Nonlinear Shear Response of Cantilever Reinforced Concrete Shear Walls with Floor Slabs by Stephen Sterling Mercer B.Eng, Dalhousie University, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2012 © Stephen Sterling Mercer, 2012 ii Abstract The nonlinear shear behaviour of cantilever reinforced concrete shear walls is complex and not fully understood. Design assumptions often oversimplify the wall response and can yield results which do not reflect the true response of the shear wall. One such assumption is analyzing the wall ignoring the effects from multiple floor slabs connected to the wall over its height. Floor slabs can provide a significant increase in wall shear capacity. This thesis examines the nonlinear shear response of walls, including the effect of floor slabs as a wall-slab system, through state-of-the-art nonlinear finite element analysis. Finite element slab models were developed to emulate the 3D slab effect within a 2D analysis environment: a high-end pseudo 3D model for in-depth slab analysis and a simple 2D slab layer model for typical wall analysis. The slab effects are explored through a parametric study varying the wall size, concrete strength, axial load, horizontal steel ratio and the slab dimensions parallel and perpendicular to the wall. The slabs were found to act like large external stirrups which provide additional tension capacity for the slab and limit shear cracking and failure. The slabs can significantly increase the shear capacity of lightly-reinforced walls. Using the developed slab models, the bounds of the slab effect were investigated by a parametric study with lightly to heavily-reinforced walls, with and without axial load, as well as varying the slab spacing. Within this study, the nonlinear response of isolated walls is compared to nonlinear uniformly-loaded membrane models. It is determined that although code-based shear capacity equations are fairly accurate, the membrane models can underestimate the shear stiffness and over predict the ductility. This study also reveals that tightly spaced slabs can increase up to 3 times the isolated wall capacity for walls with minimum horizontal steel, whereas there is little effect for walls with horizontal steel above 1%. Finally, methods were developed to predict the nonlinear shear stress-strain response of isolated walls and the peak shear capacity of wall-slab systems. iii Table of Contents Abstract .......................................................................................................................................... ii Table of Contents ......................................................................................................................... iii List of Tables ................................................................................................................................ vi List of Figures .............................................................................................................................. vii Acknowledgements .................................................................................................................... xiii Introduction .............................................................................................................. 1 Chapter 1 - Finite Element Analytical Tools .............................................................................. 6 Chapter 2 - 2.1 Membrane Element Theory.......................................................................................... 6 2.1.1 Modified Compression Field Theory ............................................................... 6 2.1.2 Disturbed Stress Field Model ........................................................................... 7 2.2 VecTor2 Finite Element Implementation and Verification ....................................... 8 2.2.1 VecTor2 Experimental Verification ................................................................. 8 2.2.2 2D Element Implementation .......................................................................... 10 2.3 Pre and Postprocessing Tools ..................................................................................... 10 2.3.1 Formworks ..................................................................................................... 10 2.3.2 Augustus ......................................................................................................... 11 2.3.3 Custom Postprocessors ................................................................................... 12 2.4 Pseudo 3D Slab Model................................................................................................. 12 2.4.1 Original Pseudo Model .................................................................................. 13 2.4.2 Modified Pseudo Model ................................................................................. 17 2.4.3 Pseudo Slab Model Issues .............................................................................. 20 2.4.4 Shear Critical Walls for Model Comparison .................................................. 25 2.4.5 Comparison of Pseudo 3D Slab Models ........................................................ 27 2.4.6 Optimal Pseudo 3D Slab Model ..................................................................... 34 Influence of Floor Slabs on Shear Walls .............................................................. 35 Chapter 3 - 3.1 Walls for Analysis ........................................................................................................ 35 3.2 Slab Effect on Shear Walls ......................................................................................... 37 3.2.1 General Response ........................................................................................... 37 3.2.2 Internal Shear Wall Damage .......................................................................... 40 3.2.3 Horizontal Restraint ....................................................................................... 43 iv 3.2.4 Strain Profile Over Wall Height ..................................................................... 44 3.3 Slab Behaviour ............................................................................................................. 49 3.3.1 Cracking ......................................................................................................... 49 3.3.2 Force-Deformation Response ......................................................................... 52 3.3.3 Internal Stresses ............................................................................................. 54 Approximate 2D Model of Slabs ........................................................................... 59 Chapter 4 - 4.1 Development of 2D Model........................................................................................... 59 4.2 Parametric Slab Size Test ........................................................................................... 62 4.3 Capturing 3D Slab Effects .......................................................................................... 65 4.3.1 Wall Cracking and Damage ........................................................................... 65 4.3.2 Simplified Flow of Forces .............................................................................. 67 4.3.3 Force-Deformation Plot ................................................................................. 69 4.4 Modifications................................................................................................................ 