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Seismic demands on gravity-load columns of reinforced concrete shear wall buildings Bazargani, Poureya 2014

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SEISMIC DEMANDS ON GRAVITY-LOAD COLUMNS OF REINFORCED CONCRETE SHEAR WALL BUILDINGS by POUREYA BAZARGANI B.Sc. Shiraz University, 2006 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2014 © Poureya Bazargani, 2014 ii  Abstract In shear wall buildings, walls serve as the seismic force resisting system while the gravity-load system consists of columns that are primarily designed to carry the weight of the building through frame action and are not detailed for seismic ductility. Design codes require the gravity-load system to be checked for deformation compatibility as the building deforms laterally. The process of checking the columns for adequate deformability still requires more work. In addition to flexural deformations, components such as shear strain and rotation of the foundation contribute significantly to lateral deformations in the wall plastic hinge zone. Shear strains in flexural shear walls are analytically shown to be a result of large vertical tensile strains in areas with inclined cracks. Based on this theory, a simple design-oriented method for estimating shear strain profile of flexural shear walls is formulated, the accuracy of which is verified against experimental results from works of other researchers. Rotation of shear wall foundations is studied through performing about 2000 Nonlinear Time-History Analysis (NTHA) considering the nonlinear interaction between the foundation and the underlying soil. Behaviour of shear walls accounting for foundation rotation is explained with emphasis on relative wall to foundation strengths. A simple method for obtaining the monotonic foundation moment-rotation response is formulated which is then used in a simple step-by-step method for estimating foundation rotation in a given shear wall building. Curvature demand on columns pushed to a given wall deformation profile is studied using a structural analysis algorithm specifically designed for the task. In the absence of wall shear strain or significant foundation rotation, column curvature demand is found to remain close to the wall maximum curvature. Wall shear strain and foundation rotation are found to cause severe increase to column curvature demand. In a parametric study on column curvature demand, parameters including wall length, column length, height of column plastic hinge zone, first storey height, fixity of the column at grade level, and the effect of members framing into the column are studied. Several simple expressions for estimating column curvature demand are derived that can be implemented in design. iii  Preface Unless otherwise noted below or in the body of the thesis, the research presented in this document is an original contribution by the author, Poureya Bazargani under the supervision of Professor Perry Adebar at the University of British Columbia, Vancouver. Research needs were identified and the research program was designed and developed under the guidance of Professor Adebar. A condensed version of ‎CHAPTER 2 is published in Adebar et al. (2012) of which I am a coauthor. A journal paper summarizing the content of ‎CHAPTER 2 was submitted to the ASCE Journal of Structural Engineering in February 2014 of which I am the author. Computer simulation results of ‎CHAPTER 3 had a direct impact on the foundation design clauses of the 2015 edition of the National Building Code of Canada (NBCC). Parts of ‎CHAPTER 3 and ‎CHAPTER 4 were published in Adebar et al. (2014) of which I am a coauthor. Figure ‎3.3 is reprinted from Anderson (2003) with written permission from D. L. Anderson. Data in Table ‎3.8, Table ‎3.9, and Table ‎3.10 was taken from Das (2001). Figure ‎3.15, Figure ‎3.16, and Figure ‎3.17 are reprinted from Negro et al. (1998) with written permission from P. Negro. Figure ‎4.1 is taken from Allotey and El Naggar (2003) and reprinted here with El Naggar’s permission. A more detailed version of the figure appears in FEMA 274. Parts of ‎CHAPTER 6 and ‎CHAPTER 7 were published in Bazargani and Adebar (2010) of which I am the author. Versions of Figure ‎5.13, Figure ‎7.2, and Figure ‎7.10 were published in Adebar et al. (2010) of which I am a coauthor.    iv  Table of Contents Abstract ......................................................................................................................................... ii Preface .......................................................................................................................................... iii Table of Contents ......................................................................................................................... iv List of Tables ................................................................................................................................ xi List of Figures ............................................................................................................................ xvi List of Symbols ....................................................................................................................... xxxiv Acknowledgements .............................................................................................................. xxxviii Dedication ............................................................................................................................... xxxix CHAPTER 1 Introduction ..................................................................................................... 1 1.1 Overview of Problem ....................................................................................................... 1 1.2 Important Elements of the Problem ................................................................................. 3 1.2.1 Wall shear strain ....................................................................................................... 5 1.2.2 Rotation of shear wall foundations ........................................................................... 6 1.3 Moment-Curvature Response of Gravity-load Columns ................................................. 7 1.4 Main Parts of the Current Study ...................................................................................... 8 1.5 Summary of Research Objectives .................................................................................... 9 1.5.1 Shear strains in flexural shear walls (‎CHAPTER 2) ................................................ 9 1.5.2 Foundation rotation of cantilever shear walls ......................................................... 10 1.5.3 Curvature demands on gravity-load columns in the plastic hinge region of shear wall buildings with flat plate floor slabs............................................................................... 10 1.6 Thesis Overview ............................................................................................................ 11 CHAPTER 2 Shear Deformation of Flexural Shear Walls ............................................... 14 2.1 Overview of the Chapter ................................................................................................ 14 v  2.2 Experimental Evidence of Wall Shear Strain from Previous Researchers .................... 15 2.3 Existing Models for Estimating Wall Shear Deformation ............................................. 18 2.4 Finite Element Analysis of Reinforced Concrete Structures Using VecTor2................ 23 2.4.1 Previous works on verification of VecTor2 ............................................................ 24 2.5 Verification of VecTor2 for Predicting Shear Strains in Walls ..................................... 25 2.5.1 Specimens RW2 and TW2 tested by Thomsen and Wallace (1995) ...................... 26 2.5.2 Specimen NTW1 tested by Brueggen (2009) ......................................................... 35 2.5.3 Summary of verification study ............................................................................... 39 2.6 Example 10-storey Rectangular Wall ............................................................................ 40 2.6.1 Calculating flexural and shear deformations .......................................................... 41 2.6.2 General observations on shear wall behaviour ....................................................... 44 2.7 A Simple Method for Estimating Average Storey Shear Strain .................................... 50 2.7.1 Shear strain of a biaxial stress RC element ............................................................ 50 2.7.2 Estimating average storey vertical strain ................................................................ 52 2.7.3 Estimating average storey strain angle ................................................................... 55 2.8 Parametric Study on Average Storey Principle Strain Angle ........................................ 60 2.8.1 Effect of vertical compressive stress ...................................................................... 61 2.8.2 Effect of vertical steel ratio ..................................................................................... 68 2.8.3 Effect of wall length ............................................................................................... 74 2.8.4 Effect of wall aspect ratio ....................................................................................... 81 2.8.5 Effect of number of floor slabs in the wall plastic hinge region ............................ 89 2.9 Shear Strain Model......................................................................................................... 92 2.9.1 Verification of the proposed shear strain model using walls considered in the parametric study.................................................................................................................... 94 2.9.2 Verification of the proposed shear strain model using real test data ...................... 95 vi  2.10 Conclusions ................................................................................................................ 99 CHAPTER 3 Nonlinear Analysis of Shear Walls Accounting for Foundation Rotation ………………….. ...................................................................................................................... 101 3.1 Introduction .................................................................................................................. 101 3.1.1 Dynamic response of foundations in the elastic range ......................................... 102 3.1.2 Existing approaches to numerical modeling of soil-structure interaction ............ 102 3.1.3 Existing design procedures for accounting for foundation rotation ..................... 105 3.1.4 Selected experiments on rotational response of foundations by other researchers ……….. .............................................................................................................................. 107 3.1.5 Anderson (2003) ................................................................................................... 108 3.1.6 Other Canadian research on shear walls with flexible foundations ...................... 111 3.1.7 Discussion ............................................................................................................. 115 3.2 Numerical Modeling and Analysis Method ................................................................. 116 3.2.1 Modeling of the shear wall ................................................................................... 117 3.2.2 Modeling of the footing ........................................................................................ 124 3.2.3 Modeling of the soil-structure interaction ............................................................ 124 3.2.4 Soil properties used in NTHA .............................................................................. 130 3.2.5 Input ground accelerations used in nonlinear time-history analysis ..................... 139 3.2.6 Soil damping ......................................................................................................... 142 3.2.7 Roadmap to the NTHA ......................................................................................... 148 3.3 Sensitivity of Wall-foundation System Response to Soil Properties ........................... 149 3.3.1 Effect of soil type.................................................................................................. 150 3.3.2 Effect of soil stiffness ........................................................................................... 153 3.3.3 Effect of soil ultimate bearing capacity ................................................................ 155 3.4 Scatter in Wall Maximum Response ............................................................................ 159 3.5 Effects of Wall Height and Mass Ratio (MR) .............................................................. 162 vii  3.6 Core NTHA .................................................................................................................. 166 3.6.1 General observations ............................................................................................ 168 3.6.2 Global drift and top wall displacement ................................................................. 176 3.6.3 Base rotation ......................................................................................................... 178 3.6.4 Interaction between shear wall and foundation strengths ..................................... 180 3.6.5 Permanent deformations in the soil ...................................................................... 183 3.6.6 Period lengthening due to rotation of the foundation ........................................... 191 3.6.7 Reduction in maximum bending moment and shear force due to rotation of the foundation ........................................................................................................................... 193 3.7 Summary and Conclusions........................................................................................... 196 CHAPTER 4 Simple Methods for Predicting the Response of Shear Walls Accounting for Foundation Rotation........................................................................................................... 199 4.1 Foundation Moment-Rotation Response ..................................................................... 200 4.1.1 Allotey and El Naggar’s method for predicting foundation moment-rotation response  ............................................................................................................................. 200 4.1.2 Soil spring backbone curves ................................................................................. 203 4.1.3 Foundation response in elastic range .................................................................... 203 4.1.4 Equivalent rectangular stress block (ERSB) concept ........................................... 206 4.1.5 Example predicted foundation moment-rotation curves ....................................... 212 4.2 Estimating Top Wall Displacement ............................................................................. 214 4.3 A Simple Method for Estimating the Displacement Profile of Shear Walls Accounting for Foundation Rotation .......................................................................................................... 230 4.3.1 Elastic displacements ............................................................................................ 231 4.3.2 Hinging shear wall ................................................................................................ 239 4.3.3 Non-hinging shear wall......................................................................................... 240 4.3.4 Prediction of foundation rotation from NTHA results ......................................... 242 viii  4.4 Summary and Conclusions........................................................................................... 244 CHAPTER 5 Moment-Curvature Response of Reinforced Concrete Gravity-load Columns ………. ....................................................................................................................... 246 5.1 Introduction .................................................................................................................. 246 5.2 Moment-Curvature Behaviour of Reinforced Concrete Columns ............................... 247 5.2.1 Probable compressive axial load on gravity-load columns accompanied by seismic forces .................................................................................................................................. 247 5.2.2 Column cross-section aspect ratios ....................................................................... 250 5.2.3 Sectional analysis procedure ................................................................................. 252 5.2.4 Moment-curvature analysis results ....................................................................... 253 5.2.5 A simple approximate approach ........................................................................... 255 5.3 Effect of Creep on Column Moment-Curvature Response .......................................... 257 5.4 Effect of Damage to the Column on Moment-Curvature Response ............................ 260 5.5 Neutral Axis Depth of Gravity-load Columns at Failure ............................................. 264 5.6 Summary and Conclusions........................................................................................... 270 CHAPTER 6 Structural Analysis of Gravity-load Columns Connected to Shear Walls with Flat Plate Floor Slabs ....................................................................................................... 272 6.1 Introduction .................................................................................................................. 272 6.2 Literature Review on the Behaviour of Gravity-load Column under Combined Axial Compression and Flexural Loading ........................................................................................ 273 6.2.1 Ibrahim and MacGregor (1996) ............................................................................ 273 6.2.2 Lloyd and Rangan (1996) ..................................................................................... 274 6.2.3 Legeron and Paultre (2000) .................................................................................. 275 6.2.4 Bae and Bayrak (2003) ......................................................................................... 276 6.2.5 Bae and Bayrak (2008) ......................................................................................... 276 6.2.6 Discussion and Summary ..................................................................................... 278 ix  6.3 Inelastic Curvature Concentration in Gravity-Load Columns ..................................... 279 6.4 Nonlinear Structural Analysis Procedure ..................................................................... 