
Figure 2-36: Hysteretic response of one-storey one-bay frame reinforced using PE SMA (Khaloo and Mobini, 2011)
The seismic performance of concrete shear walls reinforced using PE SMA was investigated analytically using the FE software package ABAQUS (Ghassemieh et al., 2012). Different configurations were examined in which PE SMA were used in addition to conventional steel reinforcement. The ratio of PE SMA-to-steel reinforcement ranged from 0 to 100%. The results of the pushover analysis in terms of the base shear (V) and the drift (Δa/hw, where Δa is the applied displacement and hw is the height of the shear wall) of the different configurations are shown in Figure 2-37. It was found that the inclusion of PE SMA increased the strength of the wall, decreased residual displacement, and decreased the stiffness when the percentage of SMA became more than 50%. Using 100% of SMA resulted in percentage reduction of 98.7% and 57.8%
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in the residual displacement and the stiffness while the strength increased by 2.4% as compared with the shear wall reinforced entirely with steel.
Figure 2-37: Base shear-drift relationships of the shear walls with different amount of SMA reinforcement (Ghassemieh et al., 2012)
Ghassemieh et al. (2013) examined the behaviour of coupled shear walls reinforced with PE SMA analytically using ABAQUS. Vertical PE SMA bars extended along the height of the shear wall and horizontal PE SMA bars were also included in the coupling beams. Different percentages of SMA and steel bars were considered in order to optimize the behaviour of the coupled shear walls. Both static and dynamic time history analyses were conducted. The results of the former are similar to those observed by Ghassemieh et al. (2012) while the result of the dynamic analysis is shown in Figure 2-38. The time responses of two configurations (100% SMA and 100% steel) subjected to Koyna earthquake are shown in the figure. It is noted that the red curves refer to the 100% steel while the green curve refers to the 100% SMA. Substituting steel with SMA results in 46% and 83% reduction in the maximum and the residual displacements, respectively, as compared with the 100% steel coupled shear wall. It was, therefore, determined 64
that more research is needed to optimize the ratio of SMA-to-steel reinforcement due to economic considerations.
Figure 2-38: Tip deflection of the coupled shear wall reinforced using steel or SMA subjected to Koyna earthquake (Ghassemieh et al., 2013)
2.6 Summary The background and the literature review necessary for conducting the research were discussed thoroughly in this chapter. An overall discussion of the suitability of using SMA as a reinforcing material in concrete structures was discussed in the context of the DDBD method. It was concluded that reinforcing using SMA can show adequate seismic performance if high energy dissipation of the system can be achieved. The different methods proposed for relocating the plastic hinge away from the face of the column were also discussed. The major drawback of these systems is the inaccuracy in locating the centre of rotation, which is necessary for determining the drift capacity of concrete structures. Moreover, the research progress in the behaviour of SSB system was discussed in detail. It was found that the SSB possesses superior performance as compared with conventional connections due to its stable hysteretic response. After that, the behaviour of SMA was discussed with emphasis on the response of PE SMA. The different factors affecting the 65
response of PE SMA were examined including, temperature, loading rate, cyclic loading, fatigue life, and size in the context of seismic applications. The main conclusions made with regard to each factor are as follows: • Temperature. It is considered to be the most significant factor due to the coupled thermal and mechanical interaction. Loading the PE SMA mechanically raises the temperature of the material which increases the transformation stresses. In a controlled environment, the increase in the temperature is associated with an increase in the dissipated energy, while in the uncontrolled environment the increase in the temperature may lead to a counter effect. • Loading rate. The effect of the loading rate can be understood in light of the temperature effect. High loading rate increases the temperature of the material which leads to an increase in the transformation stresses. At low loading rates, the temperature effect vanishes and the change in the material properties is solely due to the dynamic effect. • Cyclic loading. For trained PE SMA, cyclic loading does not have a significant effect on the properties of the material. However, significant decreases in the energy dissipated and transformation stresses associated with an increase in the permanent strain are anticipated for non-trained material. • Fatigue life. The fatigue life of PE SMA can be presented using the typical S-N curves. For the same stress ratio, the fatigue life of PE SMA is higher than that of steel reinforcement. The endurance limit of PE SMA is at strain of 0.65%. • Size effect. The surface-to-volume ratio (4/db) is the key parameter that is used to examine the size effect. The ratio decreases with increasing bar diameter. Therefore, the generated latent heat is transferred less effectively to the atmosphere, and thus, smaller amount of energy is dissipated. 66
Finally, the different applications of PE SMA with emphasis on self-cantering concrete structures were discussed. Only three experimental studies on concrete beam-column connections reinforced using PE SMA were conducted. Concrete connections reinforced using PE SMA bars were observed to experience permanent deformation due to the cracking of the concrete and slippage in the anchors. It was also concluded that none of the presented studies examined the damping properties of the PE SMA-reinforced connections although it is one of the most important parameters used in the Direct Displacement Based Design (DDBD) method. Therefore, the damping properties of the PE SMA-reinforced SSB and DSB connections proposed in this research, will be investigated thoroughly in order to bridge the research gap in this field.
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Design and Behaviour of Steel-Reinforced SSB and DSB Systems with Relocated Vertical Slots 3.1 Introduction The effectiveness of using the Single Slotted Beam (SSB) and the Double Slotted Beam (DSB) systems in relocating the centre of rotation away from the face of the column is investigated in this chapter via experimental testing. The main aspects of the design process are included in this chapter while the detailed step-by-step calculations are presented in Appendix A. The designated connections are 80% reduced-scale exterior beam-column connection at the fourth storey level in a Reinforced Concrete (RC) building located in Montréal. Two groups of connections were tested under quasi-static loading; SSB and DSB groups. In each group, the vertical slots were located at 0, 1.0 dv, and 1.7 dv from the face of the column, where dv is the effective shear depth (distance between the centroids of the top and the bottom reinforcements). The connections were analyzed and compared, in each group, in terms of the hysteretic response, cracking pattern, beam rotation, beam elongation, strain profiles, shear mechanism, and the components of deformation. After that, damage assessment of both groups is conducted using ductility-based indices, energy-based indices, and other commonly used work and stiffness indices. Finally, the relocation distance to be used in designing the SSB and the DSB systems with Pseudoelastic Shape Memory Alloy (PE SMA) is determined.
3.2 Design of Connections 3.2.1 Analysis of Forces Applied onto the Connections A seven-storey RC ductile moment-resisting office building located in Montréal was considered in this study. The elevation view and the building layout are shown in Figure 3-1 and Figure 3-2, respectively. The typical floor plan is 20 m × 20 m and the typical storey height is 3.7 m. The 68
columns were arranged such that the building would have approximately the same lateral stiffness in all directions. The dimensions of the columns and the amount of longitudinal reinforcement in the first five storeys are larger than those in the upper two storeys in accordance to the building layouts and the column cross-sections shown in Figure 3-2. The primary objective in the design of the structure was to ensure a ductile failure mechanism examined using the pushover analysis and the Capacity Spectrum Method. The ductile progression of the plastic hinges was considered to be one of the main criteria adopted in dimensioning the columns and the beam cross-sections. The beam cross-section in the first five storeys is 375 mm width × 500 mm depth and then reduced to 375 mm width × 400 mm depth in the upper two storeys. The thickness of the slab at all levels is 110 mm. The concrete compressive strength and the yield stress of the reinforcing steel are 40 MPa and 400 MPa, respectively.
Figure 3-1: Elevation view of the building with the designated connection
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The structural analysis of the building was performed using ETABs software while the applied loads were determined from the National Building Code of Canada (NBCC, 2005). The loads applied on an exterior beam-column connection located in the fourth level (circled in Figure 3-1) were then determined and scaled down to 80% in accordance to the replica scaling method (Noor and Boswell, 1992). The end-actions before and after scaling are included in Table A.1 and Table A.2, respectively, in Section A.2.
(a) Lower 5 storeys
(b) Upper 2 storeys
(c) Cross-sectional view of the columns Figure 3-2: Plan view of the building with the column identification 70
3.2.2 Connection Types and Geometries Two groups of connections were designed and examined experimentally; SSB and DSB groups. The design and the geometry of the connections in each group were the same with the only modification is the location of the vertical slots. The relocation distance, lre, defined as the distance between the face of the column to the vertical slot is represented as a fraction of the effective shear depth, dv, which equals approximately 300 mm in all connections (i.e. the distance between the top and bottom reinforcements in the beam section at the vertical slot). Three lre values were examined: 0, 1.0 dv (300 mm), and 1.7 dv (500 mm). In each group, three connections were tested as shown in Table 3-1. The connection ID is described as follows: first term (SSB or DSB) refers to the type of the system, the second term (S) refers to the reinforcing material which is steel, and the last term (0, 1.0, or 1.7) refers to the lre/dv ratio. As discussed in Section 1.2.2, the main difference between the SSB and the DSB systems is that only one vertical slot is made into the beam at the bottom side in the former while two slots (upper and lower) are made in the latter. The depth and the width of the slot in the SSB system are defined by Sd and Sw, respectively, while the depth and the width of the upper and the lower slots in the DSB system are defined by Sd,u, Sw,u, Sd,l, and Sw,l, respectively, as shown in Figure 3-3. It is noted that the beam section within the lre region is referred to as ‘stub region’ thereafter. The mechanics of deformation of the SSB and the DSB connection systems are different than that of conventional connection, and hence, the design considerations are different. The designs of the beam, the column, and the joint are discussed in the following sections while the detailed step-by-step calculations are included in Section A.3, and will be referenced in the text when needed.
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Table 3-1: Testing matrix of the SSB and the DSB systems Group SSB
DSB
Connection ID SSB-S-0 SSB-S-1.0 SSB-S-1.7 DSB-S-0 DSB-S-1.0 DSB-S-1.7
Figure 3-3: Definition of parameters in the SSB and the DSB systems
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lre (mm) 0 1.0 dv (300) 1.7 dv (500) 0 1.0 dv (300) 1.7 dv (500)
3.2.3 Beam Design 3.2.3.1 Flexural Design In the SSB system, the flexural strength is governed by the amount and the strength of the bottom reinforcement since the top reinforcement is designed to remain elastic, while the flexural strength in the DSB system is governed by the design of the top and bottom reinforcements. In this research, the negative and the positive moments in the SSB and the DSB systems are designed to be approximately the same. It is noted that positive bending, in this context, refers to the bending moment that induces compression forces at the top fibre and tensile forces at the bottom fibre, while the negative bending results in the counter effect. The flexural strength of the concrete beam is governed by the moment capacity at the critical section. The inclusion of the vertical slot acts as an artificial crack, and thus, the beam section at the vertical slot represents the centre of rotation (i.e. the critical section). The force distributions at the slot section in the SSB and the DSB systems subjected to positive and negative moments are shown in Figure 3-4. The letters F, c, bd, bw, dst, and dsb refer to applied force, depth of the neutral axis, depth of the beam, width of the beam, distance from the top fibre to the centroid of the top reinforcement, and distance from the top fibre to the centroid of the bottom reinforcement, respectively, while the subscript letters c, st, and sb, refer to the concrete, top reinforcement, and bottom reinforcement, respectively.
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Figure 3-4: Force distribution in the SSB and the DSB systems under positive and negative moments
In the SSB system, the force distribution under positive moment is the same as that in the conventional design of RC members. Under negative moment, the compression forces are induced in the bottom fibre of the concrete section at the top of the slot (referred as concrete hinge hereafter). The bottom reinforcement is designed to yield in tension and compression while the top reinforcement is designed to remain elastic under high applied drifts. Two key design parameters are considered in order to achieve the desired elastic behaviour of the top reinforcement; the depth of the slot, Sd, and the ratio of the amount of the top reinforcement to the amount of the bottom reinforcement, Ast/Asb. The strain in the top reinforcement increases with increasing Sd. Thus, it was recommended to limit the value of Sd to 0.25dsb (Au, 2010). However, this limit can be relaxed
74
in the case of high Ast/Asb ratios. In a prior research study, the Ast/Asb ratio was found to range between 2.0 and 2.5 and is achieved by using Grade 300 steel for the bottom reinforcement and Grade 500 steel for the top reinforcement (Au, 2010). In this research, Sd was set to 0.28 ds for practical purposes due to the large amount of steel that needs to be placed in the concrete hinge. However, the relatively high Sd value was compensated by the increase of the Ast/Asb ratio, which was set to 3.0 and was achieved by using 2-15M bottom reinforcement and 4-20M top reinforcement (Grade 440). In the DSB system, the beam is designed to have similar positive and negative moment capacities. Therefore, the dimensions of the upper and the lower slots in Figure 3-3 were set equal and the ratio Ast/Asb was set to equal one. A portion of the concrete hinge will be subjected to compression forces, and thus, the stresses induced in the bottom and the top reinforcement will not be the same. The magnitude of the compression forces in the concrete hinge depends on its dimensioning and the width of the cracks developed when the connection is subjected to lateral loading. Each of the top and the bottom reinforcement consisted of 2-15M steel reinforcement (Grade 440). Ignoring the contribution of the diagonal reinforcement (further discussed in Section 3.2.3.2) in the positive bending resisting mechanism and the contribution of both the concrete hinge and the diagonal reinforcement in the negative bending resisting mechanism in the SSB connections was found to provide conservative estimates of the bending capacities as recommended by Au (2011) because the diagonal reinforcement and the concrete in compression are located close to the neutral axis, and thus, the aforementioned contributions were ignored in the design process detailed in Section A.3.1.1 Similarly, the contributions of the concrete and the diagonal reinforcement in the DSB systems are ignored for simplicity purposes. The moment of 75
resistance, Mb,r, the nominal moment, Mb,n, and the probable moment, Mb,p (moment corresponding to the overstrength of the beam section) of the beams in the SSB and DSB systems are included in Table A-4 and Table A-5, respectively. It is found that Mb,r of the SSB system under positive and negative bending is 46.9 kN·m and 42.1 kN·m, respectively, while it is 42.7 kN·m for the DSB system. The maximum negative moment obtained from the analysis of the building (refer to Table A.3) is 44.8 kN·m. Even though Mb,r of the SSB system under negative bending and of the DSB system is lower than the required moment, the difference is insignificant (difference of 5.4%) and the moments are comparable.
3.2.3.2 Shear Design The inclusion of the vertical slots in the beam will result in a disturbed region “D-region” in which the stresses across the beam section are not uniform. Understanding the flow of forces and the deformation mechanism in this region will help in designing the shear reinforcement. Therefore, simplified linear elastic 2D Finite Element Models (FEMs) were developed and used to examine the flow of forces in the connections subjected to bending in order to help in the design of the shear reinforcement. The six connections included in Table 3-1 were modeled. The models intend to provide a general representation of the deformation mechanism in the connections and do not intend to provide accurate results in term of load and deflection. Only longitudinal steel (in the beams and columns) and concrete sections were considered in the model. The steel was modeled as a linear elastic material with a modulus of elasticity of 200 GPa. The concrete was also modeled as a linear elastic material with a modulus of elasticity, Ec, equal to 28.5 GPa based on the design equation proposed in CSA-A23.3-04 (2004): E c = 4500
f c'
(MPa)
Equation 3-1 76
'
where f c is the concrete compressive strength in MPa.
The geometry of the connection and the vertical slots along with the quantities of the top and the bottom reinforcements in the SSB and the DSB systems are similar to those determined in Section 3.2.3 and A.3.1. The concrete section and the steel reinforcement were modeled using 4node plane strain and beam elements, respectively. The tip of the beam was subjected to an upward 10 mm deflection to induce a positive moment and downward to induce a negative moment. The column was subjected to 405 kN (determined based on the analysis of the structure included in Section A.3.2). The column was also hinged at the top and the bottom sides. As mentioned earlier, these models intend to provide a general behaviour and not accurate results, thus, mesh sensitivity analysis was not performed and the size of the mesh was chosen such that to provide adequate level of accuracy. The bottom reinforcement in the SSB connections and the bottom and top reinforcement in the DSB connections were debonded from the surrounding concrete using the Tube-to-Tube elements for a distance of 0.5 bd as discussed in Section 3.3 (The detailed procedure of modeling using Tube-to-Tube elements is discussed in Section 6.2). The visual representation of the deformation mechanism was obtained by plotting the minimum principal strain at the integration points in each element in the concrete section. The minimum principal strain indicates the magnitude and the direction of the compressive strains flowing in the concrete section. Thus, the load path can be detected accurately and used as a base for designing the shear reinforcement. The visualization of the principal compressive strains under positive and negative bending for the SSB connections is shown in Figure 3-5 and Figure 3-6, respectively.
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(a) Connection SSB-S-0
(b) Connection SSB-S-1.0
(c) Connection SSB-S-1.7 Figure 3-5: Visualization of the compressive strains in the SSB connections under positive bending 78
(a) Connection SSB-S-0
(b) Connection SSB-S-1.0
(c) Connection SSB-S-1.7 Figure 3-6: Visualization of the compressive strains in the SSB connections under negative bending 79
It can be seen that the inclusion of the vertical slot disturbs the distribution of the compressive strains within the beam and the joint sections. Under positive bending, the SSB system behave in a similar manner to conventional connections in which the flexural compressive forces are located at the upper part of the section while the flexural tensile forces are located at the lower part of the section. Moving the vertical slot away from the face of the column in the SSB-S 1.0 and SSB-S-1.7 connections, distributes the compressive strains to a larger region proportional to the relocation distance. High strain values are observed in the concrete section on both sides of the unbonded bottom reinforcement. Relocating the vertical slot moves the high strain region away from the joint, and hence, maintain the integrity of the concrete at the joint. Under negative moment, the behaviour is characterized by high disturbance of the strain distribution within the beam section. The bottom reinforcement and portion of the concrete fibres at the concrete hinge contribute to the negative bending moment. Compression struts connecting the forces at the edge of the unbonded bottom reinforcement and the lower portion of the concrete hinge are observed to develop. In the SSB-S-0 connection, the flexural compressive forces are transferred to the joint via two paths. One path transfers the compression forces from the bottom reinforcement to the top left corner of the joint, while the other path transfers the compressive forces from the lower portion of the concrete hinge to the top left corner of the joint. Thus, two diagonal compression struts are formed in this case as opposed to the behaviour under positive bending in which only one diagonal compression strut is developed connecting the two opposite corners of the joint (refer to Section 3.2.5 and Figure 3-9 for more details). However, relocating the vertical slots (in connections SSB-S-1.0 and SSB-S-1.7) seems to be advantageous in this situation since moving the vertical slot away from the face of the column allows for the development of a single diagonal compression strut in the joint similar to the behaviour in 80
conventional connections, and thus, design guidelines of conventional joints can be used to design the connections. Diagonal compression struts are developed transferring the compression forces from the lower portion of the concrete hinge to the bottom right corner of the joint in the stub regions of the SSB-S-1.0 and SSB-S-1.7 connections. It is noted that the slopes of these diagonal compression struts decrease with increasing lre distance. The visualization of the minimum principal strains (compressive strains) in the DSB system under positive bending are shown in Figure 3-7. The observations made with regard to the behaviour in the positive bending are similar to those under the negative bending due to the symmetric nature of the system. High levels of compressive strains are induced in the concrete hinge. Diagonal compression struts are formed transferring the forces from the edge of the unbonded top reinforcement in the beam section and the upper portion of the concrete hinge. Similar to the behaviour of the SSB system, two diagonal compression struts are formed in the joint when the slots are located at the face of the column (Connection DSB-S-0), while only a single diagonal compression strut is formed when the vertical slots are relocated (Connections DSB-S-1.0 and DSB-S-1.7). Also, diagonal compression struts are formed in the beams transferring the forces from the edge of the unbonded reinforcement to the concrete hinge. In the DSB-S-1.0 and the DSB-S-1.7 connections, additional diagonal concrete compression struts are formed transferring the forces from the concrete hinge to the corner of the joint. It is thus concluded that the behaviour of the SSB under negative bending and the behaviour of the DSB system under positive and negative bending are similar with the exception that the slopes of the diagonal compression struts in the latter are higher in the joint and lower in the beam section as compared with the former system.
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(a) Connection DSB-S-0
(b) Connection DSB-S-1.0
(c) Connection DSB-S-1.7 Figure 3-7: Visualization of the compressive strains in the DSB connections under positive bending 82
Based on the discussion presented above, the beams in the SSB and the DSB connections are divided into three regions in terms of the detailing of the stirrup reinforcement as shown in Figure 3-8. In this research, the recommendation on the length of each region is adopted form Paulay and Priestley (1992). Even though the recommendations by Paulay and Priestley (1992) were based on the behaviour of beams with cross-diagonal reinforcement as a mean for relocating the plastic hinges (discussed in Section 2.3. Also, see Figure 2-5), their recommendations are applicable to the SSB and the DSB systems based on the observations made from the FE analysis. Region III (does not exist in the SSB-S-0 and DSB-S-0 connections) extends for 0.5bd to the left of the vertical slots while Region II extends for 1.5bd. Region I extends from the edge of Region II to the tip of the beam. Region I is a non-critical region in which the conventional design procedure of shear reinforcement is adopted. However, special detailing is required in Regions II and III. The special detailing of the stirrups adopted in this study is based on the anti-buckling requirements for the design of RC members in CSA-A23.3-04 (2004) (refer to Section A.3.1.2 for the detailed shear design of the SSB and the DSB systems). It is noted that the area of the stirrups right at the face of the vertical slot is doubled in order to accommodate the high shear demand at this region in both the SSB and the DSB systems. Diagonal reinforcements are also used to enhance the shear transfer mechanism near the slots as shown in Figure 3-8. In the SSB system, the diagonal reinforcement passes through the concrete hinge and extends from the top to the bottom concrete fibres while it extends from the concrete hinge to the top and the bottom concrete fibres in the DSB system. The concrete compression strut that extends between the concrete hinge and the corner of the joint may induce premature failure since the beam in this region acts like a deep beam. Thus, it was decided to
83
include distributed longitudinal reinforcements (4-10M) in this region in order to better resist the shear force and to control the cracking pattern.
Figure 3-8: Shear design regions and details in the SSB and the DSB systems
3.2.4 Column Design The design methodology of the column members in the SSB and DSB connections is the same as in conventional systems. Therefore, the amount of the longitudinal and transverse reinforcements
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in the columns were designed according to CSA-A23.3-04 (2004). The detailed design procedure and the calculations are included in Section A.3.2.
3.2.5 Joint Design The design of the shear reinforcement in the joint is considered to be one of the most critical aspects in the seismic design of RC structure. This is due to the fact that the shear deformation is not a reliable source of energy dissipation, and hence, it needs to be maintained at low levels (preferably in the elastic range). Based on the observations made with regard to the behaviour of the connections in Figure 3-5, Figure 3-6, and Figure 3-7, the applied forced onto the joint along with the deformation inside the joint is presented schematically in Figure 3-9 (the notations are described in the List of Symbols). Under positive bending, the design of the joints in all connections except DSB-S-0 follow the conventional design, while under negative bending, the design of the joints in all connections except SSB-S-0 and DSB-S-0 follow the conventional design. This is due to the fact that relocating the vertical slots away from the face of the column results in the formation of a single diagonal compression strut connecting the concrete hinge and the corner of the joint. However, placing the vertical slots at the face of the column in the DSB system leads to the formation of two diagonal compression struts; one transferring a portion of the compression forces from the compression reinforcement to the upper opposite corner of the joint, S1, while the other transfers the compression forces in the concrete hinge to the upper opposite corner of the joint, S2. The design of the joint when the slot is located at the face of the column in the SSB system is the same as the conventional design if subjected to positive moment, while it is similar to the design of DSB-S-0 if subjected to negative moment. The difference in the behaviour
85
between DSB-S-0 and SSB-S-0 is the steepness of the concrete struts. Concrete strut S2 is steeper in the former due to the fact that the concrete hinge is placed at the middle of the beam section. The forces in the joint are transferred via two mechanisms; compression strut and truss mechanisms (Paulay and Priestley, 1992). In the former, forces are transferred via concrete, while in the latter the forces are transferred via the transverse steel reinforcement. For design purposes, only the joint shear value needs to be determined in the joint in order to be able to calculate the amount of the required transverse steel. The detailed calculations of the joint shear forces and the amount of required transverse steel in the joints of the SSB and the DSB systems are included in Section A.3.3. It is noted that the beam forces correspond to those developed at the overstrength moment.
(a) Vertical slot at the face of the column
(b) Relocated vertical slots
Figure 3-9: Joint shear mechanism in the SSB and DSB connections
3.2.6 Anchorage and Buckling of the Longitudinal Reinforcement Several researchers pointed to additional design concerns that need to be taken into consideration in designing SSB connections with vertical slots located at the face of the column (Ohkubo and Hamamoto, 2004; Au, 2010, Byrne and Bull, 2012). Based on the observations and discussion 86
presented in Sections 3.2.3, the design considerations of the SSB system are the same as those for the DSB system. These design considerations include the anchorage and buckling of the longitudinal reinforcement. The anchorage of the reinforcement is one of the critical aspects in the design of SSB and DSB systems. In the conventional design of RC connections, the adequacy of the anchorage performance is taken into account by limiting the dimensions of the joint. The larger the bar diameter, the larger is the width of the joint. Unlike the conventional design, the reinforcement yields in compression, and thus, severe deterioration of the concrete in the joint and an increase in the development length are expected. However, moving the location of the plastic hinge away from the face of the column increases the anchorage length, and hence, design limits on the width of the joint can be relaxed. In this research, the design limits in CSA-A23.3-04 (2004) on the joint width were implemented and then the anchorage performance was investigated in the experimental and the analytical program to examine the adequacy of these limits in the design of SSB and DSB systems. The high compressive strains induced in the reinforcement passing through the slot may result in buckling the reinforcement before attaining its ultimate strength. In this research, the antibuckling transverse reinforcement was designed in accordance to CSA-A23.3-04 (2004) and the adequacy of the design procedure was examined afterwards experimentally and analytically.
3.3 Prediction of Rotational Capacity Low-cyclic fatigue failure of the bottom reinforcement in the SSB system and in the top and the bottom reinforcements in the DSB systems is a very critical design consideration due to the high curvature demands induced at the location of the vertical slots. Unlike the conventional design 87
of RC connection, the reinforcement at the slot is expected to yield in both tension and compression, and hence, high strain amplitudes are expected to develop during the positive and negative bending. The high strain amplitude fatigue (low-cyclic fatigue) will result in a reduction in the fatigue life of the reinforcement, consequently, brittle failure of the reinforcement may occur. Therefore, the rotational capacity of the systems is governed by the fatigue life of the steel reinforcement passing through the slots, and hence, predicting the rotational capacity depends highly on the level of accuracy in determining the fatigue life of the reinforcement. The acceptance criteria for moment frames based on the structural testing code (ACI T1.1 01, 2001) specifies that the structure should at least sustain a drift of 0.035 rad. This limit was set as the criterion for designing the maximum rotation capacity of the connections. In this research, a simplified rotational capacity prediction model is developed in order to determine the adequacy of the SSB and the DSB in satisfying the drift limit specified by the ACI T1.1-01 (2001) code. The components of the prediction model are shown in Figure 3-10, while the assumptions involved in developing it are as follows: 1. The beam behaves like a rigid element fixed at the joint with a rotational spring located at the vertical slots (i.e. the vertical slots represents the location of the plastic hinge). 2. The length of the plastic hinge equals to the summation of the width of the slot, Sw, and the length of the steel tube, lst. 3. The strain ratio, Sε (Sε = εmin/εmax), in all connections equals -1, where εmin and εmax are the minimum and the maximum strain in the cycle, respectively.
88
Figure 3-10: Schematic drawing of the rotational capacity prediction model
It is, therefore, concluded that the model assumes that the centre of rotation is located exactly at the vertical slot and moving the vertical slots away from the face of the column leads to relocating the rotational spring. Given the above assumptions, the prediction procedure is described as follows: •
Step 1: Determine the strain in the steel In order to determine the fatigue life, and hence, the rotational capacity of the connections,
the strain in the steel reinforcement passing through the vertical slots needs to be determined. The curvature at the rotational spring, ψro, is calculated as follows:
ψ ro =
θ=
θ
Equation 3-2
(Sw + lst )
Δa ls
Equation 3-3
where Δa is the applied displacement at the tip of the beam, θ the rotation angle, and ls is the shear span. 89
The strain in the steel, εs, is calculated as the product of the ψro and distance between the neutral axis and the bottom reinforcement, lo, as shown in Figure 3-10:
ε s = ψ ro ⋅lo
Equation 3-4
Substituting Equation 3-2 in 3-4 yields the following expression:
εs =
θ
(Sw +lst )
⋅lo
Equation 3-5
The distance lo equals approximately 1.2 Sd and 0.5bd for the SSB and the DSB systems, respectively, while lst is set to 0.5 bd. Based on the above assumptions, the value of Sw needs to satisfy the following relationship: S w ≤ θ ⋅lo
Equation 3-6
The second term in Equation 3-6 represents the decrease in the width of the slot under the positive and the negative bending. In order to achieve a drift of 0.035 rad, the Sw needs to be greater or equal to 12.6 mm and 7.0 mm for the SSB and DSB systems, respectively. Therefore, it was determined to use a slot width of 25 mm in both systems. This will provide a safety factor of 2.0 and 3.6 for the SSB and the DSB systems, respectively.
•
Step 2: Determine the fatigue life at each strain amplitude Predicting the strain amplitude history in the reinforcement is considered to be a difficult
task. In fact, earthquake resisting structures are subjected to non-uniform vibrations and earthquake motions. In other words, the fatigue lives in the SSB and the DSB systems depend highly on the loading regime. The loading regime adopted in this study is developed in accordance with the ACI T1.1-01 (2001) code as shown in Figure 3-15 (the rationale of the loading regime is 90
discussed in Section 3.5). Then, Equation 3-5 is used to calculate εs. After that, the fatigue life, Nf, is calculated at each strain amplitude using the following power expression (Manson, 1953; Coffin, 1954):
ε a = β c (2 N ) γ c
Equation 3-7
where εa is the strain amplitude in the steel reinforcement (equals εs in case of Sε= -1), βc and γc are scaling and exponent coefficients in the fatigue life prediction model, respectively, and N is the number of cycle. Based on the results of an extensive experimental testing, it was found that for an Sε ratio of -1 and a yield stress of 400 MPa, the parameters βc and γc equal 0.112 and -0.433, respectively, (Brown and Kunnath, 2000).
•
Step 3: Determine the cumulative fatigue damage Complex damage assessment rules were developed based on continuum damage mechanics
in order to account for the damage initiation and progression in steel subjected to low cyclic fatigue (refer to Fatemi and Yang (1998) for a comprehensive review of these models). These rules aim at providing an accurate representation of the interaction between the strain amplitude, stress range, and the loading history. The purpose of the damage assessment outlined in this section is to provide a preliminary insight into the fatigue life of the steel reinforcing bars. Given the level of complexity associated with implementing the continuum damage mechanics rules and the intend of this particular damage assessment, the simple linear damage, DL , rule by Miner (1945) is used to predict the low-cyclic fatigue failure of the reinforcing steel. In addition to its simplicity, this rule was proven to yield accurate results in predicting the low cycle fatigue lives of reinforcing steel bars in concrete structures subjected to flexural loading (Brown and Kunnath, 2000). The 91
model simply assumes a constant work absorption per cycle. Therefore, the energy accumulation leads to a linear summation of damage. Failure occurs when DL = 1.
DL =
ni N fi
Equation 3-8
where ni and Nfi are the number of cycles and the fatigue life, respectively, at a certain strain amplitude. Based on the prediction procedure presented above, the maximum εs, Δa, and θ attained in the designed connections are included in Table 3-2. It can be seen that relocating the vertical slots will increase the strain in the steel reinforcement and the rotational capacity while it will lead to a decrease in the applied displacement. All connections except SSB-S-0 and SSB-S-1.0 satisfy the drift limit of 0.035 rad by ACI T1.1-01 (2001). Table 3-2: Predicted steel strain and rotational capacity of the steel-reinforced connections
Connection εs Δa (mm) θ (rad)
SSB-S-0 0.031 50 0.030
SSB-S-1.0 0.038 45 0.033
SSB-S-1.7 0.036 40 0.035
DSB-S-0 0.026 80 0.048
DSB-S-1.0 0.028 70 0.052
DSB-S-1.7 0.028 60 0.052
3.4 Geometry, Detailing, and Construction of the Connections
Based on the analysis, design, and detailing presented in the above sections, the final design of the SSB and the DSB connections are shown in Figure 3-11 and Figure 3-12, respectively. The detailed procedure used in fabricating the connections including, building formwork, fixing steel cages, and concrete casting are included in Section A.5. The ancillary test results including concrete compression tests and steel coupon tension tests are presented in detail in Section A.5.
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Figure 3-11: Details of the steel-reinforced SSB connections (all dimensions are in mm)
93
Figure 3-12: Details of the steel-reinforced DSB connections (all dimensions are in mm)
94
3.5 Test Setup and Loading Regime
In RC structures subjected to lateral loading, the movement of the beam at the mid-span is restrained (pinned) and the movement of the lower end of the column is restrained (pinned) as shown in Figure 3-13. Thus, the drift is calculated as the ratio of the horizontal movement of the column top end to the column height. For practicality reasons, it is more convenient to have pin connections at the top and the lower ends of the column while applying the quasi-static cycling at the tip of the beam (Figure 3-13 (c)). In this case, the drift angle is defined as the ratio of the beam tip displacement to the length of the shear span. The test setup for all connections is shown in Figure 3-14.
Figure 3-13: Selection of connection test setup: (a) drift behaviour in a single storey, (b) drift behaviour in exterior connection, and (c) actual drift behaviour of the tested connections (Hanson and Conner, 1967)
95
Figure 3-14: Test setup of the beam-column connections (all dimensions are in mm)
The testing regime consisted of two stages; the application of a vertical load onto the column, and the application of quasi-static displacement at the tip of the beam. In the first stage, a compression load of 405 kN was applied to the top of the column using a 500 kN capacity hydraulic jack reacting against a cross-head I-steel beam. The value of the compression load was determined based on scaling the applied vertical compression load obtained from the 7-storey RC building 96
(refer to Sections A.2 and A.3.2). The load applied at the top of the column was found to fluctuate during the application of the quasi-static loading. It was, therefore, increased approximately by 2% in order to maintain a compression load equal to or higher than 405 kN during testing. The second testing stage consisted of the application of quasi-static displacement cycles at the tip of the beam. The quasi-static cycles were applied via a 250 kN MTS 10” stroke actuator reacting against a cross-head I-steel beam. The testing regime was designed to examine the behaviour of the connections in both the elastic and the inelastic ranges. The connections were subjected to upward (positive bending) and downward (negative bending) cycles while three cycles were conducted at each displacement level in order to obtain the stabilized response. The quasi-static loading regime is shown in Figure 3-15.
50 40
Displacement (mm)
30 20 10 0 -10
2 mm
5 mm
-20 -30 -40 -50
1 mm/sec
0.25 mm/sec
Time (sec)
Figure 3-15: Quasi-static loading history
97
The first displacement cycle was conducted at 2 mm and then increased by increments of 2 mm until it reached 10 mm. After that, the increment of the displacement was increased to 5 mm up to the failure of the specimen. Two loading rates were adopted in this study; 0.25 mm/sec and 1 mm/sec. The former loading rate was implemented up to the end of the third 30 mm cycle while the latter was implemented from the first 35 mm cycle up to the failure. The loading rates were chosen such that enough data points could be recorded during the cycling of the connections. It is also noted that the loading regime is in compliance with the recommendations provided by the ACI T1.1-01 (2001) code.
3.6 Instrumentations
Four types of instrumentations were used to monitor the behaviour of the connections during testing; Linear Strain Conversion (LSC) devices, Laser displacement sensors, Strain Gauges (SG), and Digital Image Correlation Technique (DICT). The LSC devices were mounted on the beams, columns, and joints in order to measure the concrete deformations, while the Laser displacement sensors were positioned on the laboratory floor in order to measure the displacement of the beam members under positive and negative bending. The positions and the labels of the LSC and Laser devices are shown in Figure 3-16 for the SSB and DSB connections. The SGs were mounted on the longitudinal reinforcement in the beam and column, diagonal reinforcement, and stirrups in the SSB and DSB connections as shown in Figure 3-17 and Figure 3-18, respectively. The DICT was used to measure the rotation and the deformation in the joints in order to calculate the contributions of the different deformation components to the total connection drift. The DICT instrumentation consisted of digital camera, lighting system, computer, and surface texture. The surface of the concrete joints was painted first by water-based white paint 98
and left to cure as shown in Figure 3-19. After that, black paint was sprayed on the target area in order to provide the surface texture necessary to run the analysis. Images were taken every five seconds by a 16 MP Canon digital camera and stored in the connected computer. The analysis of the pictures was performed using the GeoPiv7 software.
Figure 3-16: Positions and labels of the LSC and laser devices (all dimensions are in mm)
99
Figure 3-17: Locations of the SGs in the SSB connections (all dimensions are in mm)
100
Figure 3-18: Locations of the SGs in the DSB connections (all dimensions are in mm)
101
Figure 3-19: DICT instrumentation of the SSB and DSB connections
3.7 Experimental Behaviour of the SSB System 3.7.1 Hysteretic Response The hysteretic responses of the SSB connections are shown in Figure 3-20. All connections failed by the fracture of the bottom reinforcement. The drift ratios shown in Figure 3-20 were calculated by dividing the displacement at the tip of the beam over the shear span, ls (refer to Figure 3-10 for the definition of ls) while the moments were calculated by multiplying the loads with ls. The ls of
102
the SSB-S-0, SSB-S-1.0, and SSB-S-1.7 are 1655 mm, 1355 mm, and 1155 mm, respectively. The validity of adopting this procedure in calculating the drift ratio is examined later in this section.
(a) Connection SSB-S-0
(b) Connection SSB-S-1.0 103
(c) Connection SSB-S-1.7 Figure 3-20: Hysteretic behaviour responses of the SSB connections
The SSB-S-0 and SSB-S-1.7 connections failed during the third positive bending cycle of the 55 mm displacement amplitude cycle while connection SSB-S-1.0 failed during the first positive bending cycle of the 55 mm displacement amplitude cycle. The hysteretic responses of all connections are characterized by their stable behaviour with no signs of pinching shear, which is usually experienced in conventional RC connections subjected to high drift cyclic loading. In conventional RC connections, the increase in the width of the cracks makes it difficult for the cracks to close, and hence, to provide a compression force upon the reversal in the bending moment. This action ‘pinching shear’ will, thus, lead to a reduced stiffness and significant reductions in the amount of energy dissipated during the cyclic loading (Derecho and Kianoush, 2001). In the SSB system, the bottom reinforcement provides the main source of resistance for
104
positive and negative bending while a relatively low compression force is transferred in the concrete within the concrete hinge. Therefore, the effect of the increase in the crack width on lowering the stiffness of the connections is not observed in the SSB connections. There are four common ways of determining the yielding condition of RC structures as shown in Figure 3-21. In method (a), the yielding condition represents the onset of yielding in the flexural reinforcement, while the yielding condition in the other three methods depends on the ultimate state. Since the response of the SSB and the DSB systems in the post-yielding stages experiences high stiffness, the yielding ductility will be overestimated if it was to be determined by any method that depends on the ultimate condition (Methods (b), (c), and (d)). Therefore, Method (a) was used to define the yielding condition in this research, while the ultimate condition is defined as the peak load attained under the cyclic loading.
Figure 3-21: Alternative definitions of yielding condition (reproduced from Park (1989))
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The loads at the yielding, Py, and the ultimate conditions, Pu, for the positive and the negative bending are included in Table 3-3. It is observed that moving the vertical slots away from the face of the column increased the loads at yield and ultimate conditions. The increase is higher for the negative bending than the positive bending and can be approximated as a linear trend. The connections were designed to exhibit the same moment capacity, therefore, the increase in the load for the connections with the relocated vertical slots (Connections SSB-S-1.0 and SSB-S-1.7) indicate the decrease in the shear span. Consequently, the plastic hinge is no longer located at the face of the column. Instead, it is located at the location of the vertical slot. In order to better understand this behaviour, the ratios of the moment in the connections with the relocated vertical slots (Connections SSB-S-1.0 and SSB-S-1.7) to that of the connection in which the slot is located at the face of the column (Connection SSB-S-0), M/M0, are presented in Table 3-4 at the yielding and the ultimate conditions for the positive and the negative moments. The M/M0 ratio is approximately one at yielding for the positive and the negative bending, while it decrease with increasing relocation distance at the ultimate condition. Also, the ratios at the positive bending are lower than negative bending. This can be explained by examining Figure 3-5 obtained from the FE analysis. Under positive bending, the compression forces are concentrated at the concrete hinge when the vertical slot is placed at the face of the column. However, moving the vertical slot distributes the compression force over a larger distance, and hence, the actual location of the centre of rotation is not exactly at the location of the vertical slot under positive bending. Instead, it is located in-between the face of the column and the location of the vertical slot being closer to the latter. In general, moving the vertical slots away from the face of the column succeeded in relocating the plastic hinge under positive and negative bending. However, it is more effective 106
under negative bending than under positive bending. The definition of the shear spans (length of the beam between the applied displacement and the vertical slot) is found to be reasonably accurate in determining the location of the centre of rotation and results in a maximum error of 16% at the ultimate condition for a relocating distance up to 1.7dv (500 mm).
Table 3-3: Load comparison at the yielding and the ultimate conditions of the SSB connections Yielding
Ultimate
Positive Negative Positive Py Diff.* Py Diff.* Pu Diff.* (kN) (%) (kN) (%) (kN) (%) SSB-S-0 33 29 62 SSB-S-1.0 40 21 34 17 67 8 SSB-S-1.7 48 45 48 66 75 21 * Percentage difference with respect to the SSB-S-0 connection Connection
Negative Pu Diff.* (kN) (%) 60 71 18 82 37
Table 3-4: Evaluation of plastic hinge relocation of the SSB connections Yielding Connection SSB-S-0 SSB-S-1.0 SSB-S-1.7
Positive Moment M/M0 (kN·m) 54 1.00 55 1.02 56 1.04
Ultimate
Negative Moment M/M0 (kN·m) 49 1.00 46 0.94 55 1.12
Positive Moment M/M0 (kN·m) 103 1.00 91 0.88 86 0.83
Negative Moment M/M0 (kN·m) 99 1.00 96 0.97 95 0.96
The flexural overstrength, φo, factors calculated at the ultimate condition (55 mm displacement amplitude cycle) are included in Table 3-5. It is noted that the φo factors are defined as the applied moment, M, divided by the nominal moment, Mb,n, determined by including and excluding the effect of the diagonal reinforcement in resisting the flexural forces. The nominal moment of the connections is included in Table A.3 and Table A.4 in Section A.3.1. The φo factor takes into account all possible factors that contribute to the strength exceeding the nominal value 107
including; steel hardening, variation of concrete and steel strengths from the specified ones, and the confinement effect on concrete (Paulay and Priestley, 1992).
Table 3-5: Flexural overstrength factors, φo, for the SSB connections With diagonal reinforcement Without diagonal reinforcement Connection Positive Negative Positive Negative * * * Diff. (%) Diff. (%) Diff. (%) Diff.* (%) φo φo φo φo SSB-S-0 1.53 1.58 1.74 1.61 SSB-S-1.0 1.36 -11 1.50 -4 1.55 -11 1.55 -4 SSB-S-1.7 1.29 -16 1.51 -4 1.47 -16 1.54 -4 * Percentage difference with respect to the SSB-S-0 connection
The typical values of φo as suggested by Paulay and Priestley (1992) range from 1.25 to 1.40 depending on the strength of the flexural steel, while it is recommended by the CSA-A23.304 (2004) code to use a value of 1.25. It is seen that there is a significant difference between the suggested values and the values obtained in this study. This is due to the fact that the suggested values are based on the behaviour of conventional beams in which the steel hardening of the flexural reinforcement is significantly less than that experienced in the SSB connections. Moreover, Au (2010) found that the typical values of φo of a SSB connection in which the slot is at the face of the column are typically 15% higher than those obtained from conventional connections. The average increase in φo of the SSB connections tested in this research is 17% and 26% when including and excluding the contribution of the diagonal reinforcement, respectively (calculated based on φo of 1.25 for the conventional connections as recommended by CSA-A23.304 (2004)). The reduction in the values of the φo when moving the vertical slots away from the face of the column can be understood in light of the decrease in M/Mo ratios explained earlier.
108
3.7.2 Cracking Pattern The cracking pattern at the side face of the beams is shown in Figure 3-22. The locations of the diagonal shear reinforcement extending from the concrete hinge to the bottom side of the beams as well as the steel tube used to debond the bottom reinforcement are also shown in Figure 3-22. It is noted that ‘D’ stands for diagonal. Three types of diagonal cracks were initiated in all connections; A, B, and C. The sequence and the distribution of the vertical and the diagonal cracks are discussed with respect to the nature of the applied bending in the following paragraphs. Under positive bending, the first flexural crack to develop was located at the concrete hinge. Distributed flexural cracks, initiating at the bottom face of the beams, started to develop with increasing applied displacement. In SSB-S-0 connection, flexural cracks developed in the vicinity of 1.0 bd to the right of the vertical slot. However, no cracks at the bottom face of the beams were observed in the vicinity of 1.0 bd and 0.5 bd to the right and to the left of the vertical slots, respectively, in the SSB-S-1.0 and SSB-S-1.7 connections. Therefore, it is concluded that the intensity of the flexural cracks increased in the stub region with increasing relocation distance. The diagonal cracks (A) and (B) were developed when the beams were subjected to positive bending. The schematic drawing shown in Figure 3-23 (a) illustrates the formation of the diagonal cracks (A) and (B) when the beam is subjected to upward shear (positive bending). It is shown that the beam acts like a rigid element with its centre of rotation located at the vertical slot. This behaviour is further examined by studying the rotation distribution and the strain profiles in the flexural reinforcement as will be discussed in details in Sections 3.7.3 and 3.7.5, respectively. Diagonal cracks (B) were initiated at 40 mm, 15 mm, and 20 mm, while diagonal cracks (A) were developed at 40 mm, 35 mm, and 30 mm displacement cycles for the SSB-S-0, SSB-S-1.0, and SSB-S-1.7 connections, respectively. The significant delay in the onset of cracking in the SSB-S109
0 connections is due to the advantageous behaviour of the applied compression force onto the column in enhancing the shear resistance. The tip of the diagonal crack (B) is located approximately at the neutral axis in the concrete hinge region while crack (A) runs parallel to the diagonal reinforcements. Under positive bending, tensile strains were induced in the diagonal reinforcement while a compression strut ran parallel to the diagonal reinforcement. This action created a sliding plane at which the cracks were initiated. Under negative bending, the first flexural cracks were developed at the concrete hinge. Flexural cracks distributed over the length of the beam were developed at the top face of the beam with increasing applied displacement. Diagonal cracks (C) were developed at 45 mm, 50 mm, and 45 mm displacement cycles for the SSB-S-0, SSB-S-1.0, and SSB-S-1.7 connections, respectively. Thus, moving the vertical slot away from the face of the column does not affect the onset of the diagonal cracks under negative bending since the difference between the values is statistically insignificant. The behaviour of the beam and the formation of the diagonal cracks (C) are illustrated schematically in Figure 3-23 (b). The cracks were originated near the edge of the vertical slots and extended to the top fibres of the concrete beam at approximately 45o degrees. The state of stress and strains within the beam stub is of a complex nature. Complex cracking is formed in the stub region in the SSB-S-1.0 and SSB-S-1.7 connections as indicated in Figure 3-22. It consisted of three cracks originating from the same point. The formation of this type of crack can be understood by examining the distribution of the principal compressive strain and principal tensile strains formed under positive and negative bending as discussed in Section 3.2.3.2. Given that cracks form parallel to the direction of the minimum principal strains, and the direction of the diagonal compression struts forming in the stub region were perpendicular to each
110
other under positive and negative bending, cross cracks (complex cracking) in the stub region were developed.
(a) Connection SSB-S-0
(b) Connection SSB-S-1.0
(c) Connection SSB-S-1.7 Figure 3-22: Cracking patterns at the ultimate condition in the SSB connections 111
(a) Positive bending
(b) Negative bending
Figure 3-23: Schematic representation of the diagonal cracks in the SSB connections
The non-tearing action of the SSB system is considered to be one of its main characteristics as discussed in Section 2.4. Therefore, it is highly important to examine the cracking pattern at the top side of the beams. The cracking patterns at the top side of the beams are shown in Figure 3-24. The locations of the vertical slots are also indicated in the figure. Connection SSB-S-0 experienced the least amount of cracking while connection SSB-S-1.7 exhibited the formation of high amount of non-uniform cracks (No information is provided about the crack size because no instrumentation was installed to measure the crack width). Therefore, relocating the plastic hinge in the SSB system reduced the effectiveness of the non-tearing action of the slabs attached at the top of the beams. However, the effect of relocating the plastic hinge on the non-tearing action of the slab is not included in the scope of the present research.