72 4.4.1 Concrete Tension Strength ............................................................................. 72 4.4.2 Steel Ratio Parallel to Wall ............................................................................ 73 4.4.3 Post-Yield Truss Bars for Stability ................................................................ 74 4.4.4 Ineffective Slab Region Beyond Wall End .................................................... 74 4.4.5 Reduced Stiffness ........................................................................................... 75 4.5 Matching 2D Models for Wall Behaviour Study ...................................................... 78 Shear Behaviour of Membranes and Walls ......................................................... 80 Chapter 5 - 5.1 VecTor2 Prediction of SE8 Membrane Test ............................................................. 80 5.2 Nonlinear Membrane Shear Model ........................................................................... 83 5.2.1 Other Nonlinear Models ................................................................................. 83 5.2.2 Nonlinear Membrane Shear Model ................................................................ 84 5.2.3 Prediction of Experimental and VecTor2 SE8 Test ....................................... 86 5.3 Parametric Wall Study................................................................................................ 87 5.3.1 Parameters Studied ......................................................................................... 88 5.3.2 Development of Global Shear Stress-Strain Response .................................. 88 5.3.3 Development of Local Damage Indicators .................................................... 92 5.4 Behaviour of Membranes and Walls ......................................................................... 94 5.4.1 General Response ........................................................................................... 94 5.4.2 Local Damage Indicator ................................................................................. 98 5.4.3 Global Shear Stress-Strain Response ........................................................... 100 5.5 Shear Response Model for Shear Walls .................................................................. 102 5.5.1 Shear Strength .............................................................................................. 104 v 5.5.2 Yield Strain .................................................................................................. 104 5.5.3 Shear Strain at Ultimate and Ductility ......................................................... 106 Behaviour of Wall-Slab Systems ......................................................................... 107 Chapter 6 - 6.1 Slab Spacing Parameters .......................................................................................... 107 6.2 General Wall-Slab Behaviour .................................................................................. 108 6.2.1 General Response ......................................................................................... 108 6.2.2 Flow of Forces .............................................................................................. 112 6.2.3 Local Damage .............................................................................................. 116 6.2.4 Shear Stress-Strain Response ....................................................................... 120 6.3 Shear Response Model for Wall-Slab Systems ....................................................... 126 6.3.1 Simplified Shear Stress-Strain Response ..................................................... 126 6.3.2 Wall-Slab Shear Capacity Prediction ........................................................... 128 6.3.3 Prediction Using VecTor2 Data Results ...................................................... 130 6.4 Wall-Slab System Shear Capactiy ........................................................................... 133 6.4.1 Estimate of β and θ for Wall-Slab System ................................................... 133 6.4.2 Effective Slab Force ..................................................................................... 135 6.4.3 Simple Wall-Slab Shear Model .................................................................... 138 6.5 Future Development of Shear Stress-Strain Model ............................................... 142 Conclusion ............................................................................................................. 144 Chapter 7 - References .................................................................................................................................. 146 Appendices ................................................................................................................................. 149 A.1 Single-Stage Custom Postprocessors ...................................................................... 150 A.1.1 Model Geometry Input Page .......................................................................... 151 A.1.2 Deflection Summary ...................................................................................... 152 A.1.3 Wall Damage Sheet Input – Part A ................................................................ 153 A.1.4 Wall Damage Output – Part B ....................................................................... 154 A.1.5 Strain and Stress Distribution Profiles ........................................................... 155 A.2 Multi-Stage Custom Postprocessor ......................................................................... 156 A.2.1 Geometry and Load Input Sheet .................................................................... 157 A.2.2 Pushover Curves............................................................................................. 158 A.2.3 Damage Summary .......................................................................................... 159 A.2.4 Shear Stress & Strains .................................................................................... 160 A.3 Results from Wall Only Test ................................................................................... 161 A.4 Results from the Complete Analysis Suite ............................................................. 168 vi List of Tables Table 3.1: Elastic flexural capacity. .............................................................................................. 36 Table 4.1: Sample SCW3 conversions from 3D slab geometry to 2D slab layer (in mm). .......... 61 Table 4.2: SCW3 comparison of peak base shear from 3D model with 2D slab layer. ............... 64 Table 4.3: SCW4 comparison of peak base shear from 3D model with 2D slab layer. ............... 64 Table 5.1: Summary of equations to describe trilinear backbone membrane response. ............... 86 Table 5.2: Summary of parametric wall results. ........................................................................... 94 Table 5.3: Summary of equations for modified nonlinear wall shear model.............................. 102 Table 5.4: Measured local over average shear strain ratios. ....................................................... 106 Table 6.1: Equivalent floor height relative to wall length. ......................................................... 108 Table 6.2: Summary of results from the full analysis suite. ....................................................... 109 Table 6.3: Definitions of chart labels describing failure mechanisms. ....................................... 