282 6.5 Wall Displacement Profile used in the Pushover Analysis .......................................... 285 6.5.1 Flexural deformation ............................................................................................ 285 6.5.2 Shear deformation (strain) .................................................................................... 288 6.6 A Demonstrative Example ........................................................................................... 289 6.6.1 Finite element (FE) analysis procedure ................................................................ 289 6.6.2 Bilinear wall model vs. deformation profile from FE analysis ............................ 291 6.6.3 Solution using the proposed nonlinear structural analysis method ...................... 294 6.7 Shear Strains in Gravity-Load Columns ...................................................................... 297 6.8 Number of Constant Curvature Elements Required for an Accurate Estimate of Column Curvature Demand .................................................................................................................. 299 6.9 Number of Floors Required for an Accurate Estimate of Column Curvature Demand301 6.10 Summary and Conclusion ........................................................................................ 302 CHAPTER 7 Parametric Study on Seismic Demands on Gravity-load Columns in Shear Wall Buildings with Flat Plate Floor Slabs ............................................................................ 304 7.1 Introduction .................................................................................................................. 304 7.2 Standard Parameters ..................................................................................................... 305 7.3 Wall Shear Strain (γwall) ............................................................................................... 306 7.3.1 Simple methods for estimating column curvature demand due to imposed wall deformation in the presence of wall shear strain ................................................................ 310 7.4 Column Length ............................................................................................................ 316 7.5 Wall Length (lpw*)........................................................................................................ 322 7.6 Height of Column Plastic Hinge Zone (lpc*) ................................................................ 325 7.7 Effect of Damage to the Column on its Curvature Demand ........................................ 328 x  7.7.1 Simple methods for estimating curvature demand of damaged columns due to imposed wall deformation in the presence of wall shear strain .......................................... 336 7.8 Taller First Storey ........................................................................................................ 339 7.9 Fixity of the Column at the Base ................................................................................. 347 7.9.1 Simple methods for estimating column curvature demand due to imposed wall deformation with column continuing below grade level .................................................... 360 7.10 Inter-storey Drift....................................................................................................... 367 7.11 Effect of Members Framing into the Column on its Curvature Demand ................. 372 7.12 Effect of Foundation Rotation .................................................................................. 379 7.13 Summary and Conclusions ....................................................................................... 380 CHAPTER 8 Summary of Contributions and Recommendations for Future Work ... 382 8.1 Overview of Contributions........................................................................................... 382 8.2 Shear Strains in Plastic Hinge Region of Flexural Shear Walls .................................. 382 8.3 Rotation of Shear Wall Foundations ............................................................................ 384 8.4 Deformation Demands on Gravity-load Columns ....................................................... 386 8.5 Recommendations for Future Work ............................................................................. 388 Bibliography .............................................................................................................................. 390 Appendix A       Rotation of Shear Wall Foundations………...……..……………..……….397 Appendix B     Calculations for Probable Seismic Compressive Axial Force on Gravity-load Columns based on Provisions of NBCC 2005……………………………………….…….…617 Appendix C     Calculations for Probable Seismic Compressive Axial Force on Gravity-load Columns based on Provisions of ASCE 7-05……………………………..……………….…619 Appendix D    Mathematical Presentation of the Nonlinear Structural Analysis Algorithm used to Analyze Gravity-load Columns under Imposed Lateral Deformations…………..621  xi  List of Tables Table ‎2.1 Properties of 10-storey rectangular walls used to study the effect of compressive axial stress on average storey principle strain angle ............................................................................. 62 Table ‎2.2 Properties of 10-storey flanged walls used to study the effect of compressive axial stress on average storey principle strain angle ............................................................................. 63 Table ‎2.3 Analysis results: 10-storey rectangular walls with various compressive axial stresses (Note: values of ‘c’ reported were used to back-calculate the average storey strain angle and do not represent the actual concrete compression depth of the wall at the given global drift).......... 65 Table ‎2.4 Analysis results: 10-storey flanged walls with various compressive axial stresses (Note: values of ‘c’ reported were used to back-calculate the average storey strain angle and do not represent the actual concrete compression depth of the wall at the given global drift).......... 66 Table ‎2.5 Properties of 10-storey rectangular walls used to study the effect of amount of distributed vertical reinforcement in the web region on average first storey principle strain angle. ...................................................................................................................................................... 69 Table ‎2.6 Properties of 10-storey flanged walls used to study the effect of amount of distributed vertical reinforcement in the web region on average first storey principle strain angle............... 70 Table ‎2.7 Analysis results: 10-storey rectangular walls with various amounts of distributed vertical reinforcement in the web region (Note: values of ‘c’ reported were used to back-calculate the average storey strain angle and do not represent the actual concrete compression depth of the wall at the given global drift). .................................................................................. 71 Table ‎2.8 Analysis results: 10-storey flanged walls with various amounts of distributed vertical reinforcement in the web region (Note: values of ‘c’ reported were used to back-calculate the average storey strain angle and do not represent the actual concrete compression depth of the wall at the given global drift). ....................................................................................................... 72 xii  Table ‎2.9 Properties of rectangular walls used to study the effect of wall length on the average storey principle strain angle. ......................................................................................................... 76 Table ‎2.10 Properties of flanged walls used to study the effect of wall length on the average storey principle strain angle. ......................................................................................................... 77 Table ‎2.11 Rectangular walls: analysis results on the effect of wall length on average storey principle strain angle (Note: values of ‘c’ reported were used to back-calculate the average storey strain angle and do not represent the actual concrete compression depth of the wall at the given global drift). .................................................................................................................................. 78 Table ‎2.12 Flanged walls: analysis results on the effect of wall length on average storey principle strain angle (Note: values of ‘c’ reported were used to back-calculate the average storey strain angle and do not represent the actual concrete compression depth of the wall at the given global drift). ............................................................................................................................................. 79 Table ‎2.13 Properties of rectangular walls used to study the effect of wall aspect ratio on the average principle strain angle. ...................................................................................................... 82 Table ‎2.14 Properties of flanged walls used to study the effect of wall aspect ratio on the average principle strain angle. .................................................................................................................... 83 Table ‎2.15 Rectangular walls: analysis results for the effect of wall aspect ratio on the average principle strain angle (Note: values of ‘c’ reported were used to back-calculate the average storey strain angle and do not represent the actual concrete compression depth of the wall at the given global drift). .................................................................................................................................. 84 Table ‎2.16 Flanged walls: analysis results for the effect of wall aspect ratio on the average principle strain angle (Note: values of ‘c’ reported were used to back-calculate the average storey strain angle and do not represent the actual concrete compression depth of the wall at the given global drift). .................................................................................................................................. 85 Table ‎2.17 Analysis results for the effect of number of slabs in the plastic hinge region of wall 15STRW-L10 on the average principle strain angle (Note: values of ‘c’ reported were used to xiii  back-calculate the average storey strain angle and do not represent the actual concrete compression depth of the wall at the given global drift). ............................................................. 91 Table ‎3.1 Specifications of nonlinear 10-storey shear walls. ..................................................... 120 Table ‎3.2 Factored, nominal, and probable bending strengths of the four nonlinear walls considered in the NTHA (note: wall factored bending strength was calculated using material strength reduction factors of 0.65 and 0.85 for concrete and the reinforcing steel respectively). .................................................................................................................................................... 121 Table ‎3.3 Values of Rw calculated using the various definitions of wall bending strength........ 121 Table ‎3.4 Input parameters for the concrete material model. ..................................................... 123 Table ‎3.5 Input parameters for the steel material model. ........................................................... 123 Table ‎3.6 QzSimple1 material constants for clay and sand type soils. ....................................... 127 Table ‎3.7 Soil properties used by previous researchers to study foundation rotation. ............... 130 Table ‎3.8 Typical values for soil elastic modulus – Data from Das (2001). .............................. 131 Table ‎3.9 Typical values of soil Poisson’s ratios – Data from Das (2001). ............................... 131 Table ‎3.10 Typical values of internal friction angle of soils – Data from Das (2001). .............. 132 Table ‎3.11 Soil properties obtained through personal communication with Dr. Wijewickreme. .................................................................................................................................................... 132 Table ‎3.12 Soil properties used in NTHA. ................................................................................. 134 Table ‎3.13 Soil stiffness properties needed to calculate the initial elastic rotational stiffness of the foundations as modeled in the NTHA. ................................................................................. 139 Table ‎3.14 Summary of nonlinear time-history analysis preformed in this chapter. ................. 149 Table ‎3.15 Soil properties used in parametric study on soil properties. ..................................... 150 xiv  Table ‎3.16 Length of square footings modeled in the Core NTHA ........................................... 167 Table ‎3.17 Residual top wall displacement and maximum soil compressive displacements for wall 10R13 on Clay with Rf=3.2 from NTHA using the 10 spectrally-matched records. .......... 184 Table ‎4.1 Equivalent rectangular stress block parameters for Clay and Sand type soils. .......... 212 Table ‎5.1 Typical ratios of compressive axial load on gravity-load columns from seismic load-case to that from gravity load-case based on provisions of a) NBCC 2005, and b) ASCE 7-05. .................................................................................................................................................... 249 Table ‎5.2 Typical ratios of compressive axial load on gravity-load columns from seismic load-case to compressive strength of the gross concrete cross-section assuming 28-day concrete compressive strength of 60 MPa based on provisions of a) NBCC 2005, and b) ASCE 7-05. .. 250 Table ‎5.3 Summary of curve-fitting results for normalized moment-curvature response .......... 256 Table ‎7.1 Standard parameters for: a) gravity-load column, b) shear wall. ............................... 305 Table ‎7.2 Numerical summary of pushover analysis results for columns of various cross-sectional lengths at the point of column failure (no wall shear strain, no column damage)....... 320 Table ‎7.3 Numerical summary of column pushover analysis results at the point of column failure for the standard column connected to walls of various lengths (no wall shear strain, no column damage)....................................................................................................................................... 324 Table ‎7.4 Numerical summary of pushover analysis results at column failure for various values of lpc* (no wall shear strain, no column damage). ...................................................................... 327 Table ‎7.5 Summary of pushover analysis results demonstrating the effect of damage on column drift capacity: a) no wall shear strain, b) wall shear strain included. ......................................... 334 Table ‎7.6 Summary of pushover analysis results for columns with various first storey heights at failure for: a) undamaged column, b) damaged column (no wall shear strain). ......................... 343 xv  Table ‎7.7 Summary of pushover analysis results for columns with various first storey heights at failure for: a) undamaged column, b) damaged column (wall shear strain included). ............... 346 Table ‎7.8 Summary of global drift capacity of the fixed-base standard column. ....................... 355 Table ‎7.9 Summary of global drift capacity of the standard column with three basements ....... 356 Table ‎7.10 Summary of first storey drift capacity of the fixed-base standard column. ............. 368 Table ‎7.11 Summary of first storey drift capacity of the standard column with three basements. .................................................................................................................................................... 369              xvi  List of Figures Figure ‎1.1 Gravity-load columns connected to the shear walls with closely spaced floor slabs. ... 1 Figure ‎1.2 Schematic plan view of a shear wall building with elongated perimeter columns. ...... 3 Figure ‎1.3 Deformation demands on gravity-load column due to plastic hinging of shear wall –column pushed to same lateral deformation as wall at flood slab levels. Floor slabs idealized as short-span rigid links due to their high in-plane stiffness............................................................... 4 Figure ‎1.4 Deformation demands on the gravity-load column due to wall shear strain - column pushed to the same lateral deformation of the wall at flood slab levels. ........................................ 6 Figure ‎1.5 Deformation demands on gravity-load column due to rotation of shear wall foundation – column pushed to the same lateral deformation as wall at flood slab levels.  Floor slabs idealized as short-span rigid links due to their high in-plane stiffness. ................................. 7 Figure ‎1.6 Example column moment-curvature response. ............................................................. 8 Figure ‎1.7 Main parts of the current study. .................................................................................... 9 Figure ‎2.1 FE model for specimen RW2 in FormWorks (diagonal truss elements were modeled solely to simulate the procedure used by Thomsen and Wallace to measure average storey shear strain and have negligible stiffness). ............................................................................................ 27 Figure ‎2.2 2D and 3D views of the 2D FE model for specimen TW2 in FormWorks. ................ 29 Figure ‎2.3 FE analysis results on specimen RW2 at 2% drift: a) cracking pattern, b) FE vs. observed displacement profile during testing, c) curvature profile, and d) shear strain profile. .. 30 Figure ‎2.4 Measurement of average panel shear distortion by Thomsen and Wallace (1995). ... 32 Figure ‎2.5 Comparison of average panel shear strain from FE model and test results for specimen RW2. ............................................................................................................................................. 33 xvii  Figure ‎2.6 Comparison of average panel shear strain from FE model and test results for specimen TW2. ............................................................................................................................................. 34 Figure ‎2.7 Proportionality of average first storey shear strain and curvature for specimens RW2 and TW2. ...................................................................................................................................... 34 Figure ‎2.8 2D and 3D schematic views of FE model for specimen NTW1 in FormWorks. ........ 36 Figure ‎2.9 Strong link between curvature and shear strain profiles at various global drift levels observed during testing of specimen NTW1 with flange in tension. ........................................... 37 Figure ‎2.10 VecTor2 prediction of curvature and shear strain profiles of specimen NTW1 with flange in tension at 1.5% global drift. ........................................................................................... 