112
Figure 3-24: Cracking pattern at the top side of the SSB connections
The cracking patterns at the bottom side of the beams as well as the locations of the vertical slots are shown in Figure 3-25. Non-uniform cracks were formed in the vicinity of the stub region while flexural cracking took place in the remaining portion of the beam. It is noted that the horizontal cracks formed in the stub region were developed suddenly as a consequence of the fracture of the bottom reinforcement. The fractures of the reinforcing bars took place inside the vertical slots as shown in Figure 3-26. The fracture of the bar in the SSB-S-1.7 connection was associated with a secondary failure of the concrete cover as shown in Figure 3-27 in which a large chunk of concrete failed as a consequence of the fracture of the second bar. Upon load reversal, the bar buckled in the vicinity of two stirrups as shown in the figure.
113
Figure 3-25: Cracking pattern at the bottom side of the SSB connections
Figure 3-26: Bar fractures in the SSB connections
114
Figure 3-27: Secondary failure in the SSB-S-1.7 connection
The cracking patterns in the joint are also investigated in order to examine the effect of moving the vertical slot on the shear deformation within the joint. The cracking patterns in the joints are shown in Figure 3-28. Diagonal crack (B) formed in the joint of the SSB-S-0 connection at a displacement cycle of 40 mm. No cracks were developed within the joint of the SSB-S-1.0 connection, while vertical hairline cracks were formed in the SSB-S-1.7 connection. The difference in the performance of the SSB-S-1.0 and the SSB-S-1.7 connections can be explained in light of the length of the stub region. The beam with a large relocation distance acts like a conventional beam but with a disturbed region in the neighbourhood of the vertical slot. Consequently, the cracks within the joints are similar to those in conventional RC connections.
115
(a) Connection SSB-S-0
(b) Connection SSB-S-1.0
(c) Connection SSB-S-1.7 Figure 3-28: Cracking pattern at the joint in the SSB connections
3.7.3 Beam Rotation The effectiveness of relocating the plastic hinge by moving the vertical slot away from the face of the column is examined by plotting the rotation profile along the beam members for the positive and negative bending as shown in Figure 3-29. The displacement values obtained from the LSC devices mounted on the top and the bottom concrete fibres (see Figure 3-16) were used to calculate the beam rotation using Equation B.1. It is also noted that the positive rotation values refer to positive bending while the negative rotation values refer to negative bending. The rotation profiles 116
are plotted for four displacement amplitude cycles that were chosen to examine the elastic, the post-yielding behaviour, and the ultimate condition. The rotation profiles in the SSB-S-0 connection under positive and negative bending are approximately identical. The maximum beam rotations at ultimate (55 mm displacement amplitude cycle) were 0.028 rad and -0.031 rad under positive and negative bending, respectively. The rotations attained at the location of the slot under positive and negative bending represent 90% and 91%, respectively, of the beam total rotation at the ultimate cycle. This indicates that the section rotations are approximately constant along the beam member except at the stub and the vertical slot regions (i.e. the beam behaves like a rigid element with its centre of rotation located at the location of the vertical slot). Examining Figure 3-29 (b) and (c) yields that moving the vertical slot away from the face of the column spread the plasticity over a larger region, yet, the maximum rotation was attained at the location of the vertical slot. At the ultimate cycle, the rotations in the SSB-S-1.0 connection were 0.031 rad and -0.038 rad under positive and negative bending, respectively, while the rotations in the SSB-S-1.7 connection were 0.034 rad and -0.044 rad under positive and negative bending, respectively. The rotations at the ultimate cycle under negative bending are 23% and 29% higher than the rotations under positive bending in SSB-S-1.0 and SSB-S-1.7 connections, respectively. This indicates the more significant contribution of the stub region in the beam rotation under positive bending as compared with negative bending.
117
0.05
6 mm 20 mm 40 mm 55 mm
Beam rotation (rad)
0.03
0.01
-0.01
-0.03
-0.05 0
150
300
450
600
750
900
1050
1200
1050
1200
Distance from column face (mm)
(a) Connection SSB-S-0 0.05
Beam rotation (rad)
0.03
0.01
-0.01
6 mm 20 mm 40 mm 55 mm
-0.03
-0.05 0
150
300
450
600
750
900
Distance from column face (mm)
(b) Connection SSB-S-1.0
118
0.05
Beam rotation (rad)
0.03
0.01
-0.01
6 mm 20 mm 40 mm 55 mm
-0.03
-0.05 0
150
300
450
600
750
900
1050
1200
Distance from column face (mm)
(c) Connection SSB-S-1.7 Figure 3-29: Rotation profiles of the beams in the SSB connections
The rotation attained at the location of the vertical slot represents 67% and 32% of the total rotation at the ultimate cycle for SSB-S-1.0 and SSB-S-1.7 connections, respectively, under positive bending while it represents 75% under negative bending for both connections. Therefore, it is concluded that the rotation at the location of the vertical slot under positive bending decreases in a linear manner with increasing relocation distance, while it drops by 18% under negative bending regardless of the relocation distance. These conclusions comply with the observations made with regard to the cracking pattern discussed in Section 3.7.3. Based on the beam rotation profiles discussed in this section, the best relocating distance that successfully relocates the plastic hinge under positive and negative bending is 1.0 dv (300 mm).
119
3.7.4 Beam Elongation The change in the beam length is examined followed by studying the beam elongation in a RC frame in this section. The change in the beam length was calculated using Equation B.2 by averaging the total deformation of the top and the bottom concrete fibres of the beams. The change in the beam length with respect to increasing displacement amplitudes under positive and negative bending is shown in Figure 3-30. The change in the beam length increased under positive bending while it decreased under negative bending with increasing displacement amplitude. The positive values of the change in the length indicate the elongation of the beam member, while the negative values indicate the shortening of the member. At the ultimate conditions, the change in the beam length under positive bending was 4 mm (0.2%), 5 mm (0.3%), and 6 mm (0.4%), and under negative bending it was -3 mm (-0.2%), -4 mm (-0.2%), and -3 mm (-0.2%) for the SSB-S-0, SSBS-1.0, and SSB-S-1.7 connections, respectively. Thus, the change in the length of the beams is very small and it tends to increase with increasing relocation distance. This can be further understood by examining the elongation in a RC frame in which vertical slots are made in the beam from both ends. Under earthquake excitation, the frame will move laterally, and hence, one end will experience a negative moment while the other will experience a positive moment as shown in Figure B.1. The elongation of the member is thus the summation of the change in length at the two beam ends as expressed in Equation B.3. In this study, the elongations of the SSB connections can be calculated by summing the change in the length in both the positive and the negative bending as plotted in Figure 3-31. In general, the beam elongation increased with increasing displacement amplitude. At the ultimate displacement, the elongation was 1 mm (0.1%), 1 mm (0.1%), and 2 mm (0.1%) for the SSB-S-0, SSB-S-1.0, and SSB-S-1.7 connections, respectively.
120
0.36
SSB-S-0 SSB-S-1.0 SSB-S-1.7
4 2
0.24 0.12
0
0.00
-2
-0.12
-4
-0.24
-6
Change in length (%)
Change in length (mm)
6
-0.36 -80
-60
-40
-20
0
20
40
60
80
Displacement amplitude (mm)
Figure 3-30: Change in length in the SSB connections 2.5
0.15
SSB-S-0 SSB-S-1.0 SSB-S-1.7
0.12
1.5
0.09
1
0.06
0.5
0.03
0
0.00 0
10
20
30
40
Displacement amplitude (mm)
Figure 3-31: Beam elongation in the SSB connections
121
50
60
Elongation (%)
Elongation (mm)
2
It can be seen that the elongations of the SSB-S-1.0 and SSB-S-0 connections are approximately identical, which indicates the efficiency of the SSB-S-1.0 connection in maintaining low levels of elongation. The elongation of the SSB-S-1.7 is, however, approximately double that in the other connections. It is, thus, concluded that the increase in the relocating distance beyond a distance equivalent to 1.0 dv (300 mm) increases the elongation of the beam in RC frames significantly.
3.7.5 Strain Profile in Longitudinal Reinforcement The effectiveness of the plastic hinge relocation technique is further examined in this section by studying the strain profiles in the longitudinal reinforcements in the beams. The strain profiles in the bottom reinforcement at four displacement amplitudes under positive and negative bending are shown in Figure 3-32. The peak strains were located in the debonded region, which is equal to the width of the vertical slot and the steel tube (225 mm). In this region, the strain was approximately constant. The average strain in the debonded region under positive bending was 0.033 (12.4 εy, where εy is the yield strain), 0.029 (11.0 εy), and 0.014 (5.4 εy), while it was -0.034 (-12.7 εy), 0.039 (-14.5 εy), and -0.050 (-18.7 εy), under negative bending in the SSB-S-0, SSB-S-1.0, and SSB-S-1.7 connections, respectively. It is, therefore, seen that relocating the vertical slots away from the face of the column by distances equivalent to 1.0 dv and 1.7 dv reduced the peak strain at failure by 12% and 58%, respectively, under positive bending. However, it tended to increase the peak strain at failure by 15% and 47%, respectively, under negative bending. These observations indicate the superior performance of the SSB system in relocating the plastic hinges under negative bending as compared with positive bending. Furthermore, the strain profiles were approximately symmetric in the SSB-S-0 connection under positive and negative bending. However, the strain 122
profiles in the SSB-S-1.0 and SSB-S-1.7 connections were not uniform under positive bending and experienced higher strain values under negative bending as compared with the behaviour in the SSB-S-0 connection. Examining the strain profiles also yields that moving the vertical slots reduced the strain values at the face of the column under negative bending (111% and 80% at the ultimate cycle in the SSB-S-1.0 and SSB-S-1.7 connections, respectively). However, it was less effective in reducing the strain values under positive bending (49% and 34% in the SSB-S-1.0 and SSB-S-1.7 connections, respectively). It is also noted that the strain values at the face of the column under negative bending in the SSB-S-1.0 connection were below yielding unlike the strain values in the SSB-S-1.7 connection. The reduction of the strain values at the face of the column led to less bond deterioration inside the joint, and hence, reduced the shear distortion when subjected to cyclic loading. The increase in the strain values at the face of the columns under positive bending in the connections with relocated vertical slots was due to the less effectiveness in relocating the plastic hinge under positive bending which was discussed in details in Sections 3.7.1 and 3.7.3. It is observed that the strain values of the bottom reinforcement at the middle of the joint in all connections were below yield under positive and negative bending due to the enhanced anchorage performance provided by the applied vertical load on the concrete columns. Based on the above discussions, the best relocating distance that successfully relocate the plastic hinge under positive and negative bending is 1.0 dv (300 mm).
123
Strain, ε (mm/mm)
0.02
7.4
Yielding
0.0
0
-0.02
-7.5
6 mm 20 mm 40 mm 55 mm
-0.04
-0.06 -440
-220
-14.9
Normalized strain, ε/εy
14.9
0.04
Solid line - Positive bending Dashed line - Negative bending
0
220
440
660
880
-22.4 1100
Distance from the column face (mm)
(a) Connection SSB-S-0
Strain, ε (mm/mm)
0.02
14.9
7.4
Yielding
0
0.0
-0.02
-0.04
-7.5
6 mm 20 mm 40 mm 55 mm
-0.06 -440
-220
-14.9 Solid line - Positive bending Dashed line - Negative
0
220
440
660
880
Distance from the column face (mm)
(b) Connection SSB-S-1.0
124
-22.4 1100
Normalized strain, ε/εy
0.04
0.04
14.9
Strain, ε (mm/mm)
0.02
7.4
Yielding
0
0.0
-0.02
-0.04
-7.5
6 mm 20 mm 40 mm 55 mm
-0.06 -440
-220
-14.9
0
220
440
660
880
Normalized strain, ε/εy
Solid line - Positive bending Dashed line - Negative bending
-22.4 1100
Distance from the column face (mm)
(c) Connection SSB-S-1.7 Figure 3-32: Strain profiles of the bottom reinforcement in the SSB connections
The strain profiles in the top reinforcement at four displacement amplitudes under positive and negative bending are shown in Figure 3-33. In all connections, the strain values in the top reinforcements were positive along the beam member under positive and negative bending. This indicates that the top reinforcement was located below the neutral axis (in the tension zone). It is also observed that the peak values in the three connections were located at the locations of the vertical slots. The peak strain values at failure (55 mm displacement cycle) under positive bending were 0.0038 (1.7 εy), 0.0057 (2.5 εy) and 0.0069 (3.1 εy) while they were 0.0048 (2.1 εy), 0.0098 (4.4 εy), and 0.0122 (5.5 εy) under negative bending in the SSB-S-0, SSB-S-1.0, and SSB-S-1.7 connections, respectively. It is, therefore, seen that relocating the plastic hinge away from the face of the column by distances equivalent to 1.0 dv and 1.7 dv increased the peak strain at failure by 125
50% and 82%, respectively, under positive bending and 104% and 154%, respectively, under negative bending. Thus, the strain values at the location of the plastic hinge increased with increasing relocation distance. Under positive bending, the increase in the strain is a result of the increase in the permanent deformation due to the extensive cracking at the top fibre in the connections with relocated vertical slots as shown in Figure 3-22. However, the increase in the strains under negative bending at the location of the slot is attributed to the decrease in the neutral axis depth at the location of the vertical slot. Under positive bending, the strain values exceeded the yield strain for displacement amplitudes higher than 40 mm in all connection at the locations of the vertical slots. However, the strains exceeded the yield strain for amplitude cycles higher than 40 mm, 30 mm, and 20 mm for the SSB-S-0, SSB-S-1.0, and SSB-S-1.7 connections, respectively, under negative bending. The difference in the onset of the yielding strain under positive and negative bending is attributed to the same reasons discussed previously about the behaviour at the ultimate load. Based on the above discussion, it is concluded that moving the vertical slot away from the face of the column succeeded in relocating the plastic hinge away from the face of the column. Under positive bending, the plastic hinge was located in the vicinity of the stub region, while it was located at the vertical slot under negative bending. The strains at the top reinforcement exceeded the yielding strain only at the location of the vertical slot after being subjected to high amplitude displacement cycles.
126
0.013
5.8
Strain, ε (mm/mm)
0.009
Solid line - Positive bending Dashed line - Negative bending
4.0
0.007 0.005
4.9
3.1 2.2
Yielding
0.003
1.3
0.001
0.4
-0.001 -440
-220
0
220
440
660
880
Normalized strain, ε/εy
6 mm 20 mm 40 mm 55 mm
0.011
-0.5 1100
Distance from the column face (mm)
(a) Connection SSB-S-0 0.013
5.8
Strain, ε (mm/mm)
0.009
Solid line - Positive bending Dashed line - Negative bending
4.0
0.007 0.005
4.9
3.1 2.2
Yielding
0.003
1.3
0.001
0.4
-0.001 -440
-220
0
220
440
660
880
Distance from the column face (mm)
(b) Connection SSB-S-1.0
127
-0.5 1100
Normalized strain, ε/εy
6 mm 20 mm 40 mm 55 mm
0.011
0.013
5.8 Solid line - Positive bending Dashed line - Negative bending
4.9
6 mm 20 mm 40 mm 55 mm
0.009 0.007
4.0 3.1
0.005
2.2
Yielding 0.003
1.3
0.001
0.4
-0.001 -440
-220
0
220
440
660
880
Normalized strain, ε/εy
Strain, ε (mm/mm)
0.011
-0.5 1100
Distance from the column face (mm)
(c) Connection SSB-S-1.7 Figure 3-33: Strain profiles of the top reinforcement in the SSB connections
3.7.6 Shear Mechanism 3.7.6.1 Shear in the Beam The shear behaviour is studied by examining the distortion angle obtained from the LSC devices, the strains in the stirrups near the slot, and the strains in the diagonal reinforcement as shown in Figure 3-34. It was found that δbd,2 provided approximately a zero reading for all stages of loading while δbd,1 provided fluctuating readings as the beams were subjected to positive and negative bending. This indicates the complex nature of the shear distortion in this region, and thus, the shear distortion angle was calculated based on the readings obtained from δbd,1. The detailed calculation procedure is presented in Section B.3. The envelopes of the moment versus the shear distortion relationships are plotted in Figure 3-35. Overall, the shear distortion under positive bending was 128
approximately zero (the variation under positive bending is due to the scatter nature inherited in testing the specimens) while it tended to increase with increasing moment when subjected to negative bending. This can be understood in light of the discussion provided in Section 3.7.2 with regard to the cracking pattern when the beam members were subjected to positive and negative bending. As shown in Figure 3-23 (b), diagonal crack (C) formed when the beam was subjected to negative bending while it closed when subjected to positive bending. Therefore, the shear distortion in the beam section was mainly due to the opening of the diagonal crack (C) under negative bending. In general, the shear distortion under negative bending is insignificant (below -0.4%), and thus, it can be concluded that the beam sections behaved in almost an elastic manner in shear. Also, it is concluded that relocating the vertical slot away from the face of the column does not affect the shear mechanism in the beam section to the right of the slot since the behaviour of the three connections is comparable as shown in Figure 3-35. However, it results in increased distortion as a result of the increase in the applied loading.
Figure 3-34: Locations of the LSCs and strain gauges used to examine the shear behaviour in the beams of the SSB connections
129
Figure 3-35: Beam shear distortion in the SSB connections
The strain values in the stirrups and the diagonal reinforcement at the peak displacement cycles are plotted against the applied moment as shown in Figure 3-36 and Figure 3-43, respectively. It is also noted that the strain gauge attached to the stirrup in the SSB-S-1.0 connection was malfunctioned under negative bending as indicated in Figure 3-36. In all connections, the moment-strain relationships in the stirrups and the diagonal reinforcement follow bilinear trends in which the slope of the relationships in the post-yielding stage are significantly lower than that prior to yielding. It is noted that the strain values in all connections are below yielding, and thus, an elastic behaviour was maintained. The strain trends in the stirrups are approximately identical under positive and negative bending unlike the behaviour observed in the diagonal reinforcement in which the strains induced under positive bending are significantly higher than negative bending. 130
Figure 3-36: Moment versus strain in the beam stirrup of the SSB connections
Figure 3-37: Moment versus strain in the diagonal reinforcement of the SSB connections
131
3.7.6.2 Shear in the Joint The shear in the joint is examined by studying the shear distortion obtained from the readings of the LSCs and the strains in the stirrups within the joints as represented in Figure 3-38. As discussed in Section 3.2.5, the flow of the compression strut in the joint of the SSB system when the vertical slot is placed at the face of the column is of a complex nature. However, it was proposed that moving the vertical slot away from the face of the column will result in a mechanism of deformation within the joint that is similar to the conventional beam-column connections. This hypothesis is examined in this section.
Figure 3-38: Locations of the LSCs and strain gauges used to examine the joint behaviour of the SSB connections
The applied moment is plotted against the shear distortion angle as shown in Figure 3-39. The detailed calculation procedure is included in Section B.5. The shear distortion at ultimate was -0.01 %, 0.03%, and 0.15%, under positive bending and - 0.1%, -0.1%, and -0.1% under negative bending for the SSB-S-0, SSB-S-1.0, and SSB-S-1.7 connections, respectively. It is noted that the low shear distortion in the SSB-S-0 connection under positive bending is not due to the enhanced performance. However, it is due to the increase in the permanent deformation in the joint when subjected to negative bending as a result of the diagonal cracking in the joint as shown in Figure
132
3-28. Comparing the responses of the three connections yields that connection SSB-S-1.0 attained superior performance in terms of symmetric response under positive and negative bending and low shear distortion (equal or below 0.1%). It is, therefore, concluded that relocating the vertical slot away from the face of the column by 1.0 dv results in an improved joint response. The moment-strain relationships of the top, middle, and bottom stirrups in the joints at the peak displacement cycles are plotted in Figure 3-40. In all connections, the strain values were below yielding and increased with increasing relocation distance. This increase is due to the increase in the seismic shear force associated with relocating the plastic hinge away from the face of the column. It is also noted that the top and bottom stirrups are located close to the top and the bottom longitudinal reinforcements, and hence, the strain induced in these stirrups was partially due to their contribution in resisting the loads transferred via the flexural reinforcements as suggested by Paulay and Priestley (1992). However, the strains induced in the middle stirrup were generated due to the formation of the diagonal compression struts (refer to Figure 3-9).
Figure 3-39: Joint shear distortion in the SSB connections 133
(a) Top stirrup
(b) Middle stirrup
134
(c) Bottom stirrup Figure 3-40: Envelopes of the moment-joint strain relationships in the SSB connections
The strain values in the middle stirrups in the SSB-S-1.0 and SSB-S-1.7 connections are 1.1 and 1.0 times higher under positive bending and 7.6 and 60.2 times higher under negative bending as compared with the strains in the SSB-S-0 connection. It is, therefore, observed that relocating the vertical slots away from the face of the column by a distance 1.0 dv (300 mm) results in a minor increase in the joint strain as compared with a relocating distance of 1.7 dv (500 mm).
3.7.7 Components of Deformation The member contributions to the deformation of the tested connections are examined in this section. The procedure adopted in this analysis is based on the work by Alcocer and Jirsa (1991). However, it was modified in this research in order to account for the relocation of the centre of 135
rotation by moving the vertical slot away from the face of the column. The detailed steps of calculating the components of deformation using the DICT method are included in Section B.5. The member contributions including beam bending, column bending, and joint distortion under positive and negative bending are illustrated in Figure 3-41 with respect to increasing displacement amplitude. The percentage contributions of the aforementioned components at the ultimate cycle for the tested connections are also included in Table 3-6. It is noted that the results of the 2 mm and 4 mm amplitude cycles were discarded from the analysis since the deformations within the joint were very small and could not be captured accurately using the DICT technique. With increasing displacement amplitude, the contribution of the beam bending tended to increase while the contributions of the column bending and the joint distortion decreased in all connections. It is also observed that the contributions of the column bending and the joint distortion in the connections SSB-S-1.0 and SSB-S-1.7 were higher than connection SSB-S-0. 100
Joint distortion
Member contribution (%)
Column bending
80
60 Beam bending
40
20
0 -60
-40
-20
0
20
Displacement amplitude (mm)
(a) Connection SSB-S-0 136
40
60
100
Joint distortion
Member contribution (%)
Column bending
80
60 Beam bending
40
20
0 -60
-40
-20
0
20
40
60
40
60
Displacement amplitude (mm)
(b) Connection SSB-S-1.0 100
Joint distortion
Member contribution (%)
Column bending
80
60 Beam bending
40
20
0 -60
-40
-20
0
20
Displacement amplitude (mm)
(c) Connection SSB-S-1.7 Figure 3-41: Member contributions to the total applied displacement in the SSB connections 137
Table 3-6: Member contributions of the SSB connections at the ultimate cycle Connection SSB-S-0 SSB-S-1.0 SSB-S-1.7
Beam bending Positive Negative (%) (%) 87 91 81 84 84 79
Column bending Positive Negative (%) (%) 9 6 14 10 9 17
Joint distortion Positive Negative (%) (%) 4 3 5 6 7 4
In order to evaluate the effect of the relocation distance, lre, on the effectiveness of the SSB technique in relocating the centre of rotation, the ratio of θ2/θ1 (refer to Figure B-3 for the definitions of θ2 and θ1) is plotted against the positive and the negative displacement amplitude cycles in Figure 3-42 for the SSB-S-1.0 and SSB-S-1.7 connections. Under positive bending, the ratios were insensitive to the magnitude of the imposed displacement. However, the ratios tended to increase linearly in the SSB-S-1.0 connection and in a bilinear shape in the SSB-S-1.7 connection with increasing displacement amplitude under negative bending. At the ultimate cycle, the ratios were 5.4 and 2.9 under positive bending and 17.0 and 8.9 under negative bending in the SSB-S-1.0 and SSB-S-1.7 connections, respectively. These observations are in agreement with the conclusions made in Section 3.7.1 with regard to the superior performance of the SSB system in relocating the centre of rotation under negative bending than positive bending. The θ2/θ1 can be considered as an indicator to quantify the effectiveness of the SSB system in relocating the centre of rotation, and thus, it can be argued that the effectiveness of having the vertical slot at 1.0 dv (300 mm) away from the face of the column is 1.9 times that if the distance to the vertical slot is increased to 1.7 dv (500 mm) under both positive and negative bending.
138
20
SSB-S-1.0 SSB-S-1.7
Rotation ratio, θ2/θ1
16
12
8
4
0 -60
-40
-20
0
20
40
60
Displacement amplitude (mm)
Figure 3-42: Comparison of the rotation ratios (θ2/θ1) of the SSB-S-1.0 and SSB-S-1.7 connections
3.8 Experimental Behaviour of the DSB System 3.8.1 Hysteretic Response The hysteretic responses of the connections in the DSB group are shown in Figure 3-43. The moment and the rotation were calculated in a similar manner as described in Section 3.7.1. The connections failed by the fracture of the steel reinforcement at the vertical slots. In connection DSB-S-0, the fracture took place in the bottom reinforcement, while it took place at the top reinforcements in the DSB-S-1.0 and DSB-S-1.7 connections. Connections DSB-S-0, DSB-S-1.0, DSB-S-1.7, failed during the third positive bending cycle of the 80 mm amplitude cycle, the second negative bending cycle of the 85 amplitude cycle, and the second negative bending cycle of the 75 amplitude cycle, respectively.
139
(a) Connection DSB-S-0
(b) Connection DSB-S-1.0
140
(c) Connection DSB-S-1.7 Figure 3-43: Hysteretic behaviour of the DSB connections
The fat and stable hysteretic response indicate that moving the vertical slots away from the face of the column did not result in degrading the responses of the connections. Similar to the SSB connections, no signs of pinching shear were observed. This is due to the fact that the stiffness response of the DSB system is based mainly on the longitudinal steel and not on the concrete. Therefore, the cracking of the concrete at the concrete hinge does not have a significant effect on the stiffness of the connections even under high drift ratios. The loads at the yielding and the ultimate conditions (Py and Pu, respectively) for the positive and the negative bending of the connections are included in Table 3-7. The loads at the yielding and the ultimate conditions increased approximately in a linear manner with increasing relocation distance. However, the increase in the load values was higher for the negative bending
141
as compared with the positive bending. The increase in the Py and Pu values when the vertical slots were moved away from the face of the column indicate a reduction in the shear span distance. Consequently, the DSB technique succeeded in relocating the plastic hinge away from the face of the column. In order to further evaluate the effectiveness of using the DSB technique in relocating the centre of rotation, the ratios of the moments in the connections with relocated vertical slots (connections DSB-S-1.0 and DSB-S-1.7) to the moment of the connection in which the vertical slot is located at the face of the column (connection DSB-S-0) are included in Table 3-8 at the yielding and the ultimate conditions for both the positive and the negative bending (M/M0). In general, the M/M0 ratios at yielding and ultimate conditions under positive and negative bending are approximately equal to one. Therefore, it can be concluded that the centre of rotation was successfully located at the centre of the vertical slots using the DSB system under positive and negative bending. Therefore, the definition of the shear spans (length of the beam between the applied displacement and the vertical slot – refer to Figure 3-3) is found to be reasonably accurate in determining the drift ratios.
Table 3-7: Load comparison at the yielding and the ultimate conditions of the DSB connections Yielding Ultimate Positive Negative Positive Negative Connection * * * Py Diff. Py Diff. Pu Diff. Pu Diff.* (kN) (%) (kN) (%) (kN) (%) (kN) (%) DSB-S-0 37 30 58 45 DSB-S-1.0 41 11 36 20 68 17 57 27 DSB-S-1.7 53 43 45 50 82 41 68 51 * Percentage difference with respect to the DSB-S-0 connection
142
Table 3-8: Evaluation of plastic hinge relocation of the DSB connections Yielding Connection DSB-S-0 DSB-S-1.0 DSB-S-1.7
Positive Moment M/M0 (kN·m) 61 1.00 55 0.90 62 1.02
Ultimate
Negative Moment M/M0 (kN·m) 50 1.00 49 0.98 52 1.04
Positive Moment M/M0 (kN·m) 96 1.00 92 0.96 95 0.99
Negative Moment M/M0 (kN·m) 75 1.00 77 1.03 79 1.05
The flexural overstrength, φo, factors calculated at the ultimate condition (80 mm, 85 mm, and 75 mm displacement amplitude cycles for connections DSB-S-0, DSB-S-1.0, and DSB-S-1.7, respectively) are included in Table 3-9. It is noted that the φo factors are defined as the applied moment, M, divided by the nominal moment, Mn. Two values of Mn were considered; one excludes the contribution of the diagonal reinforcement (50 kN·m) and the other one includes the contribution of the diagonal reinforcement (61 kN·m) as determined in Section A.3.1.1. The φo factors decreased with increasing relocation distance. This is due to the fact that the plasticity spread over a larger area when the vertical slots were relocated, and thus, the overstrength of the steel and the concrete materials were attained over a larger area. The average increase in φo of the DSB connections is 13% and 36% higher than the value of 1.25 proposed by CSA-A23.3-04 (2004) when including and excluding the contribution of the diagonal reinforcement, respectively. Thus, the φo factors excluding or including the diagonal reinforcement are higher than the values attained in conventional beam-column connections. The high φo factors experienced in the DSB connections is due to the large curvature demands induced in the steel reinforcement in the neighbourhood of the vertical slots which allowed for significant hardening of the flexural reinforcement, and hence, it led to an increase in the φo factor.
143
Table 3-9: Flexural overstrength factors, φo, for the DSB connections With diagonal reinforcement Positive Negative * Diff. Diff.* φo φo (%) (%) 1.58 1.23 1.52 -4 1.27 3 1.56 -1 1.30 6
Connection DSB-S-0 DSB-S-1.0 DSB-S-1.7
Without diagonal reinforcement Positive Negative * Diff. Diff.* φo φo (%) (%) 1.90 1.49 1.83 -4 1.54 3 1.88 -1 1.56 5
It is observed that the moment capacity under negative bending was lower than that under positive bending for all connections despite the symmetric nature of the beam sections. The variation of the positive moment to the negative moment ratio (M+/M-) with increasing displacement amplitude is shown in Figure 3-44. It is seen that the ratio dropped to approximately 1 at yielding and then increased with increasing displacement amplitude. The ratio was the highest for DSB-S-0, then DSB-S-1.0 and DSB-S-1.7 connections. There are two main reasons for this difference in the ultimate loads and moments under positive and negative bending; the cyclic properties of the flexural reinforcement and the self-weight of the beam element.
Ratio, M+/M-
1.5
1.25
DSB-S-0 DSB-S-1.0 DSB-S-1.7
1
0.75 0
20
40
60
80
100
Displacement amplitude (mm)
Figure 3-44: Variation of the M+/M- ratio with the increase in the displacement amplitude 144
The hysteretic responses of the DSB connections are highly dependent on the hysteretic characteristics of the reinforcing steel since the latter represents the primary flexural resisting element in the connection (due to the minor contribution of the concrete). Therefore, examining the cyclic behaviour of the top and the bottom steel reinforcements at the debonded region (total length of the slot plus the steel tube) is very crucial in understanding the difference in the responses between the positive and the negative bending. Uniaxial testing of reinforcing steel bars is usually conducted either under controlled strain or controlled stress conditions. However, a combination of both conditions is experienced in the debonded region in the DSB system. The reduction in the yield stress after a load reversal, which increases with the enlargement of the plastic strain of the last excursion, results in the decrease in the strength under negative bending (Yu, 2006). At each amplitude cycle, the connections were initially subjected to positive bending followed by negative bending. Therefore, the yielding of the bottom reinforcement in tension was commenced first while at this stage the top reinforcement is still elastic. Upon the reversal of loading, the reduction of the yield stress along with the Bauschinger effect caused a reduction in the stress capacity of the bottom reinforcement in compression. Consequently, the capacity of the beam under negative bending was reduced as compared with that under positive bending. Prior to the application of the quasi-static cyclic displacement, the self-weight of the beam member (5.0 kN) induced a moment at the centre of rotation (i.e. the vertical slot). However, the moment due to the self-weight decreased with increasing relocation distance due to the reduction in the shear span (8.3 kN·m, 6.8 kN·m, and 5.8 kN·m for the DSB-S-0, DSB-S-1.0, and DSB-S1.7 connections, respectively). This explains the reduction in the difference between the positive and the negative bending moments with increasing relocation distance. In other words, the effect of the self-weight was reduced as the vertical slot was moved away from the face of the column. 145
3.8.2 Cracking Pattern The cracking patterns at the side face of the beam members at failure are shown in Figure 3-45. The location of the diagonal shear reinforcements and the locations of the steel tubes used to debond the flexural reinforcements are also shown in the figure. It is noted that the letter “D” stands for diagonal.
(a) Connection DSB-S-0
(b) Connection DSB-S-1.0
(c) Connection DSB-S-1.7 Figure 3-45: Cracking patterns at the ultimate condition in the DSB connections 146
The first cracks initiated at the concrete hinge in all connections. Flexural cracks developed at the bottom and top sides of the beam as the displacement amplitude was increased. However, no flexural cracks were observed to develop for a distance of approximately 1.0 bd (where bd is the beam depth) to the right of the vertical slots in all connections. Also, no flexural cracks were observed to develop within 0.75 bd and 0.5 bd to the left of the vertical slots in DSB-S-1.0 and DSB-S-1.7 connections, respectively. Therefore, moving the vertical slot away from the face of the column led to the initiation of more flexural cracks in the vicinity of the stub region. Two types of diagonal cracks were initiated in the connections; A and B. Diagonal cracks (A) refer to the cracks that initiates at the right side of the concrete hinge and extended to the upper and/or the lower face of the beam at approximately 45o angle, while diagonal cracks (B) refer to the cracks that developed to the left side of the concrete hinge and extended at a 45 angle. Diagonal cracks (A) and (B) were initiated in connections DSB-S-0 and DSB-S-1.0 while only diagonal crack (A) was initiated in connection DSB-S-1.7. It should be noted, however, that diagonal cracks (B) in connection DSB-S-0 are not shown in Figure 3-45 since they developed at the joint (refer to Figure 3-47). In the DSB-S-0 and DSB-S-1.0 connections, the diagonal cracks (A) initiated only at the lower end of the concrete hinge parallel to the diagonal steel reinforcement while they were initiated at a distance of approximately 0.25 Sd,u and 0.33 Sd,l (refer to Figure 3-3 for the definition of Sd,u and Sd,l) from the concrete hinge in DSB-S-1.7 connection. The diagonal cracks (A) initiated at displacements of 40 mm, 25 mm, and 30 mm, in connections DSB-S-0, DSB-S-1.0, and DSBS-1.7, respectively, under positive moment and initiated at a displacement of 35 mm in connection DSB-S-1.7 under negative moment. The initiation of the diagonal cracks at lower displacement levels in the connections with relocated vertical slots is due to the increase in the applied loads as a result of the reduction in the shear span. 147
In DSB-S-0 connection, the diagonal cracks (B) initiated from the edges of the concrete hinge extending through the concrete joint and the column as shown in Figure 3-45 (a) while they extended from the lower and the upper sides of the beam edges in DSB-S-1.0 connection(Figure 3-45 (b)). The diagonal cracks (B) were initiated at 35 mm and 20 mm in DSB-S-0 and DSB-S1.0 connections, respectively, under positive bending, while they were initiated at 50 mm and 20 mm in DSB-S-0 and DSB-S-1.0 connections, respectively, under negative bending. It is seen, therefore, that the onset of the initiation of diagonal cracks (B) in DSB-S-0 connection took place at higher displacement amplitude in both positive and negative bending. This is due to the fact that the applied compression force in the column helped in enhancing the shear resistance in the joint. The cracking pattern at the top and the side of the concrete beams for a distance of 1.5 bd (600 mm) from the face of the column are shown in Figure 3-46. As compared with DSB-S-0 connection, a few horizontal cracks initiated in the beam stub running parallel to the top longitudinal reinforcement in DSB-S-1.0 and DSB-S-1.7 connections. Thus, in general, moving the vertical slots away from the face of the column led to the initiation of horizontal cracks running parallel to the flexural reinforcement. Also, the intensity of the flexural cracks at the top side was the same as the one observed at the sides of the beams (refer to Figure 3-45). It is also noted that the cracking pattern and distribution at the bottom side of the beam were similar to that at the top side. The cracking patterns in the joints are shown in Figure 3-47. No diagonal cracks were observed in DSB-S-1.0 and DSB-S-1.7 connections unlike the behaviour in DSB-S-0 connection. However, a few vertical cracks were observed to develop in DSB-S-1.0 and DSB-S-1.7 connections at displacement amplitudes of 45 mm and 35 mm, respectively. Therefore, it is
148
concluded that relocating the vertical slots away from the face of the column, and hence relocating the centre of rotation, reduced the cracking intensity in the joint significantly.
(a) Connection DSB-S-0
(b) Connection DSB-S-1.0
(c) Connection DSB-S-1.7 Figure 3-46: Cracking pattern at the top side in the DSB connections
3.1
149
(a) Connection DSB-S-0
(b) Connection DSB-S-1.0
(c) Connection DSB-S-1.7 Figure 3-47: Cracking pattern at the joint in the DSB connections
3.8.3 Beam Rotation The rotation profiles under positive and negative bending along the beam length are shown in Figure 3-48. At the ultimate cycles, the total rotations of the beams were 0.047 rad, 0.060 rad, and 0.060 rad under positive bending while they were -0.050 rad, -0.062 rad, and -0.061 rad under negative bending in the DSB-S-0, DSB-S-1.0, and DSB-S-1.7 connections, respectively. The increase in the rotation at ultimate in the connections with the relocated vertical slots (DSB-S-1.0 and DSB-S-1.7) as compared with the connection having the vertical slot at the face of the column (DSB-S-0) is due to the increase in the rotation demand associated with the reduction in the shear 150
span as discussed in Section 3.3. The average values of the rotation under positive and negative bending along the beam are considered in examining the effectiveness of moving the vertical slot in relocating the centre of rotation due to the symmetric nature of the connections. The average rotations at the vertical slots represent 94%, 79%, and 79% of the total rotation at the ultimate cycle in the DSB-S-0, DSB-S-1.0, and DSB-S-1.7 connections, respectively. Thus, it is observed that the rotation at the vertical slot dropped by 15% when the vertical slot was moved away from the face of the column and it is independent from the relocation distance. This is due to the contribution of the stub region to the total rotation of the beams as the vertical slot is moved away from the face of the column. In DSB-S-1.0 and DSB-S-1.7 connections, the rotation in the stub region represent 18% and 14% of the total rotation at the ultimate cycle. Overall, the beams behaved like rigid elements with their centre of rotation located at the vertical slot. Also, the increase in the relocation distance did not influence the effectiveness of using the DSB technique in relocating the centre of rotation, and hence, the plastic hinge region. 0.08
6 mm 20 mm 40 mm 60 mm 80 mm
0.06
Rotation (rad)
0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 0
150
300
450
600
750
900
Distance from the column face (mm)
(a) Connection DSB-S-0 151
1050
1200
0.08
6 mm 20 mm 40 mm 60 mm 80 mm 85 mm
0.06
Rotation (rad)
0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 0
150
300
450
600
750
900
1050
1200
1050
1200
Distance from the column face (mm)
(b) Connection DSB-S-1.0 0.08 0.06
Rotation (rad)
0.04 0.02 0 -0.02 -0.04
6 mm 40 mm 75 mm
-0.06
20 mm 60 mm
-0.08 0
150
300
450
600
750
900
Distance from the column face (mm)
(c) Connection DSB-S-1.7 Figure 3-48: Rotation profiles of the beams in the DSB connections 152
3.8.4 Beam Elongation The elongation of the RC beams in the DSB connections is examined in this section. The change in the beam length is plotted against the increase in the displacement amplitude for the positive and negative bending as shown in Figure 3-49. The beam sections elongated under both positive and negative bending as expected (due to the symmetric cross-section). The average elongation in the beam sections under positive and the negative bending was 2 mm (0.1%), 3 mm (0.2%), and 3 mm (0.2%) for the DSB-S-0, DSB-S-1.0, and DSB-S-1.7 connections, respectively. Thus, it can be seen that moving the vertical slots away from the face of the column led to the increase in the permanent elongation in the beam sections. This is due to the fact that relocating the plastic hinge away from the face of the column increase the curvature demand at the slot sections and spreads the plasticity in the steel reinforcement as well as the cracking in the concrete over a larger length. In other words, the increase in the strains induced in the bottom and the top reinforcements lead to the increase in the elongation of the beam section, as compared with the connection in which the vertical slots are placed right at the column face. The elongation of the connections in the RC frame is plotted in Figure 3-50 against the increase in the displacement amplitude. The beam elongation increased with increasing displacement amplitude. At ultimate, the elongation of the DSB-S-0, DSB-S-1.0, and DSB-S-1.7 connections was 4 mm (0.2%), 5 mm (0.3%), and 6 mm (0.4%), respectively. Therefore, it is concluded that increasing the relocation distance increase the permanent deformation and the elongation in the beam member in a typical RC frame.
153
4
0.24
DSB-S-0 DSB-S-1.0 DSB-S-1.7
3
0.21 0.18
2.5
0.15
2
0.12
1.5
0.09
1
0.06
0.5
0.03
0 -100
Change in length (%)
Change in length (mm)
3.5
0.00 -75
-50
-25
0
25
50
75
100
Displacement amplitude (mm)
Figure 3-49: Change in length in the DSB connections 0.36
6
DSB-S-0 DSB-S-1.0 DSB-S-1.7
0.30
4
0.24
3
0.18
2
0.12
1
0.06
0
0.00 0
20
40
60
80
Displacement amplitude (mm)
Figure 3-50: Beam elongation in the DSB connections
154
100
Beam elongation (%)
Beam elongation (mm)
5
3.8.5 Strain Profile in Longitudinal Reinforcement The strain profiles of the bottom reinforcement at several displacement amplitudes under positive and negative bending are shown in Figure 3-51. The strain peaks were located in the debonded region and they were approximately constant. The average strains in the debonded region under positive bending were 0.028 (10.4 εy), 0.035 (13.0 εy), and 0.032 (11.9 εy) while the maximum compressive strains under negative bending were -0.016 (-6.0 εy), -0.032 (-11.9 εy), and -0.025 (9.3 εy) in the DSB-S-0, DSB-S-1.0, and DSB-S-1.7 connections, respectively. Relocating the plastic hinge away from the face of the column by distances equivalent to 1.0 dv (300 mm) and 1.7 dv (500 mm) increased the peak strain at failure by 25% and 14%, respectively, under positive bending and 100% and 56%, respectively, under negative bending. This indicates that the peak increase in the strains under positive and negative bending was attained when the plastic hinge is moved away from the face of the column by 1.0 dv (300 mm). Moreover, the strain profiles were approximately symmetric in the DSB-S-0 connection under positive and negative bending, while the strain profiles in the DSB-S-1.0 and DSB-S-1.7 connections were not symmetric due to the increase in the strains at the face of the column. Moving the vertical slots reduced the strain values at the face of the column by 53% and 39% under positive bending and by 130% and 97% under negative bending for the DSB-S-1.0 and DSB-S-1.7 connections, respectively. Despite the symmetric nature of the DSB connection, the effectiveness of reducing the strains at the face of the columns under negative bending was superior to that under positive bending. This behaviour can be explained in light of the higher loads applied under positive bending as described in detail in Section 3.8.1. Also, it is observed that the effectiveness of reducing the strains at the face of the column in the DSB-S-1.0 connection was higher than that in the DSB-S-1.7 connection, and thus, it is concluded that the effectiveness of 155
reducing the strain at the face of the column is highest when the vertical slots are located at 1.0 dv (300 mm) from the column face. The strains at the middle of the joint in all connections were below yielding under positive and negative bending due to the enhanced anchorage performance provided by the vertical load applied to the concrete columns as noted in Section 3.7.5. Based on the strain profiles in the bottom reinforcement, it is found that the best relocating distance that attains the highest strain values in the debonded region and the lowest strain values at the face of the column is 1.0 dv (300 mm).
0.05
18.6
Strain, ε (mm/mm)
0.03
30 mm 70 mm
14.9 11.1
0.02
7.4
0.01
3.7
Yielding
0
0.0
-0.01
-3.7
-0.02
-7.5 Solid line - Positive bending Dashed line - Negative bending
-0.03 -0.04 -440
-220
0
220
440
660
880
Distance from the column face (mm)
(a) Connection DSB-S-0
156
-11.2 -14.9 1100
Normalized strain, ε/εy
10 mm 50 mm 80 mm
0.04
0.05
18.6
10 mm 30 mm 50 mm 70 mm 85 mm
Strain, ε (mm/mm)
0.03 0.02
14.9 11.1 7.4
0.01
3.7
0
0.0
-0.01
Yielding
-0.02 -0.03
-3.7 -7.5 -11.2
Solid line - Positive bending Dashed line - Negative bending
-0.04 -440
-220
0
Normalized strain, ε/εy
0.04
220
440
660
880
-14.9 1100
Distance from the column face (mm)
(b) Connection DSB-S-1.0 0.05
18.6
10 mm 30 mm 50 mm 70 mm 75 mm
Strain, ε (mm/mm)
0.03 0.02
14.9 11.1 7.4
0.01
3.7
0
0.0
-0.01
-3.7
Yielding -0.02 -0.03
-7.5 -11.2
Solid line - Positive bending Dashed line - Negative bending
-0.04 -440
-220
0
220
440
660
880
-14.9 1100
Distance from the column face (mm)
(c) Connection DSB-S-1.7 Figure 3-51: Strain profiles of the bottom reinforcement in the DSB connections 157
Normalized strain, ε/εy
0.04
The strain profiles in the top reinforcement at multiple displacement amplitudes under positive and negative bending are shown in Figure 3-52. The peak average strain values at failure in the debonded region under positive bending were -0.018 (6.7 εy), -0.032 (11.9 εy) and -0.020 (7.5 εy) while they were 0.032 (11.9 εy), 0.037 (13.8 εy), and 0.032 (11.9 εy) under negative bending in the DSB-S-0, DSB-S-1.0, and DSB-S-1.7 connections, respectively. It is, therefore, seen that relocating the plastic hinge away from the face of the column by distances equivalent to 1.0 dv (300 mm) and 1.7 dv (500 mm) increased the peak strain at failure by 78% and 11%, respectively, under positive bending and 16% and 0%, respectively, under negative bending. Similar to the behaviour of the bottom reinforcement, the peak strains were attained when the plastic hinge was moved away from the face of the column by 1.0 dv (300 mm) under positive and negative bending. Moreover, the strain profile was approximately symmetric in all connection. Unlike the behaviour of the bottom reinforcement, the strains in the top reinforcement were more effectively reduced at the face of the column when the vertical slots were relocated. Moving the vertical slots reduced the strain values at the face of the column by 110% and 93% under positive bending and by 94% and 96% under negative bending for the DSB-S-0, DSB-S1.0, and DSB-S-1.7 connections, respectively. The reduction in the strain values at the face of the column is, thus, insensitive to the vertical slot relocation distance. Similar to the strains in the bottom reinforcement, the strains at the middle of the joint in top reinforcement were below yielding. Based on the above discussions, it is concluded that the best relocating distance that attains the highest strain values in the debonded region is 1.0 dv (300 mm).
158
0.05
18.6
Strain, ε (mm/mm)
0.03
30 mm 70 mm
14.9 11.1
0.02
7.4
0.01
3.7
Yielding
0
0.0
-0.01
-3.7
-0.02
-7.5 Solid line - Positive bending Dashed line - Negative bending
-0.03 -0.04 -440
-220
0
220
440
660
880
Normalized strain, ε/εy
10 mm 50 mm 80 mm
0.04
-11.2 -14.9 1100
Distance from the column face (mm)
(a) Connection DSB-S-0 0.05
18.6
10 mm 30 mm 50 mm 70 mm 85 mm
Strain, ε (mm/mm)
0.03 0.02
14.9 11.1 7.4
0.01
3.7
0
0.0
-0.01
Yielding
-0.02 -0.03
-3.7 -7.5
Solid line - Positive bending Dashed line - Negative bending
-0.04 -440
-220
0
220
-11.2 440
660
880
Distance from the column face (mm)
(b) Connection DSB-S-1.0
159
-14.9 1100
Normalized strain, ε/εy
0.04
0.05
18.6
10 mm 30 mm 50 mm 70 mm 75 mm
Strain, ε (mm/mm)
0.03 0.02
14.9 11.1 7.4
0.01
3.7
0
0.0
-0.01
-3.7
Yielding
-0.02 -0.03
-7.5 Solid line - Positive bending Dashed line - Negative bending
-0.04 -440
-220
0
Normalized strain, ε/εy
0.04
-11.2
220
440
660
880
-14.9 1100
Distance from the column face (mm)
(c) Connection DSB-S-1.7 Figure 3-52: Strain profiles of the top reinforcement in the DSB connections
3.8.6 Shear Mechanism 3.8.6.1 Shear in the beam The shear deformation in the neighbourhood of the vertical slots is studied in this section by examining the shear distortion angle obtained from the LSCs, the strains in the stirrup, and the strains in the diagonal reinforcements as indicated in Figure 3-53. The moment versus the shear distortion at the peaks of the displacement cycles is plotted in Figure 3-54. Overall, the trends of the shear distortion are almost symmetric under positive and negative bending. However, the maximum values under negative bending were lower than under positive bending due to the lower peak loads applied under negative bending as discussed in details in Section 3.8.1. The shear distortion under both positive and negative bending was positive, which 160
indicates that the permanent shear deformation increased with increasing applied moment. However, the shear distortion in all connections was insignificant (below 0.3%). The DSB-S-1.0 connection experienced the highest shear, then DSB-S-1.7 and DSB-S-0 connections. This is due to the fact that the DSB-S-1.0 connection experienced the highest displacement at failure (85 mm) as compared with the other connection. It is, therefore, observed that the increase in the shear distortion in the beam is due to the increase in the applied load and displacement as the vertical slot is relocated away from the face of the column.