120 Table 6.4: Constant variables for wall-slab shear model. ........................................................... 130 Table 6.5: SCW3 wall-slab system shear capacity using VecTor2 data. .................................... 131 Table 6.6: SCW4 wall-slab system shear capacity using VecTor2 data. .................................... 131 Table 6.7: TSE8 wall-slab system shear capacity using VecTor2 data. ..................................... 132 Table 6.8: SCW3 wall-slab system shear capacity using β and θ from the General Method. .... 134 Table 6.9: SCW4 wall-slab system shear capacity using β and θ from the General Method. .... 134 Table 6.10: TSE8 wall-slab system shear capacity using β and θ from the General Method..... 134 Table 6.11: Summary of average slab horizontal stress. ............................................................ 138 Table 6.12: SCW3 wall-slab system shear capacity from simplified model. ............................. 139 Table 6.13: SCW4 wall-slab system shear capacity from simplified model. ............................. 139 Table 6.14: TSE8 wall-slab shear capacity from simplified model. ........................................... 140 Table 6.15: Summary of variables and values for simplified wall-slab shear model. ................ 141 vii List of Figures Figure 1.1: Isolated wall-slab system.............................................................................................. 1 Figure 1.2: Slabs limit crack propagation. ...................................................................................... 3 Figure 2.1: Fundamental theory basis for the MCFT (based on Vecchio & Collins, 1986). .......... 7 Figure 2.2: Comparison of experimental and VecTor2 analytical prediction for extreme wall conditions. Modified from Palermo & Vecchio (2007), © Journal of Structural Engineering, by permission. ............................................................................ 9 Figure 2.3: Comparison of stiffness and ductility of the concrete strengths used in analysis. ..... 11 Figure 2.4: Orientations of the wall and slab as 2-dimensional planes perpendicular to one another. ..................................................................................................................... 13 Figure 2.5: Plane sections remain plane principle holds still holds approximately true............... 14 Figure 2.6: Steps to develop the pseudo slab model by rotating the slab vertically. .................... 15 Figure 2.7: Comparison of node numbering scheme and resulting bandwidth (indicator of computational time). ....................................................................................................... 16 Figure 2.8: Illustration of negligible vertical truss stiffness that only allows horizontal wall displacements to be transferred to the slab. ............................................................ 18 Figure 2.9: Exploded plan slab view of wall and slab nodes connected by truss bars. ................ 18 Figure 2.10: Schematic of in-plane truss bar configurations (the wall and slab nodes coincide). .............................................................................................................. 19 Figure 2.11: Pseudo 3D model cannot capture the slab interaction with wall flanges. ................ 21 Figure 2.12: Plan view to compare the pseudo model effects on symmetrical slab cracking, SCW3 with 17 x 12 m slab at Δtop = 105 mm. ................................................ 22 Figure 2.13: Original pseudo model nodal displacements. .......................................................... 23 Figure 2.14: Comparison of relative nodal displacements for various truss bar configurations, SCW3, Δtop = 90 mm. ............................................................................ 24 Figure 2.15: Wall properties of SCW1. ........................................................................................ 26 Figure 2.16: Shear failure in SCW1 with cracking and yielding in diagonal struts, Vbase = 7043 kN & Δtop = 120 mm. .................................................................................. 27 Figure 2.17: Comparison of pseudo model response with total and shear pushover curve, SCW3 with slab: 17 x 12 m. ........................................................................................... 28 viii Figure 2.18: SCW3 comparison of derived shear pushover response for each model. ................ 30 Figure 2.19: SCW4 comparison of derived shear pushover response for each model. ................ 31 Figure 2.20: Horizontal strain with small slab, SCW3 Δtop = 70 mm. .......................................... 33 Figure 3.1: Wall geometry for analysis......................................................................................... 35 Figure 3.2: Pushover curves without a slab and with a slab at mid-height. .................................. 38 Figure 3.3: Wall crack pattern near failure with and without a mid-height slab, SCW3: Δtop = 100 mm, SCW4: Δtop = 42 mm................................................................. 40 Figure 3.4: Internal damage indicating crushing and yielding failure, SCW3 Δtop = 100 mm. .... 41 Figure 3.5: Internal damage indicating crushing and yielding failure, SCW4 Δtop = 42 mm. ...... 42 Figure 3.6: Average horizontal wall strain from mid-height slabs, SCW3: Vbase = 7750 kN, SCW4: Vbase = 3360 kN. ......................................................... 43 Figure 3.7: Biaxial strain state for elements with significant horizontal strains and limited vertical strains. .................................................................................................... 44 Figure 3.8: Vertical shear strain profile with select shear strain layer profiles, SCW3 Vbase = 7750 kN. .................................................................................................. 45 Figure 3.9: Vertical curvature profile with select vertical strain layer profiles, SCW3 Vbas e= 7750 kN. .................................................................................................. 46 Figure 3.10: Principle stresses highlight regions of stress concentration for SCW3, Δtop = 100 mm. ................................................................................................................ 47 Figure 3.11: Principle stress regions with and without mid-height slab for SCW4, Δtop = 42 mm. .................................................................................................................. 48 Figure 3.12: Simplified diagram of the stress field pattern in shear critical walls with and without mid-height floor slabs. ....................................................................................... 49 Figure 3.13: Stages of slab cracking along the wall pushover curve. ........................................... 50 Figure 3.14: Progression of slab cracking patterns for SCW4. ..................................................... 51 Figure 3.15: Force-deformation plots for SCW3. ......................................................................... 53 Figure 3.16: Force-deformation plot for SCW4. .......................................................................... 54 Figure 3.17: In-plane internal slab stress with the SCW4 slab, Δtop = 40 mm. ............................. 55 Figure 3.18: Simplified diagram of the resisting mechanism and forces within the slab. ............ 57 Figure 4.1: 3D visualization of 2D slab model. ............................................................................ 