38 Figure ‎2.11 VecTor2 predictions of deformation components of specimen NTW1 with flange in tension at a) 0.5%, b) 1.0%, c) 1.5%, and d) 2.0% global drift. ................................................... 39 Figure ‎2.12 FE model of the 10-storey rectangular wall in FormWorks: a) elevation, b) lateral load modeled as support displacements at the top, and c) fixed support at the bottom. ............... 41 Figure ‎2.13 10-storey rectangular wall a) cracking pattern, b) deformation profile, c) curvature profile, and d) shear strain profile at 2% global drift. .................................................................. 42 Figure ‎2.14 10-storey rectangular wall strains across the length at 2% global drift: a) shear strain, b) vertical strain, and c) horizontal strain at first storey mid-height. ........................................... 45 Figure ‎2.15 Comparison of shear strain profiles obtained from nodal displacements and average element shear strain. ..................................................................................................................... 46 Figure ‎2.16 10-storey rectangular wall concrete stresses across the length at 2% global drift: a) shear stress, b) vertical stress, and c) horizontal stress at first storey mid-height. ....................... 47 Figure ‎2.17 10-storey wall average shear strains at various global drift levels. ........................... 48 Figure ‎2.18 10-storey rectangular wall average shear strain at various maximum tensile strains. ...................................................................................................................................................... 49 xviii  Figure ‎2.19 Definition of strain axes for a bi-axial stress reinforced concrete element. .............. 50 Figure ‎2.20 Mohr’s circle for strains of a biaxial stress element with near zero horizontal strain. ...................................................................................................................................................... 51 Figure ‎2.21 Linearly varying vertical strain assumption (i.e. plane sections remain plane). ....... 52 Figure ‎2.22 Concrete compression depth of the 10-storey rectangular wall section. ................... 53 Figure ‎2.23 Moment-curvature response of the 10-storey rectangular wall section. ................... 54 Figure ‎2.24 10-storey rectangular wall element strain angles at 2% global drift for: a) 1st, b) 2nd, and c) 3rd storeys. .......................................................................................................................... 56 Figure ‎2.25 10-storey rectangular wall average storey strain angles from VecTor2. ................... 57 Figure ‎2.26 Estimating shear strain of the 10-storey rectangular wall at various global drift levels using average storey strain angles obtained from VecTor2. ......................................................... 58 Figure ‎2.27 Estimating shear strain of the 10-storey rectangular wall at various maximum tensile strains at the base using average storey strain angles obtained from VecTor2. ........................... 59 Figure ‎2.28 Effect of compressive axial stress level on average first storey strain angle of 10-storey rectangular walls. ............................................................................................................... 67 Figure ‎2.29 Effect of compressive axial stress level on average first storey strain angle of 10-storey flanged walls. ..................................................................................................................... 67 Figure ‎2.30 Effect of amount of distributed reinforcement in the web region of average first storey strain angle of 10-storey rectangular walls. ....................................................................... 73 Figure ‎2.31 Effect of amount of distributed reinforcement in the web region of average first storey strain angle of 10-storey flanged walls. ............................................................................. 74 Figure ‎2.32 Effect of length of rectangular walls on the average storey principle strain angle. .. 80 Figure ‎2.33 Effect of length of flanged walls on the average storey principle strain angle. ........ 80 xix  Figure ‎2.34 Effect of aspect ratio of rectangular walls on the average principle strain angle. ..... 86 Figure ‎2.35 Effect of aspect ratio of flanged walls on the average principle strain angle. ........... 86 Figure ‎2.36 Effect of aspect ratio of rectangular walls on 1st storey shear to flexural drift ratio. 87 Figure ‎2.37 Effect of aspect ratio of flanged walls on 1st storey shear to flexural drift ratio. ...... 87 Figure ‎2.38 Influence of aspect ratio of rectangular walls on 1st storey shear deformation. ........ 88 Figure ‎2.39 Influence of aspect ratio of flanged walls on 1st storey shear deformation. .............. 88 Figure ‎2.40 Crack pattern of wall 15STRW-L10 with a) three, b) two, and c) one floor slab in the wall plastic hinge region. .............................................................................................................. 90 Figure ‎2.41 Effect of number of slabs in the plastic hinge region of wall 15STRW-L10 on the average principle strain angle. ...................................................................................................... 91 Figure ‎2.42 Summary of average strain angles from parametric study. ....................................... 93 Figure ‎2.43 Verification of the proposed shear strain model for predicting average shear strain in a) first storey and b) second storey of walls considered in the parametric study. ........................ 95 Figure ‎2.44 Estimates of average plastic hinge shear strain observed in tests by other researchers: a) flexure-dominated walls, and b) walls governed by formation of a shear failure mechanism. 98 Figure ‎3.1 Distribution of vertical stiffness underneath the foundation as per guidelines on FEMA 356 – Figure from FEMA 356. ....................................................................................... 104 Figure ‎3.2 (a) Idealized elasto-plastic load-deformation behavior for soils (b) Uncoupled spring model for rigid footings – Figure from FEMA 273. ................................................................... 107 Figure ‎3.3 Drift ratio versus foundation R value: (a) 7-storey structure on rock foundation, (b) 15-storey structure on rock foundation, (c) 30-storey structure on rock foundation, and (d) 30-storey structure on clay foundation - Figure from Anderson (2003). Note: Values of R in the figure legends represent the ratio of the elastic moment to the wall bending strength............... 110 xx  Figure ‎3.4 Schematic view of 2D modeling of shear walls with a flexible foundation. ............ 117 Figure ‎3.5 Cross-section of nonlinear 10-storey shear walls (dimensions in millimetres) – bending takes place about the X-X axis. .................................................................................... 119 Figure ‎3.6 Plastic hinge zone bending moment-curvature envelopes of the four nonlinear 10-storey shear walls. ....................................................................................................................... 122 Figure ‎3.7 Non-dimensional backbone curves defining the QzSimple1 material used for soil springs. ........................................................................................................................................ 126 Figure ‎3.8 Comparison of stiffness and strength properties of Clay used in this study with values used for clay type soils by other researchers. ............................................................................. 135 Figure ‎3.9 Comparison of stiffness and strength properties of the three types of Sand used in this study with values used for sand type soils by other researchers. ................................................ 135 Figure ‎3.10 Uniform stress block used to calculate foundation overturning capacity. .............. 136 Figure ‎3.11 Calibrating the stiffness of the Qzsimple1 clay material for Clay. ......................... 137 Figure ‎3.12 Calibrating the stiffness of the Qzsimple1 sand material for the three types of Sand. .................................................................................................................................................... 138 Figure ‎3.13 a) Pseudo acceleration, and b) displacement response spectra from the 10 modified (spectrally-matched) ground motions with 5% critical damping used in the NTHA. ................ 141 Figure ‎3.14 Pseudo acceleration response spectra of the 10 uniformly-scaled ground motions with 5% critical damping used to justify the use of spectrally-matched ground motions in NTHA. .................................................................................................................................................... 142 Figure ‎3.15 Elastic equivalent damping based on the initial elastic stiffness observed in the TRISEE tests – Figure from Negro et al. (1998). ....................................................................... 143 Figure ‎3.16 Equivalent damping based on the secant stiffness observed in the TRISEE tests – Figure from Negro et al. (1998). ................................................................................................. 144 xxi  Figure ‎3.17 Rotational stiffness for the two sand specimens from the TRISEE tests - Figure from Negro et al. (1998). ..................................................................................................................... 144 Figure ‎3.18 Average of top displacement envelopes of wall 10R20 with a 12.5 m square foundation with various soil damping ratios. ............................................................................. 146 Figure ‎3.19 Time-history response of wall 10R20 with a 12.5 m square foundation from spectrally-matched a) EQ7, and b) EQ3 for various levels of damping in the soil. ................... 148 Figure ‎3.20 Backbone curve of soil springs in monotonic compression for the 5 types of soil introduced in Table ‎3.15. ............................................................................................................ 151 Figure ‎3.21 Foundation bending moment-rotation response of a 12.5 m square foundation on the 5 types of soil introduced in Table ‎3.15. .................................................................................... 151 Figure ‎3.22 Average of top displacement and global drift envelopes of wall 10R20 with a 12.5 m square foundation on various soil types subjected to spectrally-matched ground motions. ....... 152 Figure ‎3.23 Average of maximum soil compressive displacement profiles for a 12.5 m foundation on various soil types subjected to spectrally-matched ground motions. .................. 153 Figure ‎3.24 Backbone curve of the soil springs in monotonic compression used to investigate the effect of soil stiffness at a constant soil strength. ....................................................................... 154 Figure ‎3.25 Foundation bending moment-rotation response of a 12.5 m square foundation on soils with various stiffnesses but the same strength. .................................................................. 154 Figure ‎3.26 Average of top displacement and global drift envelopes of wall 10R20 with a 12.5 m square foundation on soils with various stiffnesses but the same strength subjected to spectrally-matched ground motions............................................................................................................. 155 Figure ‎3.27 Backbone curve of the soil springs in monotonic compression used to investigate the effect of soil ultimate bearing capacity at constant soil stiffness. .............................................. 156 Figure ‎3.28 Foundation bending moment-rotation response of a 12.5 m square foundation on soils with various ultimate bearing capacities but the same stiffness. ........................................ 157 xxii  Figure ‎3.29 Average of top displacement and global drift envelopes of wall 10R20 with a 12.5 m square foundation on soils with various ultimate bearing capacities but the same stiffness subjected to spectrally-matched ground motions. ...................................................................... 158 Figure ‎3.30 Average of maximum soil compressive displacement profiles for various soil ultimate bearing capacities from spectrally-matched ground motions. ...................................... 159 Figure ‎3.31 Comparison between the scatter in top displacement of wall 10R13 from NTHA using spectrally-matched and uniformly-scaled ground motions on a) Dense Sand, and b) Rock. .................................................................................................................................................... 161 Figure ‎3.32 Average of maximum global drift and top displacement from NTHA using spectrally-matched ground motions for mass ratios of 0.4 and 0.6 for a) 5-storey, b) 10-storey, and c) 20-storey Elastic walls. .................................................................................................... 165 Figure ‎3.33 Average of displacement envelopes from 10 NTHA using spectrally-matched ground motions for walls a) 10R13, and b) 10R27 on Medium Sand along with top displacements from RSA using various effective stiffnesses. .................................................................................... 169 Figure ‎3.34 Average of inter-storey drift envelopes from 10 NTHA using spectrally-matched ground motions for walls a) 10R13, and b) 10R27 on Medium Sand. (Note: base rotation values are plotted at h=0 and values of average interstory drift are plotted at the top of the storey.) ... 170 Figure ‎3.35 Average of bending moment envelopes from 10 NTHA using spectrally-matched ground motions for walls a) 10R13, and b) 10R27 on Medium Sand along with base bending moment estimates from RSA using various effective stiffnesses. .............................................. 171 Figure ‎3.36 Average of shear force envelopes from 10 NTHA using spectrally-matched ground motions for walls a) 10R13, and b) 10R27 on Medium Sand along with base shear force estimates from RSA using various effective stiffnesses. ............................................................ 172 Figure ‎3.37 Average of maximum foundation rotations (θb) from 10 NTHA using spectrally-matched ground motions for walls a) 10R13, and b) 10R27 on Medium Sand plotted on the foundation moment-rotation envelope, along with rotations at which the calculated bending moment capacity of the foundation is mobilized (θoc). ............................................................... 173 xxiii  Figure ‎3.38 Average of maximum soil compressive displacement envelopes from 10 NTHA using spectrally-matched ground motions for walls a) 10R13, and b) 10R27 on Medium Sand. .................................................................................................................................................... 175 Figure ‎3.39 Average of maximum global drift and top wall displacement from NTHA using spectrally-matched ground motions for a) wall 10R13, and b) wall 10R27. .............................. 177 Figure ‎3.40 Average of maximum base rotations from NTHA using spectrally-matched records for walls 10R13 and 10R27. ....................................................................................................... 179 Figure ‎3.41 Average of maximum a) first storey displacement, and b) top displacement of walls on various foundation sizes on Clay from NTHA using spectrally-matched records. (Note: results for elastic walls are plotted as Rw=1.0) ........................................................................... 181 Figure ‎3.42 Average of maximum a) base rotation, b) first storey interstory drift, c) top storey interstory drift, and d) global drift of walls on various foundation sizes on Clay from NTHA using spectrally-matched records. (Note: results for elastic walls are plotted as Rw=1.0) ......... 182 Figure ‎3.43 Average of maximum soil compressive displacement a) at the toe, and b) underneath the centreline of the foundation for walls on various foundation sizes on Clay from NTHA using spectrally-matched records. (Note: results for elastic walls are plotted as Rw=1.0) ................... 183 Figure ‎3.44 Maximum soil compressive displacement and residual foundation displacement of wall 10R13 on Clay with Rf=3.2 from NTHA using spectrally-matched EQ2 and EQ6 ........... 185 Figure ‎3.45 Bearing stress distribution underneath wall 10R13 on Clay with Rf=3.2 from NTHA using spectrally-matched EQ2 and EQ6 at a) the end of NTHA, and b) time of occurrence of maximum soil compressive displacement. ................................................................................. 186 Figure ‎3.46 Displacement time-histories of the soil element a) underneath wall CL, and b) at foundation toe of wall 10R13 on Clay with Rf=3.2 from NTHA using spectrally-matched EQ2 and EQ6. ..................................................................................................................................... 187 xxiv  Figure ‎3.47 Vertical reaction time-histories of the soil element a) underneath wall CL, and b) at foundation toe of wall 10R13 on Clay with Rf=3.2 from NTHA using spectrally-matched EQ2 and EQ6. ..................................................................................................................................... 189 Figure ‎3.48 Average of maximum soil compressive displacement at a) foundation toe, and b) wall centreline from the Core NTHA for walls 10R13 and 10R27. ........................................... 190 Figure ‎3.49 Increase in effective period of the wall-foundation system due to foundation rotation for wall a) 10R13, b) 10R17, c) 10R20, and d) 10R27. ............................................................. 192 Figure ‎3.50 Reduction in wall maximum bending moment demand due to foundation rotation observed in the Core NTHA for walls 10R13 and 10R27. ......................................................... 194 Figure ‎3.51 Reduction in wall base shear demand due to foundation rotation observed in NTHA .................................................................................................................................................... 195 Figure ‎3.52 Ratio of shear to moment reduction factors for walls 10R13 and 10R27 obtained from the Core NTHA. ................................................................................................................. 196 Figure ‎4.1 Schematics of the different states of foundation moment-rotation response - Figure from Allotey and El Naggar (2003). ........................................................................................... 201 Figure ‎4.2 Elastic response limit of foundations: a) liftoff occurring prior to nonlinear soil behaviour (low axial load), and b) nonlinear soil behaviour occurring prior to liftoff (high axial load). ........................................................................................................................................... 