Figure 3-53: Locations of the LSCs and strain gauges used to examine the shear behaviour in the beams of the DSB connections The strain values in the stirrups and the diagonal reinforcement obtained from the experimental testing are plotted against the applied moment in Figure 3-55 and Figure 3-56, respectively. It is also noted that the strain in the diagonal reinforcements were averaged since the values of the diagonal reinforcement were approximately the same. The strains in the stirrups increased linearly with increasing applied moment until the yielding moment was reached after which the slope of the moment-strain relationships decreased. The moment-strain relationships in the diagonal steel reinforcements follow a parabolic trend. It is also observed that the responses are symmetric under positive and negative bending. However, the values at the ultimate moment 161
under positive bending are higher than those under negative bending due to the higher moment capacity experienced in the former as discussed in detail in Section 3.8.1.
Figure 3-54: Beam shear distortion in the DSB connections
Figure 3-55: Moment versus strain in the beam stirrup of the DSB connections
162
Figure 3-56: Moment versus strain in the diagonal reinforcement of the DSB connections
3.8.6.2 Shear in the Joint The shear in the joint is examined by studying the shear distortion obtained from the readings of the LSC devices and the strains in the stirrups within the joints as represented in Figure 3-38. The moment versus joint strain relationships at the peak displacement cycles of the DSB connections are plotted in Figure 3-57. It is observed that relocating the vertical slots did not affect the shear distortion under negative bending while it reduced the shear distortion under positive bending. However, the DSB-S-1.0 attained the lowest positive shear distortion, and thus, it is concluded that relocating the vertical slot away from the face of the column by 1.0 dv (300 mm) results in an improved joint response.
163
Figure 3-57: Joint shear distortion in the DSB connections
The moment-strain relationships of the top, middle, and bottom stirrups in the DSB joints at the peak displacement cycles are plotted in Figure 3-58. It is noted that all the strain values are elastic (below yield strain). Similar to the behaviour in the SSB connection, the strains in the top and the bottom stirrups contributed in resisting the horizontal forces transmitted via the flexural reinforcement, and thus, did not provide accurate representation of the joint shear behaviour. Unlike the behaviour of the DSB-S-1.0 and DSB-S-1.7 connections, compressive strains were induced in the DSB-S-0 connection in the middle stirrup under positive and negative bending. This can be explained by examining the flow of forces in the joint of the DSB connections as shown in Figure 3-9. The slope of the S2 diagonal compression strut is higher in the DSB connection than the SSB connection, and hence, the horizontal component of the S2 strut induced compressive strains in the middle stirrup. It is concluded, therefore, that relocating the vertical slots results in a
164
joint shear mechanism similar to that in conventional connections in which the middle stirrup experiences tensile strains under positive and negative bending.
(a) Top stirrup
(b) Middle stirrup
165
(c) Bottom stirrup Figure 3-58: Envelopes of the moment-joint strain relationships in the DSB connections
3.8.7 Components of Deformation The member contributions under positive and negative bending were averaged (due to the symmetry of the beam members) and plotted against the displacement amplitudes as shown in Figure 3-59. The percentage contributions of the beam, column, and joint at the ultimate cycles (80 mm, 85 mm, and 75 mm displacement amplitude cycle for the DSB-S-0, DSB-S-1.0, and DSBS-1.7 connections, respectively) are also included in Table 3-10. It is noted that the analysis results of the 2 mm and 4 mm amplitude cycles were discarded from the analysis since the deformations within the joint were very small and could not be captured accurately using the DICT technique. In all connections, the contribution of the beam bending increased while the contributions of the column bending and joint distortion decreased with increasing displacement amplitude.
166
As discussed in Section 3.8.1, relocating the plastic hinge increased the applied load. Therefore, the contribution of the elastic column bending and the joint distortion to the total displacement were anticipated to increase as well. This effect can be seen in Figure 3-59 and Table 3-10 in which the connections with relocated vertical slots (DSB-S-1.0 and DSB-S-1.7 connections) experienced higher contributions of the column bending and joint distortion, as compared with the connection in which the vertical slot is placed at the face of the column (DSBS-0 connection).
Joint distortion
100
Member contribution (%)
Column bending
80
60 Beam bending
40
20
0 0
20
40
60
Displacement amplitude (mm)
(a) Connection DSB-S-0
167
80
100
100
Joint distortion
Member contribution (%)
Column bending
80
60
40
Beam bending
20
0 0
20
40
60
80
100
80
100
Displacement amplitude (mm)
(b) Connection DSB-S-1.0
Member contribution (%)
100
Joint distortion Column bending
80
60
40 Beam bending
20
0 0
20
40
60
Displacement amplitude (mm)
(c) Connection DSB-S-1.7 Figure 3-59: Member contributions to the total applied displacement in the DSB connections 168
Table 3-10: Member contributions of the DSB connections at the ultimate cycle Connection DSB-S-0 DSB-S-1.0 DSB-S-1.7
Beam bending (%) 93 89 87
Column bending (%) 5 7 9
Joint distortion (%) 2 4 4
The ratio of θ2/θ1 is plotted against the displacement amplitude cycles in Figure 3-60 for the DSB-S-1.0 and DSB-S-1.7 connections. In both connections, the ratios are constant at 4 up to yielding after which they increase in an almost linear manner with increasing displacement amplitude. However, the slope of the DSB-S-1.0 connection is higher than that of the DSB-S-1.7 connection. At the ultimate cycle, the ratios are 11.4 and 8.2 in the DSB-S-1.0 and DSB-S-1.7 connections, respectively. It is, therefore, concluded that the effectiveness of the DSB system in relocating the plastic hinge increase with increasing displacement amplitudes and decrease with increasing relocation distance. As discussed in Section 3.7.7, the θ2/θ1 ratio can be considered as an indicator to quantify the effectiveness of relocating the centre of rotation, and thus, it can be stated that the effectiveness of having the vertical slot at 1.0 dv (300 mm) away from the face of the column is 1.4 times that if the distance to the vertical slot is increased to 1.7 dv (500 mm).
169
12
Rotation ratio, θ2/θ1
10 8 6 4
DSB-S-1.0 DSB-S-1.7
2 0 0
20
40
60
80
100
Displacement amplitude (mm)
Figure 3-60: Comparison of the θ2/θ1 ratios of the DSB-S-1.0 and DSB-S-1.7 connections
3.9 Comparison and Damage Assessment The beam-column connections in RC structures are desired to maintain high levels of energy dissipation with small, if any, degradation in the strength. Therefore, it is crucial to examine the level of damage in the SSB and DSB connection systems due to cyclic loading. There are two approaches on how to assess the seismic structural damage in the context of RC buildings; local and global assessments. The former deals with the evaluation of the damage induced in the plastic hinges in the structures while the latter is more complex and deals with the overall damage assessment of the structure using the weighted average of the damage indices relative to each component in the structure (DiPasquale and Cakmak, 1990). In this research, only the local damage is assessed. Moreover, the damage assessment methods and indices adopting the local approach are subdivided into two categories; ones that require the determination of the monotonic response 170
of the member in addition to the cyclic behaviour and others that require the behaviour under cyclic loading only. Examples of the former type are the indices by Park and Ang (1985), Stephens and James (1987), and Mehanny and Deierlein (2001). Due to the lack of the experimental data on the responses of the SSB and DSB systems under monotonic loading, the damage assessment is conducted using only the damage evaluation methods that do not require the monotonic response of the member. Furthermore, these damage evaluation methods are grouped into three categories; ductility-based, energy-based, and other indices. The damage assessment of the SSB and DSB connections are evaluated and compared in the following sections.
3.9.1 Ductility-Based Damage Assessment The conventional definition of ductility is the ability of the structure to sustain inelastic deformation without loss in strength capacity. This definition assesses both the energy and the deformation of the structure. In steel structures, for instance, the high ductility implies high deformation and energy dissipation as a result of the plastic property of steel. However, in concrete structures reinforced using elastic materials like FRP, high ductility implies high deformation with low energy dissipation (Oudah and El-Hacha, 2014). It is, therefore, important to be able to distinguish between the deformation ability of the structure and its energy dissipation capacity. In this research, the ductility of the connections is evaluated using three ductility indices. Newmark and Rosenblueth (1974) proposed a ductility index defined as the ratio of the maximum deformation with that at yielding. The ductility in this context is evaluated using the rotation at the vertical slots and the displacement at the tip of the beam. The rotation ductility, µθ, and the displacement ductility, µ∆, are expressed as follows:
171
i y
Equation 3-9
i y
Equation 3-10
where θi is the rotation at cycle i, θy the rotation at yielding, Δi the beam-tip displacement at cycle i, and Δy is the displacement at yielding. The second index is the cumulative ductility index, which is defined as the summation of the ductility indices from each cycle, i (Mahin and Lin, 1983). This ductility index is also evaluated in terms of the beam rotation at the location of the vertical slot and the beam-tip displacement. The cumulative rotation ductility index, μθ,cum, and the cumulative displacement ductility index, μΔ,cum, are evaluated as follows:
i i 1 y
Equation 3-11
i y
Equation 3-12
N
,cum N
,cum i 1
where N is the number of cycles to failure. The third index is the hysteretic ductility index, μhys, developed by Mahin and Bertero (1976). This index relates the total hysteretic energy to that dissipated by an equivalent elastoplastic system:
hys
Ecum 1 Py y
Equation 3-13
where Ecum is the cumulative energy. 172
It is noted that the average values of θy, Δy, and Py under positive and negative bending (included in Table 3-11) were used in calculating the ductility indices in Equations (3-11 – 3-13). The ductility indices are evaluated using the aforementioned expressions at the ultimate cycle. The assessment results are included in Table 3-12 along with the percentage differences with respect to the SSB-S-0 and the DSB-S-0 connections in the SSB and DSB groups, respectively.
Table 3-11: The average values of θy, Δy, and Py in the SSB and DSB connections Connection θy (rad) Δy (mm) Py (kN)
SSB-S-0 0.0030 9 33
SSB-S-1.0 0.0033 11 37
SSB-S-1.7 0.0024 13 48
DSB-S-0 0.0074 16 34
DSB-S-1.0 0.0064 14 38
DSB-S-1.7 0.0072 18 49
In the SSB group, the μθ and μΔ are reduced when the vertical slot is moved away from the face of the column. However, the reduction in the μΔ is more than μθ due to the fact that Δy increases with increasing relocation distance unlike θy (refer to Table 3-11). This observation is also applicable for the μθ,cum and μΔ,cum indices. The decrease in the μhys indicates that the increase in the energy of the equivalent elasto-plastic system (PyΔy) is higher than that of the actual cumulative energy (Ecum) when the vertical slots are moved away from the face of the column (i.e. the first term in Equation 3-13 decreased with increasing lre). In the DSB group, the ductility indices of the DSB-S-1.0 connection are higher than those of the DSB-S-0 and DSB-S-1.7 connections. This is because DSB-S-1.0 connection failed at higher displacement amplitude as compared with the other two connections (refer to Section 3.8.1), and thus, both the energy and the ductility were enhanced. The ductility indices of the DSB-S-1.7 connection is the lowest except for μθ since the rotation at the vertical slot in the DSB-S-1.7 at ultimate is approximately 1.1 higher than that in the DSB-S-0 connection. Therefore, it can be 173
concluded that DSB-S-1.0 connection shows a superior performance in terms of ductility assessment as compared with the other connections. It is also noted that the μΔ in all SSB and DSB connections is greater than 4.0 which corresponds to the minimum ductility required to design a ductile moment-resisting frame in accordance to the CSA-A23.3-04 (2004) seismic design provisions.
Table 3-12: Ductility-based assessment of the steel-reinforced SSB and DSB connections Group
Connection
μθ
Diff.* (%)
μΔ
Diff.* (%)
μθ,cum
Diff.* (%)
μΔ,cum
Diff.* (%)
μhys
Diff.* (%)
SSB-S-0 9.0 6.5 156 115 216 SSB-S-1.0 7.5 -17 5.2 -19 119 -24 82 -28 152 -30 SSB-S-1.7 8.4 -7 4.2 -35 151 -3 66 -43 109 -50 DSB-S-0 6.2 5.2 147 129 233 DSB-S-1.0 7.6 23 6.1 18 189 28 156 20 321 38 DSB DSB-S-1.7 6.7 8 4.2 -16 134 -9 95 -27 150 -36 * The difference is calculated with respect to the SSB-S-0 and DSB-S-0 connections in the SSB and DSB groups, respectively. SSB
3.9.2 Energy-Based Damage Assessment Several energy-based damage assessment approaches and indices are used to evaluate the seismic damage of RC beam-column connections. Three representative energy measures are used to assess the damage in the SSB and the DSB connections: cumulative energy, Ecum, energy index, Ein, and the equivalent viscous damping, ξeq. The Ecum is simply the cumulative hysteretic energy calculated as the summation of the areas enclosed by the hysteretic loops: N
Ecum Ei
Equation 3-14
i 1
where Ei is the energy per cycle i.
174
The Ein index proposed by Hwang (1982) is defined as follows:
k Ein Ei i i k y y i 1 N
2
Equation 3-15
where ki is the stiffness of the load versus displacement relationship in cycle i, ky the stiffness of the load versus displacement relationship at yielding, Δi the beam-tip displacement at cycle i, and Δy is the displacement at yielding.
The ξeq is also used to evaluate the energy dissipation capacity of the connections and to assess the potential damage. It is a property of the system and depends on its mass and stiffness (Chopra, 2007). The ξeq represents the simplest form of damping since the governing differential equation of motion is linear and it can be evaluated using simple experimental test. The ξeq ratio is calculated as follows:
eq
Ecum 1 4 (w / wn ) Eel
Equation 3-16
where Eel is the elastic energy, w the loading frequency, and wn is the natural frequency.
According to Equation 3-16, the ξeq depends on the frequency at which the test is conducted. It is recommended to conduct the test at a frequency near the resonance frequency since the response of the system is most sensitive to damping at that frequency (Chopra, 2007). Due to the difficulty of conducting the cyclic testing at high frequencies, it has been a common practice to exclude the (w/wn) term from equation 3-16 and to use the modified form as follows:
eq
1 Ecum 4 Eel
Equation 3-17 175
The values of the energy-based damage evaluation methods at the ultimate displacement cycles are included in Table 3-13 along with the percentage differences with respect to the values of the SSB-S-0 and the DSB-S-0 connections in the SSB and DSB groups, respectively. It is noted that the average values of ky in Equation 3-15 are 3.6 kN/m, 3.5 kN/m, and 3.7 kN/m, 2.2 kN/m, 2.7 kN/m, and 2.8 kN/m, for the SSB-S-0, SSB-S-1.0, SSB-S-1.7, DSB-S-0, DSB-S-1.0, and DSBS-1.7 connections, respectively.
Table 3-13: Energy-based assessment of the steel-reinforced SSB and DSB connections Ecum Diff.* Ein Diff.* Diff.* ξeq (kN·m) (%) (kN·m) (%) (%) SSB-S-0 65 525 0.333 SSB SSB-S-1.0 59 -9 391 -26 0.316 -5 SSB-S-1.7 69 6 720 37 0.274 -18 DSB-S-0 120 728 0.258 DSB DSB-S-1.0 171 43 1235 70 0.263 2 DSB-S-1.7 131 9 1025 401 0.230 -11 * The difference is calculated with respect to the SSB-S-0 and DSB-S-0 connections in the SSB and DSB groups, respectively. Group
Connection
The envelopes of the Ecum are plotted with respect to the displacement ductility of the SSB and the DSB groups as shown in Figure 3-61. It is also noted that the damage assessment is plotted with respect to the displacement ductility index hereafter as opposed to the rotation capacity as recommended by the Direct Displacement Based Design (DDBD) (Priestley et al., 2007). From Table 3-13 and Figure 3-61, it is seen that the cumulative energy tends to increase exponentially with increasing displacement ductility. However, the rate of increase in the DSB group is higher than that in the SSB group (i.e. the rate of energy dissipation is higher in the former group than the latter group). In the SSB group, the connections attained approximately the same amount of energy dissipation at failure but at different ductility displacement cycles. In the DSB group, however, the 176
DSB-S-1.0 connection attained a significantly higher energy dissipation at ultimate as compared with the DSB-S-0 and DSB-S-1.7 connections. This is due to the fact that the former connection failed at higher displacement amplitude (85 mm) while the latters failed at lower displacement amplitudes (80 mm and 75 mm for the DSB-S-0 and DSB-S-1.7 connections, respectively). Also, it is observed that the rate of increase in energy dissipation increases approximately in a linear manner with respect to the increase in the relocation distance, lre, in the SSB system. In the DSB system, the rate of change is insensitive to the lre distance up to 1.0 dv (300 mm) after which it tends to increase as can be observed in the behaviour of the connections in Figure 3-61 (b). 200
SSB-S-0 SSB-S-1.0 SSB-S-1.7
175 150
Cumulative energy, Ecum (kN·m)
Cumulative energy, Ecum (kN·m)
200
125 100 75 50 25 0
DSB-S-0 DSB-S-1.0 DSB-S-1.7
175 150 125 100 75 50 25 0
0
1
2
3
4
5
6
7
0
Displacement ductility, μ∆
(a) Group SSB
1
2
3
4
5
6
7
Displacement ductility, μΔ
(b) Group DSB
Figure 3-61: Cumulative energy versus displacement ductility relationships of the SSB and DSB connections
The envelopes of Ein are plotted against the displacement ductility as shown in Figure 3-62. The behaviour of the tested connections is similar to that observed when evaluating the cumulative energy but with different slopes. This behaviour can be understood by evaluating the variables in Equation 3-15. The expression consists of the multiplication of three terms; the cumulative energy, 177
the stiffness ratio, and the displacement ductility. The first term behaves as illustrated in Figure 3-61 while the second term increases insignificantly when lre is increased. However, the third term controls the behaviour since moving the vertical slot away from the face of the column increases the displacement at yielding. Therefore, the trends of the Ein curves shown in Figure 3-62 are controlled by the third term (the square of the displacement ductility). 1400
1400
SSB-S-0 SSB-S-1.0 SSB-S-1.7
1000
DSB-S-0 DSB-S-1.0 DSB-S-1.7
1200
Energy index, Ein (kN·m)
Energy index, Ein (kN·m)
1200
800 600 400 200 0
1000 800 600 400 200 0
0
1
2
3
4
5
6
7
0
1
(a) Group SSB
2
3
4
5
6
7
Displacement ductility, μ∆
Displacement ductility, μ∆
(b) Group DSB
Figure 3-62: Energy index versus displacement ductility relationships of the SSB and DSB connections
The damping properties of concrete structures are considered to be one of the most important parameters in the seismic design using the DDBD method as discussed in Section 2.1. The variation of the ξeq with the increase in the displacement ductility of the SSB and DSB connections is plotted in Figure 3-63. Based on the values reported in Table 3-13 and the trends shown in Figure 3-63, it is observed that ξeq is approximately 0.05 in the elastic range (displacement ductility of less or equal 1), which matches the value proposed by Priestley et al. (2007). Upon reaching the yielding condition, the ξeq increases in a decreasing rate with increasing displacement ductility. It is also noted that relocating the plastic hinge does not affect the trend of 178
the ξeq evolution. However, it results in decreasing the magnitude of the ξeq at the ultimate state. The decrease in the magnitude at ultimate is approximately linearly proportioned to the increase in the relocation distance lre in the SSB connections while the damping peaks at lre of 1.0 dv (300 mm) in the DSB connections. 0.4
Equivalent viscous damping, ξeq
Equivalent viscous damping, ξeq
0.4
SSB-S-0 SSB-S-1.0 SSB-S-1.7
0.3
0.2
0.1
0
DSB-S-0 DSB-S-1.0 DSB-S-1.7
0.3
0.2
0.1
0 0
1
2
3
4
5
6
7
0
Displacement ductility, μ∆
1
2
3
4
5
6
7
Displacement ductility, μ∆
(a) Group SSB
(b) Group DSB
Figure 3-63: Equivalent viscous damping versus displacement ductility relationships of the SSB and DSB connections
3.9.3 Other Damage Assessment Indices The other damage assessment indices include the Residual Displacement Index, RDI, the Work Index, WI, and the Stiffness Index, SI. The RDI is a non-dimensional index used to evaluate the residual displacement upon the reversal of loading (Hose et al., 1999). It is defined as follows:
RDI
res y
Equation 3-18
where Δres is the residual displacement upon unloading (at zero load).
179
The WI is used to evaluate the work performed by the structure when subjected to loading (Gosain et al., 1977). It is defined as follows: N
WI i 1
Pi i Py y
Equation 3-19
where Pi is the load at cycle i.
As concluded in Sections 3.7.1 and 3.8.1 there are no signs of punching shear in the hysteretic responses of the SSB and the DSB connections. In other words, the cyclic loading does not degrade the stiffness of the structures up to yielding, and hence, the conventional methods of evaluating the stiffness degradation are not suitable. Therefore, the stiffness evaluation in this research is only concerned with the post-yield behaviour. The SI is defined as the ratio of the postyielding stiffness, ki, to the yielding stiffness, ky, as follows:
SI
ki ky
Equation 3-20
The values of the RDI, WI and SI at the ultimate displacement amplitudes are included in Table 3-14 for the connections in the SSB and the DSB groups. Also, the percentage differences with respect to values of the SSB-S-0 and the DSB-S-0 connections in the SSB and the DSB groups, respectively, are included in the table. It is noted that the values are calculated based on the average responses of the connections under positive and negative bending. The relationships between the RDI and the displacement ductility of the tested connections are shown in Figure 3-64. From Table 3-14 and Figure 3-64, it is observed that the RDI values are approximately zero in the range of 0 to 1 displacement ductility after which they tend to increase in a linear manner with the same slope. This behaviour is expected since the connections behave 180
elastically for displacement ductility below one, and thus, the residual displacement is approximately zero. Knowing that the permanent plastic strain of steel increases linearly and that the hysteretic response of the connections is governed by the hysteretic response of the flexural reinforcement (discussed in Section 3.7.1 and 3.8.1), the residual displacement increases linearly with increasing displacement ductility.
Table 3-14: Other damage assessment indices of the steel-reinforced SSB and DSB connections Diff.* Diff.* WI (%) (%) SSB-S-0 3.5 179 SSB SSB-S-1.0 2.8 -20 136 -24 SSB-S-1.7 2.0 -43 107 -40 DSB-S-0 3.2 174 DSB DSB-S-1.0 1.8 -44 229 32 DSB-S-1.7 1.1 -66 126 -28 * The difference is calculated with respect to the Control connection Group
Connection
0.170 0.206 0.205 0.266 0.244 0.319
4
Residual displacement index, RDI
Residual displacement index, RDI
4
Diff.* (%) 21 21 -8 20
SI
RDI
SSB-S-0 SSB-S-1.0 SSB-S-1.7
3
2
1
0
DSB-S-0 DSB-S-1.0 DSB-S-1.7
3
2
1
0 0
1
2
3
4
5
6
7
0
Displacement ductility, µ∆
(a) Group SSB
1
2
3
4
5
6
7
Displacement ductility, µ∆
(b) Group DSB
Figure 3-64: Residual displacement index versus displacement ductility relationships of the SSB and DSB connections
181
The WI is plotted against the increase in the displacement ductility as shown in Figure 3-65. It is seen that the WI vary in a parabolic manner with increasing displacement ductility in the SSB and DSB connections. The increase in the lre distance in the SSB connections tends to decrease WI. However, the WI increases for lre of 1.0 dv (300 mm) while it decreases for lre of 1.7 dv (500 mm) in the DSB group. These variations of the response can be understood by examining the terms in Equation 3-19. In the SSB system, the values of Pi, Py, and ∆y increase with increasing lre, and thus, WI decreases since the value of ∆i remains the same in all connections. In the DSB system, however, only the values of Pi and Py increase with increasing lre, while the values of ∆y and ∆i decrease and increase, respectively, for lre of 1.0 dv (300 mm) while they tend to increase and decrease, respectively, for lre of 1.7 dv (500 mm). Therefore, the WI decreases as the vertical slots are moved away from the face of the column in SSB system while it peaks at a relocation distance of 1.0 dv (300 mm) in the DSB system. 250
250
SSB-S-0 SSB-S-1.0 SSB-S-1.7
DSB-S-0 DSB-S-1.0 DSB-S-1.7
200
Work index, WI
Work index, WI
200
150
100
50
150
100
50
0
0 0
1
2
3
4
5
6
7
0
1
(a) Group SSB
2
3
4
5
6
Displacement ductility, µ∆
Displacement ductility, µ∆
(b) Group DSB
Figure 3-65: Work index versus displacement ductility relationships of the SSB and DSB connections
182
7
The SI versus the displacement ductility relationships are plotted in Figure 3-66 for the SSB and the DSB connections. As noted in Equation 3-20, the post yield stiffness is normalized to the stiffness at yielding, therefore, a value of SI equal to one means that the systems behave elastically. From Figure 3-66 (a), it can be seen that the post-yield stiffness in the SSB connections drops significantly to a displacement ductility of approximately 1.5 after which the rate of the degradation in the stiffness starts to plateau at SI of 0.2. However, a more gradual degradation in the post-yielding stiffness is observed in the DSB system (Figure 3-66 (b)) and tends to asymptote at SI of 0.2. It is also observed that relocating the plastic hinge (i.e. increase in the lre distance) does not affect the shape of the trends. 1
1
SSB-S-0 SSB-S-1.0 SSB-S-1.7
DSB-S-0 DSB-S-1.0 DSB-S-1.7
0.8
Stiffness index, SI
Stiffness index, SI
0.8
0.6
0.4
0.2
0.6
0.4
0.2
0
0 1
2
3
4
5
6
7
1
2
(a) Group SSB
3
4
5
6
7
Displacement ductility, µ∆
Displacement ductility, µ∆
(b) Group DSB
Figure 3-66: Stiffness index versus displacement ductility relationships of the SSB and DSB connections
3.10 Summary The design and behaviour of steel-reinforced concrete SSB and DSB systems with relocated vertical slots was investigated experimentally in this chapter in order to investigate the adequacy of the proposed systems in relocating the plastic hinges away from the face of the column. Based
183
on the cracking pattern, beam rotation, strain profiles, components of deformation, and damage assessment, it is concluded that the best relocating distance of the vertical slots is 1.0 dv (300 mm) in the SSB and DSB systems. This conclusion is used in Chapter 5 in designing the PE SMAreinforced SSB and DSB connections. It is also found that the effectiveness of the SSB in relocating the plastic hinge under negative bending is superior to that under positive bending while the effectiveness of the DSB system in relocating the plastic hinge is the same under positive and negative bending.
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Design and Performance of Self-Centering Concrete Beam-Column Connections Reinforced using PE SMA 4.1 Introduction The design and performance of SSB and DSB self-centering concrete beam-column connections reinforced using PE SMA bars are discussed in this chapter. As concluded in Chapter 3, relocating the vertical slots away from the face of the column by 1.0 dv (where dv is the effective shear depth measured as the distance between the top and bottom reinforcement in the beam section at the vertical slot) yields the optimum response of the concrete connections reinforced using steel reinforcement. It is, therefore, determined to relocate the vertical slots in PE SMA-reinforced SSB and DSB connections by a distance equivalent to 1.0 dv. First, the design aspects of the proposed connections are discussed followed by characterizing the monotonic and the cyclic behaviour of the PE SMA material experimentally with emphasis on the damping and energy dissipation properties. After that, the anchorage of the PE SMA bars into the joints is experimentally investigated by testing joint-like specimens designed to simulate the behaviour in large-scale beam-column connections. Based on the test results of the joint-like specimens, modifications of the mechanical anchors and couplers are made in order to enhance the stress and the strain attained in the PE SMA material. Finally, experimental testing of large scale-scale beam-column connections is conducted to examine the performance of the proposed design. Three connections were tested; a conventionally designed beam-column connection (Control), a SSB connection reinforced using PE SMA, and a DSB connection reinforced using PE SMA. The performance of the connections is examined and compared in terms of the hysteretic response, cracking pattern, beam rotation, beam elongation, strain profiles, shear mechanism, components of deformation, and damage assessment.
185
4.2 Testing Matrix The objective of this phase of the research program is to investigate the performance of PE SMAreinforced SSB and DSB beam-column connections. Three large-scale connections were examined experimentally; steel-reinforced conventional connection (Control), PE SMA-reinforced SSB connection (SSB-P-1.0), and PE SMA-reinforced DSB connection (DSB-P-1.0). In the latter two connections, the connection ID is described as follows: first term (SSB or DSB) refers to the type of the system, the second term (P) refers to the reinforcing material which is PE SMA, and the last term (1.0) refers to the lre/dv ratio. The design considerations of the proposed connections are discussed in the following section, while the detailed design is included in Section 4.7.
4.3 Design Considerations The main design aspects that need to be examined prior to experimentally studying the behaviour of the large-scale PE SMA-reinforced SSB and DSB concrete beam-column connections are illustrated in Figure 4-1: I. II.
Relocating distance Material properties of the PE SMA
III.
Anchorage performance
IV.
Coupler performance The first item on the aforementioned list was already determined in Chapter 3 (1.0 dv). The
material properties of the PE SMA, anchorage of the PE SMA bars into the concrete joints, and coupling them to the diagonal steel reinforcement are investigated experimentally and analytically in Sections 4.4, 4.5, and 4.6, respectively. It is noted that the thermoelastic properties, critical stress and temperature states, and transformation strain evolution properties were not examined since 186
this study is only concerned with the psuedoelastic properties of the SMA material at ambient temperature.
Figure 4-1: Design considerations in the PE SMA-reinforced SSB and DSB connections
The PE SMA bars are exposed to the surrounding environment at the locations of the vertical slots. Knowing that SMA is thermal-sensitive and exhibits a coupled thermal-mechanical behaviour, the environmental exposure including cooling/heating and freeze-thaw thermal loading may alter the mechanical properties of the SMA material. Therefore, it is an important design
187
variable (especially the effect of the ambient temperature on increasing or decreasing the Af temperature). However, the environmental exposure effect is out of the scope of the present research, and thus, it will not be further discussed in the remaining of the thesis.
4.4 Material Properties of the PE SMA Material The chemical composition of the 14.9 mm (0.585 in) diameter hot roll and thermally treated bars is included in Table 4-1 as certified by the manufacturer. The fully annealed austentic finish temperature, Af, ranges between -9o C and 0o C. Therefore, the SMA material exhibits a PE response at room temperature. The material was experimentally tested under monotonic and cyclic loading in order to determine its mechanical properties. Due to the high cost of the material (approximately $ 1,000/m), only two specimens were tested; one under monotonic loading and one under cyclic loading. It is noted that the SMA material is untrained. The length of each specimens was 300 mm. The test setup, the instrumentation, and the loading regime are discussed in the following sections.
Table 4-1: Chemical composition of the PE SMA material* Ni Ti O C N (%) (%) (ppm) (ppm) (ppm) 55.84 44.16 192 292 19 * Ni, Ti, O, C, and N stand for Nickel, Titanium, Oxygen, Carbon, and Nitrogen, respectively.
4.4.1 Test Setup, Instrumentations, and Loading Regime The setup of the monotonic and the cyclic tests of the PE SMA bars is shown in Figure 4-2. The 300 mm long PE SMA bars were gripped at both ends by two 5/8” steel prestressing wedges (the
188
gauge length is 170 mm.). The lower steel wedge reacted against a prefabricated steel frame while the upper one reacted against a steel box. The steel box was connected to a 1.0 MN 10” stroke MTS actuator reacting against a cross-head I-steel beam connected to the laboratory strong floor. The strain values were measured using three methods; Strain Gauges (SG), Linear Strain Conversion (LSC) device, and Digital Image Correlation Technique (DICT). Each specimen was mounted with two SGs at one and two thirds of the gauge length, while the LSC device measured the change in the displacement between the two gripes as shown in Figure 4-2. The clear length of the specimen (90 mm, defined as the visible length of the specimen) was painted with white and black paints in order to provide the texture that is needed for the DICT method.
Figure 4-2: Test setup and locations of the LSCs and strain gauges in the PE SMA monotonic and cyclic tests 189
The monotonic test consisted of the application of ramp displacement-control loading at a rate of 1.6 mm/sec while the cyclic test consisted of the application of tension-tension cyclic loading as illustrated in Figure 4-3. The amplitude of the stroke displacement was increased by increments of 1 mm until fracture of the bar. At each displacement amplitude, three cycles were conducted in order to examine the effect of cycling at approximately a constant strain on the behaviour of the material. Two loading rates were used in this test; 0.1 mm/sec (from zero to 10 mm displacement amplitude) and 0.2 mm/sec (from 11 mm displacement amplitude to failure). In each cycle, the specimen was unloaded to 10 kN (in order to avoid possible buckling of the specimens if unloaded to zero load) at a rate of 5 kN/sec at a load-control mode.
20 18
Loading rate: 0.1 mm/sec Unloading rate: 5 kN/sec
Loading rate: 0.2 mm/sec Unloading rate: 5 kN/sec
Displacement (mm)
16 14 12 10 8
1 mm
6 4 2 10 kN
0
Time (sec)
Figure 4-3: Loading regime of the tension-tension cyclic coupon test of the PE SMA material
190
4.4.2 Monotonic Behaviour The monotonic stress-strain relationship of the PE SMA bar is shown in Figure 4-4. The SG refers to the average strains obtained from the two strain gauges. It is also noted that the SGs failed at approximately 5% strain. The strain readings obtained from the three measurement methods are in good agreement. The percentage differences in the stress and strain obtained from both the SG and LSC device are approximately zero, which indicates that no slippage occurred inside the wedges that are used to grip the PE SMA bar. The average values of the stress at the Martensite start temperature, σMs, strain at the Martensite start temperature, εMs, stress at the ultimate load, σu, strain at the ultimate load, εu, and the Austenite modulus of elasticity, EA, are included in Table 4-2. It is noted that EA is calculated as the ratio of σMs over εMs.
Table 4-2: Material properties of the PE SMA subjected to monotonic loading σMs (MPa) 485
εMs (mm/mm) 0.00875
σu (MPa) 700
εu (mm/mm) 0.0788
EA (GPa) 55
800
σu
700
Stress, σ (MPa)
600
σMs
500
LSC SG DICT
400 300 200
EA
100
εu
εMs 0 0
0.02
0.04
0.06
0.08
0.1
Strain, ε (mm/mm)
Figure 4-4: Stress-strain relationship of the PE SMA material subjected to monotonic loading 191
The strain distribution along the normalized gauge length is plotted in Figure 4-5 for multiple stress levels. The filled symbols refer to the strain readings obtained from the DICT measurement method, while the un-filled symbols refer to the strain readings obtained from the SGs. It is observed that severe nonlinearity is experienced upon the transformation from the Austenite to the Martensitic phase at a stress of 500 MPa (passing the σMS stress in Figure 2-12). The material is composed of a single phase (Austenite) prior to reaching the σMS stress, and thus, the strain along the specimen is uniform. However, upon reaching the σMS stress, the detwinning process is initiated and the transformation from the Austenite to the Martensite is commenced, which results in a non-uniform strain variation along the specimen as the material is composed of both phases at this stage (Austenite and Martensitic). This variation decrease as the stress approached the ultimate state in which the material is solely composed of a single phase (Martensite). It is also shown that the readings from the SGs correlate well with the readings obtained from the DICT method.
Normalized gauge length
1
200 MPa 600 MPa
400 MPa 700 MPa
500 MPa
0.8
0.6
0.4
0.2 Filled symbol - Digital Image Correlation (DIC) Un-filled symbol - Strain gauge
0 0
0.02
0.04
0.06
0.08
0.1
Strain, ε (mm/mm)
Figure 4-5: Strain distribution along the gauge length of the PE SMA material subjected to monotonic loading 192
4.4.3 Cyclic Behaviour The DICT measurement method was found to provide accurate results only at the peak strains in each cycle, while the SG failed at approximately 4% strain. Therefore, only the strain readings obtained from the LSC method are used to plot the cyclic response of the PE SMA material. The specimen was subjected to 109 cycles to failure. However, only the response of five cycles is plotted in Figure 4-6 for clarity purposes. The cyclic response is evaluated by examining the effect of cyclic loading on the following parameters, while the critical stresses and strains are defined in Figure 4-7: •
Strain ratio, εmin,i/εmax,i: where εmin,i and εmax,i refer to the minimum and the maximum strains, respectively, attained at cycle i.
•
Stress ratio, σMs,i/σMs,c: where σMs,i refers to the stress at the start of the Martensitic phase at cycle i while and σMs,c refers to the stress at the start of the Martensitic phase calculated from the envelope of the cyclic response.
•
Stiffness ratio, Eeff,i/EA,c: where Eeff,i and EA,c refer to the effective stiffness in the ith cycle and the modulus of elasticity of the Austeniate phase calculated from the envelope of the cyclic response, respectively. The Eeff,i is calculated as follows: Eeff ,i =
(σ (ε
max,i max,i
− σ min,i )
− ε min,i )
Equation 4-1
where σmax,i and σmin,i are the minimum and maximum stresses, respectively, attained at cycle i. 193
•
Equivalent viscous damping, ξeq: It is calculated using Equation 4-2 (note that Equation 4-2 is the same as Equation 3-35, however, the latter one is rearranged in terms of the areas under the curve):
ξeq =
1 Ahys 4π Ael
Equation 4-2
The Ahys and Ael refer to the area enclosed by the hysteresis loop and the elastic energy, respectively.
The stress and stiffness ratios require determining σMs,c and EA,c which can be obtained from the envelope of the cyclic response of the specimen plotted in Figure 4-8. The three methods of strain measurement are used to determine the response while the cyclic parameters are included in Table 4-3. 800 700
Stress, σ (MPa)
600 500 400 300 200 100 0 0
0.02
0.04
0.06
0.08
0.1
Strain, ε (mm/mm)
Figure 4-6: Stress-strain relationship of the PE SMA material subjected to cyclic loading 194
Stress, σ (MPa)
800 700
σu
600
σmax,i
500
σMs,c
400
Eeff,i 300
σMs,i
200
EA,c
100
σmin,i εmin,i
0
Ms,c 0 ε
0.02
εu
εmax,i
εMs,i 0.04
0.06
0.08
0.1
Strain, ε (mm/mm)
Figure 4-7: Definition of the parameters used to evaluate the cyclic response of the PE SMA
800 700
Stress, σ (MPa)
600 500
LSC SG DICT
400 300 200 100 0 0
0.02
0.04
0.06
0.08
0.1
Strain, ε (mm/mm)
Figure 4-8: Envelopes of the stress-strain relationships of the PE SMA material subjected to cyclic loading
195
Table 4-3: Material properties of the PE SMA subjected to cyclic loading σMs,c (MPa) 443
εMs,c (mm/mm) 0.0070
σu (MPa) 668
εu (mm/mm) 0.0958
EA,c (GPa) 63
The four parameters are plotted against the increase in the strain as shown in Figure 4-9. The strain ratio examines the effect of cyclic loading on the increase in the permanent deformation. It can be seen that this ratio increase with increasing strain values. At failure, the strain ratio is approximately 0.5. The stress ratio decrease linearly with increasing strain with a minimum value of approximately 0.4 at failure. This observation is in agreement with prior research studies in which it was shown that cyclic loading has a significant impact on σMS (referred as yield stress) (Mao et al., 2006). The stiffness ratio is observed to decrease with increasing strain and best expressed in a power equation. It is seen that the stiffness ratio tends to plateau at a value of 0.2. The damping ratio increase with increasing strain until it reaches a peak point at a strain of approximately 0.04 (corresponding to a damping ratio of approximately 3%) as shown in Figure 4-9 (a). After that, it decreases with increasing strain and plateau at approximately 2%). The observations made with regard to the damping ratio are in agreement with the experimental testing conducted by Fugazza (2005) in which it was concluded that the maximum damping ratio is approximately 3% and it is attained at a strain of 4% when the PE SMA material is subjected to a loading frequency of 1 Hz.
196
0.6
1
Stress Ratio, σMs,i /σMs,c
Strain Ratio, εmin,i /εmax,i
0.5 0.4 y = -19.136x2 + 8.8921x - 0.1877 R² = 0.9985 0.3 0.2 0.1 0
0.8
0.6 y = -7.2013x + 1.0334 R² = 0.9741 0.4
0.2
0 0
0.02
0.04
0.06
0.08
0.1
0
0.02
Strain, ε (mm/mm)
0.04
0.06
0.08
0.1
0.08
0.1
Strain, ε (mm/mm)
(a) Strain ratio
(b) Stress Ratio
1
Equivalent viscous damping, ξeq
Stifness Ratio, Eeff,i /EA,c
0.03 0.8
0.6 y = 0.0381x-0.61 R² = 0.8692 0.4
0.2
0 -0.01
0.02
0.01
0 0.01
0.03
0.05
0.07
0.09
0
0.11
Strain, ε (mm/mm)
(c) Stiffness ratio
0.02
0.04
0.06
Strain, ε (mm/mm)
(d) Equivalent viscous damping
Figure 4-9: Evaluation of the effect of cyclic loading on the behaviour of PE SMA
The strain distribution along the gauge length of the specimen is plotted in Figure 4-10 at the peak stress values for five cycles. The filled symbols refer to the strain readings obtained from the DICT measurement method while the un-filled symbols refer to the strain readings obtained from the SGs. The location of the peak strain shifted the applied load increased. Up to a stress value of 500 MPa, the peak strain is located at 60% of the gauge length, then it shifts to 30% of
197
the gauge length for stresses higher than 500 MPa. This indicates the high level of nonlinearity in the response of the material when subjected to cyclic loading.
Normalized gauge length
1
200 MPa 600 MPa
400 MPa 678 MPa
500 MPa
0.8
0.6
0.4
0.2 Filled symbol - Digital Image Correlation (DIC) Un-filled symbol - Strain gauge
0 0
0.02
0.04
0.06
0.08
0.1
0.12
Strain, ε (mm/mm)
Figure 4-10: Strain distribution along the gauge length of the PE SMA material subjected to cyclic loading
The stress-strain relationships of conventional steel reinforcement subjected to monotonic loading and PE SMA bars subjected to monotonic and cyclic loading are compared in Figure 4-11. It is seen that the cyclic loading results in softening the response of the PE SMA material. Thus, the strain in the PE SMA subjected to cyclic loading exhibits higher strain than the PE SMA subjected to monotonic loading for a given stress value. The σMS and εMs are reduced by 9% and increased by 20%, respectively, as a result of the cyclic loading. The σu is reduced by 5% while εu is increased by 22% as a result of the cyclic loading. It is also noted that the stress overshoot that is observed in the monotonic behaviour (refer to Section 2.5.1 for more discussion about the stress overshoot) is not present in the material subjected to cyclic loading. The σMs and σu of the PE SMA 198
subjected to monotonic and cyclic loading are comparable to the yielding stress and ultimate stresses of the conventional steel reinforcement. The yield stress and the ultimate stress of conventional steel are 3 % higher and 2 % lower than those of the PE SMA bar subjected to monotonic loading. However, the modulus of elasticity of the PE SMA material is only one third of the modulus of elasticity of the conventional steel. 800 700
Stress, σ (MPa)
600 500 400 Steel - Monotonic SMA - Monotonic SMA - Cyclic
300 200 100 0 0
0.02
0.04
0.06
0.08
0.1
Strain, ε (mm/mm)
Figure 4-11: Comparison of the stress-strain relationships between steel and PE SMA bars
4.5 Anchorage Performance As indicated in Section 4.3, the anchorage performance of the PE SMA bar into the concrete joint is of a paramount importance. It is, therefore, necessary to examine the performance of the concrete joint reinforced using PE SMA bars, and conduct appropriate modifications in order to enhance the performance of the system. In this section, the performance of joint-like specimens is investigated experimentally and analytically followed by conducting the necessary modifications on the anchorage system. 199
4.5.1 Anchorage Performance in Joint-Like Specimens The joint-like specimens were designed and tested in order to achieve the following objectives: 1. Investigate the performance of the mechanical anchors in terms of the maximum strain and stress attained under cyclic loading. 2. Examine the effect of the orientation of the anchor, position of the anchor, and the level of stirrup confinement on the performance of the joint-like specimens in terms of curvature and strain distribution. 3. Compare the performance of the steel-reinforced and the PE SMA-reinforced joints. 4. Determine the most practical joint configuration (ease of fabrication) and the one that result in the minimum curvature at ultimate. A prior research study investigated the use of modified mechanical couplers for connecting PE SMA bars to steel reinforcement (Alam et al., 2010). Several arrangements were examined and the coupler was modified in order to sustain high strain and stress levels. The modified mechanical coupler is shown in Figure 4-12.
Figure 4-12: Mechanical anchors for connecting PE SMA bar to steel developed by Alam et al. (2010) 200
Nine flattened screw-locks were used to connect the PE SMA bar to the coupler. Two of the screw lines were drilled at an angle of 60o away from the centre of the coupler. Experimental testing showed that the modified anchor can sustain an average strain and stress of 6.2% and 496 MPa, respectively, at failure. However, the average slip at yielding and ultimate stresses were found to be 2.4 mm and 10.0 mm, respectively. The significant slip hinders the use of the mechanical coupler since it leads to high residual deformation when the beam-column connection is subjected to cyclic loading. It was, therefore, determined not to use the coupler by Alam et al. (2010) in the beam-column connections examined in this research. Instead, new mechanical anchors and couplers were examined in this research. The mechanical anchors and couplers were provided by a manufacturer of mechanical splices for reinforced concrete construction and are shown in Figure 4-13. The mechanical anchor shown in Figure 4-13 (a) is used to anchor the PE SMA bar into the concrete joint in the beam-column connection while the mechanical coupler shown in Figure 4-13 (b) is used to splice the PE SMA bar with the steel reinforcement in the beam. Upon the request of the manufacturer, the brand name is kept confidential. The performance of the mechanical anchors in joint-like specimens is examined in the following section, while the performance of the mechanical coupler is examined in Section 4.6.
(a) Mechanical anchor (b) Mechanical coupler Figure 4-13: Mechanical anchor and coupler used in this research 201
4.5.1.1 Design Considerations The joint-like specimens were designed to simulate the behaviour in the large scale beam-column connection. The main design criterion of the joint-like specimens is to have approximately the same Mco/Vjh ratio (where Mco is the column moment and Vjh is the horizontal shear force in the joint) as experienced in the large-scale beam-column connection. In order to determine the Mco/Vjh ratio, the column behaviour when subjected to shear and bending is examined in Figure 4-14. The transformation of forces within the joint when the vertical slots are moved away by 1.0 dv from the face of the column was found to be similar to that experienced in conventional beam-column connections (refer to Section 3.2.3). Hence, the horizontal shear force in the joint, Vjh, is calculated as follows:
V jh Tb Vco
Equation 4-3
where,
Vco
Tb zb 0.5Vb hco lco
Equation 4-4
where Tb and Cb are the tension and compression forces in the beam, respectively, Vb and Vco the beam and the column shear forces, respectively, zb the distance between the centroids of Tb and
Cb, and lco is the column span.
The column moment, Mco, is calculated as follows: M co
lco z b Vco
Equation 4-5
2
202
Figure 4-14: Column behaviour subjected to shear and bending moment
Based on the design of PE SMA-reinforced SSB and DSB connections included in Section A.4, the average Mco/Vjh ratio of the SSB-P-1.0 and DSB-P-1.0 connections is 0.2, and thus, a value of 0.2 was used in dimensioning the test specimens. The span of the specimen is, therefore, 800 mm if simply supported connections are assumed. The distance from the side of the beam section to the centroid of the bottom or the top reinforcement in the large scale connections (refer to Figure 3-11 and Figure 3-12) is approximately ¼ bw (where bw is the beam width), and therefore, a plane of symmetry cutting the column section longitudinally into two equivalent halves can be assumed. Thus, only one half was considered in designing the joint-like specimens, i.e. the width of the specimen is 150 mm.
203
4.5.1.2 Testing Matrix The performance of the joint-like specimens is investigated through experimental testing of six specimens. The testing matrix and the details of the specimens are included in Table 4-4 and shown in Figure 4-15. The experimental program was designed to achieve the four objectives stated in Section 4.3.1.