60 ix Figure 4.2: Comparison between 3D and 2D slab models for monotonic and reverse cyclic pushover response for SCW3. .............................................................................. 61 Figure 4.3: Shear pushover curves for the same slab length perpendicular to wall...................... 63 Figure 4.4: Comparison of wall damage for the 3D and 2D model, SCW3 Δtop = 100 mm. ........ 66 Figure 4.5: Comparison of wall damage for the 3D and 2D model, SCW4 Δtop = 42 mm. ......... 66 Figure 4.6: Comparison of principle tension and compression stress for 3D and 2D layer model, SCW3 Δtop = 100 mm.......................................................................................... 67 Figure 4.7: Comparison of the horizontal stress field for the 3D and 2D models, SCW3 Δtop = 100 mm...................................................................................................... 68 Figure 4.8: Equivalent slab force-wall displacement plot confirms the slab force is similar between the models. ........................................................................................................ 69 Figure 4.9: Equivalent slab force-slab strain plots........................................................................ 71 Figure 4.10: Pushover curves for full and reduced concrete tension strength. ............................. 73 Figure 4.11: Pushover curve for full and half slab steel ratio for SCW3 and SCW4. .................. 74 Figure 4.12: Simplified diagram highlighting effective and ineffective slab regions to the slab resistance. ................................................................................................................ 75 Figure 4.13: Shear pushover curve for modified slab layer. ......................................................... 75 Figure 4.14: Equivalent slab force-slab strain curve for modified 2D slab layer. ........................ 76 Figure 4.15: Equivalent total normal stress parallel to the wall, SCW3 Δtop = 100 mm. .............. 77 Figure 4.16: Pushover curve matching the 2D model to the 3D model for use in the analysis suite. .................................................................................................................. 79 Figure 5.1: Shell element tester configuration (based on Stevens et. al., 1991). .......................... 81 Figure 5.2: VecTor2 SE8 models with deformed shape. .............................................................. 82 Figure 5.3: Comparison of Model A and B at reproducing uniform shear forces. ....................... 82 Figure 5.4: VecTor2 predicts the shear stress-strain behaviour of the SE8 test. Test data from Gerin (2003), © Marc Gerin, by permission. ......................................................... 83 Figure 5.5: Different shear backbone curves (based on Peer, 2010). ........................................... 84 Figure 5.6: Simplified nonlinear trilinear regions for the membrane model. ............................... 85 Figure 5.7: Nonlinear backbone model compared against the VecTor2 membrane element. Figure a) from Gerin (2003), © Marc Gerin, by permission. ......................... 87 x Figure 5.8: Shear stress profiles in wall. The average web stress is predicted using dv from flange to flange, SCW3 B – Wall Only, Δtop = 60 mm. ......................................... 89 Figure 5.9: Comparison of shear strain measurement methods, SCW3 B - Wall Only Δtop = 60 mm. ............................................................................... 90 Figure 5.10: Predicted versus measured wall displacements over the height, SCW4 Δtop = 38 mm........................................................................................................ 91 Figure 5.11: Local yielding and crushing damage values indicate if wall is under diagonal tension or compression failure. Damage state at first yield. .......................................... 92 Figure 5.12: Relation between average, maximum and local shear strain indicates wall behaviour. ............................................................................................................... 93 Figure 5.13: Reverse cyclic analysis to determine the actual failure point on the monotonic curve. ............................................................................................................ 95 Figure 5.14: Pushover curves for the walls with 0 and 10% axial load. ....................................... 96 Figure 5.15: Cracking pattern at failure load for walls with 0% axial load. ................................. 97 Figure 5.16: Cracking pattern at failure load for walls with 10% axial load. ............................... 98 Figure 5.17: Select damage indices for all walls. Shear stress for each figure is labelled as vn (vn/fc’). ....................................................................................................... 99 Figure 5.18: Shear stress-strain response of wall and an equivalent membrane element. Solid lines are VecTor2 results and dashed lines are the membrane backbone model. ........................................................................................................................... 101 Figure 5.19: Shear stress-strain response for wall and the modified membrane nonlinear model. Solid lines are VecTor2 results and dashed lines are the membrane backbone model. ......................................................................................... 103 Figure 5.20: Localized yielding of a few elements cause a reduction in stiffness for the entire wall. .................................................................................................................... 105 Figure 6.1: Slab spacing intervals for parametric study. ............................................................ 107 Figure 6.2: Pushover curves highlighting slab effect on global wall capacity. Walls have 10% axial load. .......................................................................................... 111 Figure 6.3: Crack pattern of SCW3 B at 70 mm top displacement (slab layers in blue). ........... 112 Figure 6.4: Principle compression stress highlights shallow strut angle with closer slab spacing, SCW4 A Δtop = 50 mm. .................................................................................. 113 xi Figure 6.5: Diagram of flow of forces in wall with wide and tight slab spacing........................ 114 Figure 6.6: Variation of theta with increasing displacement for all wall pairs. .......................... 115 Figure 6.7: Comparison of internal damage for SCW3 A with same average shear strain near 0.5 mm/m. Shear stress for each figure is labelled as vn MPa (vn/fc’). ................. 117 Figure 6.8: Maximum and local shear over average shear strain ratios for the analysis suite. ... 119 Figure 6.9: Shear stress-strain response for SCW3 with failure mechanisms labelled. ............. 121 Figure 6.10: Shear stress-strain response for SCW4 with failure mechanisms labelled. ........... 123 Figure 6.11: Shear stress-strain response of TSE8 with failure mechanisms labeled. ............... 125 Figure 6.12: General behaviour and mechanisms of the wall-slab shear stress-strain response. ................................................................................................... 127 Figure 6.13: For tall walls, Vslab is analogous to Vs. ................................................................... 129 Figure 6.14: Variation in normal stress parallel to the wall in the slabs. .................................... 136 Figure 6.15: Plan of effective slab area. ..................................................................................... 