206 Figure ‎4.3 Equivalent rectangular stress block concept for soil bearing pressure. ..................... 207 Figure ‎4.4 Variation of the three key parameters of the equivalent rectangular pressure block concept with vertical compressive load for clay type soils. ....................................................... 210 Figure ‎4.5 Variation of the three key parameters of the equivalent rectangular pressure block concept with vertical compressive load for sand type soils. ....................................................... 211 xxv  Figure ‎4.6 Example: verification of the proposed equivalent uniform soil bearing pressure method for estimating moment-rotation response of a) a 19 m square footing on Clay and b) a 15 m square footing on Medium Sand (see Section ‎3.2.4 for soil specifications)........................... 213 Figure ‎4.7 Estimating top displacement of nonlinear shear walls with a flexible foundation from RSA of an equivalent elastic wall with a rotational spring at its base. ....................................... 215 Figure ‎4.8 Effective flexural stiffness of elastic 10-storey shear walls to match the average maximum top displacement of the fixed-base nonlinear walls from NTHA. ............................. 216 Figure ‎4.9 Top displacements of elastic 10-storey shear walls with appropriate effective flexural stiffness obtained from RSA with various elastic rotational spring stiffnesses at the base. ....... 217 Figure ‎4.10 Variation of effective stiffness of the elastic rotational base spring of the simplified structure with a) wall strength, and b) foundation strength for foundations on Clay. ................ 219 Figure ‎4.11 Variation of effective stiffness of the elastic rotational base spring of the simplified structure with relative foundation to wall strength for foundations on Clay. ............................. 220 Figure ‎4.12 Variation of effective stiffness of the elastic rotational base spring of the simplified structure with relative wall to foundation strength for foundations on all five types of soil. ..... 221 Figure ‎4.13 Comparison of moment-rotation responses, θoc and θb of foundations with Rf of 2.0 on Clay, Loose Sand, and Medium Sand supporting wall 10R13. ............................................. 223 Figure ‎4.14 Variation of effective stiffness of the elastic rotational spring used in RSA to estimate top wall displacement accounting for foundation rotation with relative wall to foundation strength for foundations all five types of soil. .......................................................... 224 Figure ‎4.15 Accuracy of estimates of top wall displacement of nonlinear walls with flexible foundations using Ke from the best fit exponential curve. ......................................................... 225 Figure ‎4.16 Simple bilinear trendline used for estimating equivalent stiffness of the elastic rotational spring at the base of the simplified structure. ............................................................. 226 Figure ‎4.17 Accuracy of estimates of top wall displacement of nonlinear walls with flexible .. 227 xxvi  Figure ‎4.18 Estimating fundamental period of the wall-foundation system from first mode periods of the fixed-base wall and the rigid wall with an elastic rotational spring at its base. .. 228 Figure ‎4.19 Comparison between effective rotational spring stiffnesses required to match the top wall displacement from NTHA using RSA with a linear elastic rotational spring at the base of the wall and using an effective wall-foundation system period required to give the target top wall displacement from the displacement spectrum. .......................................................................... 229 Figure ‎4.20 Major components of top displacement of shear walls with flexible foundations. . 230 Figure ‎4.21 Average of curvature envelopes from NTHA, curvature profile from RSA with the effective flexural stiffness to match average of top wall displacement envelopes from NTHA, curvature profile from RSA with appropriate effective stiffness divided by Rw, and inelastic curvatures of the fixed base walls a) 10R13, b) 10R17, c) 10R20, and d) 10R27. .................... 232 Figure ‎4.22 Estimates of the elastic component of top displacement of the four nonlinear walls with a fixed-base obtained by dividing the total top wall displacement by Rw. ......................... 234 Figure ‎4.23 Variation of moment-curvature response of wall 10R13 over its height. ............... 235 Figure ‎4.24 Estimates of the elastic component of the top displacement of the four nonlinear walls from the Core NTHA obtained using Eq ‎4.27. ................................................................. 237 Figure ‎4.25 Estimates of the elastic component of the top displacement of the four nonlinear walls from the Core NTHA obtained using Eq ‎4.28. ................................................................. 238 Figure ‎4.26 Example of a hinging wall with a flexible foundation – estimating the base rotation demand from the foundation moment-rotation response. ........................................................... 239 Figure ‎4.27 Example of a non-hinging shear wall with a flexible foundation – estimating the maximum wall bending moment demand from the foundation moment-rotation response. ...... 241 Figure ‎4.28 Estimating foundation rotation demand with top wall displacement demands obtained using Ke from a) best fit exponential curve, and b) simple bilinear trendline. ............ 243 Figure ‎5.1 Column cross-sections used in the study of moment-curvature behaviour. .............. 251 xxvii  Figure ‎5.2 Moment-curvature analysis results for Section A for axial load of 0.75 Prmax. ........ 254 Figure ‎5.3 Normalized moment-curvature response of Section A for axial load of 0.75 Prmax. 254 Figure ‎5.4 Normalized moment-curvature plots for Sections A, B, C, and D for concrete strength of 40 MPa and steel reinforcement ratio of 1% at an axial load of 0.75Prmax (see Figure ‎5.1 for definition of Sections A through D). .......................................................................................... 256 Figure ‎5.5 Effect of creep on concrete stress-strain relation. ..................................................... 258 Figure ‎5.6 Effect of creep on concrete moment-curvature response. ......................................... 259 Figure ‎5.7 Different levels of damage of column cross-section: a) undamaged section, b) concrete cover lost on both column faces and compression steel bars buckled, c) concrete cover and the outer layer of reinforcement lost. ................................................................................... 261 Figure ‎5.8 Effect of different stages of damage (cover loss and bar buckling/rupture) on moment-curvature response of a 305 x 1830 mm column section. ........................................................... 262 Figure ‎5.9 Effect of different stages of damage (cover loss and bar buckling/rupture) on moment-curvature response of a 610 x 610 mm column section.............................................................. 263 Figure ‎5.10 Seismic axial load demand as a ratio of gross concrete compressive strength. ...... 265 Figure ‎5.11 Variation of the net steel force as a ratio of section full yield strength with section compression depth. ..................................................................................................................... 267 Figure ‎5.12 Variation of the net steel force as a ratio of gross concrete strength with section compression depth. ..................................................................................................................... 268 Figure ‎5.13 Accuracy of calculation of neutral axis depth of a 1220 mm long column section at failure neglecting steel forces for concrete strength of: a) 40 MPa, b) 60 MPa, and c) 80 MPa.269 Figure ‎6.1 Moment-curvature response of a gravity-load column. (Note: damage to column includes spalling of concrete cover and buckling of outer reinforcement on compression face.) .................................................................................................................................................... 280 xxviii  Figure ‎6.2 Assumptions on inelastic curvature distribution in the column plastic hinge zone: a) undamaged column, b) damaged column. .................................................................................. 281 Figure ‎6.3 Idealized column structure: a) storey forces and displacement profile, b) shear force diagram. ...................................................................................................................................... 283 Figure ‎6.4 Elastic deformation and curvature profiles of a 20-storey shear wall. ...................... 286 Figure ‎6.5  Total displacement and curvature profiles of a 20-storey shear wall at displacement ductility of 2.0. ............................................................................................................................ 286 Figure ‎6.6 Total displacement and curvature profiles of a 20-storey shear wall at displacement ductility of 3.5. ............................................................................................................................ 287 Figure ‎6.7 Vertical deformation profile at the base of the column at 2% global drift. ............... 290 Figure ‎6.8 Flexural response from bilinear model vs. FE results at 1% global drift. ................. 292 Figure ‎6.9 Flexural from bilinear model vs. FE results at 1.5% global drift. ............................. 292 Figure ‎6.10 Flexural response from bilinear model vs. FE results at 2% global drift. ............... 293 Figure ‎6.11 Shear deformations from bilinear shear strain profile vs. FE results at: a) 1.5%, and b) 2.0% global drifts. .................................................................................................................. 294 Figure ‎6.12 Column moment-curvature response. ..................................................................... 295 Figure ‎6.13 Curvature demand prediction at global drifts of: a) 0.5%, b) 1%, c) 1.5% and d) 2%. .................................................................................................................................................... 296 Figure ‎6.14 Shear strain profiles of the shear wall and the column introduced in Section ‎6.6.1 modeled by Bohl (2006) at 2% global drift obtained from Vector2. .......................................... 298 Figure ‎6.15 Comparison of average section strain of a 610x610 mm and a 2438x305 mm column cross-section both with 2% vertical steel ratio and concrete strength of 60 MPa carrying an axial load equivalent to 0.4      at various section maximum compressive strains. ......................... 298 xxix  Figure ‎6.16 Effect of number of constant curvature elements in the base storey on column drift capacity. (Note: The column was assumed to reach its drift capacity once the column curvature capacity governed by maximum permissible compressive concrete strain of 0.0035 was reached.) .................................................................................................................................................... 300 Figure ‎6.17 Number of floors required for an accurate estimate of column inelastic drift capacity. (Note: The column was assumed to reach its drift capacity once the column curvature capacity governed by maximum permissible compressive concrete strain of 0.0035 was reached.) ....... 301 Figure ‎7.1 Actual and modelled moment-curvature responses of the undamaged and damaged sections of the standard column. ................................................................................................. 306 Figure ‎7.2 Increase in column curvature demand due to wall shear strain. ................................ 307 Figure ‎7.3 Curvature profiles of the undamaged column when curvature capacity has been reached: a) no wall shear strain, b) 100% of standard wall shear strain included. ..................... 309 Figure ‎7.4 Deformation profiles at the point of column curvature capacity: a) no wall shear strain, b) 100% of wall shear strain applied (no column damage modeled). .............................. 309 Figure ‎7.5 Effect of wall shear strain on gravity-load columns treated as support rotation at the base of the column: a) scheme of the wall-column system, b) lateral deformation of floor slabs due to uniform wall shear strain, and c) bending of the gravity-load due to support rotation. .. 311 Figure ‎7.6 Estimating column curvature demand due to imposed wall deformation from wall maximum curvature in the presence of wall shear strain. .......................................................... 313 Figure ‎7.7 Schematics of a) curvature profile of the shear wall, b) shear strain profile of the shear wall, and c) curvature profile of the gravity-load column. ......................................................... 314 Figure ‎7.8 Estimating column curvature demand due to imposed wall deformation from wall maximum curvature in the presence of wall shear strain. .......................................................... 315 Figure ‎7.9 Column cross-sections used to study the effect of column length on column curvature demand. ....................................................................................................................................... 317 xxx  Figure ‎7.10 Reduction in column global drift capacity with increase in column length (no wall shear strain, no column damage). ............................................................................................... 318 Figure ‎7.11 Comparison of responses of the shear wall and the 1.8 m column section in the wall plastic hinge zone: a) curvature profiles, b) deformation profiles. (Note: No wall shear strain and no column damage were modeled.) ............................................................................................ 319 Figure ‎7.12 Pushover analysis results for the 2.4 m and 0.6 m long column cross-sections (no wall shear strain, no column damage). ....................................................................................... 320 Figure ‎7.13 Moment-curvature response of the 2.4 m and 0.6 m long columns. ....................... 321 Figure ‎7.14 Increase in column drift capacity with wall length (no wall shear strain, no column damage)....................................................................................................................................... 323 Figure ‎7.15 Moment-curvature response of the 1.8 m long column section. ............................. 324 Figure ‎7.16 Effect of height of column plastic curvature zone on drift capacity (no wall shear strain, no column damage). ......................................................................................................... 326 Figure ‎7.17 Column curvature profiles at failure for different column plastic hinge heights (no wall shear strain, no column damage). ....................................................................................... 326 Figure ‎7.18 Comparison of deformation profiles of the shear wall and the 1.8 m column section in the wall plastic hinge region for column plastic hinge heights of 0.3 m and 1.2 m (no wall shear strain, no column damage). ............................................................................................... 327 Figure ‎7.19 Pushover analysis results of the standard column for various damage levels (no wall shear strain). ................................................................................................................................ 329 Figure ‎7.20 Comparison of wall and standard column curvature profiles at column failure: a) undamaged column, b) damage level 1, c) damage level 2 (no wall shear strain). .................... 330 Figure ‎7.21 Pushover analysis results of the standard column for various damage levels (wall shear strain included). ................................................................................................................. 331 xxxi  Figure ‎7.22 Comparison of wall and standard column curvature profiles at column failure: a) undamaged, b) damage level 1, c) damage level 2 (wall shear strain included). ....................... 332 Figure ‎7.23 Comparison of wall and column deformation profiles at the point of column failure for damage level 1: a) no wall shear strain, b) wall shear strain included. ................................. 333 Figure ‎7.24 Increase in column global drift capacity with lpc in the presence of wall shear strain (damage to the column was modeled as loss of concrete cover to the centre of reinforcement on both faces of the column). .......................................................................................................... 335 Figure ‎7.25 Schematics of a) curvature profile of the shear wall, b) shear strain profile of the shear wall, and c) curvature profile of the damaged gravity-load column ................................. 337 Figure ‎7.26 Estimating curvature demand of the damaged column from wall maximum curvature in the presence of wall shear strain. ............................................................................................ 338 Figure ‎7.27 Pushover analysis results for columns with various first storey heights (no wall shear strain, no damage of the column). .............................................................................................. 340 Figure ‎7.28 Comparison of wall and column curvatures at failure for various column first storey heights of: a) 2.7 m, b) 8.2 m, c) 13.7 m (no wall shear strain, no damage of the column). ...... 341 Figure ‎7.29 Comparison of shape of column deformation profiles at failure for various first storey heights of: a) 2.7 m, b) 8.2 m, c) 13.7 m (no wall shear strain, no damage of the column). .................................................................................................................................................... 342 Figure ‎7.30 Pushover analysis results for columns with various first storey heights (wall shear strain included, no damage of the column). ................................................................................ 344 Figure ‎7.31 Comparison of wall and undamaged column curvature profiles at failure for various column first storey heights of: a) 2.7 m, b) 8.2 m, c) 13.7 m (no wall shear strain). ................. 345 Figure ‎7.32 Column resting on a stiff basement walls: a) real structure, b) idealization. .......... 347 Figure ‎7.33 Column continuing below grade: a) real structure, b) idealization. ........................ 348 xxxii  Figure ‎7.34 Effect of number of basement floors on column drift capacity. .............................. 349 Figure ‎7.35 a) slope, and b) deformation profiles of the undamaged column with five basement floors at failure (no wall shear strain). ........................................................................................ 