Table 4-4: Testing matrix of the joint-like specimens Group
Specimen
Steelreinforced
AN-S-1 AN-S-2 AN-P-1 AN-P-2 AN-P-3 AN-P-4
PE SMAreinforced
Vertical Rein. Steel Steel PE SMA PE SMA PE SMA PE SMA
Ss (mm) 65 180 65 65 65 180
Figure 4-15: Details of the joint-like specimens
204
da (mm) NA NA 370 305 370 305
Orientation NA NA Normal Normal Flipped Normal
Two groups of specimens were tested; steel-reinforced and PE SMA-reinforced. The specimens have the same dimensions and properties except for the vertical reinforcement. Steel reinforcements were used in the former group while PE SMA bars were used in the latter. Two values of the spacing distance between the middle stirrups, Ss, were considered; 65 mm and 180 mm. The former value refers to the spacing of the stirrups at the joint in the large-scale specimens while the latter value is double the value of the stirrup spacing in the columns. Therefore, they represent two extreme values. As discussed in Section 1.1, the reinforcement anchored into the joint in the SSB and the DSB systems is subjected to high strain values under both negative and positive bending unlike the behaviour of the conventional connections. Therefore, the fan of the compression forces will develop at both sides of the anchors, and hence, the location of the anchors (defined in terms of da which is the distance from the top fibre to the bottom side of the anchor) and the orientation of the anchors are important design parameters. Two values of da were considered; 305 mm and 370 mm. For practical purposes, the anchors cannot be installed exactly at the middle of the specimen. Hence, the anchors were placed either above (305 mm) or below (370 mm) the middle steel reinforcement, while two orientations of the anchors were considered; normal and flipped. Even though low levels of strains are anticipated to develop in the longitudinal reinforcement, the reinforcement was extended out of the concrete specimens and welded into steel bolts in order to enhance its anchorage (refer to Section A.6). This was conducted due to the fact that development length of the specimens is not sufficient in accordance with the CSA-A23.3-04 (2004).
205
4.5.1.3 Test Regime and Set-up Due to the difficulty of applying tension-compression cyclic loading onto the specimens (difficulty in avoiding buckling of the PE SMA bar), only tension-tension quasi-static cyclic loading was applied. The testing regime was the same as the one discussed in Section 4.4.1 and shown in Figure 3-15. The specimens were positioned inside a prefabricated steel frame as shown in Figure 4-16.
Figure 4-16: Test setup of the joint-like specimens
The simply supported boundary conditions were achieved via the use of roller and hinge supports. The same mechanical wedges used in conducting the monotonic and cyclic testing of the PE SMA bars discussed in Section 4.4.1 were used to grip the PE SMA bar and the steel reinforcement. The wedge rested inside a steel box connected to servo-hydraulic actuator reacting against a steel frame anchored into the laboratory floor.
206
4.5.1.4 Instrumentations The specimens were instrument using LSC devices and SGs in order to monitor the behaviour during testing. The deformations in the top and the bottom concrete fibres and the deformations in the steel frame were measured using the LSC devices as shown in Figure 4-17. The strain gauges were mounted on the longitudinal reinforcement, stirrup, and vertical steel reinforcement or the PE SMA bar as shown in Figure 4-18. Each reinforcement was instrumented with one strain gauge at each position except the PE SMA bars which were instrumented with two SGs at each position.
Figure 4-17: Positions and labels of the LSC devices in the joint-like specimens
Figure 4-18: Locations of the strain gauges in the joint-like specimens 207
4.5.1.5 Test Results The test results are presented in this section followed by a detailed discussion and analytical modeling in the following section. The test results at the ultimate condition (maximum sustained load) including the load at ultimate, Pu, curvature at ultimate load, ψu, stress at ultimate load in the vertical reinforcement, σu, strain at ultimate load in the vertical reinforcement, εu, and the failure mode are included in Table 4-5. The average maximum load in the steel-reinforced specimens is 145 kN (AN-S-1 and AN-S-2) while it is 78 kN in the PE SMA-reinforced specimens (AN-P-1, AN-P-2, AN-P-3, and AN-P-4). The failure mode in the steel-reinforced specimens was fracture of the steel reinforcement as shown in Figure 4-19. However, two types of failure modes took place in the PE SMA-reinforced specimens; fracture or slippage of the PE SMA bars as shown in Figure 4-20.
Table 4-5: Test results of the joint-like specimens ψu Pu (kN) (m-1) AN-S-1 144 0.0061 AN-S-2 145 * AN-P-1 77 0.0014 AN-P-2 81 0.0028 AN-P-3 79 0.0022 AN-P-4 76 0.0007 * The δct was malfunctioned near failure
σu (MPa) 718 724 441 464 451 437
Specimen
εu (mm/mm) 0.112 0.107 0.043 0.039 0.059 0.052
Failure mode Fracture Fracture Slip Slip Fracture Fracture
It is noted that the PE SMA-reinforced specimens were drilled after the test in order to examine the failure mode as it occurred at the location of the anchors. The PE SMA bar slipped in specimens AN-P-1 and AN-P-2 while it fractured in specimens AN-P-3 and AN-P-4. As discussed in Section A.5.2, the screws penetrated into the PE SMA bar when locking them by applying the 208
required torque. Thus, a stress concentration region developed in the neighbourhood of the screws. In fact, the fracture of the PE SMA bars took place at the location of the groove created by penetrating the first screw as shown in Figure 4-20 (a). With regard to the slippage failure, the increase in the pullout strain led to significant slip inside the anchor. Longitudinal grooves therefore developed along the surface of the PE SMA bar parallel to the line of screws (Figure 4-20 (b)).
Figure 4-19: Typical failure of the Steel-reinforced joint-like specimens
(a) Fracture of the PE SMA bar
(b) Slippage of the PE SMA bar
Figure 4-20: Typical failures of the PE SMA-reinforced joint-like specimens
209
The stress-strain relationship obtained by plotting the stress and strain values at the peak cycles are shown in Figure 4-21 for the steel and PE SMA bars, while the maximum values are included in Table 4-5. It is noted that the strains were determined from the readings of the LSC devices excluding the slippage that took place inside the wedge grip. However, it was assumed that no slippage took place inside the anchors up to the maximum load (this assumption is validated in Section 4.5.2). The average maximum stress and strain were 721 ± 4 MPa and 0.110 ± 0.004, respectively, in the steel-reinforced concrete specimens while they were 448 ± 12 MPa and 0.048 ± 0.009, respectively, in the PE SMA-reinforced concrete specimens. The ratios of the stress and strain in the PE SMA bars in the joint-like specimens to the stress and strain values of the cyclic PE SMA test (included in Table 4-3) are 0.668 ± 0.017 and 0.503 ± 0.090, respectively. The drift ratio corresponding to the average maximum strain induced in the PE SMA bars is 12.2 % (determined using the simplified procedure included in Section 3.3), which is higher than the minimum acceptable value of 3.5% proposed by the ACI T1.1-01 (2001) code. It is, therefore, concluded that the current form of anchorage is suitable for the application implemented in this study. However, it was decided to modify the anchors in order to increase the attained stress and strain values. This enhancement aims at developing anchors that can be used for other civil engineering applications in which PE SMA bars experience high strain levels. The modification of the anchors are included in Section 4.5.2. The hysteretic responses of the stress-strain relationships are also shown in Figures C-1. It is noted that each specimen was subjected to at least 80 cycles. However, the response is only plotted at increments of 0.001 strain starting at strain of 0.002 for clarity purposes. It is seen that the hysteretic responses of all bars are similar to that determined using the cyclic test in Section 4.2.2 except for specimen AN-P-3 in which the permanent strain was higher than that in the other 210
specimens. This is due to relative slip inside the anchor which was not considered in calculating the strains as discussed previously. 800 700
Stress, σ (MPa)
600 500
AN-S-1 AN-S-2 AN-P-1 AN-P-2 AN-P-3 AN-P-4
400 300 200 100 0 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Strain, ε (mm/mm)
Figure 4-21: Stress-strain relationships of the vertical reinforcement in the joint-like specimens
The behaviour of the joint-like specimens is examined by plotting the curvature relationship of the tested specimen and the strains in the shear reinforcement as shown in Figure 4-22 and Figure 4-23, respectively. It is seen that the average cracking load in the steel-reinforced specimens (average stress of 38 ± 13 MPa) is 45.3% lower than the average cracking load in the PE SMA-reinforced specimens (average stress of 70 ± 2 MPa). This is due to the occurrence of splitting cracks in the former specimens. A detailed discussion about the splitting cracks and the difference in the behaviour of PE SMA and deformed steel bars is included in Section 4.4.6.
211
30
Moment (kN.m)
25 20 AN-S-1 AN-S-2 AN-P-1 AN-P-2 AN-P-3 AN-P-4
15 10 5 0 0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
Curvature, ψ (m-1)
Figure 4-22: Moment-curvature relationships of the joint-like specimens
30
Moment (kN.m)
25 20 AN-S-1 AN-S-2 AN-P-1 AN-P-3 AN-P-4
15 10 5 0 -0.0001
0
0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007
Strain, ε (mm/mm)
Figure 4-23: Moment-strain relationships of the stirrup in the joint-like specimens
212
The strain in the stirrup in the ANS-S-1 specimen is 40 % higher than that in the ANS-S-2 specimen at the ultimate load. The higher strain in the former specimen is due to the fact that the stirrup was placed closer to the vertical steel reinforcement (refer to Table 4-4 and Figure 4-15). Thus, part of the strain attained in the stirrup is due to the contribution of the stirrup in resisting the vertical load transferred by the longitudinal steel as was noticed in prior research studies (Paulay and Priestley, 1992) and noticed in the test results of the steel-reinforced SSB and DSB systems presented in Sections 3.7.6.2 and 3.8.6.2. Compression strains, however, developed in the PE SMA-reinforced specimens up to the cracking load after which the tensile strains were generated in the stirrups. Thus, compression fan originating from the anchor tend to induce compression force in the stirrups prior to cracking. After cracking, the mechanism of load transfer changed and tensile forces were transmitted by the stirrups. More elaboration on this behaviour is included in the following section. In general, the strain levels in the stirrups were lower than the elastic limits due to the low forces applied to the PE SMA-reinforced specimens.
4.5.1.6 Discussion and Analytical Modeling The responses of the joint-like specimens are examined analytically in this section in order to determine the effect of anchor orientation and location on the response of the joints. As will be presented in Section 4.5.2, the anchors were modified in order to increase their strain and stress capacities, and thus, the analytical modeling also intends to predict the behaviour of using these modified anchors in the joint region. First, the effect of the vertical reinforcement type (steel or PE SMA) on the cracking load of the joint-like specimens is examined followed by developing analytical models to predict the moment versus curvature relationships.
213
The load transfer mechanism of the mechanical anchor and the deformed bar are illustrated in Figure 4-24. The loads are resisted by bearing on the anchor heads in the mechanical anchors while they are resisted via two mechanisms in the deformed hook bars; normal stressing and bearing on the deformations (Park and Paulay, 1975). In the mechanical anchors, the forces are resisted by one and two heads when the anchor is placed in the normal orientation and the flipped orientation, respectively. In the hooked deformed bars, the forces are resisted by bearing action that consists of tangential and normal stresses as illustrated in Figure 4-25. The normal stresses act as radial splitting stresses on the concrete and counteracted by ring tension stresses in the concrete. When the radial splitting stresses exceed the tensile strength of the concrete, splitting cracks initiate and propagate into the surrounding concrete.
Figure 4-24: Force resisting mechanisms in the anchored PE SMA and deformed steel reinforcement 214
Figure 4-25: Bond and splitting components of the bearing stresses (Thompson, 2002)
4.5.1.6.1 Prediction of the Cracking Load 4.5.1.6.1.1 Steel-Reinforced Joint-Like Specimens The cracking load in the steel-reinforced joint-like specimens is governed by the splitting bond stress and not by the concrete tensile stress unlike the behaviour of PE SMA-reinforced jointlike specimens. Therefore, the splitting bond stress needs to be calculated in order to determine the cracking load of the steel-reinforced specimens. The simplified version of the expression developed by Darwin et al. (1996) used to calculate the development length of the steel reinforcement as a function of the splitting strength of the concrete cover (Equation 4-6) forms the basis of the prediction model presented in this section. It is noted that the expression is converted from the imperial units to the SI units is this research.
215
f yt 1 . 046 ' 0.25 ld 48.8 f c db c ktr db
Equation 4-6
where, k tr
9.660 0.978 d b Asp
Equation 4-7
s sp ,1n
c cm 0.5d b
Equation 4-8
where ld is the development length, ktr term representing the effect of the transverse reinforcement on bond strength, cm the minimum of either the concrete cover or the side cover, f’c the concrete compressive strength, db the bar diameter, Asp the total cross sectional area of all the transverse reinforcement which cross the potential plane of splitting, fy,h the yield strength of the transverse reinforcement, ssp,1 the maximum centre-to-centre spacing of the transverse reinforcement, and nsp is the number of bars developed along the plane of splitting.
Based on simple mechanics, the ld defined as a function of the stress in the reinforcement, fs, the bond stress, μb, and db is calculated as follows:
ld
f s db 4 b
Equation 4-9
Rearrange Equation 4-9 and substituting it in Equation 4-6:
216
c ktr 20.1 f y d b b 41.8 f y ,h 2130 f ' 0.25 c
Equation 4-10
It is concluded by Darwin et al. (1996) that the value of the (c + ktr)/db ratio indicates the failure mode. Values equal or less than 4 indicate a splitting failure of the concrete cover while values higher than 4 indicate pullout failure of the reinforcing bar. However, based on the results of the tests conducted by Mathey and Watstein (1961) and the observations made in this research, splitting failures can occur at values higher than 6 and up to 7. It was, therefore, determined not to limit the value of the (c + ktr)/db ratio since splitting failure can take place at higher values. Consequently, the value calculated using Equation 4-10 corresponds to the splitting bond stress and is used to calculate the force in the reinforcing bar, Fs, at the onset of concrete splitting. The Fs is defined as a function of the development length at the onset of concrete splitting, ld,sp, as follows;
Fs b d b l d ,sp
Equation 4-11
Due to the absence of an empirical equation that defines ld,sp, the variable is determined using the experimental results of the tested steel-reinforced specimens (AN-S-1 and AN-S-2). Assuming a linear strain variation along the embedded vertical steel reinforcement and based on the readings of the strain gauges mounted on the vertical steel reinforcement, the average value of
ld,sp at the cracking loads is 98 mm. Substituting 3, 16 mm, 484 MPa, 35 MPa, 67 m, 200 mm2, and 185 mm for n, db, fy,h, f’c,
cm, Ast, and sst in Equation 4-10 gives μb of 8.3 MPa. Substituting 97.9 mm and 8.3 MPa for ld,sp and μb in Equation 4-11 results in a tension force of 41 kN at concrete splitting. The percentage 217
error between the predicted value and the average cracking load in the AN-S-1 and AN-S-2 specimens is 8 %. Therefore, the implemented procedure provides accurate prediction of the splitting cracking load.
4.5.1.6.1.2 PE SMA-Reinforced Joint-Like Specimens The cracking load of the PE SMA-reinforced specimens can be found using the conventional moment theorem due to the insignificant contribution of the bond stresses, and hence, the absence of the splitting bond failure mechanism. The internal moment, Min, and the external moment, Mex, are calculated as follows:
M in
ft' I c yt
Equation 4-12
M ex
Fs l n 4
Equation 4-13
where f’t is the concrete tensile strength, Ic the cracked moment of inertia, yt the distance from the extreme top fibre to the centroid of the section, and ln is the moment arm.
The f’t in Equation 4-12 is calculated based on the expression by CEB-FIP (1993) code: f ' 8 f t 1.4 c 10 '
2 / 3
(MPa)
Equation 4-14
Equating Equation 4-12 and 4-13 yields the following expression for the cracking load, Fcr:
Fcr
4 f ct I c l n yt
Equation 4-15
218
Based on the gross section properties of the joint-like specimen, the Fcr calculated using Equation 4-15 equals to 74 kN. The percentage error between the predicted value and the average values in the AN-P-1 and AN-P-2, AN-P-3, and AN-P-4 specimens is 6 %. Therefore, the conventional moment theorem provides accurate results of the cracking load.
4.5.1.6.2 Prediction of the Moment-Curvature Relationships After determining the cracking loads of the steel and the PE SMA-reinforced specimens, the postcracking load-curvature relationships are determined. According to CSA-A23.3-04 (2004), the tested joint-like specimens are considered as deep flexural members since the clear span to overall depth ratio is 1.7 which is less than the limit (2.0) (Clause 10.7.1). Therefore, it is recommended to use a strut-and-tie model for the analysis instead of the conventional moment theorem as per CSA A23.3-04 Clause 11.4. In fact, the strain distribution across the depth of the specimens (shown in Figure-C.2) indicates that the specimens experience non-linearity in the strain variation as the load increases. In this research, simple strut-and-tie models are developed and used to predict the behaviour of the joint-like specimen. The schematic drawing of the compression strut, tension ties, and nodes of the strut-and-tie models are shown in Figure 4-26, while the assumptions involved in developing the model are as follows: -
The models are used to predict the behaviour of the specimens in the elastic range, and thus, it is assumed that the nodes are strong enough to resist the applied loading. In other words, the strength and the detailing of the nodes are not checked.
-
The location of the compression cord F6 is calculated based on the depth of the neutral axis obtained from the conventional moment theorem of cracked concrete sections.
-
The tension ties F1 and F4 are located at the centreline of the top reinforcement. 219
The inclination angle θ1 is set while the inclination angles θ2 and θ3 (only in specimen AN-P-3) are calculated based on the geometry of the specimens. Several limitations on the feasible inclined angle are provided by different design codes. The minimum feasible inclined angle by the Swiss code (SIA 262, 2003), European standard (Eurocode 2, 2008), and the American building code (ACI 318-08, 2008) are 26o, 22o, and 25o, respectively, while the maximum values are 64o, 59o, and 65o, respectively.
(a) AN-P-1, AN-P-2, AN-P-4
(b) AN-P-3
(c) AN-S-1 and AN-S-2
Figure 4-26: Strut-and-tie models of the joint-like specimens
Based on a preliminary analysis of the joint-like specimens and the limits on the feasible angles provided by the codes, a value of 58o was chosen for θ1. The tension force in the vertical reinforcement, P, is the summation of two components; P = P1 + P2
Equation 4-16
The force P2 in all connections equal zero except in specimen AN-P-3 in which the anchor is in a flipped orientation. The ratio of P1/P2 in specimen AN-P-3 is proportional to the ratio of the head areas A1/A2 shown in Figure 4-24, which equals to 1.37. The strain values were calculated by dividing the forces by the corresponding cross-sectional areas and the modulus of elasticity. The 220
steel modulus of elasticity, Es, are included in Table A.10 while the concrete modulus of elasticity, Ec, is calculated as follows (CSA-A23.3-04, 2004): E c 4500
f c'
(MPa)
Equation 4-17
The location of the P1 force in the steel-reinforced joint-like specimens (AN-S-1 and AN-S2) is determined based on the development length. The distance between the top fibre to the location of P1 force is expressed as follows: d a ld ,sp
ld 3
Equation 4-18
Equation 4-18 assumes that the force is located at the centroid of the stress triangle developed within the development length after the onset of the splitting crack. As described in Section 4.5.1.6.1.1, ld,sp is 98 mm, while ld is 340 mm determined from Table 3.1 in the CSA-A23.3-04 (2004). Based on the above assumptions, the material properties, and the geometry of the jointlike specimens, the analysis of the strut-and-tie models presented in the moment-curvature relationships are included in Figure 4-27 and Figure 4-28 for the steel-reinforced and PE SMAreinforced joint-like specimens, respectively. Even though the PE SMA-reinforced joint-like specimens failed prior to attaining the maximum stress and strain of the PE SMA bars, the developed model was used to predict the behaviour until the ultimate stress is achieved. As will be presented in Section 4.5.2, the anchors were modified in order to increase the stress and strain at failure, and thus, the prediction model is used to examine the behaviour of the joint-like specimens with the modified anchors. The prediction model does not take into account the distribution of the stirrups in the specimens, and thus, the predicted behaviours of the steel-reinforced specimens (AN-S-1 and
221
ANS-S-2) are similar and the predicted behaviours of the AN-P-2 and AN-P-4 are also similar. Upon cracking, the tension stiffening effect in the steel-reinforced specimens results in a smooth transition between the cracked and the un-cracked states. This action is not taken into account in the models, which explains the difference in the experimental and the analytical trends upon cracking. However, the experimental and analytical moment-curvature curves of the steelreinforced joint-like specimens, in general, are in good agreement. The post-cracking slopes of the moment-curvature curves in the PE SMA-reinforced specimens depend on the location and the orientation of the anchors. Specimen AN-P-3 has the highest post-cracking slope followed by AN-P-1 specimen, followed by AN-P-2 and AN-P-4 specimens. It is, therefore, concluded that the flipped orientation will provide the least curvature. Minimizing the curvature demand in the joint is always on the safe side. However, due to the fact that PE SMA bars in the large-scale concrete beam-column connections are subjected to high strain amplitude tension-compression cyclic loading, it was determined to choose AN-P-2 configuration since it provides more room (low da) for the compression fan to develop when the PE SMA is subjected to compression force.
222
30 AN-S-1 AN-S-2
Moment (kN.m)
25 20 15
Analytical (AN-S-1 & AN-S-2)
10 5 0 0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
Curvature, ψ (m-1)
Figure 4-27: Moment-curvature prediction of the steel-reinforced joint-like specimens
30
Moment (kN.m)
25 Analytical (AN-P-3) Analytical (AN-P-1)
20
Analytical (AN-P-2 & AN-P-4)
15
AN-P-1 AN-P-2 AN-P-3 AN-P-4
10 5 0 0
0.001
0.002
0.003
0.004
Curvature, ψ
0.005
0.006
0.007
(m-1)
Figure 4-28: Moment-curvature prediction of the PE SMA-reinforced joint-like specimens
223
4.5.2 Anchorage Modification The modifications of the anchor are presented in this section. Due to the expensive cost of the PE SMA material (approximately $1,000/m), the same PE SMA bars used in the joint-like specimens were used again in the experimental testing of the modified anchors. Therefore, the strain capacities of the PE SMA bars were reduced due to the induced permanent deformation. As concluded in Sections 4.4.2 and 4.4.3, the deformation measurements of the DICT and the LSC techniques are in good agreement with the readings obtained from the SGs, and hence, only the former two methods were used to monitor the behaviour of the specimens. The test setup and the location of the LSCs are shown in Figure 4-29.
Figure 4-29: Test setup and locations of the LSC devices in the modified anchor testing 224
One LSC was used to measure the relative displacement of the two steel reacting against the PE SMA bar and the steel anchor (δu), while one LSC was used to measure the slip in the PE SMA bar inside the anchor relative to the fixed steel frame (δd). The loading regime (Figure 4-3) and the setup of the DICT method were the same as those implemented in testing the PE SMA material. The experimental testing consisted of four phases. In the first phase, the performance of the anchors without modification is examined while different anchor modifications are implemented in the other three phases. The detailed descriptions of the four phases are as follows: -
Phase 1. The performance of the mechanical steel anchors used in testing the joint-like specimens was investigated as received from the manufacturer (no modifications to the anchor) (Figure 4-30 (a)).
-
Phase 2. Additional two lines of holes were drilled at an angle of 130° from the prefabricated line of holes. In order to avoid the premature fracture of the PE SMA bar as a result of the penetration of the screw-locks into the bar, twelve flat Grade 8 screws were installed and torqued to 70 ft·lb (95 N·m) (Figure 4-30 (b)).
-
Phase 3. Two lines of holes were drilled at an angle of 130 ° from the prefabricated line of holes similar to Phase 2. However, two types of screw-locks were used; the original screwlocks (as received from the manufacturer) and flattened screw-locks. The former were installed in the original line of holes while the latter were installed in the additional two lines of drills. All screw-locks were torqued until shearing off the heads at an average torque of 70 ft·lb (95 N·m) (Figure 4-30 (c)).
-
Phase 4. Two lines of holes were drilled at an angle of 130 ° from the prefabricated line of holes similar to Phase 2. Flattened screws described in Phase 3 were installed in all the 225
holes and torqued until shearing off the heads at an average torque of 70 ft·lb (95 N·m) (Figure 4-30 (d)).
In each phase, one specimen was tested except for Phase 4 in which two specimens were tested (Specimen A and B) in order to validate the superior performance of the modified anchor in Phase 4 as compared with the other phases. The envelopes of the stress-strain relationships of the five specimens are compared in Figure 4-31. As mentioned previously, the strain capacities of the PE SMA bars were reduced due to the cyclic testing conducted in the joint-like specimens, and hence, the strain readings shown in Figure 4-31 exclude the permanent strains induced in the PE SMA bars. This is believed to yield better comparison among the tested anchors. The failure modes are shown in Figure 4-32 while the strain and stress test results at failure are included in Table 4-6. The percentage difference in the strain and stress of the modified anchors as compared with those of the unmodified anchor (Phase 1) are also included in the table. It is seen that phase 2 yielded the lowest strain and stress values while Phase 4 showed the highest ones. The average ultimate strain and stress of specimens A and B in Phase 4 were 0.0535 ± 0.0094 and 560 ± 43 MPa, respectively. The average increase in the strain and stress capacities in Phase 4 were 99% and 16%, respectively, as compared with the unmodified anchor in Phase 1. Therefore, modifying the anchors in accordance to Phase 4 enhanced the strain and stress values attained at failure.
226
(a) Phase 1
(b) Phase 2
(c) Phase 3
(d) Phase 4
Figure 4-30: Modified steel anchors 700
Stress, σ (MPa)
600 500 400
Phase 1 Phase 2 Phase 3 Phase 4-A Phase 4-B
300 200 100 0 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Strain, ε (mm/mm)
Figure 4-31: Stress-strain relationships of the PE SMA bars in the modified anchors 227
(a) Phase 1
(b) Phase 2
(c) Phase 3
(d) Phase 4-A
(e) Phase 4-B Figure 4-32: Failure modes of the modified steel anchors
228
Table 4-6: Strain and stress test results at failure of the modified anchors Specimen Phase 1 Phase 2 Phase 3 Phase 4-A Phase 4-B
Strain (mm/mm) 0.0269 0.0225 0.0320 0.0601 0.0468
Diff. (%) -16 19 123 74
Stress (MPa) 483 400 462 590 529
Diff. (%) -17 -4 22 10
Mode of failure Slippage Slippage Fracture Fracture Fracture
The strain distributions along the clear length (223 mm for all specimens except for Phase 1 which is 178 mm) of the specimens are plotted in Figure C-3 for multiple stress levels. Similar to the behaviour observed in the monotonic and cyclic testing of the PE SMA bars in Sections 4.4.2 and 4.4.3, respectively, severe nonlinearity was experienced upon the transformation from the Austenite to the Martensitic phase (passing the σMS stress in Figure 2-12). The stress-slip relationships of the modified anchors are shown in Figure 4-33. It is noted that the slip of the PE SMA bars inside the anchors were measured using the δd LSC positioned as shown in Figure 4-29. The slope of the stress-slip relationships of all anchors are approximately the same except for the anchor in Phase 3 in which the slope of the relationship is significantly lower than the other phases. Prior to reaching σMS stress, the slip was negligible. However, it increased rapidly afterwards. This is attributed to the fact that the amount of slip is proportioned to the stiffness of the material. The screws penetrate into the material when they are torqued (see Figure 4-20), which provides resistance to slippage via bearing against the SMA. Knowing that the stiffness of the SMA during the Austenite phase is significantly higher than during the forward transformation, the tendency to resist slippage is decreased upon passing the σMS stress (the forward transformation begins once the σMS is reached), and hence, high amount of slippage took place in the forward transformation process as compared with the Austenite phase.
229
700
Stress, σ (MPa)
600 500 400
Phase 1 Phase 2 Phase 3 Phase 4-A Phase 4-B
300 200 100 0 0
1
2
3
4
5
Slip, s (mm)
Figure 4-33: Stress-slip relationships of the PE SMA bars in the modified anchors
At failure, the slips were 2.3 mm, 3.4 mm, 1.0 mm, 4.7 mm, and 1.9 mm in specimens Phase 1, Phase 2, Phase 3, Phase 4-A, and Phase 4-B, respectively. In the anchor proposed by Alam et al. (2010) (refer to Section 4.5), the slip attained at a strain value similar to that experienced in Phase 4-A specimen (6.0 %), is 1.9 times that experienced in the proposed anchor. It is, therefore, concluded that the proposed anchor in Phase 4 shows superior performance as compared with the anchor proposed by Alam et al. (2010) in terms of slippage inside the anchor. Hence, the modified anchors were used in large-scale SSB and DSB beam-column connections examined experimentally in Section 4.10.
230
4.6 Coupler Performance Mechanical steel couplers are used to connect the steel diagonal steel reinforcement to the PE SMA bar as described in Section 4.3 and shown in Figure 4-1. Mechanical couplers as shown in Figure 4-13 (b) are used in this research. Upon the request of the manufacturer, the brand name is kept confidential. In essence, the anchorage mechanism of the mechanical coupler and anchor, shown in Figure 4-13, is the same (one lines of identical four screws). Therefore, the recommendations obtained from Section 4.5.2 in terms of the need of modifying the mechanical anchor is also applicable for the mechanical coupler. However, due to the difficulty of drilling two additional lines of holes into the coupler as had been done in Phase 4 in Section 4.5.2, only one additional line of holes was drilled as shown in Figure 4-34. Flattened screws described in Phase 4 were installed in all the holes and torqued until shearing off the heads at an average torque of 70 ft·lb (95 N·m). The test setup, loading regime, and specimen instrumentations are similar to those described in Section 4.5.2. Two specimens were tested in order to validate the performance of the modified coupler.
Figure 4-34: Modified steel coupler
231
The envelopes of the stress-strain relationships of the PE SMA bars in the modified coupler are shown in Figure 4-35. Specimen A failed by fracture of the PE SMA bar near the screw while Specimen B failed by excessive slippage of the PE SMA bar as shown in Figure 4-36. The ultimate strains were 0.0510 and 0.0580 while the ultimate stresses were 581 MPa and 533 MPa in specimens A and B, respectively. The average strain and stress at ultimate were 0.0545 ± 0.005 and 557 ± 34 MPa, respectively. The average strain in the PE SMA bar in the modified coupler was 1.9 % higher while the average stress was 0.5 % lower than that in the PE SMA bar in the modified anchors in Phase 4. It is, therefore, seen that the strain and stress capacities are approximately the same in the modified anchor and couplers. The strain distributions along the clear length of the specimens are plotted in Figure C-4 for multiple stress levels and at ultimate. The same observations with regard to the strain distribution made in Sections 4.4.2, 4.4.3 and 4.5.2 are applicable to the strain distribution in the PE SMA bars in the modified couplers. The stress-slip relationships of the modified coupler are shown in Figure 4-37. The relationships in both specimens are approximately the same in which they follow bi-linear trends. The slopes of the curves prior to the transition to the Martensitic phase are significantly less than that after the onset of the forward transformation (refer to Section 4.5.2 for the explanation of this phenomenon). The average slip at ultimate is 4.0 ± 0.1 mm.
232
700 600
Stress, σ (MPa)
500 400 300
Specimen A Specimen B
200 100 0 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Strain, ε (mm/mm)
Figure 4-35: Stress-strain relationships of the PE SMA bars in the modified couplers
(a) Specimen A
(b) Specimen B
Figure 4-36: Failure modes of the modified steel couplers
233
700 600
Stress, σ (MPa)
500 400
Specimen A Specimen B
300 200 100 0 0
1
2
3
4
5
Slip, s (mm)
Figure 4-37: Stress-slip relationships of the PE SMA bars in the modified coupler
4.7 Geometry, Detailing, and Construction of the Connections The final designs of the three connections included in Table 4-4 are shown in Figure 3-12 based on the design and analysis of the SSB and DSB PE SMA-reinforced concrete beam-column connections discussed in the above sections. The detailed construction procedure of the connections including; building formwork, fixing steel cages, and concrete casting are included in Section A.5.2. The ancillary test results including concrete compression tests and steel coupon tension tests are presented in detail in Section A.5
234
(a) Control connection
(b) Connection SSB-P-1.0
235
(c) Connection DSB-P-1.0 Figure 4-38: Details of the Control, SSB-P-1.0, and DSB-P-1.0 connections
4.8 Test Setup and Loading Regime The test setup and the loading regime implemented in this phase are similar to those implemented in testing the steel-reinforced SSB and DSB connections included in Section 3.5.
4.9 Instrumentations The instrumentations used to monitor the behaviour of the connections are similar to those used to monitor the behaviour of the connections in Section 3.6 including; LSC devices, Lasers displacement sensors, SGs, and DICT. The LSC devices were mounted on the beams and the joints in order to measure the concrete deformations, while the Lasers displacement sensors were positioned on the laboratory floor in order to measure the displacement of the beam members under 236
positive and negative bending and to calculate the sliding shear in the concrete hinge in the PE SMA-reinforced connections (Laser sensors 1 and 2). The positions and the labels of the LSC and Laser devices are shown in Figure 4-39. The SGs were mounted to the longitudinal reinforcement in the beam and column, diagonal reinforcement, and stirrups as shown in Figure 4-40 in order to measure the strains induced in the reinforcement. The instrumentation of the DICT method is the same as the one described in Section 3.6 and shown in Figure 3-19.
(a) Control connection
(b) Connections SSB-P-1.0 and DSB-P-1.0
Figure 4-39: Positions and labels of the LSC and laser devices
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Figure 4-40: Locations of the strain gauges in the Control, SSB-P-1.0, and DSB-P-1.0 connections 238
4.10 Experimental Behaviour of the Self-Centering Connections 4.10.1 Hysteretic Response The hysteretic responses of the connections are shown in Figure 4-41. The moment and the drift ratios were calculated as discussed previously in Section 3.7.1. The validity of the calculating method of the drift ratio is examined in detail in Section 4.10.3. The Control connection failed by crushing of the concrete followed by the fracture of the steel reinforcement in the beam at the face of the column (plastic hinge location), while the SSB-P-1.0 connection experienced a fracture of the PE SMA bar near the anchor. The DSB-P-1.0 connection, however, did not experience any drop in the load nor did show any signs of failure up to the maximum stroke of the actuator (±120 mm). Therefore, it was determined to stop the test, readjust the height of the actuator, and subject the connection to only positive bending in order to get advantage of the full stroke length of the actuator (240 mm). In this test (referred thereafter as the positive bending test), the connection failed by the fracture of the bottom PE SMA bar at the location of the anchor at 60 mm (4.4% drift) followed by buckling of the top PE SMA bar at 165 mm (12.2% drift), and fracture of the top PE SMA bar at 189 mm (13.9% drift). Due to the fact that the fracture of the PE SMA bars in the positive bending test took place at lower displacement (60 mm and 4.4% drift) than that attained in the positive-negative test (±120 mm and ±8.9%), the ultimate condition is, therefore, defined with respect to the positive-negative bending test results. However, the loading regime, hysteretic response, and failure of the DSB-P-1.0 connection subjected to the positive bending test are included in Section C.2 for reference. The fracture of the steel reinforcement and the PE SMA bar in the Control and the SSB-P-1.0 connections took place during the third positive bending cycle of the 75 mm displacement amplitude cycle.
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The Control connection experienced the typical hysteretic response of conventional beam column connections (Figure 4-41 (a)). The load carrying capacity increased with increasing drift until the maximum capacity was reached. After that, a graduate decrease in the capacity was noticed due the widening of the cracks and the development of sliding shear plane at the beam plastic hinge (located at the face of the column). These actions led to the so called ‘pinching shear’ which was discussed in detail in Section 3.7.1. Consequently, the stiffness of the system decreased upon the increase in the drift. The hysteretic response of the SSB-P-1.0 connection is characterized by its self-centering ability in which the permanent deformation upon unloading was insignificant, as compared with the behaviour of the conventional beam-column connection (Figure 4-41 (b)). The low permanent deformation also indicates that insignificant slippage took place inside the anchors and the top reinforcement remained elastic. It is seen that the loading stiffness is higher than the unloading stiffness under positive and negative bending, and thus, the hysteretic response of the connection experienced a ‘leaf-like’ shape. The difference in the ultimate load and the stiffness between the positive and negative bending is due to the difference in the tensile properties of the PE SMA and steel reinforcement, namely, the stiffness and the yield stress. Under positive bending, the tensile resistance is provided by the bottom reinforcement (PE SMA) while it is provided by the top reinforcement (steel) under negative bending. It is also important to note that the decrease in the stiffness upon reloading is not due to the pinching shear effect but it is due to the decrease in the stiffness of the PE SMA bars upon the application of cyclic loading (examined thoroughly in Section 4.4.3).
240
(a) Connection Control
(b) Connection SSB-P-1.0
241
(c) Connection DSB-P.10 Figure 4-41: Hysteretic response of the Control, SSB-P-1.0, and DSB-P-1.0 connections
Similar to the behaviour of the SSB-P-1.0 connection, the hysteretic response of the DSBP-1.0 connection experienced a leaf-like shape characterized by a considerable energy dissipation capacity, yet, insignificant permanent deformation upon unloading (Figure 4-41 (c)). The difference in the load capacity between the positive and the negative bending is due to the fact that the connection was subjected originally to its own self-weight which induced negative bending on the beam (refer to Section 3.8.1 for more discussion about the effect of self-weight on the response of beam-column connections). Therefore, it is observed that the SSB-P-1.0 and the DSB-P-1.0 connections achieved the intended self-centering ability. Further examination of the cyclic behaviour of the two connections is conducted in the subsequent sections.
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Due to the absence of an explicit yielding point in the PE SMA bars, the yielding condition of the SSB-P-1.0 and DSB-P-1.0 connections cannot be determined using the conventional methods described in Figure 3-21. Instead, the method for determining the yielding condition of the PE SMA-reinforced connection is illustrated in Figure 4-42. The yield point of the equivalent bilinear system with reduced stiffness is found as the intersection of two lines with slopes determined such that the energy excluded and included are approximately the same. The yielding point of the Control connection is also determined using the same approach for consistency purpose.
Figure 4-42: Definition of the yielding point in the Control, SSB-P-1.0, and DSB-P-1.0 connections
The loads at the yielding, Py, and the ultimate conditions, Pu (defined as the maximum attained load), for the positive and the negative bending are included in Table 3-3. At yielding, the difference in Py is insignificant under positive bending. However, considerable difference in the response is observed under negative bending. This is due to the self-weight effect discussed previously in Section 3.8.1. At the ultimate condition, all connections experienced approximately 243
the same ultimate load under positive bending. However, significant difference exists under negative bending. The significant increase in the capacity of the SSB-P-1.0 connection under negative bending as compared with positive bending is due to the higher yield force in the top reinforcement as compared with that in the PE SMA bars as discussed previously in this section.
Table 4-7: Load comparison at the yielding and the ultimate conditions in the Control, SSB-P1.0, and DSB-P-1.0 connections Yielding Ultimate Positive Negative Positive Negative Connection * * * Py Diff. Py Diff. Pu Diff. Pu Diff.* (kN) (%) (kN) (%) (kN) (%) (kN) (%) Control 40 38 51 47 SSB-P-1.0 44 10 47 24 44 -14 68 45 DSB-P-1.0 44 10 30 -21 56 10 39 -17 * Percentage difference with respect to Control connection
The flexural overstrength, φo, factors calculated at the ultimate condition (75 mm, 75 mm, and 125 mm cycles for the Control, SSB-P-1.0, and DSB-P-1.0 connections, respectively) are included in Table 3-9. The φo are defined as the ratio of M/Mb,n, where M and Mb,n are the applied moment and the beam nominal moment capacity, respectively. The values of Mb,n are included in Table A-8, Table A-9, and Table A-10. The φo factors in the Control and the DSB-P-1.0 connections under negative bending are lower than positive bending despite the symmetric nature of the connections due to the self-weight effect. However, SSB-P-1.0 connection attained higher φo values under negative bending. This behaviour indicates that ignoring the concrete contribution and the tensile forces in the diagonal reinforcement when calculating Mb,n yields very conservative results under negative bending.
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Table 4-8: Flexural overstrength factors, φo, for the DSB connections Connection Control SSB-P-1.0 DSB-P-1.0
Positive φo 1.45 1.24 1.72
% Diff. -15 19
Negative φo % Diff. 1.35 2.14 59 1.19 -12
4.10.2 Cracking Pattern The cracking pattern at the side face of the beams at ultimate is shown in Figure 4-43. The location of the diagonal shear reinforcement, PE SMA bars, and the steel couplers are also included in the figure. It is noted that ‘D’ stands for ‘diagonal’. Three types of diagonal cracks were initiated in the SSB-P-1.0 connection; A, B, and C while only the diagonal crack (A) was developed in connection DSB-P-1.0. The description of the diagonal cracks A, B, and C is as discussed in Section 3.7.2. Under positive bending, the first flexural cracks to develop were located at the face of the column in the Control connection and in the concrete hinge in the SSB-P-1.0 and DSB-P-1.0 connections. Distributed flexural cracks, initiating at the bottom face of the beams, started to develop with increasing displacement in the Control connection while only a single flexure crack developed near the steel coupler in the SSB-P-1.0 and the DSB-P-1.0 connections. This is due to the negligible bond strength of the PE SMA bars, which act as an unbonded reinforcement. Thus, no transfer of stresses is achieved between the bar and the surrounding concrete, i.e. no development of flexural cracks. The diagonal cracks (A) and (B) in the SSB-P-1.0 connection were developed at displacements of 40 mm and 15 mm, respectively, when the connection was subjected to positive bending. The mechanism of formation of the diagonal cracks (A) and (B) are the same as discussed
245
in Section 3.8.1 and illustrated in Figure 3-23 (a). In the DSB-P-1.0 connection, the diagonal cracks (A) initiated at 30 mm displacement cycle only at the lower end of the concrete hinge parallel to the diagonal steel reinforcement.
(a) Connection Control
(b) Connection SSB-P-1.0
(c) Connection DSB-P-1.7 Figure 4-43: Cracking patterns at the ultimate condition in the Control, SSB-P-1.0, and DSB-P1.0 connections 246
Under negative bending, the first flexural cracks developed at the face of the column in the Control connection and in the concrete hinge in the SSB-P-1.0 and DSB-P-1.0 connections. Flexural cracks distributed over the length of the beam in the Control and the SSB-P-1.0 connections were developed at the top face of the beam with increasing applied displacement. However, only a single flexural crack was developed at the top face of the DSB-P-1.0 connection. The formation of distributed flexural cracks at the top fiber of the SSB-P-1.0 connection is due to fact that the tensile forces under negative bending are resisted by the top steel reinforcement which is fully bonded to the surrounding concrete, unlike the behaviour of the DSB-P-1.0 connection in which the PE SMA bar has a negligible bond strength. The diagonal crack (C) in the SSB-P-1.0 connection formed at a displacement cycle of 50 mm at an angle of 45o as schematically shown in Figure 3-23 (b). The cracking patterns at the top side of the beams are shown in Figure 4-44 as well as the location of the vertical slots in the SSB-P-1.0 and DSB-P-1.0 connections. The Control connection experienced typical flexural cracking pattern. The cracking pattern in the SSB-P-1.0 connection, however, is not uniform in the neighbourhood of the vertical slot in which horizontal cracks parallel to the top reinforcement were developed. The DSB-P-1.0 connection experienced the least amount of cracking. Only a single horizontal crack ran parallel to the top reinforcement and a flexural crack near the steel coupler were developed. Therefore, if a construction joint is made in the attached concrete slab at the location of the vertical slot as described in Section 1.2.2, the DSB system will achieve a superior non-tearing action as compared with the SSB and the conventional systems.
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Figure 4-44: Cracking pattern at the top side of the Control, SSB-P-1.0, and DSB-P-1.0 connections
The cracking patterns at the bottom side of the beams and the locations of the vertical slots are shown in Figure 4-45. The concrete was severely damaged near the face of the column in the Control connection. The length of the damaged concrete, which is approximately equal to bd (400 mm), indicates the length of the plastic hinge. The bottom longitudinal reinforcement experienced severe buckling at the ultimate condition. Minimal cracking took place in the SSB-P-1.0 and the DSB-P-1.0 connections near the vertical slots. It is noted that the horizontal crack in the SSB-P1.0 connection developed suddenly upon the fracture of the PE SMA bar. Finally, it should be noted that no cracking was observed in the joints of the connections. This is attributed to the low loads applied onto the connections as compared with the steel-reinforced counterparts.
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Figure 4-45: Cracking pattern at the bottom side of the Control, SSB-P-1.0, and DSB-P-1.0 connections
4.10.3 Beam Rotation The rotation profiles of the beam members are shown in Figure 4-45. The rotation of the beam in the Control connection is examined at two conditions; ultimate load and ultimate deformation. The ultimate load refers to the maximum capacity of the connection while the ultimate deformation refers to the failure of the connection by steel fracture. The load and displacement at the ultimate load condition were 51 kN (83 kN·m) and 55 mm (3.3%), respectively, under positive bending and 47 kN (78 kN·m) and 50 mm (3.0%), respectively, under negative bending. The load and displacement at the ultimate deformation were 46 kN (76 kN·m) and 75 mm (4.5%), respectively, under positive bending and 40 kN (66 kN·m) and 75 mm (4.5%), respectively, under negative bending. The total beam rotations at the ultimate load were 0.035 rad and -0.026 rad under positive and negative bending, respectively. However, the beam rotations at ultimate deformation were 249
0.044 rad and -0.045 rad under positive and negative bending, respectively. The rotations within 200 mm (0.5bd) from the face of the column under positive and negative bending represent 96% and 83%, respectively, of the total beam rotation at the ultimate load condition. This distance corresponds to the plastic hinge distance recommended by Paulay and Priestley (1992), and hence, it is seen that most of the beam rotation was attained within this region. Examining Figure 4-46 (b) and (c) shows that most of the beam rotation was attained at the location of the vertical slots. In the SSB-P-1.0 connection, the total rotations at the ultimate load under positive and negative bending were 0.054 rad and -0.038 rad, respectively, while the rotations attained at the vertical slot represent 90% and 102% of the total rotation, respectively. Similarly, the rotations in the DSB-P-1.0 connection at the ultimate load under positive and negative bending were 0.087 rad and -0.089 rad, respectively, while the rotations attained at the vertical slot represent 99 % and 100 % of the total rotation. It is, therefore, observed that the beams in both connections behaved like rigid elements with their centre of rotation located at the vertical slot. It is noted that a portion of the beam member in the SSB-P-1.0 connection experienced positive bending even though the beam was subjected to negative bending. This is due to the fact that the percentage contribution of the beam section at the vertical slot to the total beam rotation exceeded 100 %. It is also observed that the rotation of the DSB system is significantly higher than the SSB system and the conventional connection. The rotations under positive and negative bending were approximately the same in the Control and the DSB-P-1.0 connections while a considerable difference existed in the response of the SSB-P-1.0 connection under positive and negative bending. The total beam rotation under positive bending is 1.4 times that under negative bending in the SSB-P-1.0 connection which indicates that an additional source of deformation, which is sliding shear, contributed to the applied 250
displacement. This issue is further discussed in Section 4.10.6.1 when examining the shear deformation in the SSB system. 0.1
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(c) Connection DSB-P-1.0 Figure 4-46: Rotation profiles of the beams in the Control, SSB-P-1.0, and DSB-P-1.0 connections
4.10.4 Beam Elongation The change in the beam length is plotted in Figure 4-47 (also presented in terms of the percentage change in length with respect to the original length of the beam member) against the increase in the displacement amplitude for the positive and negative bending. The beam in the Control connection elongated under positive and negative bending as anticipated due to the increase in the permanent deformation associated with the increase in the strains induced in the top and bottom reinforcement. The behaviour of the SSB-P-1.0 and DSB-P-1.0 connections is similar to the conventional counterparts discussed in Section 3.7.4 and 3.8.4, respectively. In the SSB-P-1.0 connection, the beam elongated under positive bending and shortened under negative bending due to the yielding of the bottom reinforcement under both positive and negative bending 252
while the top steel remained elastic. In the DSB-P-1.0 connection, the trends of the change in the beam length follow the same trends as the Control connection but at flatter slopes. Under positive and negative bending, the increase in the beam length is due to the increase in the cracking and the yielding of the diagonal reinforcement at the concrete hinge. The elongation of the beam members of the SSB and DSB connections in RC frame is plotted against the displacement amplitude in Figure 4-48. The slope of the elongation curves is highest for the Control connection and lowest for the SSB-P-1.0 connection. At the ultimate load condition, the elongation of the Control, SSB-P-1.0, and DSB-P-1.0 connections is 11 mm (0.6%), 0 (0%), and 8 mm (0.5%), respectively. This indicates the superior performance of the SSB-P-1.0 in maintaining zero elongation as compared with the DSB-P-1.0 connection in which the elongation of beam was reduced by 27% with respect to the Control connection. This conclusion is very important in the context of concrete frames subjected to lateral loading since the minimal
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Figure 4-47: Change in length in the Control, SSB-P-1.0, and DSB-P-1.0 connections 253
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Figure 4-48: Beam elongation in the Control, SSB-P-1.0, and DSB-P-1.0 connections
4.10.5 Strain Profile in the PE SMA Bars The strain gauges mounted on the PE SMA bars malfunctioned during testing, and thus, it is was decided to examine the strain behaviour via the LSC devices mounted on the top and the bottom fibres. The strain values obtained using the proposed method represent the average values along the bars in the SSB-P-1.0 and DSB-P-1.0 connections due to the fact the PE SMA bars act as unbonded reinforcement. The envelopes of the moment versus strain relationships of the bottom and top reinforcements at critical sections (at the face of the column in the Control connection and at the vertical slots in the SSB-P-1.0 and DSB-P-1.0 connections) are shown in Figure 4-49 and Figure 4-50, respectively, while the strain values at ultimate are included in Table 4-9.