137 Figure A1.1: Representative image of model geometry and input page. .................................... 151 Figure A1.2: Representative image of deflection summary page. .............................................. 152 Figure A1.3: Representative image of input element numbering and location for damage calculations. ..................................................................................................... 153 Figure A1.4: Wall damage output indicating cracking and values of crushing/yielding............ 154 Figure A1.5: Stress and strain profiles output. ........................................................................... 155 Figure A2.1: Model geometry and input page. ........................................................................... 157 Figure A2.2: Pushover curves from the VecTor2 data to verify with Augustus postprocessor. 158 Figure A2.3: Representative image of damage summary sheets for selected load stages. ......... 159 Figure A2.4: Sample of averaged shear strain profiles over the wall height. ............................. 160 Figure A3.1: Summary of data results for SCW3 A - Wall Only. .............................................. 162 Figure A3.2: Summary of data results for SCW3 B - Wall Only. .............................................. 163 Figure A3.3: Summary of data results for SCW4 A - Wall Only. .............................................. 164 Figure A3.4: Summary of data results for SCW4 B - Wall Only. .............................................. 165 Figure A3.5: Summary of data results for TSE8 A - Wall Only. ............................................... 166 Figure A3.6: Summary of data results for TSE8 B - Wall Only................................................. 167 Figure A4.1: Summary of data results for SCW3 A - 1.5:1 Ratio. ............................................. 169 Figure A4.2: Summary of data results for SCW3 A - 1.0:1 Ratio. ............................................. 170 xii Figure A4.3: Summary of data results for SCW3 A - 0.5:1 Ratio. ............................................. 171 Figure A4.4: Summary of data results for SCW3 B - 1.5:1 Ratio. ............................................. 172 Figure A4.5: Summary of data results for SCW3 B - 1.0:1 Ratio. ............................................. 173 Figure A4.6: Summary of data results for SCW3 B - 0.5:1 Ratio. ............................................. 174 Figure A4.7: Summary of data results for SCW4 A - 1.5:1 Ratio. ............................................. 175 Figure A4.8: Summary of data results for SCW4 A - 1.0:1 Ratio. ............................................. 176 Figure A4.9: Summary of data results for SCW4 A - 0.5:1 Ratio. ............................................. 177 Figure A4.10: Summary of data results for SCW4 B - 1.5:1 Ratio. ........................................... 178 Figure A4.11: Summary of data results for SCW4 B - 1.0:1 Ratio. ........................................... 179 Figure A4.12: Summary of data results for SCW4 B - 0.5:1 Ratio. ........................................... 180 Figure A4.13: Summary of data results for TSE8 A - 1.5:1 Ratio. ............................................ 181 Figure A4.14: Summary of data results for TSE8 A - 1.0:1 Ratio. ............................................ 182 Figure A4.15: Summary of data results for TSE8 A - 0.5:1 Ratio. ............................................ 183 Figure A4.16: Summary of data results for TSE8 B - 1.5:1 Ratio. ............................................. 184 Figure A4.17: Summary of data results for TSE8 B - 1.0:1 Ratio. ............................................. 185 Figure A4.18: Summary of data results for TSE8 B - 0.5:1 Ratio. ............................................. 186 xiii Acknowledgements I would like to thank my supervisor Dr. Adebar who helped guide and shape my research and pushed me to thoroughly develop my thesis. I would also like to thank Jennifer and my family for their constant support, encouragement and for keeping me grounded. Many thanks to my friends and colleagues, especially Jeff, who provided helpful input as well as a break from the thesis when needed. Finally, I would like to thank NSERC and UBC for without their financial support this thesis would not be possible. 1 Introduction Chapter 1 - In the design of reinforced concrete shear walls, shear is an important and complex issue. The nonlinear shear response of walls is not fully understood and often gross approximations are made in the analysis. In typical shear walls, flexure dominates the response and design while the shear response often does not govern the design. However, in certain circumstances the shear demand can be significantly larger than expected. Considering shear failures are often assumed to be brittle, this is cause for concern. A couple studies have found the shear demand in walls could be much higher than the designed shear capacity, suggesting that walls may experience shear failure during a seismic event. However, these studies tend to neglect that the shear walls are a component of a wall system, most notably a wall-slab system where the wall is surrounded by large concrete floor slabs at every level (see Figure 1.1). These slabs can act like external stirrups providing additional shear capacity. Therefore, the extra shear demand in walls may be counteracted by the extra shear capacity from the wall-system. The objective of this thesis is to quantify the shear capacity and response from the wall-slab system. This study does not investigate the separate issue of the slab outrigger effect where the floor slabs transfers load to the exterior columns; only isolated walls surrounded by are investigated. Figure 1.1: Isolated wall-slab system. 2 This study will focus on typical high-rise reinforced concrete core walls which form the seismic force resisting system. The typical design of the shear wall consists of obtaining the overturning moments and shear demands by performing a linear response spectrum analysis (RSA). The inelastic demands are obtained by scaling the RSA results by reduction factors. However, some studies have shown by performing a nonlinear time history analysis (NLTHA) that the shear demands can significantly exceed the demands from RSA. This effect is known as dynamic shear magnification and is the result of nonlinear higher mode effects from THA. However unlike the flexural response, the nonlinear shear response of walls is not fully understood and often gross approximations are made in the analysis. As the result, there are key 2 issues with dynamic shear magnification: first, the shear demand is often higher because the shear behaviour is assumed linear and second, most analysis neglects the additional shear capacity from the wall- slab system. The issue of dynamic shear magnification has been known and studied for a while. In the concrete design code CSA A23.3-04, the designer is required to account for the shear demand include the magnification of shear due to inelastic effects of higher modes. However, there is no guidance on how to account for this magnification. In contrast, the New Zealand and European codes have provided amplification factors wv and ε respectively (Rutenberg & Nsieri, 2006). A study by Yathon (2011) has also recently developed equations to amplify the shear demand for Canadian codes where the shear demand can be magnified more than 2 times. Increasing the code demands can inflate construction costs; therefore it is crucial to determine if amplifying the shear demand is necessary. A recent study assessed