350 Figure ‎7.36 Pushover analysis results for the standard column continuing 3 levels below grade (no wall shear strain). ................................................................................................................. 351 Figure ‎7.37 Curvature profiles of the standard column with 3 basement levels at failure: a) undamaged column, b) damaged column (no wall shear strain). ............................................... 352 Figure ‎7.38 Pushover analysis results for the standard column continuing 3 levels below grade (wall shear strain included). ........................................................................................................ 353 Figure ‎7.39 Curvature profiles of the standard column with 3 basement levels at failure: a) undamaged column, b) damaged column (wall shear strain included). ..................................... 354 Figure ‎7.40  Curvature profiles of the undamaged standard column with three basement levels at failure: a) no wall shear strain applied (global drift = 1.67%), b) 50% of wall shear strain applied (global drift = 1.83%), c) 100% of wall shear strain applied (global drift = 1.51%)..... 357 Figure ‎7.41 Effect of column length on global drift capacity of the undamaged standard column. .................................................................................................................................................... 359 Figure ‎7.42 Effect of percentage of applied wall shear strain on global drift capacity of the undamaged standard column. ..................................................................................................... 359 Figure ‎7.43 Effect of taller first storey on global drift capacity of the undamaged standard column. ....................................................................................................................................... 360 Figure ‎7.44 Schematic curvature profile of a column continuing below grade level in a building with rigid basement walls. .......................................................................................................... 361 Figure ‎7.45 Estimation of column curvature demand due to imposed wall deformation with column continuing below grade for several basement levels (wall shear strain included). ....... 363 xxxiii  Figure ‎7.46 Estimating column curvature demand due to imposed wall deformation with column continuing below grade for several basement levels (wall shear strain included). .................... 365 Figure ‎7.47 Estimation of column curvature demand due to combined effects of the imposed wall deformation with column continuing for several basement levels and column damage .... 366 Figure ‎7.48 Relationship between inter-storey drift and global drift at column failure: all data points plotted. ............................................................................................................................. 370 Figure ‎7.49 Relationship between inter-storey drift and global drift at column failure for all data points with lpw*=8.2m and first floor height of 2.74m. .............................................................. 370 Figure ‎7.50  Relationship between inter-storey drift and global drift for various wall plastic hinge lengths (first floor height=2.74m)............................................................................................... 371 Figure ‎7.51 Relationship between inter-storey drift and global drift for various first storey heights (wall plastic hinge length=8.2m).................................................................................... 371 Figure ‎7.52 Thin slabs framing into a relatively stiff gravity-load column. ............................... 373 Figure ‎7.53 Stiff beams framing into a relatively flexible gravity-load column. ....................... 374 Figure ‎7.54 Validation of the proposed method for estimating additional column curvature accounting for the effect of framing members. .......................................................................... 377 Figure ‎7.55 Validation of the proposed method for estimating additional column axial load demand from shear forces induced in the members framing into the column. ........................... 379     xxxiv  List of Symbols                   a : Depth of uniform stress block underneath the foundation at factored soil bearing capacity in Chapter 3, depth of uniform stress block underneath the foundation in Chapter 4Ag: Gross concrete cross-sectional areaAvg. Max. : Average of the maximum response quantity from each EQB : Foundation widthb : Column widthbw: Wall widthc : Wall ultimate concrete compression depth in Chapter 2, soil compression depth underneath the foundation in Chapter 3, concrete compression depth in Chapter 5C : QzSimple1 material type factorCr: QzSimple1 material type factorCt: Creep coefficientE : Soil modulus of elasticitye : Eccentricity of compressive load on foundationEct: Concrete secant stiffnessEI : Column effective flexural stiffnessEIg: Flexural stiffness of gross cross sectionEs: Elastic modulus of reinforcing steelEsh:Reinforcing steel strain hardening stiffness in VecTor2esh:Reinforcing steel strain at the onset of strain hardeningeθy: Foundation rotational stiffness embedment factorfc: Concrete stressfʹc: 28-day concrete compressive strengthfcr: Concrete cracking strength in tensionFu: Reinforcing steel ultimate tensile strengthFy: Reinforcing steel yield strengthG : Soil shear modulus of elasticityG0: Small-strain soil shear modulus of elasticityGeff: Effective soil shear modulus of elasticityH : Storey heighth1:First storey heighthf: Floor heightHst: Wall uniform storey heightHw: Total wall heighthw: Wall heightIe: Wall gross second moment of inertialIg: Wall effective second moment of inertialIy:Foundation second moment of inertiaxxxv                      k : Factor relating wall shear strain to curvatureke: Initial elastic stiffness of QzSimple1 soil springsKe: Effective foundation rotational stiffnessKoc:Secant foundation rotational stiffness to the point with Moc and θockθ: Soil reaction modulusKθy: Initial elastic foundation rotational stiffness accounting for foundation embedmentKʹθy: Initial elastic foundation rotational stiffness resting on the soil surfaceL : Foundation lengthl : Column lengthL0: Clear span of members framing  into the columnle: Length of heavily reinforced regions at the ends of rectangular wallslpc: Plastic hinge length of damaged column with uniform inelastic curvatures over the heightlpc*: Plastic hinge length of undamaged column with linearly varying inelastic curvatures over the heightlpw*: Wall plastic hinge length with linearly varying inelastic curvatures over the heightlw: Wall lengthLw: Shear wall lengthlw: Wall lengthm : Wall lumped mass at floor slab levelsM : Bending momentM85:Foundation overturning strength assuming a uniform stress block with bearing pressure of 0.85qult at the "toe" of the foundationMb: Bending moment at column baseMelastic: Bending moment at the onset of inelastic behaviourMn: Nominal bending strengthMoc: Overturning capacity of the foundation assuming uniform stress block at factored soil bearing capacityMRSA: Maximum bending moment demand of the wall with a fixed base calculated from response spectrum analysis using 70% of the uncracked flexural stiffnessMy: Wall probable flexural strength accounting for steel strain hardeningN : Number of storeysn : QzSimple1 material type factorP : Compressive axial force carried by the shear wall uniformly distributed over the height of the wall in Chapter 3, compressive load on foundation in Chapter 4, compressive axial load on cross-sectionPf:Factored compressive axial load demandPi: Column storey forcePns: Net steel forcePr max: Maximum factored compressive axial resistancePs: Axial load associated with seismic forcesq : Uniform bearing pressure underneath the foundation under gravity load (service) condition in Chapter 3, soil bearing pressure in Chapter 4qa: Allowable uniform bearing pressure underneath the foundation under gravity load (service) conditionqf: Factored toe bearing pressure used to size the foundation for overturningqmax: Maximum soil bearing pressure at the "toe" of the foundationxxxvi                      qult: Ultimate soil bearing capacity used in nonlinear dynamic analysisqunif.: Uniform bearing pressure of soil equivalent rectangular stress blockRf: Ratio of maximum wall bending moment at base from RSA to foundation overturning capacityRw: Ratio of maximum wall bending moment at base from RSA to wall bending moment strength at bases : Uniform foundation settlement due to compressive axial load on shear wallTestimate: Estimated first period of vibration of the wall-foundation system using square root of sum of squares of fundamental periods of the fixed base wall and the flexible foundation supporting a rigid walltf: Wall flange thicknessTf: First mode period of the wall-foundation system with the wall assumed to be rigidTmodel: First mode vibration period of the wall-foundation system from modelTs: First mode period of wall-foundation structuretw: Wall web thicknessTw: First mode period of the fixed-base wallV : Shear force in members framing into the column induced due to framing actionVi: Column storey shear forceVsSoil shear wave velocitywf: Wall flange widthZ : Compressive displacement of QzSimple1 soil springsZ50: Displacement at which 50% of the ultimate soil bearing capacity is mobilizedZe: Elastic compressive displacement of QzSimple1 soil springsZmax: Maximum soil compressive displacement at the "toe" of the foundationZp: Plastic compressive displacement of QzSimple1 soil springsα : Soil equivalent rectangular stress block pressure factor in Chapter 4, ratio of column flexural stiffness to that of the members framing into it in Chapter 7α1: Equivalent rectangular concrete stress block factorαw: Wall effective second moment of inertia factorβ : Soil equivalent rectangular stress block compression depth factorβ1: Equivalent rectangular concrete stress block factorγ : Soil equivalent rectangular stress block maximum compressive displacement factorγavg: Wall average first storey shear strainγmax: Wall maximum shear strain at the baseγxy: Shear strain of a single reinforced concrete element under biaxial stressesΔ1:Envelope of 1st storey displacementΔ10:Envelope of 10th storey displacementΔ20Envelope of 20th storey displacementΔ5Envelope of 5th storey displacementΔi:Wall displacement at the top of ith storeyΔs1: Shear deformation at the top of the first storeyε*cen.: Tensile strain at wall centroidεc: Concrete strainεʹc:Concrete strain at fʹcεcmax: Maximum allowable concrete strainxxxvii                      εcr: Concrete cracking strain in tensionεtmax: Wall maximum tensile strainεv: Average storey vertical tensile strainεx: Vertical tensile strainθ : Principle strain angle measured from wall vertical axis in Chapter 2, column rotation at grade level in Chapter 7θ1: Column rotation occurring in first basement storeyθ2: Column rotation occurring in second basement storeyθb: Base rotationθelastic: Foundation rotation at the onset of inelastic behaviourθoc: Base rotation at which the foundation overturning capacity is mobilizedθp: Total inelastic rotation in the wall plastic hingeθy: Base rotation at which wall yield bending strength is reachedν : Soil Poisson's ratioρ : Mass density of soil in Chapter 3, reinforcing steel area as a ratio of gross concrete area in Chapters 2 & 5φ : Curvatureφavg: Wall average storey curvatureΦc: Concrete material strength reduction factorφd: Total column curvature demandφmax: Wall maximum curvatureφplc: Curvature at onset of column plastic behaviourφs: Seismic column curvature demandφy: Wall yield curvaturexxxviii  Acknowledgements I would like to extend my sincerest gratitude to my research supervisor, Professor Perry Adebar, for his ongoing guidance during the entire course of my graduate studies at UBC. His expert knowledge of reinforced concrete structures, extensive experience in engineering research and invaluable mentorship where key to development and completion of this work. I am sincerely grateful to Drs. Donald Anderson and Ron DeVall for their time and insightful feedback as members of the supervisory committee. I would also like to thank Drs. Anderson and DeVall along with James Mutrie for their expert guidance in the research on rotation of shear wall foundations. Their informed input was detrimental to pointing the research on foundation rotation of shear walls in the right direction and helping problem-solve my way around obstacles along the way. Decisions on modeling assumptions for the soil-structure interaction component used to study the rotation of shear wall foundations was made with the expert advice of Drs. Dharma Wijewickreme, Mahdi Taiebat, and John Howie to whom I am truly grateful. I would also like to extend my acknowledgements to Ernest Naesgaard for his crucial input into the decision making on soil properties used for studying foundation rotation in shear wall buildings and sharing his knowledge and experience in professional geotechnical engineering practice with me. I like to thank my colleagues at the Civil Engineering Department at UBC especially Jeff Yathon for providing me with his RSA code written in MATHLAB. I am humbly grateful to my aunt Dr. Nooshin Bazargani and her husband Rahim Faghihi for their ongoing encouragements and unconditional support. Last but certainly not the least, I would like to thank my family, my mother Zohreh Seifaei, my father Naser Bazargani, and my sisters Parisa and Bahar for their selfless emotional support and encouragement, for believing in my potentials, and for being there for me whenever I needed them.  xxxix  Dedication   To my family to whose support I owe this work              1  CHAPTER 1 Introduction 1.1 Overview of Problem Reinforced concrete shear wall buildings are a common form of construction for mid-rise and high-rise buildings in Canada and many other countries around the world. Figure ‎1.1 shows a simple sketch of the major structural components of a typical shear wall building. The shear walls are the designated seismic-force-resisting-system (SFRS) of the building and are typically designed to resist the entire lateral seismic force demands on the building. The shear walls are also detailed to be sufficiently ductile for the expected displacement demands on the building.             Figure ‎1.1 Gravity-load columns connected to the shear walls with closely spaced floor slabs.  2  The structure surrounding the shear walls, which includes floor slabs, beams and columns is referred to as the gravity-load-resisting system (GLRS). The GLRS is typically not designed to resist any seismic forces because it is usually much more flexible than the SFRS; but must be able to tolerate the deformations of the building due to the design earthquake. When the GLRS consists of long-span flat plate slabs and slender columns as shown in Figure ‎1.1, the seismic displacements of the building with stiff shear walls will usually not put any significant demands on the very flexible GLRS. In Canada, gravity-load columns on the perimeter of shear wall buildings are often elongated in cross-section as shown in Figure ‎1.2 because such columns offer several architectural and structural benefits. Elongated columns can be more easily hidden inside partitions and therefore block less of the view from the windows. Elongated columns also reduce the slab span which in turn reduces the minimum slab thickness required for deflection control. Elongated columns however are much less flexible about the strong axis of bending making the GLRS much less flexible. The lateral flexibility of a column reduces as the level of axial compression applied to the column increases. Thus the gravity-load columns are the least flexible near the base of the building. This is exactly where the shear walls are expected to experience the largest inelastic deformation demands and where the special ductile detailing is provided in the shear walls. In Canada, gravity-load columns typically do not contain any ductile detailing – no confinement reinforcement and no anti-buckling ties. From a very simple perspective, it seems very inappropriate to provide extensive ductile detailing in shear walls that are subjected to relative low levels of axial compression stress, while providing no ductile detailing in the wall-like columns that are subjected to much higher levels of axial compression stress and are directly tied to the shear walls by the numerous closely-spaced floor slabs. Accurately determining the seismic deformation demands on the gravity-load columns over the critical plastic hinge region of shear wall buildings and ensuring that the demands are less than the capacity of the columns is the main subject of this thesis.   3          Figure ‎1.2 Schematic plan view of a shear wall building with elongated perimeter columns.  1.2 Important Elements of the Problem Significant previous work has been done to relate the global seismic demand on shear wall buildings to the deformation demands at the base of shear walls. Dezhdar (2012) conducted numerous nonlinear response history analyses of shear wall buildings to establish the effective flexural rigidity EIe to be used in a linear dynamic (response spectrum) analysis to determine the maximum horizontal displacement at the top of a shear wall. He also reaffirmed that the method developed by Adebar et al. (2005) can be used to accurately determine the maximum curvature at the base of a shear wall from the maximum horizontal displacement at the top of the wall. Bohl and Adebar (2011) conducted numerous nonlinear finite element analyses to determine the exact profile of inelastic curvatures at the base of shear walls. Thus the flexural deformation demands on shear walls over the plastic hinge region at the base of the building are well known. The gravity-load columns are connected to the shear walls by closely spaced floor systems. The floor systems may be thin flat plate slabs as shown in Figure ‎1.1, or may include beams. When the floor system includes beams, the demands on the gravity-load columns may be increased to due frame action (transfer of bending moments from the floor to the columns); however, as the 4  slabs and beams crack, they will become more flexible in bending. A simple lower-bound model of the floor system is to ignore the out-of-plane bending stiffness and assume the floor acts a rigid in-plane link that imposes the lateral deformation of the shear walls on the gravity-load columns at the floor levels. Such a simple model would be very accurate for flat plat floor slabs that are connected to elongated columns bending about the strong axis. Figure ‎1.3 shows a cartoon view of a gravity-load column connected to a shear wall which is hinging at its base. Note that the slabs have to bend in order to achieve deformation compatibility. However, because the out-of-plane flexural stiffness of the slabs is relatively low, the slabs function mainly as a rigid links imposing the wall deformation profile onto the column. Bending of the slabs is idealized as concentrated hinges. The span of the slab is shown exaggeratedly short.          