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Figure 4-49: Moment-strain relationships of the bottom reinforcement at the critical section in the Control, SSB-P-1.0, and DSB-P-1.0 connections 100 75
Control SSB-P-1.0 DSB-P-1.0
Moment (kN.m)
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Figure 4-50: Moment-strain relationships of the top reinforcement at the critical section in the Control, SSB-P-1.0, and DSB-P-1.0 connections
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Table 4-9: Ultimate strain values at the critical sections in the Control, SSB-P-1.0, and DSB-P1.0 connections Connection Control SSB-P-1.0 DSB-P-1.0
Bottom reinforcement Positive Negative bending bending 0.0545 0.0038 0.0155 -0.0156 0.0175 -0.0103
Top reinforcement Positive Negative bending bending 0.0038 0.0538 0.0083 0.0112 -0.0103 0.0211
It is observed that the strains induced in the top and the bottom reinforcements in the Control and the DSB-P1.0 connections were approximately symmetric under positive and negative bending due to the symmetric nature of the beam cross-sections unlike the behaviour in the SSBP-1.0 connection. The top and bottom reinforcements experienced tension under positive and negative bending in the Control connection. The top reinforcement in the SSB-P-1.0 connection also experienced tension under positive and negative bending. This indicates that the steel reinforcement was located below and above the neutral axis during the positive and negative bending, respectively, of the SSB-P-1.0 connection. Based on the behaviour shown in Figure 4-49 and Figure 4-50 and based on the values included in Table 4-9, it is seen that the strains induced in the PE SMA bars are comparable in the SSB-P-1.0 and DSB-P-1.0 connections. The strain profile along the top steel reinforcement in the SSB-P-1.0 connection is shown in Figure 4-51 at four displacement amplitudes under positive and negative bending. The strain profiles under positive and negative bending experienced similar trends. However, the strain values under negative bending were higher than those under positive bending. The strain values in the top reinforcement were positive along the beam member under positive and negative bending. This behaviour is similar to that experienced in the steel-reinforced SSB connections examined in Section 3.7.5. Even though the top reinforcement was designed to remain elastic during the cyclic 256
loading, it is seen that the strain values exceed yielding at the vertical slot. However, yielding of the steel reinforcement took place at high displacement amplitudes.
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Figure 4-51: Strain profiles in the top reinforcement in the SSB-P-1.0 connection
4.10.6 Shear Mechanism 4.10.6.1 Shear in the Beam The shear behaviour is studied by examining the distortion angle obtained from the LSCs, the strains in the stirrups near the slot and the strains in the diagonal reinforcement as shown Figure 4-52. In the SSB-P-1.0 and DSB-P-1.0 connections, the shear distortion angle obtained from the LSCs was examined at two locations (A and B), while it was examined at only one location in the Control connection. The calculation of the shear distortion obtained from the diagonal LSC devices is included in Section B.3.
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Figure 4-52: Locations of the LSCs and strain gauges used to examine the shear behaviour in the beams of the Control, SSB-P-1.0, and DSB-P-1.0 connections
The envelopes of the moment versus the shear distortion at locations A and B are plotted in Figure 4-53 and Figure 4-54, respectively. It is noted that the shear distortion in the Control connection at the face of the column is included in both figures. It is observed that the beam shear distortion in the plastic hinge region in the Control connection was significantly higher than that experienced in the SSB-P-1.0 and DSB-P-1.0 connections under positive and negative bending (in the order of 18 times higher than the PE SMA-reinforced connections). It is also, noticed that the shear distortion in the PE SMA-reinforced connections was approximately zero during all stages of loading. This is due to the fact that most of the shear distortion took place at the concrete hinge in the PE SMA-reinforced connections due to the sliding shear mechanism. The contribution of 258
the sliding shear mechanism in the total applied displacement is investigated in detail in Section 5.4.
Figure 4-53: Beam shear distortion at location A in the Control, SSB-P-1.0, and DSB-P-1.0 connections
Figure 4-54: Beam shear distortion at location B in the Control, SSB-P-1.0, and DSB-P-1.0 connections 259
The moment versus the strain in the stirrups in Control, SSB-P-1.0, and DSB-P-1.0 connections is shown in Figure 4-55 while the moment versus the strain relationships in the diagonal reinforcements in the PE SMA-reinforced connections is shown in Figure 4-56. It is noted that the strain in the diagonal reinforcements in the DSB-P-1.0 connection were averaged since the values of the diagonal reinforcement are approximately the same (discussed in Section 5.3). In the Control connection, the stirrup experienced significant increase in the strain values upon the commencement of yielding in the longitudinal reinforcement. However, a gradual increase in the strains is observed in the PE SMA-reinforced connections. The strains at ultimate in the Control connection were 2.8 and 2.4 times those experienced in the SSB-P-1.0 and DSB-P-1.0 connections, respectively, under positive bending, while they were 4.2 and 3.4 times those experienced in the SSB-P-1.0 and DSB-P-1.0 connections, respectively, under negative bending. Therefore, the stirrup in the Control connection yielded, while it remained elastic in the PE SMA-reinforced connections.
Figure 4-55: Moment versus strain in the beam stirrup of the Control, SSB-P-1.0, and DSB-P1.0 connections 260
Figure 4-56: Moment versus strain in the diagonal reinforcement of the SSB-P-1.0 and DSB-P1.0 connections
The responses of the moment versus strain in the diagonal reinforcement relationships were symmetric under positive and negative bending. It is also observed that insignificant difference exists in the response of the PE SMA connections. The average strain at ultimate in both connections was 0.001, and thus, the diagonal reinforcing bar remained elastic during the test.
4.10.6.2 Shear in the Joint The joint shear is examined by calculating the shear distortion angle obtained from the diagonal LSCs mounted on the joint and the strains in the middle stirrup obtained from the strain gauge as illustrated in Figure 4-52. The moment versus shear distortion relationships of the connections are plotted in Figure 4-57. At ultimate, the shear distortion was 0.2% under both positive and negative bending in the Control connection, while it was approximately zero in the PE SMA-reinforced connections. The Control connection experienced a bilinear relationship in which the slope prior 261
to yielding was significantly higher than that after yielding. However, the relationships of the SSBP-1.0 and DSB-P-1.0 connections were elastic. It is, therefore, concluded that the PE SMAreinforced concrete beam-column connections possess linear elastic joint distortion behaviour unlike the conventional connection. The envelopes of the moment-strain relationships of the middle stirrups in the joints are shown in Figure 4-58. The strains induced in all connections were below yielding, and thus, the stirrups remained elastic under positive and negative bending. The relationships in the Control and the DSB-P-1.0 connections were approximately the same under positive and negative bending due to the symmetric nature of the connections. However, higher strains were induced in the stirrup in the SSB-P-1.0 connection under negative bending as compared with positive bending. This behaviour is due to the unsymmetrical nature of the SSB-P-1.0 connection in which the capacity of the connection under negative bending was 1.55 higher than that under positive bending.
Figure 4-57: Joint shear distortion in the SSB connections
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Figure 4-58: Envelopes of the moment-joint strain relationships of the middle stirrup in the Control, SSB-P-1.0, and DSB-P-1.0 connections
4.10.7 Components of Deformation The member contributions under positive and negative bending were averaged and plotted against the displacement amplitudes as shown in Figure 4-59, while the percentage contributions of the beam, column, and joint at the ultimate cycles are included in Table 4-10. In the Control connection, the contribution of the beam bending increased while the contributions of the column bending and joint distortion decreased with increasing displacement amplitude. However, the member contributions of the beam bending, column bending, and joint distortion were insensitive to the applied displacement in the SSB-P-1.0 and DSB-P-1.0 connections. It is also observed that the member contributions of the SSB-P-1.0 and the DSB-P-1.0 connections were approximately the same as shown in Table 4-10. From Table 4-10 and Figure 4-59, it is concluded that SSB and DSB connections reinforced using PE SMA bars succeeded in reducing the contribution of joint 263
distortion and column bending since the center of rotations were moved away from the face of the columns.
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120
Displacement amplitude (mm)
(c) Connection DSB-P-1.0 Figure 4-59: Member contributions to the total applied displacement in the Control, SSB-P-1.0, and DSB-P-1.0 connections
Table 4-10: Member contributions of the Control, SSB-P-1.0, and DSB-P-1.0 connections at the ultimate cycle Connection Control SSB-P-1.0 DSB-P-1.0
Beam bending (%) 73 96 96
Column bending (%) 19 2 2
Joint distortion (%) 8 2 2
The θ2/θ1 ratio is plotted against the displacement amplitude cycles in Figure 4-60 for the SSB-P-1.0 and DSB-P-1.0 connections. It is noted that the rotation values are averaged under positive and negative bending. The rotation ratios increased approximately in linear manners with increasing applied displacement in both connections. However, the rotation ratio of the SSB-P-1.0 connection was lower than that in the DSB-P-1.0 connection for the same displacement amplitude. 265
At the ultimate cycle, the ratios were 35 and 52 in the SSB-P-1.0 and DSB-P-1.0 connections, respectively. Based on the trends shown in Figure 4-60 and the values at ultimate, it is concluded that the DSB system is more effective in relocating the centre of rotation as compared with the SSB system. This conclusion is in agreement with the observations made with regard to the behaviour of the steel-reinforced SSB and DSB connections. It is also concluded that the effectiveness of the SSB connection in relocating the centre of rotation under positive bending is lower than that under negative bending unlike the behaviour of the DSB connection in which the effectiveness of relocating the centre of rotation is equivalent. 60
SSB-P-1.0 DSB-P-1.0
Rotation ratio, θ2/θ1
50 40 30 20 10 0 0
20
40
60
80
100
120
Displacement amplitude (mm)
Figure 4-60: Comparison of the θ2/θ1 ratios of the SSB-P-1.0 and DSB-P-1.0 connections
The average rotation ratios in the SSB-P-1.0 and DSB-P-1.0 were 2.2 and 3.3 times higher than that in the steel-reinforced SSB and DSB counterparts, respectively (the rotation ratios correspond to values calculated at the failure displacement of the steel-reinforced SSB and DSB 266
connections due to low-cycle fatigue). Thus, the effectiveness of relocating the centre of rotation of the PE SMA-reinforced SSB and DSB systems is higher than that in the steel-reinforced counterparts.
4.11 Comparison and Damage Assessment The behaviours of the connections are compared and the damage is assessed in terms of ductilitybased damage indices, energy-based damage indices, and other damage assessment measures. The damage indices used in examining the performance are described in detail in Section 3.9.
4.11.1 Ductility-Based Damage Assessment The ductility-based damage indices include the rotation ductility index, µθ, displacement ductility index, µ∆, cumulative rotation ductility index, µθ,cum, cumulative displacement ductility index, µ∆,cum, and hysteretic ductility index, µhys. The µθ, µ∆, µθ,cum, and µ∆,cum indices were calculated using Equations 3-9, 3-10, 3-11, 3-12, and 3-13, respectively. It is noted that the values used in calculating the ductility indices in the aforementioned equations were the average values of the responses under positive and negative bending, and thus, the average values of θy, ∆y, Py were determined and included in Table 4-11 while the results of the ductility-based damage assessment indices are included in Table 4-12. Despite the high drift capacities of the PE SMA-reinforced connections, the damage of the non-structural components controls the allowable drift, and hence, it is common for design codes to specify design drift limits of 0.02 to 0.025 (Priestley et al., 2007). However, ductility yields a more representative deformation measure as it controls the structural damage of the system (the damage is limited by the strain values in the critical cross-section). The ductility-based damage 267
assessment indices of the self-centering connections are significantly less than the Control connection as indicated in Table 4-12. This is due to the low stiffness of the connections reinforced using PE SMA bars, which results in high displacement and rotation at yielding. As such, the low ductility capacities are considered to be one of the main drawbacks of the PE SMA-reinforced connections. Future research should be devoted towards avoiding the premature fracture of the PE SMA bars by enhancing the performance of the mechanical anchors and couplers. This will increase strain capacity at failure, and consequently, increase the ductility capacity of the connections.
Table 4-11: The average values of θy, ∆y, Py in the Control, SSB-P-1.0, and DSB-P-1.0 connections Connection θy (rad) ∆y (mm) Py (kN)
Control 0.0088 15 39
SSB-P-1.0 0.0228 32 34
DSB-P-1.0 0.0286 39 40
Table 4-12: Ductility-based damage assessment of the Control, SSB-P-1.0, and DSB-P-1.0 connections Diff.* Diff.* Diff.* µ∆ µθ,cum µ∆,cum (%) (%) (%) Control 5.2 5.4 124 128 SSB-P-1.0 2.4 -54 2.4 -56 60 -52 49 DSB-P-1.0 3.1 -40 3.1 -43 111 -10 120 * The difference is calculated with respect to the Control connection Connection
µθ
Diff.* (%) -62 -6
µhys 172 27 36
Diff.* (%) -84 -79
4.11.2 Energy-Based Damage Assessment Three energy measures are used to assess the damage in the connections: cumulative energy, Ecum, energy index, Ein, and the equivalent viscous damping, ξeq. The Ecum, Ein, and ξeq were calculated using Equations 3-14, 3-15, and 3-17, respectively, while the results of the energy-based damage 268
assessment are included in Table 3-13 along with the percentage differences with respect to the Control connection. It is noted that the average values of ky in Equation 3-15 are 2.7 kN·m, 1.1 kN·m, and 1.0 kN·m for the Control, SSB-P-1.0, and DSB-P-1.0 connection, respectively.
Table 4-13: Energy-based assessment of the Control, SSB-P-1.0, and DSB-P-1.0 connections Ecum Diff.* Ein Diff.* (kN·m) (%) (kN·m) (%) Control 96 420 SSB-P-1.0 28 -71 68 -84 DSB-P-1.0 54 -44 132 -69 * The difference is calculated with respect to the Control connection Connection
ξeq 0.371 0.068 0.061
Diff.* (%) -82 -84
The envelopes of the cumulative energy of the SSB-P-1.0 and DSB-P-1.0 connections are plotted with respect to the displacement ductility in Figure 4-61 (a) and (b). In each figure, the behaviour of the PE SMA-connections is compared with the Control connection and the steelreinforced SSB and DSB counterpart connections. From Table 4-13 and Figure 4-61, the cumulative energy relationships increase exponentially with increasing displacement ductility, and the rate of the cumulative energy trends of the SSB-P-1.0 and DSB-P-1.0 connections are approximately the same. Overall, the PE SMA-reinforced and steel-reinforced DSB connection exhibit superior performance in terms of higher total cumulative energy as compared with the SSB counterparts. The envelopes of the Ein of the SSB-P-1.0 and DSB-P-1.0 are plotted against the displacement ductility index and compared with the Control connection and the steel-reinforced counterparts as shown in Figure 4-62. From Table 4-13 and Figure 4-62, it is observed that the Ein increase exponentially with increasing displacement ductility index. All connections have approximately the same trend except for the DSB-S-1.0 connection which exhibits a higher rate of 269
increase. Overall, the PE SMA-reinforced SSB and DSB connections experience significantly lower Ein values as compared with the steel-reinforced counterparts and the Control connection. 200
Cumulative Energy, Ecum (kN.m)
Cumulative energy, Ecum (kN.m)
200
Control SSB-S-1.0 SSB-P-1.0
175 150 125 100 75 50 25 0
Control DSB-S-1.0 DSB-P-1.0
175 150 125 100 75 50 25 0
0
1
2
3
4
5
6
7
0
1
2
Displacement ductility, µ∆
3
4
5
6
7
Displacement Ductility, µ∆
(b) Connection DSB-P-1.0
(a) Connection SSB-P-1.0
Figure 4-61: Comparison of the cumulative energy versus displacement ductility relationships of the Control, SSB-P-1.0, and DSB-P-1.0 connections
1400
1400
Control SSB-S-1.0 SSB-P-1.0
1000
Control DSB-S-1.0 DSB-P-1.0
1200
Energy Index, EI (kN.m)
Energy index, EI (kN.m)
1200
800 600 400 200
1000 800 600 400 200
0
0 0
1
2
3
4
5
6
7
0
Displacement ductility, µ∆
(a) Connection SSB-P-1.0
1
2
3
4
5
6
7
Displacement Ductility, µ∆
(b) Connection DSB-P-1.0
Figure 4-62: Comparison of the energy index versus displacement ductility relationships of the Control, SSB-P-1.0, and DSB-P-1.0 connections
The damping properties, which are considered to be very important parameters in the DDBD method, are evaluated by plotting the equivalent viscous damping with respect to the 270
displacement ductility index as shown in Figure 4-63. Based on the values reported in Table 4-13 and the relationships shown in Figure 4-63, the steel-reinforced SSB connection exhibit approximately the same trend as the Control connection, while the steel-reinforced DSB connection experience lower rate of change as compared with the Control connection. It is also observed that the relationships of the PE SMA-reinforced SSB and DSB connections are approximately the same in which the damping drops to a value of 0.05 at the onset of yielding and then gradually increases with increasing displacement ductility index. Overall, reinforcing concrete connections using PE SMA material reduce the equivalent viscous damping significantly as compared with the steel-reinforced counterparts (average reduction of 78% at ultimate) and the Control connection (average reduction of 82% at ultimate). 0.40
0.35
Equivalent viscous damping, ξeq
Equivalent viscous damping, ξeq
0.40
Control SSB-S-1.0 SSB-P-1.0
0.30 0.25 0.20 0.15 0.10 0.05 0.00
0.35
Control DSB-S-1.0 DSB-P-1.0
0.30 0.25 0.20 0.15 0.10 0.05 0.00
0
1
2
3
4
5
6
7
0
Displacement ductility, µ∆
(a) Connection SSB-P-1.0
1
2
3
4
5
6
7
Displacement Ductility, µ∆
(b) Connection DSB-P-1.0
Figure 4-63: Comparison of the equivalent viscous damping versus displacement ductility relationships of the Control, SSB-P-1.0, and DSB-P-1.0 connections
As far as the DDBD of concrete structures is concerned (Figure 2-1), the low ductility and damping properties of the PE SMA-reinforced connections can be overcame by reducing the period of the structure, which can be obtained via increasing its stiffness. The increase in the
271
stiffness can be achieved by using high-modulus SMA bars, increase the dimensions of the concrete beam, and/or increase the amount of the reinforcing bars. The effect of the plastic hinge properties, obtained from the examined connections, on the response of concrete structures represents a rich topic for future research.
4.11.3 Other Damage Assessment Indices In this section, the Residual Displacement Index, RDI, the Work Index, WI, and the Stiffness Index, SI are evaluated. The RDI, WI, and SI were calculated using Equations 3-18, 3-19, and 3-20, respectively. However, the ki values in this section correspond to the average stiffness of the system measured from the origin to the point of interest unlike the definition used in evaluating SI in the steel-reinforced SSB and DSB connections included in Section 3.9.3. This is due to the fact that the PE SMA-reinforced SSB and DSB connections and the Control connection experience degradation in stiffness unlike the behaviour of the steel-reinforced SSB and DSB connections. The damage assessment results are included in Table 4-14 and compared with respect to the values of the Control connection. It is noted that the values were calculated based on the average responses under positive and negative bending.
Table 4-14: Other damage assessment indices of the Control, SSB-P-1.0, and DSB-P-1.0 connections Diff.* Diff.* WI (%) (%) Control 3.06 142 SSB-P-1.0 0.15 -95 79 -44 DSB-P-1.0 0.20 -93 114 -20 * The difference is calculated with respect to the Control connection Connection
RDI
272
SI 0.229 0.634 0.388
Diff.* (%) 177 69
The relationships between the RDI and the displacement ductility of the connections are shown in Figure 4-64. From Table 4-14 and Figure 4-64, it is observed that the RDI values in the steel-reinforced SSB and DSB connections are approximately zero prior to yielding after which they increase in a linear manner, at the same slope. However, the RDI index is maintained at very low levels in the PE SMA-reinforced SSB and DSB connections. At the ultimate condition, the average decrease in the RDI index is 92.5% and 94.3% as compared with the steel-reinforced SSB and DSB connections and the Control connection, respectively. It is, therefore, concluded that the PE SMA-reinforced SSB and DSB connection possess an effective self-centering ability as the residual displacement is insignificant during all stages of testing. 4
Residual displacement index, RDI
Residual displacement index, RDI
4
Control SSB-S-1.0 SSB-P-1.0
3
2
1
0
Control DSB-S-1.0 DSB-P-1.0
3
2
1
0 0
1
2
3
4
5
6
7
0
Displacement ductility, µ∆
(a) Connection SSB-P-1.0
1
2
3
4
5
6
7
Displacement ductility, µ∆
(b) Connection DSB-P-1.0
Figure 4-64: Comparison of the residual displacement index versus displacement ductility relationships of the Control, SSB-P-1.0, and DSB-P-1.0 connections
The variation of WI with respect to the increase in the displacement ductility index is plotted in Figure 4-65. From the assessment results included in Table 4-14 and Figure 4-65, it is observed that the WI increase exponentially with increasing displacement ductility index. The relationships of the steel-reinforced SSB and DSB connections are approximately the same as the
273
Control connection. The rate of increase in the WI index in the PE SMA-reinforced SSB and DSB connections is, however, higher than that in the steel-reinforced counterparts and the control connection. It is also observed that the decrease in the WI index for the PE SMA-reinforced DSB connection is less than that in the SSB counterpart. It is, therefore, concluded that the PE SMAreinforced DSB connection experience superior performance as compared with the PE SMAreinforced SSB connection in terms of the WI index. 250
250
Control SSB-S-1.0 SSB-P-1.0
Control DSB-S-1.0 DSB-P-1.0
200
Work Index, WI
Work index, WI
200
150
100
50
150
100
50
0
0 0
1
2
3
4
5
6
7
0
Displacement ductility, µ∆
(a) Connection SSB-P-1.0
1
2
3
4
5
6
7
Displacement Ductility, µ∆
(b) Connection DSB-P-1.0
Figure 4-65: Comparison of the work index versus displacement ductility relationships of the Control, SSB-P-1.0, and DSB-P-1.0 connections
The comparisons of the SI versus the displacement ductility relationships are plotted in Figure 4-66. The behaviour of the steel-reinforced counterparts are not included due to the difference in the definition of stiffness degradation as discussed previously. From Table 3-14 and Figure 4-66, it is observed that the stiffness of all connections degrade gradually with increasing displacement ductility index at approximately the same rate. However, the stiffness degradation at ultimate in the Control connection is higher than that experienced in the PE SMA-reinforced SSB and DSB connections.
274
2.5
Control SSB-P-1.0 DSB-P-1.0
Stiffness Index, SI
2
1.5
1
0.5
0 0
1
2
3
4
5
6
7
Displacement Ductility, µ∆
Figure 4-66: Comparison of the stiffness index versus displacement ductility relationships of the Control, SSB-P-1.0, and DSB-P-1.0 connections
4.12 Summary The design and behaviour of PE SMA-reinforced SSB and DSB connections was examined in this chapter. Anchoring the PE SMA bars to the concrete at the joint and splicing them to the steel reinforcement was examined experimentally and analytically. Furthermore, the mechanical steel anchors and couplers were modified in order to minimize the slip and increase the stress and strain capacities of the PE SMA bars at failure. The performance of reduced-scale joint-like specimens was examined in order to determine the optimum position of the steel anchor, within the joint, in terms of construction practicality and low curvature demand. Based on the results of Chapter 3 and the behaviour of the anchorage and coupling systems, the final design of the PE SMAreinforced SSB and DSB systems was conducted and large-scale specimens were tested. Test results based on the hysteretic response, cracking pattern, and beam rotation indicated the superior performance of the proposed systems in achieving a self-centering behaviour, and hence, minimize the amount of permanent damage. Based on the evaluation of the components of deformation, it is concluded that the PE SMA-reinforced SSB and DSB systems experienced superior performance
275
in relocating the centre of rotation away from the face of the columns as compared with the steelreinforced counterparts. Even though one of the objectives of this research is to achieve high energy dissipation of the proposed systems, the damage assessment analysis indicated that the use of the PE SMA bars led to significant reductions in the damping properties.
276
Analytical Prediction Model 5.1 Introduction Analytical prediction models of load-deflection relationships of the steel-reinforced and the PE SMA-reinforced SSB and DSB connections are developed and validated against the experimental test results in this chapter. The models predict only the pushover response of the connections and are not concerned with the cyclic response. However, the formulation of the developed model can be extended to account for the cyclic response in future research. The formulation of a newly developed shear deformation model is also developed and presented in this research in order to better understand the sliding shear mechanism in reinforced concrete structures in general and in the SSB and DSB connections in particular. The models were also used to conduct a parametric study of the PE SMA-reinforced connections.
5.2 Development of the Load-Deflection Prediction Model Several deformation mechanisms contribute to the beam-tip displacement, namely; beam bending, stub bending, column bending, joint shear, rigid-beam rotation, and sliding shear. Based on the analysis results of the steel-reinforced and the PE SMA-reinforced SSB and DSB connections (refer to Chapters 3 and 4), it was found that the beam shear deformation in the neighbourhood of the vertical slot is very small compared to the other deformation mechanisms, and thus, it is ignored in the development of the load-deflection prediction model. However, the sliding shear at the concrete hinge is believed to contribute to the beam-tip deformation, and hence, it was included in the prediction model. The applied load is governed by the moment of the beam at the vertical slot (Equation 5-1) while the total displacement is the summation of the displacements due to the various deformation mechanisms (Equation 5-2). The contributions of 277
the five deformation components are described in detail in the following sections. The calculation of V is included in Section 5.2.1.
V=
M ls
Equation 5-1
ΔT = Δb2 +Δstub +Δco + Δ js +Δrig +Δss
Equation 5-2
where V, M, ls, ΔT, Δb2, Δstub, Δco, Δjs, Δrig, and Δss, are the applied load, moment at the vertical slot, shear span, total displacement, displacement due to beam bending in the shear span, displacement due to beam bending in the stub region, displacement due to column bending, displacement due to joint shear, displacement due to rigid body deformation, and displacement due to sliding shear, respectively.
5.2.1 Beam Bending
Due to the presence of a debonded region around the steel reinforcement passing through the vertical slots in the SSB and DSB connections and due to the significantly low bond strength of the PE SMA bars, the common assumption of plane sections before deformation remaining plane after deformation is not valid. Therefore, the moment-rotation formulation was adopted instead of the conventional moment-curvature formulation. The procedure implemented in predicting the beam bending contribution is based on previous work by Au (2010). However, it is modified in the context of the present research as needed. Rotation at the critical section (at the location of the vertical slot), θb,vs, is first imposed, followed by iterating the depth of the neutral axis until equilibrium of forces is achieved. Thus, Δb2 is calculated as follows:
278
Δb2 = θb,vs ⋅ls
Equation 5-3
In order to calculate V, the equilibrium of forces at the vertical slot needs to be achieved. The strains in the bottom reinforcement, diagonal reinforcement, top reinforcement, and concrete hinge are calculated based on the axial deformations induced at the critical section while a fibre model is used to predict the tension-stiffening in the concrete hinge. The moment capacity of the + , the moment capacity of the SSB system under negative SSB system under positive bending, M SSB − , and the moment capacity of the DSB system, M DSB , are calculated using Equation bending, M SSB
5-4. n
F (
d −
d )
+
F (
d −
F )
+
Fts , n ( d sb − d ts,n ) +
st sd sb sd st sb n=1 F c (d sb − (1 − β ) ⋅ c) M = n F ( d − d ) + F ( d − F ) + F ( d − d ) +
sd sd ts , n st sb sb ts,n st sb n=1 F c (d sb − hch + (1 − β ) ⋅ (hch − c))
+ M SSB
Equation 5-4
− M SSB or M DSB
where Fc, Fst, and Fsd are the forces in the concrete, top reinforcement, and diagonal reinforcement, respectively, dsb, dst, and dsd the distances from the top concrete fibre to the centroid of the bottom, top, and diagonal steel reinforcements, respectively, Fts,n and dts,n the tension-stiffening force in the concrete fibre nc and the distance from the top fibre to the centroid of the concrete fibre nc, respectively, β the centroid of the compression force in the concrete hinge measured from the centroid of the section, hch the depth of the concrete hinge, and c is the depth of the neutral axis.
279
Based on the induced rotation at the vertical slot section, θb,vs the axial deformation in the bottom reinforcement, Δsb, diagonal reinforcement, Δsd, top reinforcement, Δst, and the concrete hinge, Δch, are calculated as follows:
Δsb = θb,vs ⋅(dsb −c)
Equation 5-5
Δsd = θb,vs ⋅(dsd −c)
Equation 5-6
Δst = θb,vs ⋅(dst −c)
Equation 5-7
θ b,vs ⋅ c Δ ch = θ b,vs ⋅ (hch − c )
+ M SSB
Equation 5-8
− M SSB or M DSB
The strain in the bottom reinforcement, εsb, is calculated by dividing Δch by the length of the unbonded region, lub , as follows;
ε sb =
Δ sb lub
Equation 5-9
The strains in the diagonal reinforcement, εsd, and top reinforcement, εst, are calculated by dividing the axial deformations by 4/3 lpe as recommended by Au (2010), where lpe is the strain penetration length:
ε sd =
Δ sd 4 3 l pe
Equation 5-10
ε st =
Δ st 4 3 l pe
Equation 5-11
280
The lpe is calculated using the expression proposed by Paulay and Priestley (1992) as a function of the yield stress, fy (MPa), and the bar diameter, db (mm). (mm)
l pe = 0.022 f y d b
Equation 5-12
It is also noted that the 4/3 factor in Equations 5-10 and 5-11 was determined from experimental test results of SSB connections with vertical slots located at the face of the column (Au, 2010). However, it is assumed that this factor is also applicable for the SSB and DSB connections with relocated vertical slots. The strain in the concrete, εc, is calculated by a simplified expression proposed by Au (2010) as expressed in Equation 5-13. The factor α is a factor to modify the strain penetration length, which ranges between 2/3 and 4/3 for the top reinforcement. An average value of 1.0 is used for α.
εc =
Δc α ⋅ l pe
Equation 5-13
5.2.1.1 Constitutive Model for Concrete Because vertical stirrups are not installed at the critical section (concrete hinge) due to practicality reasons, an unconfined concrete model is used to model the concrete at the critical section. It is also necessary to use a continuous function of the concrete stress-strain relationship in order to facilitate analytical integration of the stress-strain curve which is involved in determining the β factor. Thus, the model by Todeschini et al. (1964) is used to model the concrete compressive behaviour at the critical section as shown in Figure 5-1 and described in the following equations: fc =
(
2 f c' ⋅ ε c ε c'
(
1+ εc ε
)
' 2 c
)
Equation 5-14
281
c' 1.71
f c' Ec
Equation 5-15
where fc is the concrete stress, f 'c the concrete compressive strength in MPa, ε'c the concrete strain at f 'c, and Ec is the modulus of elasticity of concrete in MPa.
Concrete stress, fc
f'c
f't
ε'c Concrete strain, εc
Figure 5-1: Concrete stress-strain relationships used in the analytical model
The Ec is calculated as follows (CSA-A23.3-04, 2004): E c 4500 f c'
(MPa)
Equation 5-16
The tensile concrete stress-strain relationship is assumed to follow a bilinear relationship as shown in Figure 5-1. The concrete tensile strength, f 't, is calculated as a function of f’c (MPa) as follows (CEB-FIP, 1993):
f ' 8 f t 1.4 c 10 '
23
(MPa)
Equation 5-17 282
The concrete strain at f 't, ε’t, is taken as 0.002 as recommended by Gedling et al. (1986).
5.2.1.2 Constitutive Model for Steel The nonlinear response of strain hardening of the steel reinforcing bars is modeled using the expressions proposed by Menegotto and Pinto (1977) that relates the normalized average strain,
ε*, to the average stress in the reinforcement, fs, as shown in Figure 5-2 and expressed in the following equations:
ε∗ f s = f y ⋅ (1 − b ) ⋅
1 + ε ∗RB
(
ε∗ =
Equation 5-18
ε εy
RB = Ro −
ξ=
)
1 RB
+ b ⋅ ε ∗
Equation 5-19
a1 ⋅ ξ a2 + ξ
Equation 5-20
ε max − ε min εy
Equation 5-21
where b is a parameter that defines the strain-hardening slope, RB a factor that rules the steepness of the Bauschinger effect in the stress-strain curve, ξ the mean strain normalized to the yield strain updated following a strain reversal, εmax the maximum strain in the cycle, εmin the minimum strain in the cycle, εy the yield strain, Ro the value of the parameter RB during first loading, and ɑ1 and ɑ2 are experimentally determined parameters.
283
Steel stress, fs
Normalized strain, ε*
Figure 5-2: Steel stress-strain relationships used in the analytical model
The reinforcement that passes through the vertical slots (bottom reinforcement in the SSB connection and the top and bottom reinforcements in the DSB connections) are subjected to high amplitude cyclic loading in tension and compression, and thus, it is believed that isotropic hardening plays a major role in increasing the stress levels attained in the reinforcement. Therefore, the increase in the tension stress due to isotropic hardening, fs,ih, is calculated using the expression proposed by Filippou et al. (1983) which is based on shifting the yield asymptote:
f s,ih f y a3 max a3 a4 y
Equation 5-22
where ɑ3 and ɑ4 are the experimentally determined parameters (ɑ3 and ɑ4 are 0.01 and 7, respectively, as recommended by Filippou et al. (1983)).
284
The stress calculated using Equation 5-22 is added to the stress values obtained from Equation 5-18 only for the reinforcement that passes through the vertical slot. This is due to the fact that the diagonal reinforcement and the top reinforcement (in the case of the SSB connection only) experienced low levels of strains in the experimental testing of the steel-reinforced and PE SMA-reinforced SSB and DSB connections.
5.2.2 Stub Bending
The contribution of the stub region in the total displacement is calculated as follows:
Δ stub = θ b1 ⋅ lre
Equation 5-23
where θb1 and lre are the rotation of the stub region and the relocation distance, respectively.
Due to the complex nature of the flow of forces within the beam stub, and hence, the complexity encountered in computing the rotation demand, the rotation due to the bending of the stub region is determined empirically as a function of the rotation due to beam bending at the critical section (i.e. the vertical slot). The rotation ratio Rθ, calculated as θb1/θb2 (where θb1 and θb2 are the flexure rotations of the stub and the beam members, respectively) is used to evaluate the rotation of the stub. The experimentally obtained relationships of the Rθ versus θb1 are shown in Figure 5-3. It is noted that lower values of Rθ indicate lower amount of rotation at the stub. It can be seen that the SSB system responded differently under positive and negative bending. The Rθ ratio under positive bending increased significantly as the vertical slot was moved away from the face of the column (refer to Section 3.7.3 for detailed discussion). However, the responses under positive and negative bending of the DSB system were approximately the same as expected due to
285
the symmetric nature of the system. It is also noted that the Rθ ratio of the PE SMA-reinforced connections was significantly lower than that of steel-reinforced counterparts. In order to be able to incorporate the stub bending into the analytical model developed in this research, the relationships shown in Figure 5-3 need to be idealized. For all connections except DSB-P-1.0, the relationships can be idealized as quadri-linear relationships as shown in Figure 5-4. Connection DSB-P-1.0 experienced approximately zero Rθ for all levels of rotation, and thus, it is assumed that the stub remains rigid during all stages of loading. The parameters of the proposed model shown in Figure 5-4 are obtained by best fitting the curves using the experimental values. The values of the constants are included in Table 5-1. It is also noted that the values included in Table 5-1 can be used for interpolation and extrapolation in cases in which different relocating distances are used in the model.
SSB-S-1.0 SSB-S-1.7 DSB-S-1.0 DSB-S-1.7 SSB-P-1.0 DSB-P-1.0
2 Solid line - Positive bending Dashed line - Negative bending
Ratio, Rθ
1.5
1
0.5
0 -0.1
-0.06
-0.02
0.02
0.06
0.1
Rotation, θb1 (rad)
Figure 5-3: Rotation ratio (Rθ) versus the stub rotation (θb1) experimentally obtained relationships
286
Ratio, Rθ
θr,u
Eθ
θr,f
θb1,y
θb1,u
θb1,f Rotation, θb1 (rad)
Figure 5-4: Parameter definition of the rotation ratio (Rθ) versus the stub rotation (θb1)
Table 5-1: Parameters of the proposed rotation ratio (Rθ) versus the stub rotation (θb1) relationships Type Steel-reinforced SSB Steel-reinforced DSB SMA reinforced SSB Bending Positive Negative Positive Negative lre* 1.0 1.7 1.0 1.7 1.0 1.0 ** ε y (S w + l st ) / l o θb1,y θb1,u 2.0 2.0 2.0 2.5 2.5 1.0 1.0 θb1,f 4.0 3.0 4.0 5.0 5.0 3.0 6.0 θr,u 0.700 1.700 0.525 0.235 0.465 0.120 0.380 /1.50 /1.50 /2.25 /2.00 /2.00 /4.00 θ r ,u θ r ,u θ r ,u θ r ,u θ r ,u θ r ,u θ r ,u /4.00 θr,f 0 189.7 0 3.9 -1.5 0 2.1 Eθ * where lre is the relocation distance ** where Sw, lst, and lo are the width of the slot, length of the steel tube, and distance between the neutral axis and the bottom reinforcement, respectively. 5.2.3 Column Bending
The Δco and the rotation due to the column bending, θco, derived via the moment-area theorem, are calculated as follows: 287
Δ co = θ co ⋅ (l s + l re )
θ co =
Equation 5-24
(lco − d v )2Vco
Equation 5-25
8Ec I e
where lco is the column span, dv the effective shear depth, Vco the column shear force, and Ie is the effective moment of inertia.
The Vco is calculated as a function of the tension force in the bottom steel, Fsb, as follows based on equilibrium of forces in exterior beam-column connection:
Vco =
Fsb ⋅ d v + 0.5Vb ⋅ hco lco
Equation 5-26
where hco is the depth of the concrete column.
The effective moment of inertia, Ie, is calculated using the following expression by Branson and Metz (1965): M
I e = cr M co
M cr =
M co =
3 M
⋅ I g + 1− cr M co
3
⋅ I c ≤ I g
Equation 5-27
ft' ⋅ I g
Equation 5-28
y t
Mb 2
Equation 5-29
288
where Mcr, Mco, and Mb are the cracking moment, the moment applied onto the column, and the beam moment, respectively, Ig and Ic the gross and cracked moments of inertia, respectively, and
yt is the distance from the extreme top fibre to the centroid of the section.
5.2.4 Joint Shear
The Δjs is calculated as follows:
Δ js = θ js ⋅ (l s + l re )
θ js =
Equation 5-30
Vhj,avg
Equation 5-31
kv
where θjs, Vhj,avg, and kv are the rotation due to joint shear, average horizontal shear force in the joint, and joint shear stiffness, respectively.
The cracking concrete shear stress and the cracked shear stiffness of the joint need to be determined in order to calculate the contribution of the joint shear in the deformation of the connections. The cracking concrete shear stress, vc, is calculated using the following expression proposed by Paulay and Priestley (1992) that accounts for the effect of axial compression force,
Pɑ, in enhancing the shear strength:
vc = 4vb
Pa Ag fc'
v b = 0.2
f c'
(MPa)
Equation 5-32
(MPa)
Equation 5-33
where Ag is the gross-sectional area and vb is a shear stress factor.
289
The kv can be cracked or un-cracked depending on the level of force in the joint. The un cracked, kv,unc and cracked, kv,cr shear stiffness of the joint are calculated using Equation 5-34 and 5-35, respectively. The kv,cr is assumed to be elastic and it is formulated based on the conventional truss analogy. This assumption is valid due to the low levels of shear distortion experienced in the SSB and DSB connections described in Sections 3.7.6.2, 3.8.6.2, and 4.10.6.2.
kv,unc =
0.4Ecbwd f
Equation 5-34
k v ,cr =
ρ v sin 4 α θ sin 4 βθ ⋅ (cot α θ + cot βθ )2 E s bw d sin 4 α θ + nρ v sin 4 βθ
Equation 5-35
where ρv is the volumetric ratio of the transverse reinforcement steel (Av/(sspbwsinβθ)), Av the amount of the transverse reinforcement, ssp the stirrup spacing, αθ the crack angle, βθ the stirrup angle, n the modulus ratio (Es/Ec), bw the width of the concrete cross-section, d the effective depth, and f is the factor that allows for the non-uniform distribution of the shear stresses (1.2 for rectangular section, 1.0 for T and I-sections).
As concluded from the experimental results in Chapter 3, moving the vertical slots away from the face of the column alters the flow of forces within the joint, and thus, the difference in the response is taken into account in this model. The critical horizontal joint sections of the SSB and DSB systems under positive and negative bending are shown in Figure 5-5. The average horizontal shear force in the joint, Vjh,avg, is calculated in accordance to Equation 5-36 while the horizontal shear weights, wjh,1, are calculated using the dimensions of the concrete beam sections:
290
n
V jh,avg =
V
jh,n
Equation 5-36
n=1
w jh,1V jh,1 + ... + w jh,nV jh,n
Figure 5-5: The critical horizontal shear forces in the joints of the SSB and DSB systems
5.2.5 Rigid-body Rotation
Rigid-body deformation at the critical section contributes to the total rotation of the reinforced concrete members (Park and Ang, 1985; Alsiwat and Saatcioglu, 1992). The Δrig is calculated as a function of the rigid-body rotation, θrig, as follows:
Δrig = θrig ⋅ls
Equation 5-37
The rigid-body deformation in the steel-reinforced connections is composed of the bar extension and hook slippage while it is composed of the anchor slippage in the PE SMA-reinforced
291
connections. The analytical modelling of the rigid body rotation for the steel-reinforced and the PE SMA-reinforced connections is described as follows:
5.2.5.1 Steel-Reinforced Connections The deformation of the hooked reinforcement when subjected to cyclic loading is insignificant as compared with the contribution of the bar extension to the rigid body rotation, and hence, the hook deformation is ignored in this model (Alsiwat and Saatcioglu, 1992). Therefore, the θrig is calculated as a function of only the bar extension, δext, as follows:
θrig =
δ ex
Equation 5-38
(ds −c)
where ds and c are the distance from the concrete fibre to the centroid of the reinforcement and the depth of the neutral axis, respectively.
The accumulation of strains along the length of the steel reinforcement results in the extension of the reinforcement. The model by Alsiwat and Saatcioglu (1992) was used to calculate the extension of the bars. The strain distribution in the bar needs to be constructed in order to calculate the bar extension by integrating the strains (Figure 5-6). Upon the commencement of strain hardening at high drift, the strain profile is subdivided into four regions; elastic, yield plateau region, strain-hardening, and pull-out cone region. The bar extension, δext, is calculated as follows:
δext = 0.5ε y ⋅ Le +0.5(ε sh + ε y )⋅ Lyp +0.5(ε s + ε sh ) ⋅ Lsh + ε s ⋅ Lpc
292
Equation 5-39
where Le, Lyp, Lsh, and Lpc are the lengths of the elastic, yield plateau, strain hardening, and pullout cone regions, respectively, and εsh is the steel strain corresponding to the end of the yielding plateau.
Figure 5-6: Bar extension model by Alsiwat and Saatcioglu (1992); (a) reinforcing bar embedded in concrete; (b) stress distribution; (c) strain distribution; (d) bond stress between concrete and steel
The Le is calculated based on the equilibrium shown in Figure 5-6 while the average elastic bond stress, ue, and the development length, ld, are calculated as proposed in ACI 408.1R-90 (1990): 293
fsdb 4u e
Le =
f y db
ue =
ld =
Equation 5-40
Equation 5-41
4l d
440Ab f y ≥ 300 (mm) K f c' 400
Equation 5-42
where Ab is the area of the reinforcement in mm2, while fy and f’c in Equation 5-42 are in MPa.
The K coefficient in Equation 5-42 takes into account the effect of confinement steel, concrete cover, and bar spacing, and it is equal to 3db. The length of the yielding plateau region forms in a small segment of the bar due to the localized nature of steel yielding. Crushing of the concrete keys between the deformations of the steel reinforcement is anticipated at this stage, and thus, the Lyp can be accurately calculated based on the frictional bond stress, uf, expressed by Pochanart and Harmon (1989):
L yp =
Δf s d b 4u f
Equation 5-43
f c' SL ⋅ u f = 5.5 − 0.07 (MPa) H L 27.6
Equation 5-44
where SL and HL are the clear spacing and height of lugs on the bar, respectively, and f’c in Equation 5-44 is in MPa.
The Lsh is also calculated using Equation 5-43 due to the fact that the bond transfer mechanism (friction) is the same for the yielding plateau region and the strain-hardening region. 294
However, the Δfs in Equation 5-43 in this case is taken as the difference in the steel stress between the current load stage and the beginning of the strain-hardening stage. When subjected to high strain levels, a pullout cone forms in the concrete cover region. It refers to a constant stress and strain region which develops when the concrete cover at the loaded end of the embedded bar breaks loose in tension. Thus, the Lpc equals to the concrete cover length.
5.2.5.2 PE SMA-Reinforced Connections The bond between the PE SMA and the concrete is negligible, and thus, the SMA bar experienced constant strain and stress along the gauge length measured from the steel anchor embedded in the joint to the steel coupler connected with the diagonal reinforcement. Consequently, the extension of the SMA bar is taken into account in the moment-rotation formulation discussed in section 5.2.1 while only the slippage of the SMA bar in the steel anchor and coupler, δan, contributes to the rigidbody rotation. The θrig is, therefore, calculated as follows:
θ rig =
δ an
Equation 5-45
(d s − c )
The stress-slip relationships of the modified anchor and coupler shown in Figure 4-33 (Phase 4) and Figure 4-37, respectively, are used to develop and stress-slip prediction relationships. The proposed stress-slip prediction relationships of the steel anchors and coupler are compared with the experimentally obtained results as shown in Figure 5-7. The constitute model of the stress-slip relationship is expressed as follows:
ESMA,s s as ln s + bs
σ =
0 ≤ s ≤ sy
Equation 5-46
s > sy
295
where ESMA,s is the elastic modulus of the stress-slip relationship of SMA, s and sy are the slip and the slip at yielding of the SMA bar, respectively, ɑs the slope of the stress versus slip relationships, and bs is the intercept of the stress versus slip relationship with the y-axis. The ESMA,s is 1993 MPa/mm and 6163 MPa/mm for the steel anchor and coupler, respectively. The ɑs and bs are 60 MPa/mm and 481 MPa, respectively, for the steel anchor while they are 49 MPa/mm and 474 MPa, respectively, for the steel coupler. The sy is 0.18 mm and 0.07 mm for the steel anchor and coupler, respectively. Equation 5-46 is, therefore, used to solve for s in order to calculate the cumulative slip of the anchors, δan, which contributes to the rigid-body
700
700
600
600
500
500
Stress, σ (MPa)
Stress, σ (MPa)
rotation.
400
Specimen A Specimen B Model
300 200 100
400
Specimen A Specimen B Model
300 200 100
0
0
0
1
2
3
4
5
0
1
Slip, s (mm)
(a) Steel anchor
2
3
4
5
Slip, s (mm)
(b) Steel coupler
Figure 5-7: Comparison of experimental and analytical stress-slip relationships
5.2.6 Sliding Shear
The shear transfer mechanism at the concrete hinge in the SSB and DSB systems is of a complex nature. In fact, the sliding shear mechanism is anticipated to take place in this region due to the relatively larger forces that need to be transferred from the concrete beam to the concrete stub.
296
There are several shear strength and/or deformation models developed to predict the behaviour of RC members subjected to shear forces. However, none of the existing models is capable of predicting the shear deformation accurately at the concrete hinge as will be explained later in detail. The need for accurate modeling of the degradation in the shear capacity at the concrete hinge was the primary motivation for the development of a new shear-deformation theory in this research called the Two Distinct Element (TDE) theory. The newly developed model is formulated and validated in this research, then used to predict the shear deformation at the concrete hinge in the SSB and DSB connections. In the following section, a critical review of the existing shear deformation theories is conducted followed by introducing the proposed TDE theory highlighting its main features. After that, the formulation of the model including the equilibrium, compatibility, and constitutive models are discussed. Then, the model is validated by comparing the predicted shear response obtained using the TDE model with experimental results of RC shear walls, subjected to combined axial and lateral loading, found in the literature. The validation of the model is, however, included in Appendix D. Finally, the TDE model is used to predict the contribution of the sliding shear at the concrete hinge in the SSB and DSB connections to the total applied displacement at the tip of the concrete beam.
5.2.6.1 Critical Review The truss analogy developed by Ritter (1899) and Mӧrsch (1908) was the first approach used to estimate the amount of transverse steel reinforcement in concrete structures. It takes into account the flexural-shear interaction in a simple form. The model consists of diagonal compression strut, transverse reinforcement (stirrups), and tension and compression chords. Most of the research was 297
conducted in the so-called “B region” in which the stress and strain distributions are uniform (Kupfer, 1962; Dilger, 1966). There are two types of truss models; continuum and discrete. The former assumes that the transverse steel is smeared throughout the entire member while the latter treats the transverse steel as discrete elements with finite spacing. The shear stiffness obtained from the discrete model converges to the shear stiffness obtained from the more common continuum model (Kim and Mander, 2007). Both the discrete and the continuum models can have constant crack angle (B-region) or variable crack angle (D-region). Following are the limitations on the shear stiffness calculated using the different truss models described above:
• The shear stiffness of the cracked concrete member is elastic. • The transverse steel is assumed to be smeared throughout the entire concrete member in the continuum model which may lead to overestimating the shear capacity of the section when the stirrup spacing is relatively large.