the seismic performance of a shear wall using Canadian design procedures. They defined the dynamic shear magnification as the ratio of shear force determined from the code’s linear RSA to the shear force determined from THA (Boivin & Paultre, 2010). This ratio, βV, was determined to be approximately 1.5 for the cantilever wall system whereas the New Zealand magnification factor wv for the same building is 1.7. However, this study utilized a linear shear model with a stiffness based on the effective shear area. If shear behaviour was linear and brittle, this approximation would be sufficient but real structures can crack and yield during seismic events. Shear cracking and yielding reduces the stiffness which therefore reduces the shear force demand. Conversely, their linear shear model is much stiffer and over predicts 3 the shear demand. Therefore an accurate nonlinear shear model is required to emulate the response of a real structure. The second issue with dynamic shear magnification is the wall-slab system is not considered. In many cases, a shear wall is located in the center of the floor slab which contains a significant amount of steel and can act like a horizontal clamp on the wall. It is difficult to have a shear failure at this location simply because the failure must spread through the wall and the large slab. Consider a typical high-rise shear wall 9m long with floor spaced at 2.7 m, a diagonal crack is very likely to cross a floor slab which engages the slab (Figure 1.2). The slabs must influence the shear capacity but this has yet to be quantified. Ideally, there would be shear wall tests with slabs where the shear capacity with and without slabs can clearly be observed. However, there are only a limited number of shear wall tests with typical floor slabs and almost no tests that fail in shear. A few of the more notable tests are discussed below. Figure 1.2: Slabs limit crack propagation. A large-scale test of a 7-storey shear wall test with slabs was performed at the large shake table at the University of California at San Diego. The structure represented a residential structure complete with a rectangular wall surrounded with concrete slabs 200 mm thick spaced at 2.74 m (Restrepo, Conte, & Panagiotou, 2010). The wall was supported by piers on either side to provide lateral and torsional stability. The wall was subjected to various ground motions which resulted in an inelastic response. During the tests, the wall formed a flexural hinge at the base, which satisfied the design methodology, and the failure occurred from a vertical lap splice failure at the base (Marios Panagiotou, 2008). Even though the study tested a wall-slab system, the wall experienced very little shear demand; the peak shear recorded was 1.54 MPa, a shear ratio vf/fc’ 4 of 0.047. Therefore, little can be learned from the test regarding the slab effect on the wall shear capacity. Another notable study is the test performed by Thomsen and Wallace (2004) where T-shaped walls with slabs was subjected to reverse cyclic loading. The walls were heavily instrumented including diagonal wire potentiometers to measure shear deformations. During one of the cycles with the web in compression, the applied shear exceeded the predicted nominal shear capacity while not exceeding the flexural capacity. However ultimate failure was the result of the vertical bars in the web buckling due to insufficient transverse steel. Since the vertical bars buckled, it is not possible to determine if the wall would have failed in shear. In addition, the observed shear strength above the predicted shear capacity could have been the result of an inaccurate shear capacity prediction. Again, this test does not provide answers regarding the slab effect on the wall shear capacity. Shake table tests were also performed by Ghorbanirenani et al. (2010) on two reinforced concrete shear walls with a focus on higher modes. The approximate half-scale cantilever walls were connected to external artificial masses through small slabs at each floor. These tests followed analysis from Tremblay et al., (2008) which determined the shear demands would exceed the design shear capacity. These results were also confirmed with the nonlinear concrete finite element program VecTor2 which also predicted the design base shear would be exceeded, but to a lesser extent. Even though the design base shear was exceeded, a shear failure was not observed in the analysis which could mean the design base shear prediction is low or shear failure is prevented by some other mechanism. The preliminary shake table tests did not exhibit shear failure or significant shear damage. The tests also revealed the concrete contribution could be up to 25% above the nominal shear capacity. However, these tests were not specifically assessing the slab effect and actually ignored the contribution from the slabs. Considering this, these tests cannot indicate whether the additional wall shear capacity is from poor shear capacity prediction or if the slabs are providing some benefits. The final notable study is of a 1:3 scale shake table test of a 5-storey H-shaped wall with slabs as part of the ECOLEADER research project (Fischnger, Isaković, & Kante, 2006). The thin, lightly-reinforced walls are effectively two T-shaped walls connected by coupling beams surrounded by floor slabs. The wall system was subjected to bi-directional ground motions and 5 the wall eventually experience shear failure in the T-wall webs. This test may provide an indication of the shear capacity of walls with slabs considering the coupling beam experienced little to no shear cracking. However there are several issues that diminish the validity of the results. First, the shear failure occurred in a coupled wall which behaves differently than cantilever walls. Second, the thin walls (60 mm thick) were connected to thicker, heavily- reinforced slabs along with inconsistent bar detailing which does not replicate realistic configurations. Finally, limited data on the shear failure was obtained because shear failure was not expected in the test. Even though this wall does achieve shear failure in a shear wall with slabs, unfortunately there is no useful data from the test. The issues discussed above all lead to the purpose of the current work: to investigate the influence of floor slabs on shear walls failing in shear. One of the lessons learned from the studies above is the only conclusive method to determine the slab effect are multiple tests comparing the shear behaviour with and without slabs. Another lesson is the walls must have sufficient flexural capacity (and detailing) such that shear failure is achieved. This is relatively difficult to ensure since there are many failure mechanism that can form prior to shear failure and it is a challenge to prevent all others from occurring. Finally, detailed measurements of the shear strains and stress within the wall are required to capture the shear response. To satisfy these conditions, a state-of-the-art finite element analysis is performed studying the behaviour of shear-critical walls. A benefit to analytical work is that many tests can be performed with different parameters without having to design and build multiple test specimens. The analysis will vary parameters such as horizontal reinforcement ratio, axial load and different slab spacing. Several models that can emulate the 3D slab effect in a 2D analysis are developed and presented. In addition, this study will provide an indication the validity of existing nonlinear shear models to describe wall behaviour. By also analyzing the failure of walls without slabs, a comparison of the shear behaviour of walls and equivalent membranes is performed. At the end of the work, techniques are presented which describe the shear response of walls and wall-slab systems. 