Figure ‎1.3 Deformation demands on gravity-load column due to plastic hinging of shear wall –column pushed to same lateral deformation as wall at flood slab levels. Floor slabs idealized as short-span rigid links due to their high in-plane stiffness.  Large inelastic rotation due to inelastic curvatures in the plastic hinge region at the base of the wall causes lateral displacement at the top of the first storey that the second floor slab will impose on the gravity-load column. If the column is not flexible enough to tolerate the 5  displacement demand (e.g. because it is a large elongated column subjected to a large axial compression force), the imposed displacement may result in damage of the column in the form of concrete cover spalling, buckling of the outer layer of reinforcement, and fracturing of reinforcement subjected to tensile strains. Such damage to a gravity-load column may significantly reduce the axial load carrying capacity of the column. If the seismic performance and life-safety of the building are to be ensured, the level of damage to the gravity-load columns will have to be controlled. For this purpose, the seismic demand on the column has to be calculated and compared against the column deformation capacity. Factors that are expected to influence the deformation demands on the gravity-load columns include the level of deformation of the shear walls and length of the shear wall, which will influence the inelastic wall curvature profile, the spacing of the floor slabs, and the support conditions at the base of the column. Factors that are expected to influence the deformation capacity of the gravity load columns include the level of applied axial compression, geometry of the cross section, arrangement of reinforcement and the concentration of damage in the column. 1.2.1 Wall shear strain Top displacement of typical cantilever shear walls is caused primarily by flexural deformations of the walls. Shear deformations of the wall usually contribute a negligible portion to the top wall displacement. On the other hand, the shear deformation in flexural walls tends to be concentrated in the plastic hinge region near the base. Such shear deformation may contribute a very significant amount to the horizontal displacement of the shear walls at a short distance up from the base of the walls.  Figure ‎1.4 shows a cartoon view of a gravity-load column connected to a flexural shear wall with significant shear strain in the plastic hinge region. Wall shear strain produces significant lateral deformation at the second floor slab level that the floor imposes on the gravity load columns. If the column does not experience any shear deformation, the increased lateral displacement of the wall will increase the flexural damage to the columns. The effect of wall shear strain on column deformation demands needs to be considered.  6           Figure ‎1.4 Deformation demands on the gravity-load column due to wall shear strain - column pushed to the same lateral deformation of the wall at flood slab levels.  1.2.2 Rotation of shear wall foundations Another possible source of significant deformation in cantilever shear wall buildings is the rigid body movement of the shear wall due to foundation rotation. While the foundation is usually assumed to be fixed at the soil level in the analysis of a building, shear wall foundations will rotate due to the applied overturning moment. The rotation is due to foundation uplift (separation of foundation from underlying soil on the tension side) and compressive displacement of soil at the “toe” of the foundation. Shown in Figure ‎1.5 is a cartoon view of a gravity-load column connected to a shear wall that experiences rigid body rotation due to rotation of the foundation. Although the shear wall is undeformed, it displaces horizontally and the floor slabs again impose that lateral displacement on the gravity-load columns at floor slab levels which may cause damage to the column.   7           Figure ‎1.5 Deformation demands on gravity-load column due to rotation of shear wall foundation – column pushed to the same lateral deformation as wall at flood slab levels.  Floor slabs idealized as short-span rigid links due to their high in-plane stiffness.  1.3 Moment-Curvature Response of Gravity-load Columns The flexural behaviour of gravity-load columns is best represented by its moment-curvature response in combination with a model to account for the distribution of inelastic curvatures over the height of the column. An example moment-curvature response of a gravity-load column is shown in Figure ‎1.6. The onset of column nonlinear behaviour due to concrete cracking, the point of peak bending strength, and the column curvature capacity as governed by maximum compression strain of unconfined concrete are among the information that can be obtained from the moment-curvature response. Throughout this thesis, column curvature will be used to quantify the flexural deformation demands and flexural deformation capacity of the gravity-load columns.   8           Figure ‎1.6 Example column moment-curvature response. 1.4 Main Parts of the Current Study Figure ‎1.7 presents a “road map” to determining the seismic deformation demands on gravity-load columns. The first step is to define the lateral displacement profile of the shear walls. The flexural deformations of the shear walls are known from previous research. However very little is known about the shear deformation of shear walls and thus this is studied in detail in ‎CHAPTER 2. Also, limited information is available on estimating the rotation of shear wall foundations so this is studied in ‎CHAPTER 3 and ‎CHAPTER 4.  Once the displacement profile of the shear walls is known, the curvature demands on the gravity-load columns must be determined accounting for the nonlinear bending moment-curvature response of the columns. A study on the bending moment- curvature response of typical gravity-load columns is presented in ‎CHAPTER 5, while the simplified analysis method that was developed for shear wall – rigid link – gravity-load column systems is presented in ‎CHAPTER 6. In the third and final step, the effect of various wall and column parameters on column curvature demand is studied.  9               Figure ‎1.7 Main parts of the current study.  1.5 Summary of Research Objectives The following are the main objectives of this dissertation sorted by research topic. 1.5.1 Shear strains in flexural shear walls ( CHAPTER 2)  To better understand the mechanisms of shear strains in flexural shear walls through surveying the existing experimental literature on flexural shear walls.  To study the effect of the various parameters influencing shear strains in flexural walls using state-of-the-art experimentally-verified nonlinear finite element analysis. 10   To develop a simple method that can be used to estimate shear strains in flexural shear walls. 1.5.2 Foundation rotation of cantilever shear walls  To study the rotation of shear wall foundations through a series of Nonlinear Time-History Analysis (NTHA) considering various soil types, wall and foundation parameters and building configurations, where both the shear walls and the underlying soil are modelled as nonlinear elements (see ‎CHAPTER 3).  To develop a simple method for estimating foundation rotation in a given flexural shear wall building using the basic structural parameters and analytical tools available to the designer (see ‎CHAPTER 4) 1.5.3 Curvature demands on gravity-load columns in the plastic hinge region of shear wall buildings with flat plate floor slabs  To study the moment-curvature response of a broad range of gravity-load columns in order to better understand their nonlinear flexural behaviour (see ‎CHAPTER 5).  To develop a simple structural analysis algorithm to analyze curvature demands in gravity-load columns in buildings with flat plate floor slabs under imposed lateral shear wall deformation profiles (see ‎CHAPTER 6).  To investigate the seismic demands on gravity-load columns through a parametric study considering various column and wall parameters as well as building configurations (see ‎CHAPTER 7).  To develop simple methods for estimating maximum curvature demand on gravity-load columns pushed to the lateral deformation profile of the shear wall given the basic information available to the designer (see ‎CHAPTER 7). 11  1.6 Thesis Overview This thesis consists of eight chapters and four appendices. In terms of research topic, the thesis can be divided into three distinctive parts. The subject of shear strains in flexural shear walls is studied in ‎CHAPTER 2. ‎CHAPTER 3 and ‎CHAPTER 4 deal with rigid body movement of shear walls due to rotation of their foundation. The main objective of this thesis which is estimating seismic demands on gravity-load columns connected to flexural shear walls with flat plate floor slabs is the topic of ‎CHAPTER 5, ‎CHAPTER 6, and ‎CHAPTER 7. Results of ‎CHAPTER 2, ‎CHAPTER 3, and ‎CHAPTER 4 could then be used as an input to the methods developed in ‎CHAPTER 7 for estimating seismic curvature demands on gravity-load columns. In ‎CHAPTER 2, shear deformation of flexural shear walls is studied. The chapter begins with a literature review on experimental evidence of shear strains in flexural shear walls and the existing models for estimating shear strains in flexural shear walls. VecTor2 which uses the Modified Compression field Theory (MCFT) is chosen as the structural analysis platform and is proven to be a reliable tool for estimating shear strains in flexural shear walls by verifying it against selected experiments. A parametric study is conducted to identify critical factors that affect shear strains in the plastic hinge region of flexural shear walls. The results of the numerical study are then used towards the end of the chapter to develop a simple model for estimating shear strains in flexural shear walls the accuracy of which is then compared against experimental results. ‎CHAPTER 3 presents results of Nonlinear Time-History Analysis (NTHA) on shear walls accounting for foundation rotation. The chapter begins with a selective literature review of existing numerical techniques for modeling soil-structure interaction. Nonlinear Winkler springs are chosen as the modelling approach implemented in OpenSees in order to carry out the NTHA. Nonlinear behaviour of the wall is modelled by constructing the wall cross-section out of nonlinear concrete and reinforcing steel fibres. 10 carefully picked ground motion records are chosen and altered to provide spectra which closely match the Uniform Hazard Spectrum (UHS) for Vancouver. A pilot study is conducted on the effect of geotechnical parameters such as soil type, stiffness, and strength on the response of shear walls accounting for foundation rotation. Other parameters studied include level of radiation damping in the soil, wall height, and mass 12  ratio (the ratio of the weight of the building supported directly by the shear wall to the total building weight). The Core NTHA was then designed based on the results of the parametric study. It included 5 types of soil and 5 shear walls of various bending strengths each supported on at least 5 foundations of various overturning strengths. Results of the NTHA of ‎CHAPTER 3 are then used to better understand the interaction between shear wall and foundation strengths. ‎CHAPTER 4 utilizes the vast data obtained in ‎CHAPTER 3 to formulate a simple step-by-step procedure for estimating foundation rotation in a shear wall building. A simple method for obtaining the monotonic foundation moment-rotation response is formulated first. An effective elastic rotational spring stiffness is then proposed that can be used at the base of the wall in a Response Spectrum Analysis (RSA) to estimate the total top wall displacement accounting for foundation rotation. Towards the end, a step-by-step method for estimating rotation of a shear wall foundation simple enough to be incorporated into design procedures is formulated. The chapter concludes with verifying the accuracy of the proposed method against NTHA results of ‎CHAPTER 3. Since deformation profile of gravity-load columns is dominated by flexure, ‎CHAPTER 5 is entirely dedicated to studying the moment-curvature response of reinforced concrete gravity-load columns. To obtain a reasonable range for the probable axial load on the columns, a series of column cross-sections are designed based on provisions of NBCC 2005 and ASCE 7-05. In addition to the amount of axial load, the effect of concrete strength and steel ratio is also examined by varying those parameters over their expected range. The effect of column damage in the form of concrete cover spalling, bar buckling and fracture and creep of concrete on the moment-curvature response are studied. The chapter concludes with a discussion on methods of finding the neutral axis depth in a column cross section. The main objective of ‎CHAPTER 6 is to develop a structural analysis algorithm for analyzing curvature demands in gravity-load columns subjected to the imposed deformation profile of a wall assuming flexural stiffness of members framing into the column to be negligible compared to that of the column. At the beginning of the chapter, a selected number of publications by other researchers on the subject of plastic hinging of gravity-load columns are summarized. The concept behind development of the structural analysis procedure is then presented. Deformation 13  profile of the wall is divided into flexural and shear components with bilinear curvature and shear strain profiles assumed. Accuracy of the proposed structural analysis method is then compared against results from a sophisticated Finite Element (FE) program such as VecTor2. The structural analysis algorithm is further refined for computational efficiency towards the end of the chapter.  ‎CHAPTER 7 is primarily focused on estimating curvature demand on gravity-load columns connected to a flexural shear wall with flat plate floor slabs. A set of standard parameters are defined for the wall and the column and in each section, one of the parameters is varied keeping the rest at their standard value to investigate how that particular parameter influences column curvature demand. Parameters studied include wall shear strain, column length, wall length, height of column’s first storey, damage of the column cross-section, column plastic hinge height and flexibility of the column’s boundary condition at grade level. Several simple expressions are developed to estimate column curvature demand from wall maximum curvature using basic design parameters. At the end of the chapter, some guidance is provided on determining additional curvature demands induced in gravity-load columns due to flexural stiffness of members framing into the column to assist with decision-making on whether or not flexural stiffness of the framing members is negligible compared to that of the column. The thesis includes four appendices. Appendix A provides a detailed summary of the results of all of the NTHA carried out in ‎CHAPTER 3. Appendix B and Appendix C show calculations for estimating the probable range of axial load on gravity load columns designed to provisions of NBCC 2005 and ASCE 7-05 respectively. Mathematical representation of the structural analysis algorithm developed in ‎CHAPTER 6 to analyze column demands when pushed to the given displacement profile of a shear wall is presented in Appendix D.    14  CHAPTER 2 Shear Deformation of Flexural Shear Walls 2.1 Overview of the Chapter Although shear deformations do not contribute significantly to the total displacement at the top of a flexural shear wall, they constitute a substantial portion of the wall’s displacement profile in the plastic hinge region where flexural deformations are relatively small. Wall shear strains are shown to be at their maximum in the first storey of the wall and that they may constitute up to one-half of the total wall displacement at the first floor slab level. Displacement demands on gravity-load columns are highly affected by wall displacement at the first few floor slab levels. Hence, shear deformation of flexural shear walls needs to be determined carefully if an accurate estimate of the deformation demands on the gravity-load system is to be made. The chapter begins with a brief introduction of VecTor2, the finite element analysis software used to model the flexural and shear behaviour of reinforced concrete walls. Accuracy of shear strains predicted by VecTor2 is verified against test results by Thomsen and Wallace (1995) and Brueggen (2009). General concepts of wall shear deformation are then examined by modeling a 10-storey shear wall in VecTor2. The mechanism of formation of shear strains in the plastic hinge region of flexural shear walls is explained. Distribution of shear strain along the height of the wall is found to be similar to the curvature profile with the majority of the shear strain concentrated in the wall plastic hinge region. Towards the end of the chapter, a simple model is introduced that can accurately estimate average shear strain in each storey of a shear wall. The method is formulated such that it can be incorporated into a standard design procedure that can be used by designers. The average storey shear strains multiplied by the storey heights give the amount of shear deformation occurring in each storey which when added to the flexural deformations, gives the total wall deformation profile accounting for the correct flexural and shear deformations.  15  2.2 Experimental Evidence of Wall Shear Strain from Previous Researchers Evidence of wall shear deformation has been observed in tests on shear walls since the beginning of the last quarter of the 20th century. Among the earliest observations on wall shear strain is the report by Wang et al. (1975) on testing of four three-storey squat walls with aspect ratio of 1.27. The ultimate objective of the experimental program was to develop practical methods for the seismic design of combined wall-frame structural systems. Wang et al. reported that shear deformation was not only significant for the walls tested but dominant at the levels of the first and second storeys. Shear stiffness reduced significantly as the wall yielded in flexure causing pinching of the shear force-shear strain response similar to the bending moment-curvature response. In 1976 (Phase I) and 1979 (Phase II), Oesterle et al. (1976 & 1979) conducted two series of tests on structural walls for the Portland Cement Association (PCA).  