• The models do not consider the effect of increasing the crack width, as the load is applied, on the shear resisting mechanisms, particularly the reduction in the aggregate interlock resisting mechanism.
• The models are incapable of predicting the shear transformation mechanism in the location of plastic hinges subjected to load reversal in which the sliding shear mechanism is a more realistic representation than the truss analogy.
Rotating and fixed crack models were developed in order to better capture the shear deformation in reinforced concrete elements. As the shear load increases, the cracks have the ability to change direction inside the concrete membrane element in the rotating crack models while the cracks have the same direction in the fixed crack models regardless of the applied load. 298
The well-known Modified Compression Field Theory (MCFT), developed by Vecchio and Collins (1986), is a form of rotating crack models. The model assumes that any RC member can be represented as a concrete membrane element with smeared longitudinal and transverse steel reinforcement. It also assumes that the average principal stresses and the average principal strains have the same direction. The MCFT has been used widely by design codes and finite element analysis (Vecchio, 1989). However, experimental test results indicated the inaccuracy of the MCFT in predicting the shear deformation in concrete panels having low amounts of transverse reinforcement since the assumption of having the same direction for both the principal average stresses and strains is not accurate in this situation. Therefore, the Disturbed Stress Field Model (DSFM) (Vecchio 2000; Vecchio 2001) was developed to overcome the shortcomings of the MCFT. In the DSFM, the direction of the change in the average principal stress is allowed to lag behind the change in the principal strain direction as observed in the experiments. The DSFM introduces a discrete slip on the crack surface that accounts for the disturbance in the stress fields caused by the initiation of the cracks. Despite the advancement in modeling the shear response of RC members achieved by the MCFT and the DSFM, both models have limitations that hinder their use in the seismic design:
• The models are not capable of accurately predicting the sliding shear failure that can be experienced in RC structures subjected to load reversal.
• The assumption that the steel reinforcement is smeared throughout the concrete leads to inaccurate prediction of the shear deformation, especially between the stirrups.
• The models are not capable of predicting the shear transformation in regions where forces in the concrete web are not transformed based on the diagonal strut (compression
299
field) manner such as in plastic hinges in which the concrete web degrades significantly.
• The models do not provide a physical mathematical representation of the shear resisting mechanisms.
5.2.6.2 Proposed TDE model The structure of the shear deformation theory proposed in this research is described in this section. The theory of the TDE model is compared with the MCFT in Figure 5-8 for a RC columnfoundation sub-assemblage. The MCFT consists of one type of smeared element that contains longitudinal and transverse reinforcement while the proposed model contains two types of elements; one element contains only longitudinal reinforcement and is called the concrete element (C-element) and the other element contains both longitudinal and transverse reinforcements and is called the stirrup element (S-element). The C-element is located in the regions between the stirrups while the S-element is located at the sections that contain the stirrups. The proposed model is, therefore, called the Two Distinct Elements (TDE) shear deformation theory. The region defined by lc-e in Figure 5-8 contains the C-elements while the region defined by ls-e contains the Selements. The longitudinal reinforcements are smeared throughout the lc-e region while the longitudinal and transverse reinforcement are smeared throughout the ls-e region. One key issue with the development and the formulation of this model is the compatibility and the continuity at the surface between the two elements. The shear resisting mechanisms in the two elements are different, and thus, achieving continuity between the two elements is essential. Another important observation is that the TDE predicts a non-uniform shear deformation distribution along the length of the RC members unlike the results obtained from the MCFT. It is, therefore, believed that the 300
TDE theory idealizes the transformation of the shear forces better than the MCFT in the context of seismic design.
Figure 5-8: Schematic drawing showing the difference between the MCFT and TDE models
The sources of the shear resistance in the C-element are (1) the bond between the longitudinal steel and concrete, (2) the residual tensile strength of the concrete, (3) the dowel action of the longitudinal steel, and (4) the aggregate interlock. The sources of the shear resistance in the Selement are (1) the stirrups, (2) the bond between the longitudinal and the transverse steel and concrete, (3) the residual tensile strength of the concrete, (4) the dowel action of the longitudinal steel, and (5) the aggregate interlock. The model also distinguishes between the tension softening and the tension stiffening in order to better model the contribution of the bond between the steel reinforcement and the concrete. The tension softening is defined as the post-peak stress-strain relationship in concrete while the tension stiffening refers to the post-peak stress-strain relationship
301
in RC. Therefore, the tension stiffening is defined as the combined effect of tension softening and bond-slip behaviour encountered at the concrete steel interface (Nayal and Rasheed, 2006). Furthermore, prior research concluded that modeling using tension softening plus bond-slip provides better agreement with the experimental data of membrane RC elements subjected to shear loading (Gedling et al., 1986, Martin-Perez and Pantazopoulou, 2001). The following necessary assumptions are made in order to simplify the problem and to achieve compatibility within the elements:
• The steel reinforcements (longitudinal and transverse in the S-element and longitudinal in in the C-element) are smeared within the elements themselves.
• The cracks are parallel to the direction of the principal compressive stress. • The steel reinforcement is assumed to be anchored to the concrete, and hence, the deformation in the concrete is the same as the deformation in the steel. However, local equilibrium at the crack interface allows the steel to experience strain levels higher than the average strains experienced in the steel and the concrete in between the cracks.
• The orientation of the principal strains coincide with the orientation of the principal stresses for the cracked concrete.
• The cracks are smeared throughout the cracked concrete.
5.2.6.3 TDE Model Formulation 5.2.6.3.1 Equilibrium The forces applied to the C-element and the S-element are resisted by concrete and/or steel reinforcement as shown on the free-body diagrams in Figure 5-9. For the S-element, the shear forces are resisted by the concrete and the steel reinforcement in both the x and y-directions. The 302
shear forces are resisted by the concrete only in the x-direction while they are resisted by both the steel and the concrete in the y-direction in the C-element.
Figure 5-9: Applied forces (a) parallel to the crack surface in y-direction in the S-element, (b) parallel to the crack surface in x-direction in the S-element, (c) parallel to the crack surface in ydirection in the C-element, and (d) parallel to the crack surface in x-direction in the C-element
In order to satisfy the equilibrium requirement, the forces should sum to zero in both the x and the y-directions. The equilibrium equations for the S-element are:
n dA = f x
A
Ac
cx
dAc +
f
sx
Equation 5-47
dA s
As
303
n
y
dA =
A
f
cy
dAc +
Ac
f
sy
Equation 5-48
dA s
As
where nx and ny are the axial stresses in the x and y-directions, fcx and fcy the average concrete stresses in the x and y-directions, respectively, fsx and fsy the average steel stresses in the x and ydirections, respectively, A the area of the membrane element, Ac the area of the concrete, and As is the area of the steel reinforcement.
Ignoring the reduction in the concrete area due to the presence of the steel reinforcement yields the following equations:
nx = f cx + ρ x f sx
Equation 5-49
n y = f cy + ρ y f sy
Equation 5-50
where ρx and ρy are the volumetric ratio of the steel reinforcement in the x and y-directions.
Similarly, the equilibrium equations of the C-element are as follows:
nx = f cx
Equation 5-51
n y = f cy + ρ y f sy
Equation 5-52
Using Mohr’s circle yields the following useful relationships:
f c1 + f c 2 = f cx + f cy
Equation 5-53
f c1 − f cx = vxy tan θ
Equation 5-54
f c1 − f cy =
vxy tan θ
Equation 5-55
304
where fc1 is the concrete tensile principal stress, fc2 the concrete compressive principal stress (it is noted that these definitions of fc1 and fc2 are commonly used in structural engineering applications (Vecchio and Collins, 1986)), θ the inclination of the principal tensile stress with respect to the xaxis, and vxy is the shear stress in the x-y plane.
Substituting Equation 5-53 into Equation 5-54 yields Equations 5-56 that relates the principal concrete tensile stress and shear stress to the principal concrete compressive stress:
f c2 = f c1 − vxy (tanθ + cotθ )
Equation 5-56
It is noted that the expressions derived so far are presented in terms of the average stresses due to the assumption that the steel reinforcement are smeared throughout each element. However, the localized behaviour at the crack location is taken into consideration in the present study by including the effect of the bond-slip between the steel reinforcement and the surrounding concrete. The stress in the steel reinforcement at the crack location is higher than the average steel stresses in-between the cracks. The difference in the stresses as a result of the non-uniform distribution of stresses along the steel reinforcement is transmitted to the surrounding concrete by bond. The steel stresses at the crack location is the sum of the average steel stress determined from constitutive relationship of the steel and the stresses that result from the bond effect (Martin-Perez and Pantazopoulou, 2001): f sx,cr = f sx +
f sy ,cr = f sy +
sθx f bx d bx sθy d by
Equation 5-57
Equation 5-58
f by
305
where fsx,cr and fsy,cr are the steel stresses at the crack in the x and y-directions, respectively, dbx and dby the diameters of the steel reinforcement in the x and y-directions, respectively, fbx and fby the
bond stresses in the x and y-directions, respectively, and sθx and sθy are the average diagonal crack spacing in the x and y-directions, respectively.
The sθx and sθy are computed using the expressions by CEB-FIP (1978): l d sθx = 2 ⋅ g + x + 0.25k1 bx ρ x 10
Equation 5-59
d l sθy = 2 ⋅ g + y + 0.25k1 by ρ y 10
Equation 5-60
where g is the maximum distance from the reinforcement crossing the crack, lx and ly the maximum distances between the reinforcement in the x and y-directions, respectively (but less than 15 dbx or dby), and k1 is a factor that accounts for the surface texture of the reinforcing bar (equals 0.4 and
0.8 for deformed and plain reinforcements, respectively).
The forces in the S and C-elements at the concrete section and at the cracked section are shown in Figure 5-10. The equilibrium of the forces needs be satisfied in the x and the y-direction for the S-element:
nx + vxy tan θ = ρ x f sx,cr + ρ y f dy tan θ − vagg tan θ
Equation 5-61
ny + vxy cotθ = ρ y f sy,cr + vagg cotθ
Equation 5-62
Similarly, the equilibrium equations for the C-element:
nx + vxy tanθ = ρ x f sx,cr − vagg tanθ
Equation 5-63
306
ny + vxy cotθ = ρ y f dx cotθ + vagg cotθ
Equation 5-64
where vagg is the shear stress due to aggregate interlock.
Figure 5-10: Stresses (a) parallel to the crack surface in the S-element, (b) at the crack surface in the S-element, (c) parallel to the crack surface in the C-element, and (d) at the crack surface in Celement
5.2.6.3.2 Compatibility One of the most critical aspects in modeling the shear behaviour using the TDE model is to establish the compatibility in deformation inside the elements and at the boundary faces between the C-elements and the S-elements. Based on the assumptions outlined in Section 5.2.6.2, the strain
307
in any direction can be calculated using the following geometric relationships obtained from Mohr’s circle if the strain in the x-direction, εx-d, strain in the y-direction, εy-d, and the shear strain, γxy, are known: Equation 5-65
ε x−d + ε y−d = ε1 + ε 2
γ xy =
2⋅ (ε1 − ε x −d ) tan θ
tan 2 θ =
Equation 5-66
ε 1 − ε x−d ε y −d − ε 2 = ε 1 − ε y−d ε x−d − ε 2
Equation 5-67
ε1 =
1 (ε x−d + ε y−d ) + 1 2 2
(ε
− ε y−d ) + γ xy
2
x−d
ε2 =
1 (ε x−d + ε y−d ) − 1 2 2
(ε
− ε y−d ) + γ xy
2
x−d
2
2
Equation 5-68
Equation 5-69
where ε1 is the principal tensile strain and ε2 is the principal compressive strain (it is noted that these definitions are commonly used in structural engineering applications (Vecchio and Collins, 1986)).
5.2.6.3.3 Equilibrium and Compatibility at the Interface The S and C-elements have different shear resisting mechanisms. Therefore, it is necessary to ensure equilibrium and compatibility at the interface between the two types of elements. Unless restricted from rotation, the crack orientation in each element is different than the other. This is due to the fact that the elements are formulated based on the rotating crack model and therefore each element my experience different crack orientation depending on the state of stresses and strains in each element. It is, therefore, determined to constrain the crack angle, θ, in both elements 308
in order to be able to achieve the equilibrium of stresses at the boundary interface between the S and C-elements. In order to allow easier conversion of the analytical model, the θ is restrained by a lower bound, θlb, and an upper bound values, θub.
θ lb ≤ θ ≤ θ ub
Equation 5-70
The θlb and θub are set to 30o (Preistley et al., 1994a; Preistley et al., 1994b) and 45o (based on the conventional truss analogy), respectively. The components of stresses normal to the crack direction are summed up for S-element (Equation 5-71) and the C-element (Equation 5-72) as follows: f c1,s = sin 2 θ ⋅ (ρ x ( f sx,cr ,s − f sx,s ) + ρ y f dy,s ) + cos 2 θ ⋅ (ρ y ( f sy,cr,s − f sy,s ))
Equation 5-71
f c1,c = sin 2 θ ⋅ (ρ x ( f sx,cr ,c − f sx,c )) + cos 2 θ ⋅ (ρ x f dx,c )
Equation 5-72
where the s subscript refers to the stirrup element and the c subscript refers to the concrete element. It is seen that fc1,s in the S-element depends on the bond of the longitudinal and the transverse reinforcement and the dowel action of the transverse reinforcement, while fc1,c depends on the bond and the dowel action of the longitudinal reinforcement in the C-element. The equilibrium of the stresses at a section parallel to the crack surface requires that the stresses normal to the crack surface to be the same in the S and C-elements at the boundary interface. Substituting Equations 5-57 and 5-58 into Equations 5-71 and 5-72 and equating Equations 5-71 and 5-72 yields the following expression: c c sin 2 θ ⋅ ρ x θx f bx,s + ρ y f dy,s + cos 2 θ ⋅ ρ y θy f by,s − f c1,s =
d
by d bx
c sin 2 θ ⋅ ρ x θx f bx,c + cos 2 θ ⋅ (ρ x f dx ,c ) − f c1,c d bx
309
Equation 5-73
The bond stress in the longitudinal direction is considered to be the same in both elements due to the continuity of the longitudinal reinforcement. Thus, Equation 5-73 is further simplified as follows:
cθy sin 2 θ ⋅ (ρ y f dy ,s ) + cos2 θ ⋅ ρ y f by,s − f c1,s = cos2 θ ⋅ (ρ x f dx,c ) − f c1,c d
by
Equation 5-74
The contribution of the dowel action of the stirrup reinforcement in RC member is small compared to that of the flexural reinforcement due to the following reasons: (1) the dowel action of the flexural reinforcement is enhanced by the confinement effect of the stirrups tied to it (Park and Paulay, 1975), (2) the crack width in RC beams subjected to bending is larger at the level of the flexural reinforcement as compared with the stirrup reinforcement, (3) the dowel action is a function of the reinforcement diameter, and thus, the dowel action of the stirrup is lower than that of the flexural reinforcement due to the fact that the diameter of the stirrup is usually small (11 mm), (4) the relative slippage of the crack surfaces is small at the stirrup location, and (5) in case of the RC member subjected to axial tension, the dowel action of the stirrup is not activated. Accordingly, the contribution of the dowel action of the stirrup in the S-element can be ignored and Equation 5-74 can be simplified as follows:
ρy
cθy d by
f by ,s − f c1,s = ρ x f dx,c − f c1,c
Equation 5-75
It is seen that for the post-cracking in tension, the demand on the dowel action in the longitudinal steel minus the concrete principal tensile stress at the C-element equals the demand on the bond in the stirrup minus the concrete principal tensile stress at the S-element given that the reinforcement ratios in the longitudinal and the transverse directions are the same.
310
5.2.6.3.4 Constitutive Models The constitutive relationships of the material properties and the shear resisting mechanisms illustrated in terms of the average stress and average strain are shown in Figure 5-11. The detailed description of the constitutive models are discussed in the following sections:
f't
f'c
Increasing ε1
fp
Ec
Et GF
εp εo
εcr
(a) Concrete under compression
εu
(b) Concrete under tension
fu fb
Esh
fy
Es
εy
εu
εsh
εy
(c) Steel under tension
(d) Concrete-steel bond
vagg
fd
ε
ε
(e) Aggregate interlock
(f) Dowel action
Figure 5-11: Constitutive models used in the TDE model
311
5.2.6.3.4.1 Concrete in Compression Cracking of the concrete was experimentally shown to soften the compressive stress-strain relationship (Vecchio and Collins, 1986). The constitutive model of concrete in compression, therefore, takes in to account the softening behaviour as shown in Figure 5-11 (a). The fc2 is related to the principal concrete compressive strain, εc2, according to the following expression (Vecchio and Collins, 1986; Vecchio and Collins, 1993):
f c2 = f p
ε n ⋅ c 2 ε p
(n − 1) + ε c 2 ε p
nk
(MPa)
Equation 5-76
f p = −β d ⋅ f c' (MPa)
Equation 5-77
ε p = − β d ⋅ ε c'
Equation 5-78
n = 0.8 −
fp
Equation 5-79
17
1 k = fp 0.67 −
62
ε < ε c2 < 0
Equation 5-80
ε c2 < ε p
where fp is the reduced peak compressive strength, εp the reduced peak compressive strain, βd the strength reduction factor, f'c the concrete compressive strength in MPa, and ε'c is the peak compressive strain at f'c.
312
The βd factor is calculated as follows:
βd =
1
1 + C s ⋅ Cd
(
Equation 5-81
C d = 0.27 ⋅ ε c1 / ε c' − 0.37
1 C s = 0.55
)
Equation 5-82
MCFT
DSFM
Equation 5-83
The main difference between the MCFT and the DSFM is that the latter accounts for the slip between the cracks explicitly in the formulation of the theory through the inclusion of discrete slip model at the crack surfaces. The formulation of the present model is similar to the formulation of the MCFT in terms of the implicit inclusion of the slip between the crack surfaces. It is, therefore, determined to set Cs = 1 in the current TDE theory.
5.2.6.3.4.2 Concrete in Tension The relationship between fc1 and εc1 is defined by a bilinear curve as shown in Figure 5-11 (b). The concrete modulus of elasticity, Ec, is defined as follows: Ec =
fp
Equation 5-84
εp
The softening behaviour of the concrete in tension is determined in accordance with the procedure implemented by Martin-Perez and Pantazopoulou (2001). The softening behaviour depends on the fracture energy, GF. The f't is calculated using Equation 5-17. The GF is used to describe the area enclosed by the tensile stress and the crack opening displacement. In order to express the softening behaviour in terms of ε1, the following relationship is used to determine the ultimate tensile strain in tension, εut (Bazant, 1986; Martin-Perez and Pantazopoulou 2001):
313
ε ut =
2G F f t ' ⋅ sθ
Equation 5-85
The GF is calculated as specified by Bazant and Becq-Giraudon (2002): 0.46
f c' GF = 2.5α 0 ⋅ 0.051
0.22
−0.3
d agg w 1 + 11.27 c
(N/mm)
Equation 5-86
where αo is a constant that takes into account the type of the aggregate (1.44 for crushed or angular aggregates), dagg the maximum aggregate size in mm, and w/c is the water-cement ratio.
Thus, fc1 is computed as follows; Ec ⋅ ε 1 f c1 = f t ' − E c ⋅ ε − ε t' 0
(
)
0 ≤ ε 1 ≤ ε t'
Equation 5-87
ε t' < ε 1 ≤ ε ut ε 1 > ε ut
5.2.6.3.4.3 Steel Reinforcement The stress-strain relationship of the longitudinal reinforcement follows the relationships expressed in Section 5.2.1.2 while the stress-strain relationship of the stirrup reinforcement is assumed to have a tri-linear trend as shown in Figure 5-11 (c). The behaviour is characterized by stress at yielding, fy, stress at ultimate, fu, strain at yielding, εy, strain at ultimate, εu, strain at the end of the yielding plateau, εsh, modulus of elasticity of steel, Es, and stiffness of the strain hardening, Esh.
E s ε s fy f s = f y + E sh (ε s − ε sh )
0
0 ≤ εs ≤ εy
ε y < ε s ≤ ε sh
Equation 5-88
ε sh < ε s ≤ ε ut ε s > ε ut
314
5.2.6.3.4.4 Concrete-Steel Bond In the vicinity of the crack, the tensile stresses in the steel reinforcement are above average while they are below average in-between the cracks. This variation in the stresses along the steel reinforcement is due to the transfer of some of the tensile stresses from the steel to the concrete through bond. The bond behaviour is modeled using a simple bond stress-average axial strain relationship. By establishing equilibrium along a small segment of the reinforcing bar and assuming that the rate of change in the axial strain is approximately equal to the ratio of the average strain in the steel and crack spacing, the expression by Martin-Perez (1995) (Equation 5-89) is obtained. The bond stress-axial strain relationship is shown in Figure 5-11 (d). The bond stress increases with increasing axial strain until the onset of yielding after which the Es,t equals zero due to the yielding of the steel.
fb =
db ⋅ Es ,t ⋅ ε s 4sx
Equation 5-89
where εs is the average tensile strain in the steel reinforcement and Es,t is the tangent modulus of the stress-strain curve of the reinforcing steel.
5.2.6.3.4.5 Aggregate Interlock Aggregate particles projecting from the two surface of a crack tend to interlock and provide a sort of shear resistance when the cracked concrete member is given a shear displacement, hence, comes the name of the phenomenon as “aggregate interlock”. The interaction of the normal stresses, shear stresses, shear displacement, and crack width need to be examined in order to understand the mechanism and to develop an accurate mathematical modeling of the aggregate interlock. Two fundamental modes of behaviour are involved; overriding and irreversible deformation (Walraven, 315
1981). The former is developed when the two crack surfaces slide on top of each other while the latter is developed when the contact surface is reduced leading to high contact stresses that cause irreversible deformation of the matrix. The overriding mode contributes to the shear resistance through friction between the crack surfaces and it is influenced by the aggregate size. The irreversible deformation mode contributes to the shear resistance through the yielding of the matrix and it is affected by the concrete strength and properties. The increase of the crack width would consequently lead to the reduction in the aggregate shear resisting mechanisms, especially when the steel reinforcement yields (Vaz Rodrigues, 2012). Therefore, an accurate mathematical expression of the aggregate interlock should be a function of the principal tensile strain at the crack interface, which provides a measure of the increase in the crack width. Martiz-Perez (1995) proposed Equation 5-90 and calibrated it with experimental results in order to determine the contribution of the aggregate interlock, vagg. The effect of increasing the crack width (increasing ε1) on the aggregate interlock can be seen in Figure 5-11 (e).
v agg =
0.1342×10 −3 Ec 1+
ε1
(MPa)
Equation 5-90
0.0018
5.2.6.3.4.6 Dowel Action There are three mechanisms by which the dowel action can contribute to the shear resisting mechanism; flexure, shear, and kinking. Research studies have shown that the kinking mechanism is the primary mechanism in sliding shear, particularly when small size bars are used (Park and Paulay, 1975; Paulay et al., 1974). The modeling of dowel action is performed by assuming the steel reinforcement as beam resting on an elastic foundation as proposed by Timoshenko and 316
Lessels (1925). The contact between the beam and the elastic foundation is modeled using discrete elastic springs and, therefore, the reaction force at any point on the foundation is proportional to the deflection of the beam at that particular point. The analytical model of the dowel action carried out in this research has been implemented by previous studies in which it showed good agreement with the experimental results (Millard and Johnson, 1985; Martin-Perez and Pantazopoulou, 2001; El-Ariss, 2007). The load-deformation relationship of the dowel force is determined as follows (Millard and Johnson, 1984): kd ,i F du Fd Fdu 1 e
Equation 5-90
where Fd is the dowel force at a shear displacement of Δ at the crack, Fdu the ultimate dowel force, and kd,i is the initial stiffness of the dowel force-crack width relationship.
The Fdu is evaluated using the empirical expression proposed by Dulacska (1972): Fdu 1.27 d b2
f c' f y
(N)
Equation 5-91
The kd,i is estimated using the expression developed by Millard and Johnson (1984): k d ,i 0.166k 0f .75 d b1.75 E s0.25 (N/mm)
Equation 5-92
where db and Es are in mm and MPa, respectively.
The elastic foundation stiffness of the surrounding concrete, kf, is determined using the following empirical expression developed by Soroushian et al. (1987):
317
kf =
127 c1 d b2
f c' 3
(N/mm)
Equation 5-94
where c1 is a coefficient accounts for the clear spacing of the bars (ranging from 0.6 for a clear bar spacing of 25 mm to 1 for a larger bar spacing) while db and f’c are in mm and MPa, respectively.
The Δ is the component of the crack width normal to the steel reinforcement, wc. Therefore, the components of the shear displacement in the x-direction, Δs,x, and in the y-direction, Δs,y, are determined as follows:
Δ s,x = wc ⋅sinθ
Equation 5-95
Δ s, y = wc ⋅cosθ
Equation 5-96
wc = sθ ⋅ ε1
Equation 5-97
After determining the dowel force in accordance to the above expressions, the dowel stress, fd, is determined by dividing the Fd by the area of the steel reinforcement bridging the crack: 4 f d = Fd ⋅ 2 π db
Equation 5-98
The dowel load-deformation behaviour of steel reinforcement bridging a crack is shown in Figure 5-11 (f). The contribution of the dowel action increases with increasing crack width until it reaches the maximum dowel force. This can be explained by the fact that the exponential term in Equation 5-91 tends to approach zero as the crack width increases which, in turn, leads to approaching the ultimate dowel force. Comparing the behaviour of the dowel action (Figure 5-11 (f)) and the behaviour of the aggregate interlock (Figure 5-11 (e)) indicates that the dowel action
318
is the primary shear resisting mechanism in a concrete member after the breakdown of the aggregate interlock as the crack width increases.
5.2.6.4 Solution Scheme Five equations need to be solved (Equations 5-49, 5-50, 5-61, 5-62, and 5-70 for the S-element and Equations 5-51, 5-52, 5-63, 5-64, and 5-70 for the C-element) with two unknowns (εy and γxy) for each type of element. Therefore, the number of equations needs to be reduced in order to be able to solve the problem and to avoid the large computational effort that is needed to satisfy the five equations simultaneously, if possible. The γxy can be determined from Equations 5-62 and 5 64 for the S-element and the C-element, respectively, if the stress in the stirrup, aggregate interlock, and the dowel action are calculated. Thus, the number of equations is reduced by one. Moreover, the equilibrium of the forces in the longitudinal direction (Equations 5-61 and 5-63 for the stirrup and the concrete elements, respectively) can be ignored due to the fact that the calculation of the shear resistance is more of a concern in the transverse direction. Equations 5-49 and 5-51 for the S-element and the C-element, respectively, can be also not enforced if proper account is taken to implicitly include the effect of the axial stress confinement on the shear resistance of the elements. Equations 5-49 and 5-51 take into account the effect of axial stress on enhancing the shear resistance through the term nx. Assuming a plane strain condition, the axial stress will induce a stress in the transverse direction equivalent to αcnx (where αc is the Poisson’s ratio of concrete) if αc is assumed to be constant throughout the stages of loading. The assumption of constant αc was proven to be reasonable and provide accurate results up to high concrete stress levels (approximately 75% of f'c) (Park and Paulay, 1975). In this manner, the effect of axial stress is accounted for by the modification of ny as follows: 319
n y = α c nx
Equation 5-99
The number of equations is, thus, reduced to two in order to equal the number of unknowns in the model. During the linear elastic stage of loading (un-cracked concrete), the strains εy and γxy are determined by solving the following set of linear equations: ε y 1 − α c2 =
E c γ xy
1 + ρ y E s 0
−1
0
1 − α c 2
n y
v xy
Equation 5-100
Alternatively, the εy at the onset of cracking can be determined, with reasonable accuracy, by dividing the f't by Ec for simplicity purposes. The algorithm shown in Figure 5-12 is implemented for the stirrup and the concrete elements during the cracked stage. It is noted that εx was set as the independent variable in this model rather than γxy since it is more of a concern to examine the shear capacity as a function of the axial strain induced in the longitudinal reinforcement when RS structures are subjected to combined bending and shear.
320
Figure 5-12: Solution scheme of the TDE model for cracked concrete subjected to shear loading
5.2.6.5 TDE Model Validation The validation of the proposed model against the degradation of the shear capacity in RC shear walls subjected to combined axial compression and lateral bending is presented in Appendix D. Based on the analysis results it was concluded that the model predicted accurately the sliding shear failure mode of the tested walls, and thus, it can be applied to predict the sliding shear response of the SSB and DSB systems investigated in this research. The concrete hinge in the SSB and DSB 321
connections is composed solely of C-element due to the absence of transverse reinforcement, and thus, the Δss is calculated as a function of γxy and lc-e as expressed in Equation 5-101. The thickness of the concrete element in the SSB and DSB connections, lc-e, equals to the distance between the stirrups on both sides of the vertical slots (90 mm).
Δss = γ xy ⋅lc−e
Equation 5-101
5.3 Application and Sensitivity of the Load-Displacement model
After examining the contribution of each of the deformation mechanisms, the load and displacement were calculated using Equations 5-1 and 5-2, respectively. Prior to validating the models, the sensitivity of the prediction model to the concrete tension stiffening is examined in this section. Concrete connections subjected to cyclic loading experience cracking at the top and the bottom sides of the concrete beam, and thus, the effect of tension stiffening is commonly ignored in modeling the seismic response of concrete structures. However, pushover analysis involves the application of lateral loading at only one side of the structure. Thus, it is determined to examine the sensitivity of the proposed model to the concrete tension stiffening. The response of the SSB-S-0 connection with and without concrete tension stiffening are compared in Figure 5-13 in terms of the load-displacement relationship and the neutral axis depth. It is seen that the effect of tension stiffening is negligible as far as the load-displacement relationship is concerned. The tension stiffening effect also has a minor effect on the neutral axis depth prior to yielding of the structure. It is, therefore, determined to exclude the effect of concrete tension stiffening since it better represents the response of concrete structures subjected to cyclic loading.
322
70
Neutral axis depth, c (mm)
W/ tension stiffening W/O tension stiffening
55
Load (kN)
30 5 -20 -45 -70
W/ tension stiffening W/O tension stiffenning
60 50 40 30 20 10 0
-60
-40
-20
0
20
40
-60
60
Displacement (mm)
(a) Load-displacement comparison
-40
-20
0
20
40
60
Displacement (mm)
(b) Neutral axis depth comparison
Figure 5-13: Sensitivity of the analytical model to concrete tension stiffening
5.3 Model Validation The validation of the proposed load-deflection prediction model with the experimental results of the steel-reinforced and the PE SMA-reinforced SSB and DSB connections is examined in this section. The load comparisons between the analytically and the experimentally obtained results at ultimate are included in Table 5-2, while the pushover load-deflection responses are shown in Figure 5-14 and Figure 5-15, while the load comparison at ultimate is included in. The following observations can be made with regard to the behaviour of connections:
Even though the model provides accurate representation of the steel-reinforced connections, the accuracy of the model in predicting the response of the DSB connections is higher than that of the SSB connections.
The stiffness of the SSB-P-1.0 connection obtained from the analytical model and subjected to positive bending is higher than that obtained from the experiment. This is attributed to an unanticipated rotation that took place at the pin support of the column, which contributed to the overall rotation of the experimentally tested connection. Under 323
negative bending, the load plateaus upon reaching the σMs stress in the PE SMA bar in the analytical model. However, the load capacity of the experimentally tested connection kept increasing with the increase in the induced beam-tip displacement. Therefore, it can be seen that other sources of overstrength such as the overstrength of the top steel reinforcement are not captured accurately by the model.
The load obtained from the analytical model of the DSB-P-1.0 connection starts to decay as the displacement increases. This is due to the fact that the concrete at the concrete hinge is strained beyond the strain corresponding to the maximum concrete strength (approximately 0.0025). However, this does not indicate failure as the concrete strain at failure is in the range of 0.012 – 0.05 as proposed by Paulay and Priestley (1992).
Overall, the developed models provide accurate representation of the pushover behaviour of the hysteretic response of the steel-reinforced and PE SMA-reinforced connections.
Table 5-2: Comparison of the Analytical and Experimental loads at ultimate Load (kN) Connection SSB-S-0 SSB-S-1.0 SSB-S-1.7 DSB-S-0 DSB-S-1.0 DSB-S-1.7 SSB-P-1.0 DSB-P-1.0
Positive bending Analytical Exp. %Error 56.3 62.0 -9.2 69.5 67.4 3.1 74.9 74.8 0.1 55.5 57.8 -4.0 71.2 67.9 4.9 83.9 81.9 2.4 49.9 44.4 12.4 51.6 55.7 -7.4
324
Negative bending Analytical Exp. %Error -49.1 -60.1 -18.3 -61.8 -70.6 -12.5 -74.3 -82.4 -9.8 -46.3 -45.1 2.7 -62.0 -56.9 9.0 -75.3 -68.1 10.6 -48.8 -68.4 -28.7 -42.3 -38.7 9.3
100
100
Experimental Analytical
Experimental Analytical
75
50
50
25
25
Load (kN)
Load (kN)
75
0 -25
0 -25
-50
-50
-75
-75
-100
-100 -80
-60
-40
-20
0
20
40
60
80
-80
-60
-40
Displacement (mm)
(a) Connection SSB-S-0
20
40
60
80
50
75
100
50
75
100
100
Experimental Analytical
75 50
50
25
25
0 -25
0 -25
-50
-50
-75
-75
-100 -80
-60
-40
-20
Experimental Analytical
75
Load (kN)
Load (kN)
0
(b) Connection SSB-S-1.0
100
0
20
40
60
-100 -100
80
-75
-50
Displacement (mm)
0
25
(d) Connection DSB-S-0
100
100
Experimental Analytical
75
25
25
Load (kN)
50
0 -25
0 -25
-50
-50
-75
-75
-75
-50
-25
Experimental Analytical
75
50
-100 -100
-25
Displacement (mm)
(c) Connection SSB-S-1.7
Load (kN)
-20
Displacement (mm)
0
25
50
75
100
-100 -100
Displacement (mm)
(e) Connection DSB-S-1.0
-75
-50
-25
0
25
Displacement (mm)
(f) Connection DSB-S-1.7
Figure 5-14: Validation of the load-displacement relationships of the analytical model with the steel-reinforced SSB and DSB connections 325
80 60
60
Experimental Analytical
40
20
Load (kN)
Load (kN)
40
80
0 -20
20 0 -20
-40
-40
-60
-60
-80 -125
-75
-25
25
75
-80 -125
125
Displacement (mm)
(a) Connection SSB-P-1.0
Experimental Analytical
-75
-25
25
75
125
Displacement (mm)
(b) Connection DSB-P-1.0
Figure 5-15: Validation of the load-displacement relationships of the analytical model with the PE SMA reinforced SSB and DSB connections
The member contributions of the validated steel-reinforced and PE SMA-reinforced connections are shown in Figure 5-16. Based on the member contribution analysis, the following observations are drawn:
Steel-reinforced SSB connections. Under positive bending, the contribution of the beam bending is the largest followed by the bar slippage in the SSB-S-0 and SSB-S-1.0 connections. However, relocating the vertical slot to a distance equivalent to 1.7 dv (500 mm) negatively affect the response as the stub bending is significantly increased on the expense of the beam bending. This observation is in agreement with the experimental test results analyzed in Section 3.7. In the SSB-S-1.7 connection, the contribution of the beam bending increases until the commencement of the bar yielding. After that, the increase in the displacement is associated with the increase in the contribution of the stub region. Under negative bending, the contribution of the beam bending starts to increase after yielding in all connections while the stub bending is very small in all stages of loading.
326
Steel-reinforced DSB connections. Due the symmetric nature of the DSB systems, only the behaviour under positive bending is shown in Figure 5-16. It can be seen that all connections possess approximately the same behaviour with the exception that there is no stub bending in the DSB-S-0 connection.
PE SMA-reinforced SSB and DSB connections. The Contribution of the beam bending is significantly higher than that in the steel-reinforced connections. In both connections, the contribution of the beam bending decreases until the PE SMA bars yield. After which, the contribution of the beam bending starts to increase up to failure. It is also noted that the contribution of the joint shear in the SSB-P-1.0 connection is lower than that in the DSBP-1.0 connection due to the difference in the joint shear mechanism in both systems as discussed in Section 5.2.4.
Sliding shear. The contribution of the sliding shear in all connections is insignificant
100%
100%
90%
90%
80%
80%
70% 60% 50%
Column bending Joint shear Sliding shear Slippage Stub bending Beam bending
40% 30% 20% 10% 0%
5
10
15
20
25
30
35
40
45
50
Member contribution (%)
Member contribution (%)
(ranges between 1% and 4%). It can be, therefore, ignored for design purposes.
60% 50%
Column bending Joint shear Sliding shear Slippage Stub bending Beam bending
40% 30% 20% 10% 0%
55
Displacement (mm)
(a) Connection SSB-S-0 (Positive bending)
70%
5
10
15
20
25
30
35
40
45
50
55
Displacement (mm)
(b) Connection SSB-S-0 (Negative bending)
327
100%
90%
90%
80%
80%
70% 60% 50%
Column bending Joint shear Sliding shear Slippage Stub bending Beam bending
40% 30% 20% 10% 0%
5
10
15
20
25
30
35
40
45
50
Member contribution (%)
Member contribution (%)
100%
70% 60% 50%
Column bending Joint shear Sliding shear Slippage Stub bending Beam bending
40% 30% 20% 10% 0%
55
5
10
15
20
Displacement (mm)
25
30
35
40
45
50
55
Displacement (mm)
100%
100%
90%
90%
80%
80%
70% 60% 50%
Column bending Joint shear Sliding shear Slippage Stub bending Beam bending
40% 30% 20% 10% 0%
5
10
15
20
25
30
35
40
45
50
Member contribution (%)
Member contribution (%)
(c) Connection SSB-S-1.0 (Positive bending) (d) Connection SSB-S-1.0 (Negative bending)
70% 60% 50%
Column bending Joint shear Sliding shear Slippage Stub bending Beam bending
40% 30% 20% 10% 0%
55
Displacement (mm)
5
10
15
20
25
30
35
40
45
50
55
Displacement (mm)
100%
100%
90%
90%
80%
80%
70% 60% 50%
Column bending Joint shear Sliding shear Slippage Stub bending Beam bending
40% 30% 20% 10% 0%
Member contribution (%)
Member contribution (%)
(e) Connection SSB-S-1.7 (Positive bending) (f) Connection SSB-S-1.7 (Negative bending)
60% 50%
Column bending Joint shear Sliding shear Slippage Stub bending Beam bending
40% 30% 20% 10% 0%
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Displacement (mm)
(g) Connection DSB-S-0
70%
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Displacement (mm)
(h) Connection DSB-S-1.0
328
100%
90%
90%
80%
80%
70% 60% 50%
Column bending Joint shear Sliding shear Slippage Stub bending Beam bending
40% 30% 20% 10% 0%
5
10 15
Member contribution (%)
Member contribution (%)
100%
70% 60% 50%
Column bending Joint shear Sliding shear Slippage Stub bending Beam bending
40% 30% 20% 10% 0%
20 25 30 35 40 45 50 55 60 65 70 75
5
10 15
20 25 30 35 40 45 50 55 60 65 70 75
Displacement (mm)
Displacement (mm)
(j) Connection SSB-P-1.0 (Positive bending)
100%
100%
90%
90%
80%
80%
70% 60% 50%
Column bending Joint shear Sliding shear Slippage Stub bending Beam bending
40% 30% 20% 10% 0%
5
10 15
Member contribution (%)
Member contribution (%)
(i) Connection DSB-S-1.7
60% 50%
Column bending Joint shear Sliding shear Slippage Stub bending Beam bending
40% 30% 20% 10% 0%
20 25 30 35 40 45 50 55 60 65 70 75
Displacement (mm)
(k) Connection SSB-P-1.0 (Negative bending)
70%
10
20
30
40
50
60
70
80
90
100 110 120
Displacement (mm)
(l) Connection DSB-P-1.0
Figure 5-16: Member contributions obtained from the analytical model
5.4 Parametric Study In this section, a parametric study is conducted using the analytical model. This parametric study is concerned only with the behaviour of the PE SMA-reinforced SSB and DSB connections, in which the effect of changing different design variables on the load-displacement response is examined. The design variables including the concrete compressive strength, f’c, concrete hinge depth, hch, yield strength, fy, ratio of top-to-bottom reinforcement (only in the SSB connections),
329
Ast / Asb, amount of diagonal reinforcement, Asd, and vertical slot relocation distance, lre, are considered in this study as indicted in Table 5-3 and Table 5-4 for the SSB and DSB connections, respectively. The failure of the connection is defined as the load and displacement corresponding to the crushing of concrete or failure of the anchors, whichever takes place first. Paulay and Priestley (1992) concluded that the crushing strain in concrete structures ranges between 0.012 and 0.05, and thus, it was determined to use the average value (0.031) as the crushing concrete strain in this research. The failure of the anchorage is induced once the strain in the PE SMA bars reaches a threshold value. This value is determined from the experimental results included in Section 4.10.5. For the SSB and DSB connections, the threshold values are 0.0156 and 0.0193, respectively.
Table 5-3: Parametric Study matrix for the PE SMA-reinforced SSB connections Connection SSB-1 SSB-2 SSB-3 SSB-4 SSB-5 SSB-6 SSB-7 SSB-8 SSB-9 SSB-10 SSB-11 SSB-12 SSB-13 SSB-14 SSB-15 SSB-16 SSB-17
f’c (MPa) 30 40 50 40 40 40 40 40 40 40 40 40 40 40 40 40 40
hch (mm) bd/4 bd /4 bd /4 3bd /8 bd /2 bd /4 bd /4 bd /4 bd /4 bd /4 bd /4 bd /4 bd /4 bd /4 bd /4 bd /4 bd /4
fy (MPa) 400 400 400 400 400 300 500 400 400 400 400 400 400 400 400 400 400
330
Ast / Asb 3.45 3.45 3.45 3.45 3.45 3.45 3.45 2.59 4.31 2.59 4.23 3.45 3.45 3.45 3.45 3.45 3.45
Asb (mm2) 348 348 348 348 348 348 348 348 348 462 284 348 348 348 348 348 348
Ast Asd (mm2) (mm2) 1200 600 1200 600 1200 600 1200 600 1200 600 1200 600 1200 600 900 600 1500 600 1200 600 1200 600 1200 400 1200 1000 1200 600 1200 600 1200 600 1200 600
lre 1.0 dv 1.0 dv 1.0 dv 1.0 dv 1.0 dv 1.0 dv 1.0 dv 1.0 dv 1.0 dv 1.0 dv 1.0 dv 1.0 dv 1.0 dv 0 dv 0.5 dv 1.5 dv 2.0 dv
Table 5-4: Parametric Study matrix for the PE SMA-reinforced DSB connections Connection DSB-1 DSB-2 DSB-3 DSB-4 DSB-5 DSB-6 DSB-7 DSB-8 DSB-9 DSB-10 DSB-11 DSB-12 DSB-13
f’c (MPa) 30 40 50 40 40 40 40 40 40 40 40 40 40
hch (mm) bd /4 bd /4 bd /4 3bd /8 bd /2 bd /4 bd /4 bd /4 bd /4 bd /4 bd /4 bd /4 bd /4
fy (MPa) 400 400 400 400 400 300 500 400 400 400 400 400 400
Ast / Asb 3.45 3.45 3.45 3.45 3.45 3.45 3.45 3.45 3.45 3.45 3.45 3.45 3.45
Asb (mm2) 348 348 348 348 348 348 348 348 348 348 348 348 348
Ast Asd (mm2) (mm2) 1200 600 1200 600 1200 600 1200 600 1200 600 1200 600 1200 600 1200 900 1200 1500 1200 600 1200 600 1200 600 1200 600
lre 1.0 dv 1.0 dv 1.0 dv 1.0 dv 1.0 dv 1.0 dv 1.0 dv 1.0 dv 1.0 dv 0 dv 0.5 dv 1.5 dv 2.0 dv
5.4.1 Effect of the Concrete Strength The effect of concrete strength on the response of the SSB and DSB systems is examined in Figure 5-17. Three concrete strengths were considered for this study; 30 MPa (SSB-1 and DSB-1), 40 MPa (SSB-2 and DSB-2), and 50 MPa (SSB-3 and DSB-3). It is observed that the ultimate load linearly increased with increasing concrete strength in the SSB and DSB connections. However, the effect of concrete strength on the response of the SSB connection subjected to negative bending is negligible. It is also observed that the change in the concrete strength has negligible effect on the ultimate displacement.
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60
60
40
20
Load (kN)
20
Load (kN)
40
SSB-1 SSB-2 SSB-3
0
0
-20
-20
-40
-40
-60 -90
-60
-30
0
30
60
-60 -200
90
Displacement (mm)
(a) SSB connections
DSB-1 DSB-2 DSB-3
-150
-100
-50
0
50
100
150
200
Displacement (mm)
(b) DSB connections
Figure 5-17: Effect of concrete strength on the load-displacement behaviour of the connections
5.4.2 Effect of the Concrete Hinge Depth Three concrete hinge depths were considered in this study as shown in Figure 5-18; bd /4 (SSB-2 and DSB-2), 3bd /4 (SSB-4 and DSB-4), and bd /2 (SSB-5 and DSB-5). It is observed that changing the concrete hinge depth has no effect on the response of the SSB connection subjected to positive bending. This is due to the fact that the vertical slot in the SSB system acts like an artificial crack while the concrete top fibres are subjected to compression forces (i.e. act like a conventional beam). However, significant effect is observed in the behaviour under negative bending. As the concrete hinge depth is increased, the stiffness and the load increase. This is explained by the reduction in the lever arm of the concrete compression forces as the concrete hinge depth is increased. Also, the load-displacement behaviour follows a parabolic shape as the concrete depth is increased. This is explained by the fact that increasing the depth of the concrete hinge increases the concrete strain, and thus, the ultimate concrete strain is reached faster. The behaviour of the DSB connection is similar to the SSB connection under negative bending. However, it is noticed
332
that the increase in the concrete hinge depth significantly reduced the ultimate displacement since the concrete hinge was strained at a faster rate, and thus, crushed early in the loading stage. 80
Load (kN)
40 20
60
SSB-2 SSB-4 SSB-5
40
Load (kN)
60
80
0 -20
20 0 -20
-40
-40
-60
-60
-80 -100
-75
-50
-25
0
25
50
75
-80 -200
100
Displacement (mm)
DSB-2 DSB-4 DSB-5
-150
-100
-50
0
50
100
150
200
Displacement (mm)
(a) SSB connections
(b) DSB connections
Figure 5-18: Effect of concrete hinge depth on the load-displacement behaviour of the connections
5.4.3 Effect of the Yield Strength The effect of the yield strength is examined by changing the yield strength of the bottom, top, diagonal, and stirrup reinforcements. Three values were considered as illustrated in Figure 5-19; 300 MPa (SSB-6 and DSB-6), 400 MPa (SSB-2 and DSB-2), and 500 MPa (SSB-7 and DSB-7). It is observed that the load increases linearly with increasing yield strength in the SSB and DSB connections as illustrated in Figure 5-19. The yield strength does not change the ultimate displacement when increased to 400 MPa in the SSB connections. However, the ultimate displacement is increased by 27.4% when the yield strength is increased to 500 MPa. The ultimate displacement increased approximately in a linear manner with increasing yield strength in the DSB connections.