6 Finite Element Analytical Tools Chapter 2 - The analysis in this study is performed with the state-of-the-art nonlinear finite element analysis (FEA) program VecTor© developed at the University of Toronto. VecTor combines advanced reinforced concrete material models with the power of FEA to capture the complex behaviour of concrete structures (Wong & Vecchio, 2002). The 2-dimensional version of the VecTor suite, VecTor2, is used for analysis since it is powerful and robust. The program has been developed extensively and refined where it can accurately model the behaviour and internal wall forces of the structure. Although it cannot replace experimental testing in a laboratory, it can reasonably reproduce specimen testing while providing even more data on the behaviour; internal strains, forces and damage levels for the entire structure for every load stage. In fact with the nature of FEA, VecTor2 produces so much data that the information can be overwhelming unless the user understands how to extract the useful data. Therefore the user must understand the behaviour of structures as well as the implementation of finite element analysis and its limitations. 2.1 MEMBRANE ELEMENT THEORY At the program core, VecTor2 is based on models of the Modified Compression Field Theory (MCFT) and the Disturbed Stress Field Model (DSFM). These analytical models define the load-deformation response of the composite action of reinforced concrete elements. The models are described in further detail in the following subsections. 2.1.1 Modified Compression Field Theory The Modified Compression Field Theory (MCFT) was developed by researchers at the University of Toronto to describe the behaviour of concrete elements subjected to in-plane shear and normal stresses (F.J. Vecchio & Collins, 1986). The MCFT is the result of many tests of panel elements subjected to uniformly-loaded shear and normal stresses (Figure 2.1a). By modelling the structure with many small regions (elements), it can assume that each element is experiencing uniform loading. To describe the behaviour of the composite material, strain compatibility is assumed between the concrete and the perfectly-bonded steel (Figure 2.1b), where cracked concrete has its own distinct response from the steel response (Figure 2.1c). 7 a) Average Stress Condition b) Strain compatibility c) Distinct Stress-Strain material relationships Figure 2.1: Fundamental theory basis for the MCFT (based on Vecchio & Collins, 1986). There are many assumptions used in the MCFT including assuming stress and strain are averaged uniformly over many cracks and the cracks are uniformly distributed and rotate (Wong & Vecchio, 2002). The MCFT also assumes that the steel is distributed uniformly in the element and there is no slippage between the steel and concrete (strain compatibility). The orientation of the principle strain angle and principle stress angle is assumed as the same. As a result of these assumptions, a check must be performed to ensure that localized failure or slip does not occur at the cracks. The MCFT has been shown to accurately describe the behaviour of membrane elements and has been incorporated into the shear design methods of the Canadian codes (Bentz & Collins, 2006). However, even though the MCFT can predict the ultimate shear strength of an element, which is sufficient for code design, it is not as accurate at predicting the entire load-deformation response for certain conditions. Having an accurate load-deformation response is critical for FEA and as such the MCFT has been modified to the DSFM to account for the discrepancies. 2.1.2 Disturbed Stress Field Model The Disturbed Stress Field Theory (DSFM) is an extension of the MCFT to address fundamental issues in order to create a more robust element model. The assumption that the principle stress and strain angles coincide does not hold true to lightly-reinforced members where shear slip can occur. The result is that the MCFT generally overestimates the stiffness of these elements while it underestimates the stiffness of elements with limited strain angle rotation (Wong & Vecchio, 2002). To address this, the DSFM accounts for crack shear slip by distinguishing the strains due to concrete stress from the strains due to shear slip. This separates the orientation of the 8 principle strain and stress angle resulting in a smeared delayed rotating-crack model. The following are the generic compatibility and equilibrium equations for the DSFM. (1) (2) (3) This modification separates the total strains into the summation of the concrete net strains and shear slip strains (Equation 1). This is an important condition because the user must take the appropriate strain: total strains for the behaviour of the entire element and net strains for the behaviour of the concrete. As shown in Equation (2) and (3), the average stress condition is still the same as the MCFT, which sums the concrete and steel stress, and the total shear stress is taken by the concrete. The advantages of DSFM is that it can better predict the response of elements in outlying conditions and the results have been backed by experimental results (F.J. Vecchio, Lai, Shim, & Ng, 2001). The DSFM is applied in VecTor2 by default. 2.2 VECTOR2 FINITE ELEMENT IMPLEMENTATION AND VERIFICATION The main analysis tool to model the 3-dimensional slab effect is the 2-dimensional nonlinear finite element analysis program VecTor2. VecTor2 uses the DSFM to predict the response of an element with the power and flexibility of FEA to accurately describe the behaviour of the entire structure. VecTor2 also contains a large library of advanced concrete and steel models, allowing the user to extensively model the complex behaviour of reinforced concrete. VecTor2 has been shown to reasonably predict the behaviour of complex and unusual specimens. The following subsections will highlight the verification and implementation of VecTor2. 2.2.1 VecTor2 Experimental Verification Vector2 has been widely used as an analytical tool of reinforced concrete members due to its accurate predictions. The program has been validated against several experimental tests and correlates well to both monotonic and reverse cyclic loadings (Bohl, 2006; Ghorbani-Renani et al., 2009; Palermo & Vecchio, 2002, 2007). A study presented by Palermo and Vecchio (2007) summarizes the accuracy of the VecTor2 prediction of walls over a wide variety of conditions. In their study, slender walls with an opening and squat shear walls were tested experimentally 9 and were compared to the analytical prediction from VecTor2 (see Figure 2.2). These are both very difficult conditions to model because the walls experience strut action and stress concentration. VecTor2 is capable of modelling these extreme conditions as well as simple sectional regions. As can be seen from the figure below, VecTor2 captures the general reverse- cyclic behaviour fairly well including the load capacity and peak displacement. VecTor2 does slightly over predict the cracked stiffness but overall it is a very good match. a) Slender Wall with opening b) Squat shear wall Figure 2.2: Comparison of experimental and VecTor2 analytical prediction for extreme wall conditions. Modified from Palermo & Vecchio (2007), © Journal of Structural Engineering, by permission. There are many tests where VecTor2 has been