The objective of the testing program was to evaluate earthquake resistance of semi-slender structural walls. Two rectangular, nine barbell shaped, and two flanged walls were tested all with height to length ratio of 2.4. All specimens in Phase I carried minimal axial compressive load while those tested in Phase II had to sustain a significant axial compression between 7.3% and 13.4% of      . Even though the focus of the tests was on developing design procedures to ensure adequate flexural and shear strength and ductility and energy dissipation capacity, the report offered thought-provoking observations on wall shear deformation. Oesterle et al. observed that shear yielding (i.e. the point beyond which shear deformations started to grow rapidly) occurred simultaneously with flexural yielding and was not necessarily accompanied by yielding of shear reinforcement; in fact, significant shear deformations were observed even in walls “over-reinforced for shear”. In tests on squat structural walls with aspect ratios of 1.42 and 1.27 subjected to cyclic loading, Vallenas et al. (1979) also observed simultaneous yielding of the flexural and shear mechanisms. Despite the shear stress being constant over the height of the wall, shear yielding was only observed in areas were the flexural mechanism yielded. Vallenas et al. recognized the effect of diagonal cracks on reducing shear stiffness and increasing shear deformations of the wall. They also observed that shear deformation was almost a constant factor of flexural deformation for 16  monotonic loading while the ratio of shear to flexural deformations increased with number of load reversals and increased deformation intensity.  Shear deformation has been observed not only in squat or semi-slender walls, but also in slenderer walls whose behaviour is dominated by flexure. Shiu et al. (1981) conducted a test to verify the effect of openings in walls on their seismic behaviour. Two 6-storey walls with aspect ratios of 2.9 were tested. Shiu et al. reported that despite the response of the walls being dominated by flexure, “shear deformation was dominant in the first storey region” where the flexural rotations were quite small. Shear distortions started to decrease and flexural rotations stated to increase in higher storeys such that at the second storey level, the contributions of flexural and shear deformations to the total deformation were almost equal. Thomsen and Wallace (1995) conducted cyclic tests on two rectangular walls and two T-shaped walls with aspect ratios of 3.1 with primary objective of evaluating the effectiveness of using a displacement-based procedure for designing reinforced concrete structural walls. The walls were subjected to a sustained compressive axial load of 7% to 10% of      . Thomsen and Wallace measured shear strains in each of their specimens’ four storeys and affiliated the large shear distortion observed in the first storey of all specimens with flexural and shear cracking resulting from development of large inelastic tensile strains in the web area. However, because shear deformations constituted a relatively small portion of the wall top displacement and the focus of the test was on wall behaviour and design, Thomsen and Wallace concluded that “The exclusion of shear deformations from analytical models is not critical for slender walls”.  Dazio et al. (1999) conducted tests on 6 rectangular walls with aspect ratio of 2.3 and axial compressive loads between 5.7% and 10.8% of      . It was observed that while flexural deformations dominated the response of the walls, shear deformations were significant over the plastic hinge region of the walls. In addition, the ratio of shear to flexural deformations remained nearly constant over the entire inelastic drift range.  In tests on walls with highly-confined boundary elements where the walls had an aspect ratio of 4.0 and sustained an axial load of 0.10     , Hines (2002) also observed that ”the flexural and shear components of displacement are related to one another linearly at least at the displacement 17  peaks” and that large shear deformations were a result of the flexure-shear mechanism in the plastic hinge region. These general observations on wall shear deformation were not restricted to walls loaded in their plane of symmetry. For their two U-shaped specimens with aspect ratio of 2.8 or 2.6 subjected to multidirectional loading, Beyer et al. (2008) confirmed that “the ratio of shear to flexural displacements Δs/Δf at peak displacements remains approximately constant over the entire ductility range”; “however, the magnitude of the Δs/Δf  ratio varied strongly with the direction of loading.” They also confirmed that “the shear deformations were concentrated in the plastic hinge region at the base of the wall undergoing inelastic deformations” and that “the contribution of the shear displacements was largest when a wall section was under net tension”.  Brueggen (2009) tested a T-shaped wall with aspect ratio of 3.2 under multidirectional cyclic loading with 0.03      sustained axial compressive force and confirmed that “the larger shear deformations toward the base provide an indication of the effect that plastic hinging and flexural damage have on shear deformations and reducing shear stiffness.” Brueggen also mentioned that the ratio of shear to flexural deformations remained approximately constant over the inelastic drift range. What distinguishes Brueggen’s work from the work of other researchers is illustrating the similarity between the shapes of the curvature and shear strains profiles of the wall and recognizing the direct link between curvature and shear strain. Brueggen also emphasized on the importance of capturing the shear deformation profile of the wall and not only the total shear deformation at the top. A general trend that can be observed more or less in all of the existing experimental literature of wall shear strain is the link between formation of a flexural plastic hinge and development of large shear strains. This evidence has been stated in various forms such as flexural yielding resulting in simultaneous shear yielding, nearly constant shear deformation to flexural deformation ratio over the entire deformation range, or shear strain and curvature profiles of the flexural shear walls having the same shape. Shear strains are proven to constitute a substantial amount of the lateral deformation of flexural shear walls especially in the plastic hinge region. If the correct deformation profile of the shear wall is to be determined, the amount of shear deformation must be quantified. 18  2.3 Existing Models for Estimating Wall Shear Deformation Oesterle et al. (1984) correlated web crushing strength to the deformations within the hinging region of structural reinforced concrete walls. Oesterle et al. based their analytical model on a truss analogy with 45 degree concrete compression struts, vertical tensile reinforcement carrying flexural tension, horizontal reinforcement acting as tension ties, and flexural compression carried by both concrete and longitudinal reinforcement. Due to the complexity of the relationship between shear distortion, flexural rotation, and total drift within the plastic hinge region, an experimental approach was used to produce a formula for estimating shear distortion. An empirical equation obtained from linear regression analysis on test results reported by Oesterle et al. (1976 & 1979) was proposed for estimating the ratio of shear distortion to total drift in the plastic hinge region. Axial compressive load (N) on the wall was considered the main parameter controlling the contribution of shear distortion to total drift in the plastic hinge region as shown in the following expressions by Oesterle et al.                                          Eq ‎2.1 And                            Eq ‎2.2 where γ is the average shear distortion occurring within the hinge region and δ is the total drift ratio within the hinge region. Oesterle et al. also pointed out the strong link between shear distortion and average vertical strains resulting from curvature to conclude that “shear distortions and flexural rotations are coupled” but did not base their analytical model on this observation. Although Oesterle et al. realized the importance of focusing on inter-storey drifts within the 19  plastic hinge zone rather than just the top wall displacement and provided a simple model for predicting the interaction of flexural, shear, and axial loads, the proposed model lacks generality in that it is based on test results of fairly squat walls with aspect ratio of 2.4. Shear distortions are assumed to account for 52% of the total drift in the plastic hinge region for all walls under axial load exceeding 0.09      regardless of other properties such as wall length and concrete compression depth in the plastic hinge zone. This may result in inaccurate prediction of drifts due to shear distortion in slender walls. Although the linear relationship between the total (top) flexural and shear deformations of flexural walls had been confirmed prior to 2002, Hines (2002) was among the first to utilize this approximation to propose a simple model for estimating shear deformation of a wall from its flexural deformation. In Hines’s model, the concrete cracking pattern in the plastic hinge zone is used to derive a geometric relationship between total flexural and shear deformation of the wall. In this model, the ratio of total shear to flexural deformation is given as a constant factor of the aspect ratio of the wall whose shear transfer mechanism does not undergo severe damage. To account for the reduced shear stiffness of walls designed with inadequate shear reinforcement or walls experiencing shear failure, Hines added an empirical multiplier calculated from the ratio of the wall’s shear capacity to resist diagonal tension and compression to the applied shear force respectively. For the walls tested by Hines himself, the model in its final form is given below                   [   (        )]       Eq ‎2.3 where Δs and Δf are the total shear and flexural deformations respectively, D is the total length of the wall, L is the shear span, and the term in the bracket is the multiplier accounting for  additional shear displacement from loss of strength of the shear-carrying mechanism. Despite the simplicity of the final model, cumbersome derivations are required to formulate the proportionality constant (i.e. the 0.25 in the expression above) for the relationship between the ratio of shear to flexural displacements and aspect ratio of various walls. Furthermore, the cracking pattern of the plastic hinge region must also be known which makes Hines’s model less attractive to a designer. Even though Hines’s model captures the fundamentals of the interaction between the flexural and shear deformations of walls and provides a valuable insight into the 20  mechanisms of shear deformation, it is formulated to evaluate total top wall displacement and hence cannot be used to predict inter-storey drifts resulting from shear deformation in the plastic hinge region.  Brueggen (2009) appears to be the first to recognize the similarity between curvature and shear strain profiles due to concentration of shear strains in the plastic hinge region where large curvatures are encountered. In Brueggen’s method, shear strain is proportional to curvature with a proportionality factor called C constant over the entire inelastic drift range. The ratio of the flexural to shear stiffness of the wall at yielding is then used to calculate C. Equations below summarize Brueggen’s model.            ⁄     Eq ‎2.4 Flexural stiffness at yielding is the yield bending moment divided by the yield curvature. The shear stiffness is calculated from the expression for shear stiffness of cracked reinforced concrete beams by Park and Paulay (1975). For 45 degree cracks, this equation is given below in terms of the area ratio of shear reinforcement (ρv), steel elastic modulus, steel to concrete modular ratio (n), and width (bw) and depth (d) of the wall.                     Eq ‎2.5 This shear stiffness is multiplied by the shear span of the wall to give the desired units for C.  Brueggen’s method is capable of predicting shear strain profile of the wall and not just the total shear displacement at the top. However, calculating the shear stiffness which is needed to estimate C requires having an estimate of the crack angle best representative of the plastic hinge zone which is generally not available to the designer. Brueggen suggests assuming 45 degree cracks for simplicity. Based on her report, the model does an acceptable job of estimating shear strains assuming 45 degree cracks; but, if the actual crack angles observed during tests are used in estimating wall shear stiffness from the general expression by Park and Paulay (1975), the accuracy of the prediction of shear strains is improved. 21  Despite its elegance in relating shear strain to curvature and being “intended for use by structural engineers”, Brueggen’s method is not simple enough to be used by the designer. Its shortcoming is in estimating the wall shear stiffness using an elastic truss model, for two reasons. First is that the elastic truss model does not account for the interaction between flexural and shear deformations in reinforced concrete members. In formulating their expression for shear stiffness of cracked reinforced concrete beams, Park and Paulay assumed infinitely rigid tension and compression truss cords. This means that shear deformation is assumed to be decoupled from flexural rotation which is not a realistic assumption. In addition, the derivation of the expression assumes significant elongation of shear reinforcements and shortening of compression struts whereas in flexural walls with adequate shear (horizontal) reinforcement, horizontal strains in the web are negligible and relatively small compressive strains are observed in the compression struts.  Park and Paulay’s expression for shear stiffness is not intended to account for shear strain resulting from large vertical tensile strains and because Brueggen’s model is based on the same equation, neither does Brueggen’s model capture shear strain coming from large vertical tensile strains. Secondly, estimating shear stiffness of flexural walls from Park and Paulay’s expression requires knowledge of the probable crack angles in the plastic hinge region which is not available to the designer. Assuming a 45 degree crack angle reduces the accuracy in estimating wall shear strain and is not a viable solution to the problem. Even though the link between the magnitudes of shear strain and vertical tensile strain had been observed in tests conducted by Vallenas et al. (1979), Oesterle et al. (1984), and Thomsen and Wallace (1995), Beyer et al. (2011) were the first to utilize this experimental observation directly to formulate a model for predicting shear deformation of walls. Beyer et al.’s method uses the geometry of the Mohr strain circle for a reinforced concrete membrane element under biaxial stress in conjunction with a plastic hinge model to estimate the ratio of shear to flexural displacements at the top of the wall. Similar to Hines’s, Beyer et al.’s model assumes that the ratio of flexural to shear displacement at the top of the wall remains approximately constant for walls in which the shear transfer mechanism is not degrading in strength; an observation made in tests by Vallenas et al. (1979), Dazio (1999), and others. To incorporate this observation into the model, the contribution of horizontal strains and diagonal compression strains to shear strain of flexural walls is neglected which results in shear strain and vertical strain being proportional for a given crack angle. Furthermore, the assumption of linear variation of strain across the length of 22  the wall along with constant concrete compression depth in the plastic hinge region makes vertical strains proportional to curvature. Both flexural and shear displacement at the top of the wall are then calculated from a plastic hinge model with constant curvature equal to maximum curvature of the wall assumed over a certain height. Beyer et al.’s model for the ratio of total shear to flexural displacement is given below                    Eq ‎2.6 where εm is the vertical tensile strain at the centroid of the wall section, φ is the maximum curvature of the wall, Hn is the total height of the wall, and β is the crack angle outside the fan area in the plastic hinge zone. Despite its attractive concept, Beyer et al.’s method can only be used to estimate the total shear displacement of the wall at the top and is not formulated to give the shear strain profile or the shear deformation profile of the wall. Although the method is valuable to evaluation of shear wall behaviour, it cannot be used to estimate additional storey drift demands due to presence of shear strains in the plastic hinge region. Apart from that, the model still requires an estimate of the crack angle best representative of the fanned crack pattern within the plastic hinge zone. Beyer et al. suggest either assuming a 45 degree crack angle or estimating the crack angle from a complex equation presented by Collins and Mitchell (1997). The 45 degree crack angle assumption is too simplistic and gives inaccurate prediction of shear deformation; Beyer et al. compared predicted shear deformation of several walls against those observed during tests using the plastic hinge crack angles observed during the tests and yet, the accuracy of the prediction was not convincing. The equation given by Collins and Mitchell for predicting the crack angle requires knowledge of wall parameters that are not generally available to the designer and does not solve the problem. In summary, Hines’s and Beyer et al.’s models are not formulated to provide the shear strain profile or the distribution of shear deformation along the height of the wall. Because Oesterle et al.’s model is an empirical model based on a narrow range of wall tests all with aspect ratio of 2.4, the model lacks accuracy when used to estimate shear deformations in taller walls. The proportionality constant relating curvatures to shear strains provided Brueggen’s model is 23  derived from an expression for estimating shear stiffness of beams which does not capture shear softening resulting from presence of large tensile strains in a diagonally cracked web of flexural walls and hence, falls short of providing an accurate estimate of shear deformation. The need for a simple but accurate model for estimating shear deformation profile of flexural walls is therefore obvious.  2.4 Finite Element Analysis of Reinforced Concrete Structures Using VecTor2 VecTor2 was selected as the nonlinear finite element software for this research as it uses the state-of-the-art material models for cracked reinforced concrete subjected to axial, shear, and bending. VecTor2 uses the Disturbed Stress Field Model (DSFM) formulated by Vecchio (2000) which is a refinement of the Modified Compression Field Theory (MCFT) introduced by Vecchio and Collins (1986).  The MCFT determines the average and local strains and stresses of the concrete and reinforcement, and the widths and orientation of cracks throughout the load-deformation response of an element. Based on this information, the failure mode of the element can also be determined. The concrete model accounts for the reduction of compressive strength and stiffness due to transverse cracking and tensile straining. The reduction in concrete cracking strength due to transverse compressive stresses is also accounted for.  