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80 60
60
SSB-2 SSB-6 SSB-7
40
20
Load (kN)
Load (kN)
40
80
0 -20
20 0 -20
-40
-40
-60
-60
-80 -120
-90
-60
-30
0
30
60
90
-80 -250
120
Displacement (mm)
DSB-2 DSB-6 DSB-7
-200
-150
-100
-50
0
50
100
150
200
250
Displacement (mm)
(a) SSB connections
(b) DSB connections
Figure 5-19: Effect of yield strength on the load-displacement behaviour of the connections
5.4.4 Effect of the Bottom-to-Top Reinforcement Ratio and the Amount of Bottom Reinforcement in the SSB System Only the SSB system is considered in this study since the DSB system is designed to experience a symmetric behaviour under positive and negative bending. Three reinforcement ratios were considered in this study; 2.59 (SSB-8), 3.45 (SSB-2), and 4.31 (SSB-9) and three amounts of bottom reinforcement were considered; 284 mm2 (SSB-10), 348 mm2 (SSB-2), and 462 mm2 (SSB-11). As illustrated in Figure 5-20 (a), changing the bottom-to-top reinforcement ratio affects the response only under negative bending, while no effect is observed on the behaviour under positive bending. Under positive bending, the neutral axis is located near the top reinforcement, and thus, the stress in the top reinforcement is approximately zero. Therefore, changing the reinforcement ratio does not affect the response under positive bending. However, increasing the reinforcement ratio increases the load under negative bending. This is due to the fact that the neutral axis is located away from the top reinforcement, and thus, the top reinforcement has higher contribution towards the capacity of the system. It can be also seen that changing the amount of 334
the bottom reinforcement increases the load under positive and negative bending as shown in Figure 5-20 (b). However, the ultimate displacement is insensitive to the increase in the amount of the bottom reinforcement. 60
60
SSB-2 SSB-8 SSB-9
40
20
Load (kN)
Load (kN)
20
SSB-2 SSB-10 SSB-11
40
0
0
-20
-20
-40
-40
-60
-60 -90
-60
-30
0
30
60
90
-90
Displacement (mm)
-60
-30
0
30
60
90
Displacement (mm)
(a) Effect of the bottom-to-top reinforcement ratio (b) Effect of bottom reinforcement Figure 5-20: Effect of bottom-to-top reinforcement ratio and the amount of the bottom reinforcement on the load-displacement behaviour of the SSB connection
5.4.5 Effect of the Amount of the Diagonal Reinforcement The effect of the amount of the diagonal reinforcement is shown in Figure 5-21. Three amounts of diagonal reinforcement were considered; 400 mm2 (SSB-12 and DSB-8), 600 mm2 (SSB-2 and DSB-2), and 1000 mm2 (SSB-13 and DSB-9). It can be seen that changing the amount of the diagonal reinforcement slightly affects the response under positive bending in the SSB connections. No effect is observed in the response of the SSB connections under negative bending and for the response of the DSB connections. This is due to the fact that the neutral axis depth is located approximately at the centroid of the diagonal reinforcement when subjected to negative bending, while it is placed above the centroid of the diagonal reinforcement when subjected to
335
positive bending in the SSB connections. The diagonal reinforcement is placed near the neutral axis in the DSB connections, and thus, the strain is elastic. 60
60
SSB-2 SSB-12 SSB-13
40
20
Load (kN)
Load (kN)
20
40
0
0
-20
-20
-40
-40
-60 -90
-60
-30
0
30
60
-60 -200
90
Displacement (mm)
(a) SSB connections
DSB-2 DSB-8 DSB-9
-150
-100
-50
0
50
100
150
200
Displacement (mm)
(b) DSB connections
Figure 5-21: Effect of the amount of the diagonal reinforcement on the load-displacement behaviour
5.4.6 Effect of the Relocation Distance The load-displacement relationships of five relocation distances are shown in Figure 5-22; 0 (SSB14 and DSB-10), 0.5 dv (SSB-15 and DSB-15), 1.0 dv (SSB-2 and DSB-2), 1.5 dv (SSB-16 and DSB 12), and 2.0 dv (SSB-17 and DSB-13). All SSB connections failed by the rupture of the PE SMA, while the DSB connections failed by concrete crushing except for DSB-10 (0 dv) which failed by rupture of the PE SMA reinforcement. Increasing the relocation distance increases the load and the stiffness of the connections, and thus, the strain corresponding to the failure of the anchorage is attained at lower displacement. This is due to the fact that relocating the vertical slots away from the face of the column increases the curvature demand at the critical section. The optimum relocation distance is examined by plotting the displacement ductility index, μΔ, rotation at the vertical slot, θb,v, and the contribution of the beam bending in the shear span,
336
Δb2, to the total displacement as shown in Figure 5-23. The optimum relocation distance is defined as the distance that provides the maximum response in terms of μΔ, θb,v, and, Δb2. Due to the inconsistency of the trends shown in Figure 5-23, optimum relocation distances that satisfy the three criteria: μΔ, θb,v, and Δb2 at the same time cannot be determined. Instead, the designed relocation distance should be based on optimizing the performance of the connection with regard to one of the three properties depending on the desired design outcome. The optimum relocation of each design criterion is included in Table 5-5. 80
80
SSB-2 SSB-14 SSB-15 SSB-16 SSB-17
Load (kN)
40 20
40
0 -20
20 0 -20
-40
-40
-60
-60
-80 -100
-75
-50
DSB-2 DSB-10 DSB-11 DSB-12 DSB-13
60
Load (kN)
60
-25
0
25
50
75
-80 -200
100
-150
-100
Displacement (mm)
-50
0
50
100
150
200
Displacement (mm)
(a) SSB connections
(b) DSB connections
2.9
14.5
SSB - Positive bending SSB - Negative bending DSB
2.7
Rotation at vertical slot, θb,v
Displacement ductility index, μ∆
Figure 5-22: Effect of the relocation distance on the load-displacement behaviour
12.5
2.5
10.5
2.3
2.1
1.9
SSB - Positive bending SSB - Negative bending DSB
8.5 6.5 4.5
0
0.5
1
1.5
2
0
Relocation distance as a function of dv
(a) Displacement ductility index, μΔ
0.5
1
1.5
Relocation distance as a function of dv
(b) Rotation at vertical slot, θb,v 337
2
Contribution of beam bending in the shear span, ∆b2 (%)
94 92 90 88 86
SSB - Positive bending SSB - Negative bending DSB
84 82 0
0.5
1
1.5
2
Relocation distance as a function of dv
(c) Contribution of beam bending in the shear span to the total displacement, Δb2 Figure 5-23: Evaluation of the optimum relocating distance in the PE SMA-reinforced connections
Table 5-5: Optimum relocation distances of the PE SMA-reinforced connections Connection SSB – Positive bending SSB – Negative bending DSB
μΔ 0 0 0
θb,v 0.5 dv 1.5 dv 1.5 dv
Δb2 0 0 0.5 dv
5.6 Summary
An analytical prediction model of the steel-reinforced and PE SMA-reinforced SSB and DSB connections was developed and validated in this chapter. The model was used to predict the pushover response of the load-displacement relationship and examine the different member contributions to the connection drift. Also, a new shear deformation-based model (TDE) was developed and validated in order to examine the sliding shear behaviour of concrete structures. Based on the parametric study of the PE SMA-reinforced connections, the following conclusions can be made:
338
a. Changing the concrete hinge depth and the relocation distance have significant effects on the stiffness, ultimate load, and ultimate displacement of the connections. b. A bottom-to-top reinforcement ratio of 2.89 will yield a symmetric response under positive and negative bending in the SSB connection. c. The concrete in the concrete hinge of the DSB connections experiences high strain levels which may lead to the crushing of the concrete prior to the failure of the anchorage. This behaviour results in significant reduction of the ultimate displacement when the concrete hinge depth is increased. d. An optimum relocation distance that maximizes the displacement ductility, rotation at the vertical slot, and the contribution of beam bending into the beam displacement cannot be determined for the PE SMA-reinforced connections as the optimum relocation distances of each of the aforementioned criteria vary significantly among the connections.
339
Finite Element Simulation of the Tested Connections 6.1 Introduction Finite Element Method (FEM) models are developed to simulate the response of the tested steelreinforced and PE SMA-reinforced concrete connections using ABAQUS 6.9 software. The FEM technique provides a precise tool for examining the deformation mechanisms that cannot be predicted using the analytical model developed in Chapter 5 such as the cracking pattern, damage progression, and the effect of the width of the vertical slots on the concrete damage. In this chapter, the FEM models are only used to validate the response of the connections. However, they can be used in future research in order to conduct a parametric study concerning the design variables. The FEM simulation presented in this research intends to predict the response of the developed systems for seismic design applications, such as the design using the Capacity Spectrum Method, and thus, only the pushover response is modeled. Firstly, the model development is discussed in terms of the element types, material constitutive models, geometry, loading and boundary conditions, and mesh density. The optimum mesh density is determined based on a mesh sensitivity analysis concerning the number of concrete and steel elements. Secondly, the model is validated against the experimental test results with respect to the load-displacement response, strains in the bottom reinforcement, beam rotation, and cracking behaviour. Finally, the concluding remarks of this chapter are presented.
6.2 Model Development 6.2.1 Element Type The concrete beam-column connection and the steel plates were modeled using fully integrated eight-node linear brick elements (C3D8). The steel cage in the steel-reinforced connections, the 340
PE SMA reinforcement in the PE SMA-reinforced connections, the steel bars connecting the steel plates, and the steel tube were modeled using two-node linear beam elements (B31). The steel anchors and couplers in the PE SMA-reinforced connections were modeled using four-node tetrahedral elements (C3D4). The contact between the steel reinforcement and the steel tube was modeled using single node tube-to-tube contact element (ITT31).
6.2.2 Material Constitutive Models 6.2.2.1 Concrete The damage plasticity model was used to model the concrete response. The model requires determining the concrete stress-strain constitutive models and the damage under tension and compression. The implemented compressive and tensile constitutive models are shown in Figure 6-1. The concrete compressive stress fc varies with the concrete strain in accordance to the model developed by Todeschini et al. (1964) and as expressed in Equations 5-14 and 5-15.
Concrete stress, fc (MPa)
45
35
25
15
5
-5 -0.03
-0.02
-0.01
0
0.01
Concrete strain, εc (mm/mm)
0.02
Figure 6-1: Concrete constitutive model used in the FEM modeling
341
0.03
The concrete tensile stress, ft varies linearly until reaching the concrete tensile strength, f’t, followed by an exponential curve determined as follows (Sima et al., 2008):
f t = E ε e ' c t
α 1−
ε ε t'
Equation 6-1 −1
G ⋅E 1 α = F 'c2 − ≥ 0
2
lcbw ⋅ f t
Equation 6-2
where ε’t is the concrete strain at f’t, GF the concrete fracture energy, Ec the concrete modulus of elasticity, and lcbw is the crack bandwidth in mm.
The Ec is calculated as the secant slope of the concrete compressive stress-strain relationship in the range of 0 to 0.6 f’c. The lcbw is introduced in the model to guarantee the objectivity of the results with respect to the size of the finite element mesh and set to 10. The GF is calculated using Equation 5-86. The parameters required to define the concrete damage under compression, Dc, and tension, Dt, are shown in Figure 6-2. The relationship between the total concrete strain, εc,tot, and the concrete plastic strain, εc,p, upon unloading is expressed in the following power-type equation (Bahn and Hsu, 1998):
ε c, p = C p (ε c,tot )n
Equation 6-3
p
where Cp is a coefficient of plastic strain and np is an optimum order of proposed equation type in the concerned test data (the change of np will provide change in curvature of a curve shape for future modification).
342
Figure 6-2: Definition of the parameters used in calculating the concrete compressive and tensile damage
Bahn and Hsu (1998) recommended using constants of 0.3 and 2 for Cp and np, respectively. The unloading concrete stiffness, E’c, is then calculated using Equation 6-4 while Dc is calculated using Equation 6-5 (Figure 6-3 (a)): Ec' =
(ε
f c , un c , un
Dc = 1−
Equation 6-4
− ε c, p )
Ec Ec'
Equation 6-5
The variation of Dt with increasing concrete strain is shown in Figure 6-3 (b) and expressed as follows (Sima et al., 2008):
εc
α 1− ε ε Dt = 1− ct ⋅ e ε
ct
Equation 6-6
343
The Poisson’s ratio is 0.2 while the concrete density is 2400 kg/m3 (MacGregor, 1997). The angle of dilatancy was chosen to be 30 degree as recommended by Coronado and Lopez (2006). 1
Concrete tensile damage, Dt
Concrete concrete damage, Dc
1
0.8
0.6
0.4
0.2
0.8
0.6
0.4
0.2
0
0 0
0.005
0.01
0.015
Concrete strain, εc (mm/mm)
0.02
0.025
(a) Compressive damage
0
0.001
0.002
0.003
Concrete strain, εc (mm/mm)
0.004
0.005
(b) Tensile damage
Figure 6-3: Concrete damage versus strain relationships
6.2.2.2 Steel Reinforcement and Steel Tube The material properties of the experimentally tested coupon tension tests presented in Section A.7.2 were used in modeling the steel reinforcement. The nonlinear response of strain hardening of the bottom reinforcement in the steel-reinforced SSB and DSB connections was modeled using the proposed expressions by Menegotto and Pinto (1977) as outlined in Section 6.2.1.2 while the steel reinforcement elsewhere was modeled using bilinear relationships. The Poisson’s ratio was set to 0.3 while the steel density was 7,800 kg/m3 (korotkov et al., 2004).
6.2.2.3 PE SMA As described in Section 6.1, only the backbone behaviour (pushover) is modeled in this study, and thus, the plasticity model in ABAQUS is found to reasonably represent the behaviour of the PE 344
SMA material. The PE SMA material was modeled as a bilinear relationship based on the coupon test results of the cyclically loaded specimen (Section 4.4.3). The Poisson’s ratio and the density of the SMA material were set to 0.33 (Lagoudas and Bo, 1999) and 7,800 kg/m3 (Jackson et al., 1972), respectively.
6.2.3 Geometry
Three-dimensional (3D) FEMs were developed using ABAQUS 6.9 software. The connections have one plane of symmetry (x-y plane) cutting the connection into two halves, and thus, one half of the connection was modelled. The connection sections were sketched using 3D deformable solid extrusion and then solid cut extrude was created to model the vertical slots in the SSB and the DSB connections as shown in Figure 6-4 (note that the other connections were modeled using the same method except that the location and the dimension of the extrusion are different). The steel anchors and couplers (used in anchoring the PE SMA bars in the SSB-P-1.0 and DSB-P-1.0 connections) were also sketched using 3D deformable solid shape as show in Figure 6-5. Similarly, the steel plates were sketched using 3D deformable solid shape as shown in Figure 6-6. The steel reinforcement, steel bars (refer to the bars used to attach the steel plates on both sides of the concrete column and beam), steel tube, and the PE SMA bars were sketched using 3D wires. The concrete part, steel reinforcement parts, steel tube part, steel plate parts, and steel anchors (only in the SSB-P-1.0 and DSB-P-1.0 connection) were assembled in one assembly using edge-to-edge, instance rotate, and instance translate constraints. Partition faces were also created to tie the steel plates on the top and the bottom of the column and beam tip sections. The assembled connections are shown in Figure 6-7. It is noted that only two steel-reinforced SSB and DSB
345
connections are shown in the figure. However, the other connections were assembled using the same technique with the only difference being the location of the vertical slot.
Figure 6-4: Modeled connection (SSB-S-1.0 connection)
(a) Anchor
(b) Coupler
Figure 6-5: Modeled mechanical steel anchor and coupler
346
Figure 6-6: Modeled steel plates
The steel cage, steel tube, and steel anchors (only in the SSB-P-1.0 and DSB-P-1.0 connection) were embedded, their translation degrees of freedom are constrained by the response of the nodes, inside the concrete elements except in the debonded region (refer to Figure 3-3 for the definition of the debonded region) in the steel-reinforced connections and the PE SMA bars in the PE SMA-reinforced connections.
347
Figure 6-7: Assembled FEM models
Tube-to-tube contact elements were used to debond the steel reinforcement from the steel tube in the debonded region in the steel-reinforced connections as shown in Figure 6-8. The tubeto-tube elements are slide line contact elements in the sense that they assume that the relative motion of the steel tube and the reinforcing bar is predominantly along the line defined by the axis of the reinforcing bar and their relative rotations are assumed to be small. The contact elements were attached to the steel tube while the slide line was defined in the steel reinforcement. The
348
clearance between the inner face of the tube and surface of the steel reinforcement was set to 2 mm as illustrated in Figure 6-8. Based on the test results presented in Section 4.10.5, the bond of the PE SMA bars to concrete is negligible, and thus, the PE SMA bars were modeled as debonded reinforcement. Instead of using the conventional contact interaction methods, the tube-to-tube contact elements were used to model the interaction. The contact elements were attached to a high stiffness thin film tube while the sliding line was defined as the PE SMA bars. This method provides accurate modeling of the interaction behaviour between the PE SMA bars and the concrete. The interaction between the inner surfaces of the concrete beam at the location of the vertical slot was modeled using hard contact in order to simulate the rocking mechanism in case the vertical slot closes under negative bending (However, this action did not take place in the modeled connections since the vertical slots did not fully close upon negative bending).
Figure 6-8: Tube-to-tube contact elements in the steel-reinforced connections
6.2.4 Boundary Conditions and Loading
The boundary conditions are illustrated in Figure 6-9. The displacement in the direction perpendicular to the plane was held at zero at the plane of symmetry. The translational degrees of 349
freedom in the vertical direction at bottom plate located under the concrete column were constrained, while the translational degrees of freedom in the horizontal direction at the steel plates attached to the sides of the concrete column were constrained.
Bottom steel plate
Side steel plate
Figure 6-9: Boundary connections applied onto the connections
350
It is noted that the boundary condition applied to only one side is shown for clarity purposes. However, the same boundary condition was applied to the four steel plates attached to the sides of the column member. The loading consisted of the application of three steps as follows: • Step 1. Application of gravity load to simulate the self-weight of the connection • Step 2. Application of vertical load on the steel plate tied to the top face of the concrete column in the form of uniform stress (Figure 6-10 (a)) • Step 3. Application of the quasi-static loading in the form of a boundary condition
applied to the steel plate attached to the top of the beam tip (Figure 6-10 (b))
(a) Vertical loading
(b) Vertical displacement
Figure 6-10: Loading applied onto the modeled connections
6.2.5 Mesh Sensitivity Analysis Mesh sensitivity analysis was conducted to examine the effect of mesh size on the solution and to determine the optimum mesh density to be used in the FEM models. The mesh sensitivity analysis was conducted on the SSB-S-0 Connection subjected to positive bending for a displacement of 55 mm (corresponds to the maximum displacement obtained from the experimental testing). The analysis consisted of two stages; (1) examine the effect of the concrete element mesh size, and (2) 351
examine the effect of the steel element mesh size. In the former stage, the number of concrete elements was increased until convergence of the solution was achieved, while the number of steel elements was held constant (fine mesh was adopted for the steel elements). The concrete strain at the top beam fibre at the concrete hinge was the primary output in this stage. After determining the optimum mesh density that provided precise results of the concrete strain, the second stage of the mesh sensitivity analysis was conducted. In the second stage, the number of steel elements was increased until convergence in the strain values, in the bottom reinforcement in the debonded region, and in the load at displacement amplitude of 55 mm was achieved. Intel Core I7 – 2670QM CPU processor (2.2 GHz) and 16 GB of RAM memory was used to run the FEM analyses. The mesh sensitivity results of the first and the second stages are included in Figure 6-11 and Figure 6-12, respectively. It is seen that the concrete strain plateaued as the number of elements was increased beyond a total number of elements of 11,333. This number of elements was chosen as the base for the second stage in which the number of steel elements was changed until convergence of the bottom steel strain and the load at the tip of the beam were achieved. 0
Concrete strain (ε)
-0.002 -0.004 -0.006 -0.008 -0.01 -0.012 -0.014 -0.016 0
5000
10000
15000
20000
Total number of elements
Figure 6-11: Mesh sensitivity results of the first stage
352
25000
67 66.5
0.0357 0.0355
66
Load (kN)
Bottom steel strain (ε)
0.0359
0.0353 0.0351
65.5 65
0.0349 64.5
0.0347 0.0345 10500
11000
11500
12000
12500
13000
64 10500
13500
Total number of elements
(a) Bottom steel strain
11000
11500
12000
12500
13000
13500
Total number of elements
(b) Load
Figure 6-12: Mesh sensitivity results of the second stage
From Figure 6-12, increasing the number of elements beyond 11,774 had a negligible effect on the response values, and thus, it represents the optimum number of elements. This number of elements correspond to an approximate global element length of 35 mm and 10 mm for the concrete and the steel reinforcement, respectively (it is noted that if the edge length is not an integer multiple of the element length, ABAQUS will change the element length slightly to obtain an integer number of elements along the edge). The aforementioned element lengths were, therefore, used in modeling the other eight connections. The optimum mesh density is shown in Figure 6-13.
353
Anchor
Coupler Figure 6-13: Optimum mesh density of the modeled connections
6.3 Model Validation
The results obtained from the FEM models are compared with the experimental results under positive and negative bending with respect to the load-displacement relationships, profile of beam rotation, and profile of strain variation in the bottom reinforcement. The comparison results are presented in Table 6-1 and Table 6-2 under positive and negative bending, respectively, in terms of the load, beam rotation, and strain in the bottom reinforcement at the ultimate displacement. The percentage errors with respect to the experimental results are also included in the tables. The analysis of the comparison results included in the tables are discussed in the following paragraphs.
354
Table 6-1: Comparison of the FEM results with the experimental results under positive bending Load (kN)
Connection
SSB-S-0 SSB-S-1.0 SSB-S-1.7 DSB-S-0 DSB-S-1.0 DSB-S-1.7 SSB-P-1.0 DSB-P-1.0
FEM
Exp.
66 72 84 61 76 88 54 59
62 67 75 58 68 82 44 56
Beam rotation (rad) Error (%) 6 7 12 5 12 7 23 5
FEM
Exp.
0.033 0.038 0.041 0.050 0.063 0.062 0.053 0.093
0.028 0.031 0.034 0.047 0.060 0.060 0.054 0.087
Error (%) 18 23 21 6 5 3 -2 7
Strain in bottom reinforcement (mm/mm) Error FEM Exp. (%) 0.035 0.033 6 0.036 0.029 24 0.037 0.014 164 0.030 0.028 7 0.035 0.035 0 0.034 0.032 6 0.017 0.016 6 0.017 0.018 -6
Table 6-2: Comparison of the FEM results with the experimental results under negative bending Load (kN)
Connection
SSB-S-0 SSB-S-1.0 SSB-S-1.7 DSB-S-0 DSB-S-1.0 DSB-S-1.7 SSB-P-1.0 DSB-P-1.0
FEM
Exp.
-53 -69 -87 -43 -61 -75 -49 -48
-60 -71 -82 -45 -57 -68 -68 -39
Beam rotation (rad) Error (%) -12 -3 6 -4 7 10 -28 23
FEM
Exp.
-0.032 -0.040 -0.045 -0.054 -0.064 -0.062 -0.057 -0.094
-0.031 -0.038 -0.044 -0.050 -0.062 -0.061 -0.038 -0.089
Error (%) 3 5 2 8 3 2 50 6
Strain in bottom reinforcement (mm/mm) Error FEM Exp. (%) -0.035 -0.034 3 -0.043 -0.039 10 -0.050 -0.050 0 -0.026 -0.016 63 -0.032 -0.032 0 -0.031 -0.025 24 -0.016 -0.016 0 -0.017 -0.010 70
The pushover results of the FEM models are compared with the hysteretic responses of the steel-reinforced and the PE SMA-reinforced connections in Figure 6-14 and Figure 6-15, respectively, with respect to the load-displacement relationships. Based on the load comparison included in Table 6-1 and Table 6-2 and based on results shown in Figure 6-14 and Figure 6-15, the following observations are made. The envelope curves obtained from the FEM models are comparable with the behaviour of the experimentally tested connections. It is also observed that 355
the accuracy of the FEM models in predicting the response under negative bending for the SSB S-1.7 connection is more accurate than that under positive bending. This is due to the failure of the damage plasticity model in accurately predicting the complex shear behaviour in the stub region, which contributes to the beam total drift. The general observations made with regard to the response of the analytically modeled PE SMA-reinforced connections as compared with the experimental behaviour presented in Section 5.4 are also applicable to the response of the FEM
100
100
50
50
Load (kN)
Load (kN)
models.
0
Exp. FEM
-50
0
Exp. FEM
-50
-100
-100 -80
-40
0
40
80
-80
-40
Displacement (mm)
100
100
50
50
0
Exp. FEM
-50
-60
-20
80
20
60
0
Exp. FEM
-50
100
-100 -100
Displacement (mm)
(c) Connection SSB-S-1.7
40
(b) Connection SSB-S-1.0
Load (kN)
Load (kN)
(a) Connection SSB-S-0
-100 -100
0
Displacement (mm)
-60
-20
20
Displacement (mm)
(d) Connection DSB-S-0
356
60
100
100
50
50
Load (kN)
Load (kN)
100
0
Exp. FEM
-50
-100 -100
-60
-20
20
60
0
Exp. FEM
-50
100
-100 -100
-60
Displacement (mm)
-20
20
60
100
Displacement (mm)
(e) Connection DSB-S-1.0
(f) Connection DSB-S-1.7
80
80
40
40
Load (kN)
Load (kN)
Figure 6-14: Comparison of the experimental and FEM load-displacement response of the steelreinforced connections
0
Exp. FEM
-40
-80 -125
-75
-25
25
75
0
125
-80 -125
Displacement (mm)
(a) Connection SSB-P-1.0
Exp. FEM
-40
-75
-25
25
75
125
Displacement (mm)
(b) Connection DSB-P-1.7
Figure 6-15: Comparison of the experimental and FEM load-displacement response of the PE SMA reinforced connections
The rotation profiles in the beam members obtained from the FEM models and the experimental results are compared for the steel-reinforced and PE SMA-reinforced connections in Figure 6-16 and Figure 6-17, respectively. It is noted that the FEM beam rotation values at the ultimate displacement included in Table 6-2 and Table 6-2 are those determined at the same section as for the experimentally obtained values for consistency purposes. Overall, the FEM results are 357
in good agreement with the experimentally obtained results. However, the rotation at the stub region in numerically modeled steel-reinforced SSB connections is underestimated as compared with the experimental results. It is also observed that the FEM models provide approximately zero rotation in the stub region in the PE SMA-reinforced concrete connections which is in good agreement with the experimentally tested connections.
30 mm
55 mm
-6 mm
-30 mm
6 mm
-55 mm
0.05
0.03
0.03
Rotation (rad)
Rotation (rad)
6 mm 0.05
0.01
-0.01
30 mm
55 mm
-6 mm
-30 mm
0.01
-0.01
Solid line - FEM Dashed line - Exp.
-0.03
-0.03
-55 mm
Solid line - FEM Dashed line - Exp.
-0.05
-0.05 0
220
440
660
880
1100
1320
0
1540
220
(a) Connection SSB-S-0 30 mm
55 mm
660
880
1100
1320
1540
(b) Connection SSB-S-1.0 -6 mm
-30 mm
-55 mm
10 mm
0.05
0.1
0.03
0.06
Rotation (rad)
Rotation (rad)
6 mm
440
Distance from the column face (mm)
Distance from the column face (mm)
0.01
-0.01
-0.03
40 mm
80 mm
-10 mm
-40 mm
-80 mm
0.02
-0.02
-0.06 Solid line - FEM Dashed line - Exp.
Solid line - FEM Dashed line - Exp.
-0.05
-0.1 0
220
440
660
880
1100
1320
1540
0
Distance from the column face (mm)
(c) Connection SSB-S-1.7
220
440
660
880
1100
1320
Distance from the column face (mm)
(d) Connection DSB-S-0
358
1540
40 mm
85 mm
-10 mm
-40 mm
10 mm
-85 mm
0.1
0.06
0.06
Rotation (rad)
Rotation (rad)
10 mm 0.1
0.02
-0.02 Solid line - FEM Dashed line - Exp.
-0.06
40 mm
75 mm
-10 mm
-40 mm
-75 mm
0.02
-0.02 Solid line - FEM Dashed line - Exp.
-0.06
-0.1
-0.1 0
220
440
660
880
1100
1320
0
1540
220
440
(e) Connection DSB-S-1.0
660
880
1100
1320
1540
Distance from the column face (mm)
Distance from the column face (mm)
(f) Connection DSB-S-1.7
Figure 6-16: Comparison of the experimental and FEM beam rotation profiles in the steelreinforced connections 10 mm -40 mm
40 mm -80 mm
80 mm 120 mm
-10 mm -120 mm
0.12
Rotation (rad)
0.08 0.04 0 -0.04 -0.08
Solid line - FEM Dashed line - Exp.
-0.12 0
220
440
660
880
1100
1320
1540
Distance from the column face (mm)
(a) Connection SSB-P-1.0
(b) Connection DSB-P-1.7
Figure 6-17: Comparison of the experimental and FEM beam rotation profiles in the PE SMA reinforced connections
The strain profiles in the bottom reinforcement of the steel-reinforced concrete connections under positive and negative bending for multiple displacement amplitudes are compared in the Figure 6-18, while the load versus the strain in the bottom PE SMA bars in the PE SMA-reinforced connections are compared in Figure 6-19. Based on the profiles shown in Figure 6-18 and based on the comparison at the ultimate displacement included in Table 6-1 and Table 6-2, the following 359
observations are drawn. The FEM models provide accurate results under positive and negative bending when the vertical slots are placed at the face of the column. However, the accuracy in predicting the strain profile of the SSB connections with relocated vertical slots is reduced under positive bending. This is due to the complex nature of the force transformation in the stub region which is not accurately captured by the FEM. Furthermore, in the experimental testing, it was observed that the plastic hinge does not exactly coincide with the location of the vertical slot in the SSB connections with relocated vertical slots and subjected to positive bending unlike the behaviour of the FEM models. The PE SMA strain values shown in Figure 6-19 were obtained at the vertical slot location. Due to the fact that the PE SMA bars are debonded from the surrounding concrete, the strain profile in the SMA bars are approximately constant (there is a slight variation since the rotation is concentrated at the vertical slot section). It is seen that the strain values at ultimate obtained from the FEM and experiment are approximately the same. The stiffness of the relationships obtained from the FEM models are higher and lower than the experimental values under positive and negative bending, respectively, for the SSB-P-1.0 connection. The same observations made earlier with regard to the load-displacement comparison explain this behaviour. 0.06
0.06
6 mm 55 mm -30 mm
0.02
0.04
Strain, ε (mm/mm)
Strain, ε (mm/mm)
0.04
30 mm -6 mm -55 mm
Yielding
0 -0.02 -0.04 -0.06 -440
0
220
440
660
880
0 -0.02
-0.06 -440
1100
Distance from the column face (mm)
(a) Connection SSB-S-0
Yielding
-0.04
Solid line - FEM Dashed line - Exp.
-220
0.02
6 mm 55 mm -30 mm -220
30 mm -6 mm -55 mm 0
Solid line - FEM Dashed line - Exp.
220
440
660
880
Distance from the column face (mm)
(b) Connection SSB-S-1.0 360
1100
0.06
0.06 Solid line - FEM Dashed line - Exp.
0.02
Yielding
0 -0.02
6 mm 55 mm -30 mm
-0.04 -0.06 -440
-220
10 mm 80 mm -40 mm
0.04
Strain, ε (mm/mm)
Strain, ε (mm/mm)
0.04
30 mm -6 mm -55 mm 0
0.02
Yielding
0 -0.02 -0.04
220
440
660
880
Solid line - FEM Dashed line - Exp.
-0.06 -440
1100
-220
Distance from the column face (mm)
0.04
0.04
Strain, ε (mm/mm)
Strain, ε (mm/mm)
0.06
Yielding
0 -0.02 -0.04 -0.06 -440
-220
40 mm -10 mm -85 mm 0
440
660
880
0.02
440
660
880
1100
Yielding
0 -0.02 -0.04
Solid line - FEM Dashed line - Exp.
220
220
(d) Connection DSB-S-0
0.06
10 mm 85 mm -40 mm
0
Distance from the column face (mm)
(c) Connection SSB-S-1.7
0.02
40 mm -10 mm -80 mm
-0.06 -440
1100
10 mm 75 mm -40 mm -220
Distance from the column face (mm)
40 mm -10 mm -75 mm 0
Solid line - FEM Dashed line - Exp.
220
440
660
880
1100
Distance from the column face (mm)
(e) Connection DSB-S-1.0
(f) Connection DSB-S-1.7
80
80
40
40
Load (kN)
Load (kN)
Figure 6-18: Comparison of the experimental and FEM strain profiles in the bottom reinforcement of the steel-reinforced connections
0
Exp. FEM
-40
-80 -0.03
-0.02
-0.01
0
0.01
0.02
Exp. FEM
-40
-80 -0.03
0.03
Displacement (mm)
(a) Connection SSB-P-1.0
0
-0.02
-0.01
0
0.01
0.02
0.03
Displacement (mm)
(b) Connection DSB-P-1.7
Figure 6-19: Comparison of the experimental and FEM strain profiles in the bottom reinforcement of the PE SMA reinforced connections 361
The tensile damage in the concrete material at the integration points is found to reasonably represent the cracking pattern, and thus, the tensile damage at ultimate positive and negative bending of the steel and PE SMA-reinforced connections is visualized as shown in Figure 6-20. Following are the observations made with regard to the cracking pattern:
• Steel-reinforced SSB connections. Cracks are developed in the concrete hinge under positive and negative bending. As the vertical slots are moved away from the face of the column, flexural cracks develop in the stub region which explains the contribution of the stub region in the beam rotation. Thus, the plasticity is spread over a larger area as the vertical slot is moved away from the face of the column.
• Steel-reinforced DSB connections. The cracking in the concrete hinge and stub region is of a complex nature. The diagonal cracks (A) developed during the experimental testing (illustrated in Figure 3-45) can be clearly shown in the cracking behaviour obtained from the FEM models. It is also found that flexural cracks develop in the stub region as the vertical slot is moved away from the face of the column.
• PE SMA-reinforced SSB and DSB connections. Since the PE SMA bars are debonded from the surrounding concrete in the SSB connection, no cracking is observed in the stub region. However, flexural cracks at the top fibre are developed under negative bending due to the fact that the top steel reinforcement is bonded to the surrounding concrete. The cracking in the DSB connection took place only at the concrete hinge region due to the negligible bond of the PE SMA bars to concrete.
362
363
Figure 6-20: Concrete tensile damage visualization of the tested connections
6.4 Summary
Three-D FEM models of the steel-reinforced and PE SMA-reinforced SSB and DSB connections were developed in this chapter. The detailed procedure used in constructing the models was discussed followed by validating the FEM models by comparing the load-displacement relationships, beam rotation profiles, bottom reinforcement strain profiles, and cracking patterns with the experimental results. It was found that the accuracy of the FEM models in predicting the response of the SSB system is reduced as the vertical slot is moved away from the face of the column by 1.7 dv (500 mm). This is due to the fact that the shear deformation mechanism is of a 364
complex nature in the SSB connections. Overall, the results obtained from the FEM models were in good agreement with the experimental results. However, the analytical model developed and validated in Chapter 5 showed superior performance in terms of accurately predicting the behaviour of the connections as compared with the FEM models developed in this chapter.
365
Chapter Seven: Conclusions and Recommendations 7.1 General The use of smart materials in the seismic design of concrete structures is emerging as an alternative to conventional steel reinforcement. In this research, newly developed concrete beam-column connections reinforced using PE SMA material were examined thoroughly using experimental testing of large-scale connections, analytical modeling, and finite element simulation. It is noted that the results obtained from the experimental testing provide a representative indication regarding the performance of the specimens and not an exact behaviour. This is due to the limited number of specimens, the level of variability in the material properties (concrete, steel, and SMA), and the level of variability in the testing procedure (accuracy of maintaining a constant vertical load applied to the columns and the level of fixity of the boundary conditions). The major conclusions drawn from this study are listed below while the detailed conclusions of each phase and the recommendations for future research are described in the following sections. •
The steel-reinforced SSB and DSB systems were developed and modified in this research in order to overcome the main challenges encountered in the design process; the yield penetration into the joint, the joint cracking, and the complex joint shear deformation mechanism. These challenges were overcame by moving the vertical slot away from the face of the column, and hence, relocating the plastic hinge.
•
The PE SMA-reinforced SSB connection possess excellent self-centering behaviour under both positive and negative bending even though the SMA bar is placed only at the bottom side of the beam member. Therefore, the proposed design has a great potential in reducing the construction cost while attaining an outstanding seismic performance.
366
•
The PE SMA-reinforced DSB system sustains very high drift ratios (up to 9%), as compared with the PE SMA-reinforced SSB system and the steel-reinforced counterparts, while exhibiting an excellent self-centring behaviour. Thus, it can be used as an alternative to the PE SMA-reinforced SSB system when the structure is subjected to high drift demands.
•
The ductility and energy dissipation properties of the PE-SMA reinforced connections are low as compared with conventional connections, and hence, research advances in this area are needed.
•
The Two Distinct Element (TDE) shear deformation theory provides a more accurate prediction of the shear capacity of lightly reinforced concrete structures, especially those subjected to sliding shear, as compared with the Modified Compression Field Theory (MCFT). Based on the TDE modeling, the design of the concrete sections at the location of the concrete hinges in the SSB and DSB connections are adequate as far as the sliding shear is concerned.
•
The analytical model provides an accurate tool to predict the envelope of the loaddisplacement behaviour of the SSB and DSB connections. However, the contribution of the sliding shear can be eliminated from since it is insignificant as compared with the other deformation mechanisms.
•
Given that the percentage errors of the finite element models and the level of complexity in developing them are higher than the analytical model, the analytical model provides a more efficient design tool. The mathematical coding of the model is simple, and thus, can be implemented in software for design purposes.
367
7.2 Conclusions 7.2.1 Design of Steel-Reinforced SSB and DSB Connections •
The forces inside the joint are transformed via two diagonal compression struts when the vertical slot is placed at the face of the column. However, moving the vertical slot away from the face of the column allows for the development of a single diagonal compression strut in the joint similar to the behaviour in conventional connections.
•
The developed simplified rotational capacity prediction model can be used for design purposes to predict the rotational capacities of the systems, which are governed by the fatigue life of the reinforcement passing through the vertical slots.
7.2.2 Behaviour of Steel-Reinforced SSB and DSB Connections •
Moving the vertical slots away from the face of the column does not result in degrading the overall response of the connections.
•
In general, moving the vertical slots away from the face of the column succeeded in relocating the centre of rotation, which coincide with the location of the vertical slot. However, it is more effective under negative bending than under positive bending.
•
The average overstrength of the SSB connections with and without the contribution of the diagonal reinforcement is 16.8% and 26.4%, respectively, higher than the overstrength of the conventional connections (1.25), while for the DSB connections with and without the contribution of the diagonal reinforcement it is 17.6% and 26.4%, respectively, higher than the overstrength of the conventional connections.
•
The number of the flexural cracks developed within the stub region increases as the vertical slot is moved away from the face of the column. 368
•
Relocating the vertical slot reduces the effectiveness of the non-tearing action of the slabs attached at the top of the beams.
•
Relocating the vertical slot by a distance equivalent to 1.0 dv (300 mm) results in superior performance with regard to the minimal cracking within the joint as compared with the other relocating distances.
•
In the SSB system, the rotation at the location of the vertical slot under positive bending decreases in a linear manner with the increase in the relocation distance while it drops by 17.7% under negative bending regardless of the relocation distance. In the DSB system, the rotation at the vertical slot drops by 15.8% when the vertical slot is moved away from the face of the column and it is independent from the relocation distance. Therefore, the best relocating distance that successfully relocate the plastic hinge under positive and negative bending is 1.0 dv (300 mm).
•
In the SSB system, relocating the vertical slot to a distance equivalent to 1.0 dv (300 mm) does not increase the beam elongation. However, relocating the vertical slot beyond the aforementioned distance increases the elongation of the beam in RC frames significantly (100% increase when for a relocation distance of 1.7 dv (500 mm)). In the DSB system, increasing the relocation distance increases the permanent deformation and the elongation in the beam member similar to the behaviour in a typical RC frame.
•
The strain in the longitudinal steel reinforcement at the face of the column is reduced as the vertical slot is moved away from the face of the column which indicates the efficiency of the systems in maintaining low levels of yield penetration into the concrete joint.
369
•
Relocating the vertical slot away from the face of the column does not affect the shear mechanism in the beam. It is also concluded that relocating the vertical slot away from the face of the column by 1.0 dv (300 mm) results in improved joint shear response.
•
Based on the beam-to-stub rotation ratio (θ2/θ1), the effectiveness of having the vertical slot at 1.0 dv (300 mm) away from the face of the column is 1.86 and 1.91 times that if the distance to the vertical slot is increased to 1.7 dv (500 mm) under positive and negative bending, respectively, in the SSB system. In the DSB system, the effectiveness of having the vertical slot at 1.0 dv (300 mm) away from the face of the column is 1.4 times that if the distance to the vertical slot is increased to 1.7 dv (500 mm).
•
The ductility-based assessment indices for the SSB connections decrease when the vertical slots are moved away from the face of the column. However, they tend to peak at a relocation distance of 1.0 dv (300 mm) in the DSB system.
•
The rate of energy dissipation increases approximately in a linear manner with respect to the increase in the relocation distance in the SSB system. However, the rate of change is insensitive to the relocation distance up to 1.0 dv (300 mm) in the DSB system.
•
The evolution of the damping ratio is insensitive to relocating the centre of rotation. However, it only affects the ultimate value.
•
A more gradual degradation in the post-yielding stiffness is observed in the DSB system than the SSB system. Also, relocating the plastic hinge does not affect the shape of the trends.
370
7.2.3 Behaviour of Joint-Like Specimens, Anchor, and Coupler •
The transformation from the Austenitic to the Martensitic phases undergoes severe nonlinearity in the strain variation along the length of the PE SMA bar. However, it tends to decrease as the stress approaches the ultimate load.
•
The ultimate damping ratio of the PE SMA bar is approximately 3% and it is attained at as strain value of approximately 4%.
•
The cyclic loading results in softening the response of the PE SMA material i.e. the strain in the PE SMA subjected to cyclic loading is higher than the strain in the PE SMA subjected to monotonic loading for the same stress level.
•
The cracking load in the steel-reinforced joint-like specimens is governed by the splitting bond stress and not by the concrete tensile stress unlike the behaviour of PE SMAreinforced joint-like specimens.
•
The developed strut-and-tie model for the joint-anchorage specimen provides momentcurvature relationships in good agreement with the experimental values, and thus, it is validated.
•
Under tension-tension cyclic loading, the anchor and coupler modified in this research sustain 55.8% and 57.4% of the ultimate strain of the PE SMA bars. At ultimate, the slips in the modified anchor and coupler are 3.3 ± 2.0 mm and 4.0 ± 0.07 mm, respectively.
7.2.4 Behaviour of PE SAM-Reinforced SSB and DSB Connections •
The hysteretic responses are characterized by its self-centering ability.
•
Very limited number of cracks were developed in the connections, and thus, the non-tearing action is achieved. 371
•
The beams behaved like rigid elements with their centre of rotation located at the vertical slot.
•
The beam elongation in the SSB system is zero while the beam elongation in the DSB system is reduced by 32.1% with respect to the Control connection.
•
The beam shear distortion is approximately zero during all stages of loading.
•
The joint stirrup strain is below yielding, and thus, the stirrups remained elastic under positive and negative bending.
•
At the ultimate cycle, the beam-to-stub rotation ratios (θ2/θ1) are 34.6 and 52.0 in the SSB and DSB connections, respectively. Therefore, the DSB system is more effective in relocating the centre of rotation as compared with the SSB system.
•
The DSB connection experienced better performance in terms of all the ductility-based damage assessment measures as compared with the SSB connection.
•
The DSB connection exhibited superior performance in terms of higher total cumulative energy as compared with the SSB counterpart.
•
Overall, reinforcing concrete connections using PE SMA material reduces the equivalent viscous damping significantly as compared with the steel-reinforced counterparts (average reduction of 55.8% at ultimate) and the control connection (average reduction of 82.7% at ultimate).
•
At the ultimate condition, the average decrease in the Residual Displacement Index is 92.5% and 94.3% as compared with the steel-reinforced SSB and DSB connections and the Control connection, respectively. Therefore, the PE SMA-reinforced connections possess an effective self-centering ability.
372
•
The stiffness of the connections degrade exponentially with increasing displacement ductility index at approximately the same rate.
7.2.5 Analytical Prediction Model and Parametric Study •
The proposed Two Distinct Element shear deformation theory was proven to provide accurate prediction of the degradation of the shear capacity in reinforced concrete shear walls, especially in lightly reinforced concrete members.
•
The analytical prediction of the pushover behaviour of the steel-reinforced and PE SMAreinforced connections provide good agreement with the experimental results in terms of the load-displacement behaviour, beam rotation, strain profile, and cracking pattern.
•
The optimum bottom-to-top reinforcement ratio in the SSB system is 2.89. This ratio will result in a symmetric response under positive and negative bending. It was also concluded that changing the depth of the concrete hinge has the most significant impact on the stiffness, load, and displacement of the system.
•
An overall optimum relocation distance that maximizes the displacement ductility, rotation at the vertical slot, and the contribution of beam bending into the beam displacement of the PE SMA-reinforced SSB and DSB connections cannot be determined.
7.2.6 Finite Element Simulation •
The pushover FEM curves, beam rotation, and strain profiles are in good agreement with the behaviour of the experimentally tested connections. However, the accuracy of the response prediction is reduced as the vertical slots are moved away from the face of the columns. 373
•
The rotations at the stub region in the numerically modeled steel-reinforced SSB connections with relocated vertical slots are underestimated as compared with the experimental results.
•
The cracking pattern of the FEM models is in good agreement with the experimentally observed cracking.
7.3 Recommendations •
A relocating distance of 1.0 dv (300 mm) in both the SSB and DSB connections was found to provide an excellent seismic performance, and thus, it is recommended to use it in the design of concrete beam-column connections in which the centre of rotation is relocated.
•
The design of the vertical slot width implemented in this research accounted only for the closure of the slots as a results of the induced negative bending. In service, concrete creep of the beam member also results in the closure of the vertical slots, and thus, experimental and analytical research should be devoted towards examining the creep effect in the steelreinforced and PE SMA-reinforced connections.
•
The PE SMA material is temperature sensitive, and thus, it is recommended to investigate the environmental exposure effect on the performance of the SSB and DSB connections.
•
It is recommended to use trained PE SMA bars in order to maintain a stabilized response of the connections when subjected to cyclic loading.
•
Even though experimental testing of the modified anchors and couplers showed that the average maximum strain attained is 0.0535 and 0.0550 in the modified anchor and coupler, respectively, the maximum strain attained in the PE SMA bars in the large-scale connections is in the rage of -0.0156 to 0.0211. This can be explained by the fact that the 374
testing of the anchors and couplers was performed under tension-tension cyclic loading while they experience tension-compression cyclic loading in the large-scale connections. Therefore, the anchors and couplers should also be modified and tested under tensioncompression cyclic loading. It is also recommended to use threaded anchors welded to a steel plate rather than screwlock anchors as the stress concentration is avoided in the former type, and hence, the premature failure of the PE SMA is avoided. •
The results presented in this research study correspond to the behaviour of the developed systems at the local level. However, the global behaviour of concrete structures with SSB and DSB connections needs to be examined before being used in real-life structures. Dynamic and capacity spectrum analyses should be used to determine the suitability of the developed systems in the seismic design of concrete structures.
•
The Two Distinct Element (TDE) shear-deformation theory developed in this research can be implemented in design codes and software of to predict the behaviour of concrete structures in general, and lightly reinforced concrete members in particular.
•
The developed analytical model can be implemented in computer software for design purposes of steel-reinforced and PE SMA-reinforced SSB and DSB connections.
•
Parametric study concerning the design parameters of the connections should be conducted using the developed FEM models in order to examine the effect of slot dimensions on the local cracking behaviour of the connections.
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and Technology, 122(2), 186-191. Tobushi, H., Shimeno, Y., Hachisuka, T., and Tanaka, K. (1998). “Influence of strain rate on superelastic properties of TiNi shape memory alloy,” Mechanics of Materials, 30(2), 141 150. Tobushi, H., Takata, K., Shimeno, Y., Nowacki, W. K., and Gadaj, S. P. (1999). “Influence of strain rate on superelastic behaviour of TiNi shape memory alloy,” Proceedings of the
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Vecchio, F. J. and Collins, M. P. (1986). “The modified compression-field theory for reinforced concrete elements subjected to shear,” ACI Journal Proceedings, 83(2), 219-231. Vecchio, F. J and Collins, M. P. (1993) “Compression response of cracked reinforced concrete,”
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Division, 107(ST11), 2245-2270. Wierschem, N. E. (2009). “Superelastic shape memory alloy composite bars for reinforcing concrete structures,” M.Sc. Thesis, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA, 114 pp. Wierschem, N. and Andrawes, B. (2010). “Superelastic SMA-FRP composite reinforcement for concrete structures,” Smart Materials and Structures, 19(2), 13 pp. Witting, P. R. and Cozzarelli, F. A. (1992). “Shape memory structural dampers: Material properties, design and seismic testing,” Technical Report NCEER-92-0013, National Centre for Earthquake Engineering Research, State University of New York at Buffalo, Buffalo, NY, USA, 41 pp. Youssef, M. S., Alam, M.S., and Nehdi, M. (2008). “Experimental investigation of the seismic behaviour of beam-column joints reinforced with superelastic shape memory alloys,”
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395
Appendix A: Specimen Design, Construction, and Material Properties A.1 Introduction
The experimental program of this research consists of two phases as mentioned in Section 1.5. The first phase includes testing three SSB and three DSB connections reinforced using conventional steel reinforcement while the second phase includes testing one PE SMA-reinforced SSB connection, one PE SMA-reinforced DSB connection, and one Control connection. Joint-like specimens were also fabricated and tested to investigate the anchorage performance of the PE SMA material into the concrete joints. The analysis of the forces applied onto the connections and the scaled end-actions are first presented in the following section. After that, the design, construction, and material properties of the tested specimens are presented.