The theory utilizes a set of simplifying assumptions most important of which are uniformly distributed and rotating cracks, using average strains over a gage length including several cracks, compatibility of average concrete and reinforcement strains and negligible shear stress in reinforcement. The theory also assumes the orientation of principal average strain, θε, and that of principal average stress, θσ, to be the same. Principle tensile and compressive concrete strains can then be determined using Mohr’s circle for strains. To satisfy force equilibrium, summation of concrete and steel stresses is set equal to the applied stress resultants on the element. Local stress conditions at cracks are also considered to make sure steel reinforcement can bear the extra tensile stress carried through concrete tension stiffening elsewhere.  24  Adding constitutive relations for both concrete and steel makes the MCFT ready for FE implementation. However, VecTor2 uses a modified version of the theory called the Disturbed Stress Field Model (DSFM) introduced by Vecchio (2000). The DSFM addresses systematic deficiencies of the MCFT in predicting the response of certain structures and loading scenarios by accounting for the effect of shear slip on the state of stress and strains of 2D reinforced concrete membrane elements. Further information on the DSFM can be found in the VecTor2 manual (see Wong and Vecchio (2002)) and is therefore excluded from this discussion. In VecTor2, the constitutive relationship used for reinforcing steel in tension has an initial linear-elastic response, a yield plateau, and a linear strain-hardening phase until rupture which can be easily fitted to the measured bare bar stress-strain relationships of the reinforcement. VecTor2 uses three-node constant strain triangular elements with six degrees of freedom (DOF) and four-node plane stress rectangular elements with eight DOF to model concrete with distributed reinforcement and uses two-node truss bar elements with four DOF to model discrete steel reinforcement. 2.4.1 Previous works on verification of VecTor2 Vecchio first introduced the DSFM in 2000. The analytical method has been used by many researchers studying the behaviour of reinforced concrete structures since and has gained appreciable popularity. Among the early attempts to validate DSFM as a reliable analysis platform for reinforced concrete structures was the work published by Vecchio et al. (2001). Vecchio et al modeled tests on RC panels, beams, and shear walls using the DSFM. The DSFM was found to provide accurate estimates of strength, load-deformation response, and failure mode of the tests modeled with superior accuracy to the predictions made previously using the MCFT. Palermo and Vecchio (2004) used VecTor2 to model two flexural walls with aspect ratio of 2.4, two semi-slender walls with aspect ratio of 2.0, and two squat walls with aspect ratios less than 1.0 in order to further validate VecTor2 and the DSFM as a reliable finite element (FE) analysis tool for RC structures subjected to reverse cyclic loading. To simplify their analytical model, Palermo and Vecchio used only rectangular elements with distributed steel reinforcement to 25  construct their FE model. Despite their simple modeling procedure, the FE model accurately predicted the load-deformation response and the failure mechanism observed during the test. Palermo and Vecchio also mentioned that “The analyses indicated that slender walls, controlled by flexural mechanisms, are generally a test for reinforcement models, whereas squat walls, demonstrating shear-dominant behaviour, are a better test for concrete models.” Following the publication in 2004, Palermo and Vecchio (2007) published a summary of their work in validating VecTor2 and the DSFM. After modeling a combination of slender, slender/squat, and squat walls totalling to more than 20 wall tests, they reported that the FE element analysis provided simulations that were in substantial agreement with the test results in terms of peak strength, post-peak response, ductility, energy dissipation, and failure mechanism despite using only low-powered rectangular elements with distributed steel reinforcement to construct their models. Despite the numerous works on validating the DSFM and VecTor2 as reliable analysis tools for predicting strength, stiffness, and failure mechanism of RC structures subjected to cyclic loading, to the author’s knowledge, VecTor2 has not been verified for its accuracy in predicting various deformation components of a shear wall, specifically wall shear strain. In Section ‎2.5, shear strains obtained from VecTor2 are compared with those recorded in experiments to validate Vector2 as an appropriate analysis tool for predicting shear strains in flexural shear walls. 2.5 Verification of VecTor2 for Predicting Shear Strains in Walls A summary of the available experimental literature with measurement of shear deformation is summarized by Beyer et al (2011). It is therefore beneficial to use this reference in choosing tests to validate Vector2 for predicting shear deformation in slender shear walls. The majority of the walls tested fall in the boundary range between slender and squat walls based on their aspect ratio. For this reason, only walls with aspect ratios of 3.0 or higher are chosen which narrows the choice down to tests carried out by Thomsen and Wallace (1995), Hines (2002), and Brueggen (2009). Hines only reported the total shear deformation of the wall specimens at the top and did not measure the distribution of shear strains or shear deformations over the height of the wall 26  specimens; hence, his work cannot be used for the purpose of validating the analytical model used in this study. 2.5.1 Specimens RW2 and TW2 tested by Thomsen and Wallace (1995) Thomsen and Wallace (1995) tested two rectangular specimens and two T-shaped specimens all with an aspect ratio of 3.0 subjected to reverse cyclic loading. Only one rectangular specimen (RW2) and one T-shaped specimen (TW2) are chosen here. In this section, Specimens RW2 and TW2 are simulated in VecTor2 to validate the shear strains predicted by the FE program against those observed in the tests. The wall specimens were quarter scaled models of a 4-storey wall. The length of the wall specimens was 1220 mm and their total height was 4880 mm. Specimen TW2 had thin reinforced concrete plates constructed at floor slab levels to account for the effect of the floor slabs on crack pattern and other wall parameters. The average axial compressive stress applied to the walls during the test was 7% and 7.5% of       for specimens RW2 and TW2 respectively. See Thomsen and Wallace (1995) for further details on cyclic loading routine, instrumentation, specimen construction, and test set-up. FE models of the two specimens were constructed in VecTor2 using only rectangular elements with distributed reinforcing steel. The FE models were loaded using support displacements at the top to simulate the effect of hydraulic jacks pushing and pulling the test specimens. Stress-strain relationship for reinforcing steel was adjusted to match the bare bar characteristics reported by Thomsen and Wallace (1995).  Figure ‎2.1 shows a schematic view of the finite element model for specimen RW2 with each colour representing a different reinforcement arrangement embedded in the same concrete material with     of 31.2 MPa. Although in Thomsen and Wallace’s test report different concrete strengths were reported for different concrete pours, concrete strength of 31.2 MPa reported for the first storey of the specimen was used throughout the entire finite element model for simplicity. All of the steel reinforcement was modeled as distributed steel in reinforced concrete membrane elements. 27                Figure ‎2.1 FE model for specimen RW2 in FormWorks (diagonal truss elements were modeled solely to simulate the procedure used by Thomsen and Wallace to measure average storey shear strain and have negligible stiffness).  Because the test was conducted in displacement-control mode, the push load at the top of the FE model was simulated using uniform support displacements along the top row of nodes of the model. The bottom row of the pedestal was fixed against movement using pin supports. The black diagonal elements are extremely thin truss bars modeled to simulate the shear strain computation mechanism used in the Thomsen and Wallace’s test (explained later in the section). 28  Figure ‎2.2 shows a schematic view of the FE model for specimen TW2 in both 2D and 3D views. Thickness of the rectangular elements in the slab region was increased to simulate the effect of slabs on wall behaviour. Vertical steel ratio of the slab region was reduced accordingly such that the total amount of vertical steel in the slab region was equal to that of the adjacent regions of the wall. This ensured that the slabs did not add to flexural strength of the model. Extremely thin truss bars shown in black were again modeled to simulate the shear strain measurement mechanism used in the test report by Thomsen and Wallace. A simple displacement-control pushover analysis was performed on the FE models despite the loading routine of the actual test being reverse cyclic. This meant that the results from the pushover analysis of the FE model had to be compared with the envelope of the hysteretic response observed during the tests. Figure ‎2.3a shows a schematic of the deformed FE model at 2% lateral drift (deformation magnifier=5.0). Orientations of the red lines indicate the orientation of the cracks and their width is a coarse representation of the crack width.  To distinguish between flexural and shear components of the wall deformation, flexural deformations were computed by integrating curvatures over the height of the wall and then subtracted from the total deformation profile to obtain shear deformations. Because VecTor2 does not output curvatures, curvature profile of the wall also had to be calculated. If the slope of the straight line connecting the two nodes on the ends of a single row of nodes is considered to be the average rotation of the wall at that elevation, then the average curvature along a row of elements would be the change of rotation (slope) between the two rows of nodes which bound a row of elements. Average element curvatures calculated in this manner are plotted in Figure ‎2.3c and the flexural deformations obtained from integrating those curvatures over the height of the wall are shown in Figure ‎2.3b as the dotted line. The displacement profile of the specimen measured during the test is also plotted in Figure ‎2.3b which is in very good agreement with total displacement profile obtained from FE analysis. Subtracting flexural deformations from the total deformation profile resulted in shear deformations which were converted into average shear strains over the height of each row of elements (see Figure ‎2.3d). 29   Figure ‎2.2 2D and 3D views of the 2D FE model for specimen TW2 in FormWorks.  It is important to note the similarity in the shape of the curvature and the shear strain diagrams for specimen RW2. VecTor2 analysis showed that shear strains were concentrated in the plastic hinge region of the wall where large vertical tensile strains were encountered. This agrees with the observation reported by Thomsen and Wallace in the test report.  30                       Figure ‎2.3 FE analysis results on specimen RW2 at 2% drift: a) cracking pattern, b) FE vs. observed displacement profile during testing, c) curvature profile, and d) shear strain profile. a) 31  Similar observations were made in the case of specimen TW2 and hence, corresponding figures for specimen TW2 are excluded for brevity. As the objective of this section is to verify the accuracy of shear strains predicted using Vector2, shear strains from FE analyses are compared to those measured during the tests. Average panel shear distortion (strain) was measured using strain measurements of an X-type configuration.  Figure ‎2.4 shows the expressions that Thomsen and Wallace used to calculate shear distortion. Although this method of calculating average shear strain does not give the actual average shear strain of a panel due to the assumption of uniform curvature over the panel height, the same method is used to calculate average shear strain of the FE model to make comparison possible. This is done by modeling extremely thin truss-bar elements in X-type configuration shown in black in Figure ‎2.1 and substituting the deformed and un-deformed lengths of the truss-bars in the equation for average panel shear strain shown in Figure ‎2.4. Figure ‎2.5 compares average panel shear strains obtained from FE results against those observed in the test at five different global drift (top displacement) levels. Top displacements of the FE model were matched to those observed during the test to make the comparison easier. As shown in Figure ‎2.5, VecTor2 was able to predict the average panel shear strain of the first storey quite accurately. The accuracy of the prediction was not as impressive in the case of average panel shear strain of the second storey. However, because the magnitude of the shear strain in the second storey was much smaller and hence less significant, it was concluded that VecTor2 is able to predict flexural and shear deformation profiles of a rectangular shear wall with good accuracy. Figure ‎2.6 compares average panel shear strains obtained from FE analysis with test results for specimen TW2. VecTor2 was able to predict the average first storey shear strain with great accuracy when the flange was in compression. The accuracy of the prediction of the first storey shear strain was less accurate but still acceptable when the flange was in tension. VecTor2 predictions of the second storey average shear strains was not very accurate; however, because the magnitude of the second storey shear strain is relatively small compared to that of the first storey, the less accurate predictions of second storey shear strains can be neglected.  32               √                    √                                              Figure ‎2.4 Measurement of average panel shear distortion by Thomsen and Wallace (1995).  Figure ‎2.7 compares the relationship between average first storey shear strain and curvature obtained from test results and FE analysis. During the test, total shear deformation of the wall panels in each storey was measured using strain gages mounted on the diagonals of the wall panel in an X-type configuration. The differential change in the length of the diagonals was assumed to be solely due to shear deformation of the panel. In other words, curvature distribution over each storey height was assumed to be constant which leads to over-estimating shear deformations and strains. First storey average curvature was calculated assuming all the inelastic rotation takes place in the first storey. Total first storey deformation was divided by the storey height to give the total inelastic rotation from which average storey curvature was 33  calculated. This procedure for calculating average first storey curvature overestimates curvature because some part of the first storey total deformation is inevitably due to shear. However, for the purpose of comparison, average first storey shear strain and curvature from FE analysis were calculated using the approach taken by Thomsen and Wallace to be consistent.          Figure ‎2.5 Comparison of average panel shear strain from FE model and test results for specimen RW2.  Based on Figure ‎2.7, VecTor2 was able to predict average first storey shear strains of the two specimens with good accuracy. There is an evident correlation between curvature and shear strains plotted in Figure ‎2.7. Shear strains seem to increase with curvature nearly proportionally. Thomsen and Wallace attributed this to shear strains being a result of large vertical tensile strains combined with diagonal cracking and damage of concrete. The larger the curvatures get, the larger the tensile strains become and hence, more shear strain is generated. This strong link between shear strains and curvature is used as the basis for formulating a simple model for estimating shear strain later on in this chapter.  34            Figure ‎2.6 Comparison of average panel shear strain from FE model and test results for specimen TW2.           Figure ‎2.7 Proportionality of average first storey shear strain and curvature for specimens RW2 and TW2. 35  2.5.2 Specimen NTW1 tested by Brueggen (2009) Another experimental work that was used to verify the accuracy of shear strains predicted by VecTor2 is the work of Brueggen (2009). Brueggen presented results of multi-directional loading of two T-shaped walls as a part of her PhD dissertation. The two specimens named NTW1 and NTW2 had aspect ratios of 3.2 and 1.6 respectively which makes specimen NTW2 a squat wall. Hence, only specimen NTW1 is considered here. Despite the test specimen being subjected to multi-directional loading, test data obtained for loading parallel to the web is used here because VecTor2 is a 2D FE analysis software. Specimen NTW1 was a T-shaped wall with overall length of 2286 mm (90”), flange width of 1829 mm (72”) and equal web and flange thicknesses of 152.4 mm (6”). The overall height of the specimen was 7315 mm (288”) comprising 4 storeys. The specimen was subjected to a sustained axial compressive load of 2.7% of       in addition to its self-weight. Floor slabs were constructed at storey levels to account for the effect of presence of floor slabs on the specimen’s cracking pattern and behaviour.  FE model of the specimen was constructed in VecTor2 using only rectangular elements with distributed reinforcing steel. Width of the rectangular elements located in the floor slab region was increased to model the effect of the floor slab on stiffness and cracking pattern of the model. The FE model was loaded using support displacements at the top to simulate the effect of hydraulic jacks pushing and pulling the test specimen at the top. Stress-strain relationship for reinforcing steel was adjusted to match the bare-bar characteristics reported in the test. Perfect bond between reinforcement and concrete material was assumed. Figure ‎2.8 shows 2D and 3D schematic views of the FE element model for specimen NTW1.      36                Figure ‎2.8 2D and 3D schematic views of FE model for specimen NTW1 in FormWorks.  Brueggen measured curvature and shear strain over 4 panels along the height of the first storey, 2 panels in the second storey, and 1 panel in the third and fourth storeys each with panel height being almost equal to the storey height. Curvature and shear strain profiles calculated from measurements during testing of specimen NTW1 with flange in tension are shown in Figure ‎2.9. The detailed instrumentation layout made observation of the direct link between curvature and shear strain possible. Curvature and shear strain profiles have similar shapes. High values of shear strain are concentrated in the plastic hinge region where large inelastic curvatures are encountered. This test evidence supp