A.2 Analysis of Forces Applied onto the Connections
The applied loads were determined from the National Building Code of Canada (NBCC, 2005) for an office building. In addition to the self-weight of the structure, stresses of 2.0 kPa on all floors and 1.0 kPa only on the roof were applied as dead loads. The stresses induced by the live loads are 2.4 kP and 1.0 kPa on all office floors and on the roof, respectively. The snow induces a stress of 2.48 kPa on the roof while the velocity pressure of the wind is 0.4 kPa. The structure is founded on a stiff soil (class D). Therefore, the acceleration-based, Fa, and velocity-based site coefficients, Fv, are 1.124 and 1.360, respectively. The design spectral response acceleration, Sa, of the building is shown in Figure A-1. The natural period of the building was found to be 1.38 s obtained using a dynamic analysis conducted by ETABs. It is noted that accidental eccentricity was also taken into account in the analysis.
396
0.8
Spectral Acceleration, Sa(g)
0.7 0.6 0.5 0.4 0.3 0.2 (1.38,0.11)
0.1 0 0
1
2
3
4
5
Period, T (sec)
Figure A-1: Design spectral response acceleration
The end-actions obtained from the structural analysis performed using ETABs at the designated connection are included in Table A-1. As mentioned in Section 3.2, the connection is scaled down to 80% using the replica scaling method (Noor and Boswell, 1992). The scaled dimension of the section is the product of the dimension and the scale factor (0.8) while the scaled forces are the product of the forces and the square of the scale factor (0.64). The scaled end-actions of the designated connection are included in Table A-2 while the scaled geometry is shown in Figure A-2. The letters P and M refer to the applied load and moment, respectively, while the subscripts D, L, and E refer to dead, live and earthquake loads, respectively. The factored axial load, Pf, factored shear load, Vf, and factored moment, Mf, applied at the designated beam-column connection are included in Table A-3 for the following applicable load combinations (the load combinations that include the wind load are excluded from the analysis due to the low applied wind loading.):
397
•
Caes 1: 1.25D + 1.5L
•
Case 2: 1.0D + 1.0E
•
Case 3: 1.0D - 1.0E
•
Case 4: 1.0D + 0.5L + 1.0E
•
Case 5: 1.0D + 0.5L - 1.0E
Table A-1: End-actions obtained from the analysis at the designated connection End-actions Bottom of the column at the 5th floor Top of the column at the 4th floor Beam end
PD (kN)
PL (kN)
PE (kN)
MD (kN·m)
ML (kN·m)
ME (kN·m)
VD (kN)
VL (kN)
VE (kN)
289.6
72.9
14.2
-17.3
-7.0
11.6
-9.5
-3.9
5.6
382.8
103.1
24.7
12.2
4.9
-9.2
-9.0
-3.7
6.6
-
-
-
-36.5
-14.8
26.1
-39.6
-15.0
10.6
Table A-2: Scaled end-actions obtained from the analysis at the designated connection End-actions Bottom of the column at the 5th floor Top of the column at the 4th floor Beam end
PD (kN)
PL (kN)
PE (kN)
MD (kN·m)
ML (kN·m)
ME (kN·m)
VD (kN)
VL (kN)
VE (kN)
185.3
46.7
9.1
-11.1
-4.5
7.4
-6.1
-2.5
3.6
245.0
66.0
15.8
7.8
3.2
-5.9
-5.8
-2.3
4.2
-
-
-
-23.3
-9.5
16.7
-25.3
-9.6
6.8
Table A-3: Factored axial load, shear, and moments at the designated column and beam Location of the end-actions Bottom of the column at the 5th floor Top of the column at the 4th floor Beam end
End-actions
Case 1
Case 2
Case 3
Case 4
Case 5
Pf (kN)
301.7
194.4
176.3
217.8
199.6
Mf (kN·m)
-20.6
-3.7
-18.5
-5.9
-20.7
Pf (kN)
405.2
260.8
229.2
293.8
262.2
Mf (kN·m)
14.5
1.9
13.7
3.5
15.3
Vf (N)
-46.1
-18.6
-32.1
-23.4
-36.9
Mf (kN·m)
-43.4
-6.7
-40.0
-11.4
-44.8
398
(a) Before scaling
(b) After scaling
Figure A-2: Geometry of the designated connection (all dimensions are in mm)
A.3 Design of Steel-Reinforced SSB and DSB Connections A.3.1 Beam Design
A.3.1.1 Flexural Design Based on the recommendations mentioned in Section 3.2.3.1 and the forces shown in Figure 3-4, the proposed sections at the location of the vertical slots are shown in Figure A-3. The factored moments of resistance of the beam, Mb,r, in the SSB and DSB systems are calculated as follows by ignoring the contribution of the diagonal reinforcement and the concrete hinge (as suggested in Section 3.2.3.1):
399
• For the SSB system, the moments are taken about the centroid of the compression force and the centroid of the top reinforcement for the positive and negative bending, respectively, at the vertical slot section: Positive bending: M b,r = φs Asb f y,sb (d sb − a / 2 )
Equation A-1
Negative bending: M b,r = φs Asb f y,sb (d sb − d st )
Equation A-2
+
−
where,
φs Asb f y,sb α1φ c fc'bw
a =
Equation A-3
• For the DSB system, the moment is taken about the centroid of the top reinforcement at the vertical slot section:
Mb,r = φs Asb f y,sb (dsb − dst )
Equation A-4
where ϕc and ϕs are the concrete and steel material resistance factors, respectively, Asb is the amount of the bottom reinforcement, fy,sb the yield stress of the bottom reinforcement, dsb and dst are the distances from the top concrete fibre to the centroid of the bottom and top reinforcements, respectively, a is the depth of the equivalent rectangular stress block, α1 the ratio of average stress in rectangular compression block to the specified concrete strength, f’c the concrete compressive strength, and bw is the width of the beam.
400
(a) SSB system
(b) DSB system
Figure A-3: Beam dimensions and details at the vertical slots (all dimensions are in mm)
The aforementioned design procedure is also used to calculate the factored moment of resistance, Mb,r, the nominal moment, Mb,n, and the probable moments, Mb,p of the beams in the SSB and the DSB systems under positive and negative bending. The Mb,n is calculated by setting the material resistance factors in Equations A-1 to A-4 equal to one (ϕc = ϕs = 1). The Mb,p is calculated by multiplying the yield stresses in Equations A-1 to A-4 by the overstrength factor, λo as described in Equation A-5 and setting ϕc = ϕs = 1(Paulay and Priestley, 1992):
λo = λ1 + λ2
Equation A-5
where λ1 is the ratio of the actual to the specified yield strength and the λ2 is the increase in the stress resulting from the strain hardening.
401
As recommended by Paulay and Priestley (1992), the values of λ1 and λ2 are 1.15 and 1.25, respectively. The Mb,r, Mb,n, and the Mb,p moments of resistance of the SSB and DSB systems are included in Table A-4 and Table A-5, respectively.
Table A-4: Design moments of the SSB system
Moment
Mb,r (kN·m) 46.9
Positive Mb,n (kN·m) 55.6
Mb,p (kN·m) 77.0
Mb,r (kN·m) 42.1
Negative Mb,n (kN·m) 49.5
Mb,p (kN·m) 69.4
Table A-5: Design moments of the DSB system Moment
Mb,r (kN·m) 42.7
Mb,n (kN·m) 50.2
Mb,p (kN·m) 70.3
A.3.1.2 Shear Design As discussed in Section 3.2.3.2, the beam is subdivided into three regions in terms of the shear behaviour. The shear design of SSB-S-1.7 connection under positive bending is only presented in this section since it experiences the highest shear forces induced as a result of the development of the plastic hinge at the location of the vertical slots. The conventional method of shear design is used to calculate the amount and spacing of the stirrups in region I. In regions II and III, the recommendations by Paulay and Priestley (1992) and the anti-buckling requirements CSA-A23.3 04 (2004) are implemented, respectively. The shear resistance is calculated and checked against the applied shear.
402
A.3.1.2.1 Region I The maximum factored shear force in region I, Vf,I, is 33.2 kN. The factored concrete shear resistance, Vc, is (CSA A23.3-04 Clause 11.3.4): Vc = φc λβ v
f c' bw d v = 0.65 ×1.0 × 0.18 × 40 × 300 × (0.9 × 353 ) ×10 −3 = 70.5 kN
Equation A-6
where dv and βv are the effective shear depth and factor accounting for shear resistance of cracked concrete, respectively, λ is a factor to account for the density of concrete (equals 1.0 for normal density concrete), and bw is the width of the beam.
Since Vf,I < Vc, the spacing of the stirrups is controlled by the smaller of the following two requirements: i.
Minimum spacing of stirrups (CSA A23.3-04 Clause 11.2.8.2):
ssp ≤
Av f y 0.06 f c' bw
=
200× 400
0.06× 40 ×300
= 703 mm
Equation A-7
where ssp is the stirrup spacing and Av is the amount of the transverse reinforcement within a distance ssp. ii.
Spacing limits (CSA A23.3-04 Clause 11.2.8.2):
V f < 0.125φc fc'bwdv = 0.125×0.65×40×300×0.9×356×10-3=312.4 kN
Equation A-8
Then ssp,max = smaller of 600 mm or 0.7 dv = 0.7 × 0.9 × 356 = 224.3 mm. where ssp,max maximum strirrup spacing.
403
Therefore, spacing of 224.3 mm controls. The spacing of the stirrups was, however, set to 150 mm. The calculated stirrup factored shear resistance, Vs, is as follows: Vs =
φs Av f y d v ssp
=
0.85 × 200 × 400 × 0.9 × 356 × 10 − 3 = 145.2 kN 150
Equation A-9
The factored shear resistance, Vr, is therefore:
Vr = Vc +Vs = 35.5 + 145.2 = 180.7 kN
Equation A-10
Check the maximum factored shear resistance, Vr,max:
Vr,max = 0.25φc fc'bwdv = 0.25×0.65×40×300×0.9×356×10-3 = 624.8 kN
Equation A-11
A.3.1.2.1 Region II and III
The maximum factored shear force in regions II, Vf,II, and III, Vf,III are 98.9 kN and 69 kN,
respectively. The spacing of the anti-buckling reinforcements shall not exceed the following (CSA
A23.3-04 Clause 21.3.3.2):
i.
dst / 4 = 356 / 4 = 89 mm
ii.
8dbl = 8 × 15 mm
iii.
24dbh = 24 × 10 = 240 mm
iv.
300 mm
where dbl and dbh are the diameters of the longitudinal and stirrup reinforcements, respectively.
Therefore, 90 mm was set as the spacing between the stirrups. The Vs is calculated using Equation A-9:
404
Vs =
φ s Av f y d v s sp
=
0.85 × 200 × 400 × 0.9 × 356 × 10 − 3 = 242.1 kN 90
Therefore, Vr is calculated using Equation A-10
Vr = Vc +Vs = 35.5 + 242.1 = 277.6 kN Check Vr,max using Equation A-11:
Vr,max = 0.25φc fc'bwdv = 0.25×0.65×40×300×0.9×356×10-3 = 624.8 kN
A.3.1.3 Anchorage The 90o hooks of the flexural and the diagonal reinforcement in the joint satisfy the design requirements in CSA A23.3-04 Clause 6.6.2 (Annex A). The minimum requirements and the dimensions used in the design of the connections are included in Table A-6 while the parameters A, B, and C are defined in Figure A-4. Table A-6: Detailing for standard 90o hooks for deformed bars Rebar
15 M 20 M
Minimum requirement A B C (mm) (mm) (mm) 180 220 100 240 260 120
Used in the design A B C (mm) (mm) (mm) 245 295 100 280 340 120
Figure A-4: Definition of the design parameters in a standard 90o hook for deformed bars
405
A.3.2 Column Design
The diameter of the longitudinal bars must satisfy the following requirement (CSA A23.3-04 Clause 21.5.5.6)
db ≤ jd/24 = 440 / 24 = 18 mm
Equation A-12
where jd is the depth of the joint.
Therefore, 15 mm bars where chosen in the arrangement shown in Figure A-5.
Figure A-5: Column reinforcement details (all dimensions are in mm)
The minimum and the maximum amounts of the longitudinal steel (As,min and As,max, respectively) are as follows (CSA A23.3-04 Clause 21.4.3.1):
As,min = 0.01Cd Cw = = 0.01 × 440 ×300 = 1320 mm2
Equation A-13
As,max = 0.03Cd Cw = 0.03 × 440 ×300 = 3960 mm2
Equation A-14
where Cd and Cw are the depth and the width of the column, respectively.
Even though the amount of reinforcement in the column (1200 mm2) is lower than the
As,min, it was decided that the minimum requirement can be relaxed since the axial load and the 406
flexural moment applied to the column are relatively low. The factored axial-moment interaction diagram of the column section is shown in Figure A-6. The load combinations of the end reactions at the top of column in the 4th floor are also included. The end-reactions at the bottom of the column in the 5th floor are not shown since they do not control the design of the connection. It is seen that the column section is capable of resisting the applied loading.
Factored axial load, Pr (kN)
3500
1 3 5
3000
2 4
2500
2000
1500
1000
500
0
0
50
100
150
200
250
Factored moment resistance, Mr (kN.m)
Figure A-6: Interaction diagram for column section
A.3.2.1 Strong-Column Weak-Beam Requirement The nominal moment of the columns, Mco,n, must exceed the probable moment of the beams, Mb,p in order to satisfy the strong-column weak-beam requirement:
M
co,n
= M b, p
Equation A-15
The lowest flexural resistance will occur at either the highest or lowest axial load (Case 3 and Case 4, respectively) as shown in Figure A-7. Load Case 1 is also investigated due to the fact
407
that the applied axial load used in the experimental testing is equivalent to the factored axial load determined in Case 1.
Figure A-7: Capacity design of columns for three factored load combinations
The summation of the nominal capacities of the column for the three cases are as follows: Case 1:
M
c,n
=127.9 +143.0 = 270.9 kN·m
Case 3:
M
c,n
=124.5+133.9 = 258.4 kN·m
Case 4:
M
c,n
=131.5+144.7 = 276.2 kN·m
The summation of the probable beam moment is as follows:
M
b, p
= 76.3 kN·m
Therefore, the requirement stated in Equation A-15 is satisfied. It is also noted that the beam and column shears acting at the joint are neglected for simplicity.
408
A.3.2.2 Shear Design The factored shear force is calculated as follows:
Vf =
2M co 2×38.2 = = 29.8 kN lco 2.56
Equation A-16
M b , p 76 .3 = = 38 .2 kN·m 2 2
Equation A-17
M co =
where Mco is the column moment.
The factored concrete shear resistance is calculated as expressed in Equation A-6 assuming
dv = 0.72Cd (CSA A23.3-04 Clause 11.3.6.3): Vc = φc λβ v
f c' bw d v = 0.65 ×1.0 × 0.1× 40 × 300 × 0.72 × 440 ×10 −3 =39.1 kN
Therefore, the concrete shear resistance is sufficient to resist the applied shear force. However, minimum transverse and confinement reinforcements are provided according to Equation A-7 (CSA A23.3-04 Clause 11.2.8.2):
ssp ≤
Av f y 0.06 f c' bw
=
200× 400 0.06× 40 ×300
= 703 mm
Therefore, a spacing of 700 mm controls. The area of the rectangular hoop reinforcement is as follows (CSA A23.3-04 Clause 21.4.4.2):
kn =
nl 6 = =1.5 nl − 2 6 − 2
Equation A-18
α1 = 0.85 − 0.0015 f c' = 0.85 – 0.0015×40 = 0.79
Equation A-19
β1 = 0.97 − 0.0025 f c' = 0.97 – 0.0025×40 = 0.87
Equation A-20
409
Po = α1 fc' (Ag − As ) + f y As = 0.79× 40× (300× 440− 6× 200) ×10−3 = 4133.3 kN kp =
Pf Po
=
405 4133.3
Equation A-21
Equation A-22
where α1 is the ratio of average stress in rectangular compression block to the specified concrete strength, β1 the ratio of depth of rectangular compression block to depth to the neutral axis, Po the axial strength of the column, and kp is the ratio of the factored axial load to the axial strength of the column. Therefore, the area of the confinement reinforcement is: Av = 0.2kn k p
Ag f c' sspC d Ach f yh
Equation A-23
but not less than:
Ash = 0.09 s sp C d
f c' 40
= 0.09 × 390 × ssp = 3.51ssp f yh 400
Equation A-24
For Ash = 200 mm2, ssp = 200/3.51=57 mm. The spacing of the hoops shall not exceed (CSA A23.3-04 Clause 21.4.4.3): i.
h 440 = = 110 mm 4 4
ii.
6db = 6 × 15 = 90 mm
iii.
sx =
100 + (350 − hx ) 100 + (350 − 146.7 )
= = 101 mm 3 3
Thus, the stirrups were placed at 90 mm. The length of the region in which the confinement is provided must be greater than the following limits (CSA A23.3-04 Clause 21.4.4.5): 410
i.
1.5d = 1.5× 390 = 585 mm
ii.
lco 2560 = = 427 mm 6 6
A.3.3 Joint Design
As discussed in Section 3.2.5, the joint design in the connections with relocated vertical slot follows the conventional method, while two diagonal compression struts form in the connections with vertical slots located at the face of the column. In order to make a valid comparison, the joints in all connections were designed to have the same shear resistance that is based on the design of joints in connections at which the vertical slots are placed away from the face of the columns. The factored horizontal shear forces in the joints, Vjh,f, are calculated at joint mid-section in accordance to the following expression:
Vjh, f = λc Asb f y,sb −Vcol Vco =
Equation A-25
Fst dv + 0.5Vb hc lc
Equation A-26
The λo is calculated as described in Equation A-5 while Vb corresponds to the applied probable shear force. Also, dv is set to be equal to 0.72 Cd while lco refers to the column span. Since the designated beam-column connection is located at the side of the building frame, three equaldepth beams frame into the joint. Therefore, the factored shear resistance of the joint is (CSA A23.3-04 Clause 21.5.4.1): Vr = 1.6λφ c
f c' A j = 1.6 ×1.0 × 0.65 × 40 × 300 × 440 ×10 −3 =868.2 kN
411
Equation A-27
The spacing of the transverse reinforcement according to CSA A23.3-04 Clauses 21.5.2 and 21.4.4.3 is the smallest of the following: i.
C d 440 = = 110 mm 4 4
ii.
6db = 6×15 = 90 mm
iii.
sx =
100 + (350 − hx ) 100 + (350 − 146.7 )
= = 101 mm 3 3
Thus, 90 mm controls the spacing of the transverse reinforcement in the joint. However, for practicality reasons, the spacing is reduced to 67 mm, and hence, five stirrups were placed inside the concrete joints of all connections.
A.4 Design of SSB and DSB Connections Reinforced using SMA and Control Connection
The SSB-P-1.0 and DSB-P-1.0 were designed to have the same dimensions and design specifications as the SSB-S-1.0 and DSB-S-1.0 connections, respectively, except that PE SMA bars are used instead of the steel reinforcement passing through the vertical slots. Therefore, only the flexural strength of the PE SMA-reinforced connections are different than those reinforced using steel while the beam shear strength, the column flexural and shear strength, and the joint strength for the SSB-P-1.0 and DSB-P-1.0 connexions are similar to those determined in Sections 3.2.1.2, 3.2.2, and 3.2.3, respectively, for the SSB-S-1.0 and DSB-S-1.0 connections, respectively. Based on the design procedure included in Section 3.2.1.1, the Mb,r, Mb,n, and Mb,p of the SSB-P 1.0, DSB-P-1.0, and Control connections are included in Table A-7, Table A-8, and Table A-9, respectively.
412
Table A-7: Design moments of the SSB-P-1.0 connection
Moment
Mb,r (kN·m) 41.1
Positive Mb,n (kN·m) 48.7
Mb,p (kN·m) 79.6
Mb,r (kN·m) 36.8
Negative Mb,n (kN·m) 43.3
Mb,p (kN·m) 71.4
Table A-8: Design moments of the DSB-P-1.0 connection Moment
Mb,r (kN·m) 37.4
Mb,n (kN·m) 44.0
Mb,p (kN·m) 61.5
Mb,n (kN·m) 57.9
Mb,p (kN·m) 70.0
Table A-9: Design moments of the Control connection Moment
Mb,r (kN·m) 47.9
A.5 Fabrication of Connections A.5.1 Steel-Reinforced SSB and DSB Connections
A.5.1.1 Formwork Three formworks were fabricated using 19 mm think plywood sheets. Two formworks were fabricated on a wooden table such that the formwork of the beams in the two frames were connected laterally as shown in Figure A-8 (a) while the third formwork was fabricated on the laboratory floor as shown in Figure A-8 (b). Closely spaced aluminum angles were used to attach the vertical components of the formwork to the bottom side. Horizontal holes were made into the vertical components of the formwork in order to insert 3 mm diameter steel rods that are used to provide lateral bracing for the formwork during concrete casting. The steel rods were positioned in place using steel washers and nuts. The inner sides of the formwork were sprayed with oil prior to cast of concrete in order to facilitate the removal of the connections. 413
(a) Two connected formworks
(b) Isolated formwork
Figure A-8: Fabricated wooden formworks for concrete connections
A.5.1.2 Steel Cage All connections were fabricated and constructed in the structural laboratory at the University of Calgary. The 10 M, 15 M, and 20 M reinforcing bars were provided by local supplier. The beam flexural reinforcement and the shear diagonal reinforcement were cut using a chop saw and bent using the bending machine in order to form the 90o hooks. The diagonal reinforcements were also bent at an angle of 45o two and four times in order to form the specified shape of the diagonal reinforcement in the SSB and DSB connections, respectively. The stirrups were bent using a jig specially fabricated according to the dimensions of the stirrups of the beam and the column. The 25 mm diameter steel tubes (length of 200 mm) were slid along the bottom reinforcements in the SSB connections and along the top and bottom reinforcements in the DSB connections and fixed in their pre-determined positions as shown in Figure A-9 (a). Rubber tubes were also used to seal the ends of the steel tubes in order to prevent the cement past to penetrate inside the steel tubes.
414
Strain gauges were installed at the pre-determined locations prior to assembling the steel cage. The specified maximum capacity of the strain gauges is 3%. The surface of the steel was first grinded and cleaned from dust and dirt. Then, the strain gauges were bonded to the steel surface using a bonding adhesive, left to cure, covered with a layer of wax, and finally wrapped with an electrical tape. Steel cages were assembled by tying the longitudinal and diagonal reinforcements to the beam and column stirrups using steel tie wires. The positions of the stirrups were marked on the longitudinal reinforcement of the columns. Then, all stirrups were slid along the reinforcement and fixed in their position except the stirrups located at the joint since they were kept loose in order to facilitate the installation of the flexural and diagonal reinforcements of the beam. The top beam reinforcement were then installed followed by the installation of the diagonal and bottom beam reinforcements. After that, the beam stirrups were slid and fixed at their pre-marked positions using steel ties. Finally, the column stirrups at the joint were fixed. Plastic chairs were attached to the three sides of the column and the beam in order to maintain the specified concrete cover. Steel anchors designed according to CSA-A23.3-04 (2004) were fabricated by bending two 420 mm – 15 M steel reinforcing bars at an angle of 180o and welding them to a ½ in (12.7 mm) Hex coupling nut as shown in Figure A-9 (b). The steel anchors were attached to the top side of the column cage in order facilitate lifting the specimens and positioning it under the frame. Three steel wires were also fixed to the steel cage using steel ties in order to be able to lift the connections after the cast of concrete.
415
(a) Steel tube
(b) Steel hooks
Figure A-9: Fixing the steel tube and the steel hooks
After assembling the steel cages, 25 mm thick Styrofoam were cut into the dimensions of the vertical slots (300 × 300 mm and 150 × 300 mm for the SSB and DSB connections, respectively). After that, circular holes were made into the Styrofoam at the location of the longitudinal reinforcements and then the Styrofoam sheet was cut into two pieces at the centreline of the circular holes in order to facilitate attaching them to the steel cages. The Styrofoam was attached to the steel cage as shown in Figure A-10 using duct tapes. The still cages of the steelreinforced connections and positioned inside the formworks are shown in Figure A-11.
(a) SSB connections
(b) DSB connections
Figure A-10: Fixing the Styrofoam in connections
416
(a) Connection SSB-S-0
(b) Connection SSB-S-1.0
(c) Connection SSB-S-1.7
(d) Connection DSB-S-0
(e) Connection DSB-S-1.7
(f) Connection DSB-S-0
Figure A-11: Steel cages of SSB and DSB connections reinforced using steel
417
A.5.1.3 Concrete Casting The connections were cast in two batches using ready mix concrete provided by a local concrete supplier, LAFARGE. The workability of the concrete was adjusted on site to match the specified slump value (120 mm) by adding superplasticizers. The concrete was cast first into a steel bucket lifted by the laboratory crane. The bottom of the bucket was then opened in order to pour the concrete into the formwork, while the crane was controlled such that to move the bucket along the length of the connection as shown in Figure A-12 (a). This process was repeated as necessary to cast the three connections. Electrical vibrators were also used frequently to consolidate the concrete as shown in Figure A-12 (b). Twelve 100 mm × 200 mm cylinders were also cast for each batch (refer to Section A.7.1 for concrete compressive test results). After that, the connections and the concrete cylinders were covered by thin plastic sheets for 48 hours after which the wooden formworks were disassembled.
(a) Pouring concrete
(b) Vibrating concrete
Figure A-12: Concrete casting of the connections
418
A.5.2 Control, SSB, and DSB Connections Reinforced using PE SMA
The same formworks used to cast the connections in Section 3.5.1 and the same methods of steel fixing and concrete casting were used to prepare the last set of connections. However, PE SMA bars were used in the SSB-P-1.0 and DSB-P-1.0 connections instead of conventional steel. The steel anchors, couplers, and screws were modified as described in Sections 4.5.2 and 4.6. Two and one additional lines of holes were made into the sides of the steel anchors and couplers, respectively, while the steel screws were flattened using a grinding machine. Each PE SMA bar was cut into two halves in order to match the required length (610 mm) and then fixed into the steel anchors by torqueing the screws until shearing off the heads of the screws as specified by the manufacturer (average torque value of 70 ft·lb (95 N·m)). After that, the anchored PE SMA bars were positioned inside the joint as shown in Figure A-13 and fixed to the diagonal reinforcements using the modified steel couplers as shown in Figure A-14.
Figure A-13: Fixing the anchors to the steel cage
419
Figure A-14: Fixing SMA bars to couplers and to the steel cage
The PE SMA bars were extended for a length of one bar diameter (15 mm) to the other side of the coupler as specified by the manufacturer. The screws were also torqued until shearing off the heads at an average torque of 70 ft·lb (95 N.m). After that, the Styrofoam and the steel hooks were fixed to the steel cages as described in Section 3.5.1.2. The still cages of the SSB and DSB connections reinforced using PE SMA and the Control connection positioned inside the formworks are shown in Figure A-15.
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(a) Connection SSB-P-1.0
(b) Connection DSB-P-1.0
(c) Control Connection Figure A-15: Steel cages of SSB and DSB connections reinforced using PE SMA and Control connection
A.6 Fabrication of Joint-Like Specimens
Six steel cages were fabricated according to the specimen details illustrated in Figure 4-14 while the formwork was fabricated as shown in Figure A-16. Two wooden frames were fabricated such that each frame is used to cast three specimens. Closely spaced aluminum angles were used to attach the vertical components of the formwork to the bottom side. Horizontal holes were made into the vertical components of the formwork in order to insert 3 mm diameter steel rods, which
421
were used to provide lateral bracing for the formwork. The inner sides of the formwork were sprayed with oil prior to cast of concrete in order to facilitate the removal of the connections.
Figure A-16: Fabricated wooden formworks for anchorage specimens
The specimens were cast from one batch using concrete mixed in the Structural Engineering laboratory at the University of Calgary. The specified concrete strength is 40 MPa. The workability of the concrete was adjusted to match the specified slump value (120 mm) by adding superplasticizers. Metal scoops were used to cast the concrete while Electrical vibrators were used to consolidate the concrete as shown in Figure A-17 (a) and (b), respectively. Three 100 mm × 200 mm cylinders were also cast to determine the 28 days strength of the concrete. After that, the connections and the concrete cylinders were covered by thin plastic sheets for 48 hours after which the wooden formworks were disassembled.
422
(a) Pouring concrete
(b) Vibrating concrete
Figure A-17: Concrete casting of the anchorage specimens
A.7 Material Properties A.7.1 Concrete
Concrete compression tests were conducted according to the ASTM C39/C39M (2010). Four concrete batches were cast during the course of this research. In each batch, the average 28 days concrete compressive strength of three 100 mm × 200 mm concrete cylinders was determined as included in Table A-10. The table also includes the average f’c of the concrete compressive tests conducted at testing and at 28 days. This average value is believed to be valid due to the fact that the testing of the beam-column connections, in each group, was conducted at approximately 28 days from concrete cast.
423
Table A-10: Concrete compression test results Concrete batch
A B C D
Specimen
SSB-S-0 SSB-S-1.0 SSB-S-1.7 DSB-S-0 DSB-S-1.0 DSB-S-1.7 Control SSB-P-1.0 DSB-P-1.0 Joint-like specimens
f’c at testing (MPa) 44.4 ± 4.6 42.3 ± 3.7 41.4 ± 3.4 46.9 ± 3.6 40.6 ± 3.1 42.3 ± 1.8 41.7 ± 0.2 44.7 ± 3.8 42.5 ± 4.1
f’c at 28 days (MPa)
Average f’c (MPa)
45.0 ± 4.8
43.3 ± 3.6
40.9 ± 4.2
42.8 ± 3.9
42.5 ± 4.1
43.0 ± 3.1
NA
39.0 ± 1.1
A.7.2 Steel
Steel tension coupon tests were conducted according to ASTM A370 (2010) on each type of steel reinforcement; 10 M, 15 M, and 20 M. Three steel coupon specimens were tested for each type of steel reinforcement. The nominal cross-sectional area of the 10 M, 15 M, and 20 M are 100 mm2, 200 mm2, and 300 mm2, respectively. Each specimen was instrumented with one strain gauge mounted at the centre. The length of each specimen is 508 mm (20 in) with a gauge length of (203 mm) 8 in. The loading rate was set to 1.59 mm/min (0.0625 mm/min). The steel modulus of elasticity, the yield stress, yield strain, ultimate stress, and ultimate strain are included in Table A-11. The yield stress is defined as the stress after which an increase in the strain occurs without an increase in the stress. The tensile modulus of elasticity was calculated using the chord method in which the modulus of elasticity is defined as the slope between two points on the elastic range (slope of the stress-strain curve in the 0.001 to 0.002 strain range).
424
Table A-11: Steel tension coupon test results Reinforcement Type 10 M 15 M 20 M
Es (MPa) 187,127 180,492 194,759
fy (MPa) 484.8 484.0 435.6
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εy (mm/mm) 0.00259 0.00268 0.00224
fu (MPa) 730.4 729.5 588.2
Appendix B: Deformation Analysis Methods B.1 Beam Rotation
The beam section rotation, θb, is calculated using the readings provided by the LSC mounted on the bottom and top of the beam section as follows:
θb =
δb − δt
Equation B-1
bd + d LSC
where δb and δt are bottom and top deformations obtained using LSC devices, respectively, bd is the depth of the beam, and dLSC is the diameter of the LSC (17.5 mm and 25.5 mm for the LSCs of 25 mm and 50 mm gauge lengths, respectively)
B.2 Beam Elongation
The average change in the length of the beam member, δch, is calculated as follows: δ ch =
δ tot ,b − δ tot ,t
Equation B-2
2
where δtot,b and δtot,t are the summation of the LSC readings mounted on the bottom and the top, respectively, of the beam member in the beam-column connections.
In concrete frames, one end of the beam will be subjected to a positive bending while the other is subjected to a negative bending when subjected to lateral loading as shown in Figure B-1. Therefore, the beam elongation, δel, is calculated as follows:
δ el = δ ch+ + δ ch−
Equation B-3
where δch and δch refer to the change in beam length under positive and negative bending, +
−
respectively. 426
(a) Before loading
(b) After loading
Figure B-1: Gaping opening and closing in SSB system
B.3 Beam Shear Distortion
Based on the shear deformation mechanism shown in Figure B-2, the shear distortion, γs, in the beam is calculated as follows: γs =
δ ds ⋅ Dbh 2 + Dbv 2
Equation B-4
2 Dbh ⋅ Dbv
where, δ ds =
(δ
bd ,1
+ δ bd , 2 ) 2
Equation B-5
where δds is the average displacement of the diagonal LSCs and Dbh and Dbv are the horizontal and vertical projections of the diagonal distance between the two pins connecting the LSCs to the concrete surface, respectively.
427
(a) Measured deformations
(b) Shear deformation
Figure B-2: Calculation of beam shear deformation
B.4 Joint Shear Distortion
With reference to the labels of the LSCs shown in Figure B-3 (a), the shear distortion, γs, is calculated as follows
γs =
Δs Djv
Equation B-6
where,
Δs =
δ ds, j
δ ds, j =
δ dh =
δ dv =
Equation B-7
cos R
δ jd,1 + δ jd,2 2
δ jt + δ jb 2
δ jl + δ jr 2
− δ dh − δ dv
Equation B-8
cosR
Equation B-9
sin R
Equation B-10
428
where Δs,j is the horizontal shear displacement in the joint, δds,j the modified average diagonal displacement in the joint, δdh the average horizontal expansion, δdv the average vertical expansion,
Djv the vertical projection of the diagonal distance between the two pins, and R is the angle between the diagonal LSCs and the horizontal axes.
(a) Measured deformations
(b) Horizontal expansion
(c) Vertical expansion
(d) Shear deformation
Figure B-3: Calculation of joint shear deformations
429
B.5 Components of Deformation
The deformation components of the beam-column connection are illustrated in Figure B-4 (The notations shown in the figure are described in the following paragraphs). The objective of this deformation analysis is to determine the components of deformation; beam, column, and joint distortion and to examine the effectiveness of relocating the plastic hinge by calculating the drift ratio. Beam tip displacement, Δb, is calculated as follows
Δ b = Δ b1 + Δ b 2
Equation B-11
where Δb1 is extrapolated displacement of the beam member in the l region (relocation distance and half of the joint width) and Δb2 is the beam displacement in ls region (shear span).
The Δb1 is calculated as the multiplication of the beam drift ratio in region l, θb1, by the length, L, as follows:
Δ b1 = θ b1 ⋅ L
Equation B-12
where,
θ b1 =
Δ b1,l l
Equation B-13
Δ b1,l = Δ i,l − Δ1 − Δ 2
Equation B-14
Δ1 = α j ⋅ l
Equation B-15
where Δb1,l is the displacement of the beam member in the l region shown in Figure B-4, Δi,l the total displacement at l, Δ1 the beam displacement due to the shift of the joint, Δ2 the beam displacement due to the rotation of the joint, jd is the depth of the joint, αj and βj are the rotation
430
angles of the vertical and the horizontal components of the joints with respect to the x and y-axes, respectively.
The Δb2 is calculated as the subtraction of the Δb1, the extrapolated displacement due to column deformation, Δco, and the extrapolated displacement due to joint distortion, Δs, from the applied displacement at Lt, Δi,Lt, as follows:
Δb2 = Δi,Lt − Δb1 − Δco − Δs
Equation B-16
where,
Δ co = θ co ⋅ L
θ co =
Equation B-17
2Δ co,l lco
Equation B-18
Δ co,l = Δ 3 + Δ 4 Δ3 = α j ⋅
Equation B-19
jh 2
Equation B-20
j l Δ4 = β j ⋅ co − h 2 2
Equation B-21
Δ s = θ s ⋅ Lt
Equation B-22
θ js = θ1 − θ b1 − θ co
Equation B-23
θ1 =
Δ i,l l
Equation B-24
431
where θco, θjs, θ1, θb1 are the rotations due to column deformation, joint shear deformation, total deformation in region l, and beam deformation in region l, respectively, Δco,l is the displacement due to column deformation at l, Δ3 the fictitious column displacement due to the shift of the joint, Δ4 the fictitious column displacement due to the rotation of the joint, and jh is the joint height.
Therefore, the beam, column, and joint deformations are calculated using Equations B-12, B-19, and B-23, respectively. If the vertical slot is placed at the face of the column, l is substituted by Lt in all the above equations, where Lt is the length of the beam member and half of the joint. The rotation ratio θ1 / θ2, used to examine the effectiveness of relocating the plastic hinge, is defined as the ratio of θ1 calculated using Equation B-25 to the total rotation in region ls, θ2, as follows: Equation B-25
θ 2 = θ1 + θ b 2
θb2 =
Δ b2 (Lt − l )
Equation B-26
where θb2 is the rotation due to the beam deformation in region Lt.
The only unknowns in the above calculation procedure are the αj and βj factors. The DICT method is used to measure the αj and βj factors during testing. The vertical and horizontal components of deformation of four points located at the corners of the joint are tracked and analyzed in order to calculate the αj and βj factors.
432
Figure B-4: Deformation components in the beam-column connections
433
Appendix C: Supplementary Test Results
This appendix includes supplementary test results in reference to Chapter 4. It includes additional test results of the joint-like specimens and the positive bending test of the DSB-P-1.0 connection. The discussion of the joint-like specimen test results are included in Section 4.51.5 while the discussion of the positive bending test results are included in Section 4.10.1.
C.1 Joint-Like Specimens
500
500
400
400
Stress, σ (MPa)
Stress, σ (MPa)
The hysteretic response of PE SMA bars in the joint-like specimens are shown in Figure C-1.
300
200
300
200
100
100
0
0 0
0.01
0.02
0.03
0.04
0.05
0
0.06
0.01
(a) AN-P-1
0.03
0.04
0.05
0.06
0.05
0.06
(b) AN-P-2
500
500
400
400
Stress, σ (MPa)
Stress, σ (MPa)
0.02
Strain, ε (mm/mm)
Strain, ε (mm/mm)
300
200
100
300
200
100
0
0 0
0.01
0.02
0.03
0.04
0.05
0.06
0
0.01
Strain, ε (mm/mm)
(c) AN-P-3
0.02
0.03
0.04
Strain, ε (mm/mm)
(d) AN-P-4
Figure C-1: Hysteretic response of the PE SMA bars in the joint-like specimens
434
The strain distribution across the mid-section of the joint-like specimens is shown in Figure C-2. It is noted that the strain gauges of the AN-P-3 specimen were malfunctioned near the ultimate condition. The strain gauge at the top longitudinal reinforcement in the AN-S-1 was also malfunctioned near failure. The LSC devices mounted at the top fibres of the AN-S-1 and AN-S2 specimens were detached from the concrete surface due to the extensive cracking near failure, and thus, were excluded from the analysis.
Specimen depth (mm)
-40
-120
-200
AN-S-1 (144 Kn) AN-S-2 (145 kN) AN-P-1 (72 kN) AN-P-3 (79 kN) AN-P-4 (76 kN)
-280
-360
-440 -0.0005
0
0.0005
0.001
0.0015
0.002
0.0025
Strain, ε (mm/mm)
Figure C-2: Strain distribution at the mid-span section of the joint-like specimens
The strain distributions along the clear length of the modified anchors and couplers are shown in Figure C-3 and Figure C-4, respectively. The discussion with regard to the strain distribution of the modified anchors and couplers are included in Section 4.5.2.
435
1
0.8
0.8
Normalized gauge length
Normalized gauge length
1
0.6
0.4
200 MPa 400 MPa 483 MPa
0.2
0.6
0.4
200 MPa 400 MPa
0.2
0
0
0
0.02
0.04
0.06
0.08
0
0.02
Strain, ε (mm/mm)
(a) Phase 1
0.06
0.08
0.06
0.08
(b) Phase 2
1
1
0.8
0.8
Normalized gauge length
Normalized gauge length
0.04
Strain, ε (mm/mm)
0.6
0.4
200 MPa 400 MPa 462 MPa
0.2
0.6
0.4
200 MPa 400 MPa 500 MPa 590 MPa
0.2
0
0 0
0.02
0.04
0.06
0.08
0
0.02
Strain, ε (mm/mm)
0.04
Strain, ε (mm/mm)
(c) Phase 3
(d) Phase 4-A
Normalized gauge length
1
0.8
0.6
0.4
200 MPa 400 MPa 500 MPa 529 MPa
0.2
0 0
0.02
0.04
0.06
0.08
Strain, ε (mm/mm)
(e) Phase 4-B Figure C-3: Strain distribution along the normalized gauge length of the modified anchors
436
1
0.8
0.8
Normalized gauge length
Normalized gauge length
1
0.6
0.4
200 MPa 400 MPa 500 MPa 581 MPa
0.2
0.6
0.4
200 MPa 400 MPa 500 MPa 533 MPa
0.2
0
0 0
0.02
0.04
0.06
0.08
0
0.02
Strain, ε (mm/mm)
(a) Specimen A
0.04
0.06
0.08
Strain, ε (mm/mm)
(b) Specimen B
Figure C-4: Strain distribution along the normalized gauge length of the modified coupler
C.2 Positive Bending Test of the DSB-P-1.0 Connection
The test setup and the application of the vertical load onto the column member are the same as discussed in Section 3.5. The quasi-static loading history implemented in the positive bending test of the DSB-P-1.0 connection is shown in Figure 3-15. The quasi-static cycles were applied at the tip of the beam member via a 250 kN MTS 10” stroke actuator reacting against a cross-head Isteel beam connected to the steel frame which is anchored to the laboratory strong floor. The first displacement cycle was conducted at 25 mm and then increased by increments of 25 mm until it reached 125 mm (note that the maximum displacement attained in the positive-negative cyclic bending test is 120 mm). After that, the increment of the displacement were reduced to 5 mm up to the failure of the specimen while a constant loading rate of 1.5 mm/sec was maintained throughout the testing. The hysteretic response of the connection under positive bending loading is shown in Figure C-6. The bottom PE SMA bar fractured at 60 mm (4.4% drift), followed by buckling of the top
437
reinforcement at 165 mm (12.2% drift), and then fracture of the top PE SMA reinforcement at 189 mm (13.9% drift). The cracking pattern and fracture of the top PE SMA bar at the end of the test are shown in Figure C-7. 160 1.5 mm/sec
140
Displacement (mm)
5 mm
120 100 80 60
25 mm
40 20 0
Time
Figure C-5: Quasi-static loading history of the positive bending test of the DSB-P-1.0 connection
Figure C-6: Hysteretic response of the DSB-P-1.0 connection subjected to positive bending
438
Figure C-7: Cracking and fracture of top PE SMA bar at failure in the DSB-P-1.0 connection subjected to positive bending test
439
Appendix D: Verification of the TDE model D.1 Introduction
The TDE deformation theory is validated in this section against the shear response of RC structural shear walls subjected to the combined effect of axial and lateral loading. The deformation model is then used in Chapter 5 to examine the sliding shear performance of the SSB and DSB connections.
D.2 Description of the Examined Shear Walls
The Reinforced Concrete (RC) shear walls tested by Zhang and Wang (2000) are examined in this research. The details and dimensions of the two tested specimens are shown in Figure D-1. The dimensions of the walls are 100 mm × 700 mm × 1750 mm. The steel reinforcement number and configuration in the two specimens are the same except for the boundary reinforcement (Ø 14 mm in the SW7 specimen and Ø 12 mm in the SW8 specimen). The 28 days concrete compressive strength is 33.26 MPa and 35.83 MPa for SW7 and SW8 specimens, respectively. The vertical reinforcement ratio is 2.19% and 1.72% in the S-element and C-element, respectively, (assuming that the dimension of the S-element is the same as the C-element) while the horizontal reinforcement ratio is 1.12% in both specimens. The yield stress of the Ø 6 mm, Ø 8 mm, Ø 12 mm, and Ø 14 mm are 366 MPa, 305 MPa, 432 MPa, and 405 MPa, respectively. The specimens were subjected to combined axial load and quasi-static cyclic lateral displacement at a height of 1500 mm from the top of the base foundation. The axial loads on the SW7 and SW8 specimens are 499 kN and 784 kN, respectively.
440
Figure D-1: Details of the tested shear walls (reproduced from Zhang and Wang (2000), all dimensions are in mm)
D.3 Flexure-Axial-Shear Interaction
The envelope of the lateral force versus deformation of the shear walls is controlled by the flexural response, shear response, or a combination of both. The flexural-axial interaction is determined based on the conventional flexural theory in which plane sections are assumed to remain plane before and after bending and the concrete contribution in tension is zero. The degradation of the shear capacity is evaluated using the following steps:
• Step 1: Determine the section at which the shear capacity is evaluated. • Step 2: Determine the type of elements (C-element, S-element, or both). The thickness of the C- and S-elements equal to half the stirrup spacing.
• Step 3: Determine the mesh size and the number of elements, n.
441
• Step 4: Calculate the axial strain obtained from the flexural analysis and induced in each element, εax,n.
• Step 5: Calculate the average longitudinal strain values, εax,avg, based on the values obtained in Step 4 according to the following expression:
ε ax,avg =
ε ax,n + ε ac,n +1
Equation D-1
2
• Step 6: Determine the shear stress based on the TDE model • Step 7: Sum the shear stresses of all elements to obtain the total shear capacity.
The ultimate deformation of the walls is controlled by the crushing strain of the confined concrete (flexural failure) or the ultimate shear strain (shear failure). The crushing strain, εcu, of the confined concrete is calculated using the following expression (Mander et al., 1988):
ε cu = 0.004 + 1.4
ρv f y,hε su
Equation D-2
f cc'
where ρv is the volumetric ratio of the transverse reinforcement (for rectangular sections ρv = ρx +
ρy, where ρx and ρy are the volumetric ratio of the steel reinforcement in the x and y-directions, respectively), fy,h the yield stress of the hoops, εsu the ultimate strain of the hoops, and f’cc is the confined concrete compressive stress.
442
The f’cc is assumed to be the same as f’c which is a valid assumption for concrete confined with rectangular hoops (Park and Paulay, 1975). The ultimate shear strain, γu, is determined using the following expressions developed by Gerin and Adebar (2004): vy γu = 4 − 12 ' ; γy fc
γy =
f yh Eh
+
(v
− nax ) 4v y ; + ρ v Eh Ec y
v y ≤ 0.25f c'
0≤
(v
y
− nax )
ρ v Eh
Equation D-3
≤
f yh Eh
Equation D-4
where γy and vy are the shear strain and stress at yielding of the hoop reinforcement, respectively,
nax is the axial stress, ρv the volumetric ratio of the transverse reinforcement steel, and Eh is the modulus of elasticity of the hoop reinforcement.
D.4 Analysis Results
The shear capacity of the S-element (V Capacity – S), the shear capacity of the C-element (V Capacity – C), the flexural response (M Capacity), and the experimental moment-curvature relationships (M Capacity) of the SW7 and SW8 specimens are plotted in Figure D-2. The shear capacity in the S-element increases with increasing curvature and then stabilize once the yield stress is reached. Unlike the S-element, the shear capacity in the C-element degrades with increasing curvature. The reduction in the capacity of the C-element is due to the decrease in the contribution of the aggregate interlock shear resistance as a result of crack widening. It can be seen that the shear capacity of the C-element controls the shear response of the walls. The intersection of the shear capacity curve of the C-element with the theoretical moment-curvature relationship represents the predicted ultimate strength of the wall. Therefore, the predicated ultimate strength 443
of the SW7 and SW8 shear walls is 274 kN and 303 kN, respectively. The percentage error with the respect to the experimental results is 2.0% and 2.6%, respectively. The concrete compressive crushing strain calculated using Equation D-2 for the SW7 and SW8 specimens is 1.84% and 1.74%, respectively, while the ultimate shear strain for the SW7 and SW8 specimens calculated using Equation D-3 is 0.981% and 0.977%, respectively. The analysis predicts that the shear strain in the concrete element is reached at concrete strain values at the extreme compression fibre of 0.55% and 0.54% for the SW7 and SW8 specimens, respectively. It can be seen, therefore, that the shear strain controls the ultimate deformation of the specimens. The predicted ultimate curvature for the SW7 and SW8 specimens is 0.0254 m-1 and 0.0188 m-1, respectively, while the percentage error is 15.4% and 10.3%, respectively. Based on the above analysis, the strength and the deformation were found to be controlled by the shear capacity of the
600
600
500
500
400
400
M (kN.m)
M (kN.m)
C-element.
300
V Capacity - C V Capacity - S M Capacity Experiment
200 100 0 0
0.005
0.01
0.015
0.02
0.025
300
V Capacity - C V Capacity - S M Capacity Experiment
200 100 0
0.03
0
0.005
0.01
Ψx (mm/mm)
(a) Specimen SW7
0.015
0.02
0.025
Ψx (mm/mm)
(b) Specimen SW8
Figure D-2: Moment-curvature relationships comparison between the TDE results and the experimental results of shear walls
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D.5 Conclusions
The TDE shear theory is capable of predicting the shear deformation and strength responses of various RC structural systems. The basic idea of the theory is that it distinguishes between the shear responses of the regions containing concrete and longitudinal reinforcement, and the regions that contain concrete, stirrups, and longitudinal reinforcement without losing the compatibility between the two types of regions. The model was implemented to predict the strength and deformation response of RC structural walls. It was predicted that the strength and deformation were controlled by the shear capacity of the concrete element. The model showed good agreement with the experimental behaviour. Thus, it is used to predict the sliding shear effect at the concrete hinge in the SSB and DSB connections in Chapter 5.
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