shown to reasonably predict experimental results. The prediction is not perfect, but it is the best tool available to obtain the behaviour of a wide variety of walls with reasonable certainty. Therefore, the VecTor2 analytical results will be treated similarly to experimental data; raw stress and strain data obtained throughout the structure will be collated, summarized and used for further structural analysis. However, when performing purely analytical work it is important to keep in mind that the results will not be Wall RW3 Experiment Wall RW3 Analysis Wall SW3 Analysis Wall SW4 Experiment -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 Horizontal Drift (%) Horizontal Drift (%) -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 Horizontal Drift (%) -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 Horizontal Drift (%) -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 Horizontal Drift (%) 200 100 0 -100 -200 200 100 0 -100 -200 L o a d (kN ) 120 60 0 -60 -120 120 60 0 -60 -120 L o a d (kN ) L o a d (kN ) L o a d (kN ) 10 perfect. In some cases, VecTor2 may slightly over predict the analytical response because FEA does not contain the irregularities within an experimental test. These points are important to consider when developing models from the VecTor2 analytical results. 2.2.2 2D Element Implementation The study by Palermo and Vecchio (2007) provides some suggestions on modelling specimens in VecTor2. They suggest using low-powered rectangular elements with distributed steel smeared in the element. In selecting an appropriate mesh refinement, they suggest 14-16 elements in the shortest wall direction with a maximum element aspect ratio of 1.5. Using these guidelines, all of the elements are rectangular with distributed steel including the vertical flexural reinforcement. An alternative would be to define the principle flexural reinforcement with distinct line elements known as truss bars as suggested by Wong and Vecchio (2002). However, including truss bars increases the complexity of the model and it is unnecessary considering the flexural response is limited in this study. All of the walls have 18 elements in the shortest direction and the largest element aspect ratio in the web region is 1.16 (500 x 430 mm). Note the slab thickness required some elements in the slab region to have a higher aspect ratio, but these elements were monitored to ensure unusual loading situations did not occur as a result. 2.3 PRE AND POSTPROCESSING TOOLS VecTor2 is a FEA processor that calculates the structural response but it is only one part of the FEA process. VecTor2 requires programs to pre-process and post-process the data; one program to provide the model input and another to analyze the output. 2.3.1 Formworks The program Formworks© is a preprocessor with a graphical user interface for creating and checking the 2D finite element model for VecTor2. The user can define all the necessary model parameters where Formworks then creates the necessary input text files for VecTor2. The user has the ability to define model geometry, loads, material properties, and VecTor2 analysis parameters including the various concrete models. 11 All of the default models were used in analysis except for the concrete compression response and the crack allocation process. The concrete compression response has been defined by the Propovics model for normal strength concrete. The Propovics model better captures the greater stiffness and linearity prior to the peak stress and the reduced ductility of the higher-strength concrete (Wong & Vecchio, 2002). This model is a better representation of the actual concrete response as opposed to a parabolic model. The normal strength concrete model is adequate for the current study because the concrete strength ranges between 30 and 50 MPa (see Figure 2.3). Figure 2.3: Comparison of stiffness and ductility of the concrete strengths used in analysis. 2.3.2 Augustus The postprocessor packaged with VecTor2 is Augustus© that reads the output text files from VecTor2 and graphically presents the results. The user can obtain strain, stress and damage data for every element for each load stage as well as cracking and load-deformation plots. The program is a very powerful tool since it can quickly present the data from analysis. However, there are a few limitations to the program. First, Augustus cannot plot separately the data from 2 overlapping element layers, which affects the models developed in the following section. Second, the program is limited to presenting the element data directly from the text files; it can perform very limited structural analysis. The program does have the ability to present section plots where the curvature and average shear strain can be obtained for a selected layer. However, it cannot perform bulk structural analysis such as obtaining the curvature and 0 10 20 30 40 50 60 0 1 2 3 4 5 C o m pres si v e St res s (M P a ) Compression Strain (mm/m) fc' = 50 MPa fc' = 30 MPa fc' = 37 MPa12 shear strain distribution, flexural and shear deformation or the shear strain-stress response of a whole section. This structural analysis is crucial for the current study in order to understand the behaviour of the wall-slab system. Therefore, custom post-processors were developed to perform the majority of the structural analysis. 2.3.3 Custom Postprocessors To address the shortcomings of Augustus, custom post-processors were developed using the spreadsheet software Excel©. The custom postprocessors read the VecTor2 output text files which contain the strain and stress data for each element and node displacements. By providing the program with model geometry and parameters, it is capable of performing structural analysis to describe the wall behaviour. Two main postprocessors are developed; one for in-depth analysis of a single load stage and another for global responses for the entire pushover response. The single-stage processor was primarily used to explore the complex wall-slab behaviour in Chapter 2 to 4 while the multi-stage processor was used to compare the wall responses in Chapter 5 and 6. The single load stage postprocessor can calculate the flexural and shear deformations, the curvature and shear strain distribution, wall and slab cracking pattern and damage states (including yielding and crushing indicators). The multi-stage postprocessor can calculate the pushover curves, averaged damage indicators, and shear stress-strain responses for many regions of the wall. Both these programs are fairly flexible and can be adapted to analyze a wide variety of walls and conditions. The advantage to the custom postprocessors is they can perform a great deal of analysis and present the information succinctly such that only the useful data remains. Another advantage to these programs is the data for two overlapping layers can be presented separately. The full capability of the custom postprocessors is described in Appendix A.1 and Appendix A.2. 2.4 PSEUDO 3D SLAB MODEL A good slab model must be developed to study the effects of slabs on walls. The slab effect is a 3-dimentional problem since the slabs extend in the out-of-plane direction of the wall. Creating a full 3D model is complex and computationally-intensive and is not always necessary considering it is possible to present 3-dimensional geometry in a 2-dimensional space. For 13 instance, walls with flanges are often modeled in a 2D space where the out-of-plane motion is neglected and the flanges are modeled as a thicker 2D region. The wall-slab system can be similarly simplified by only considering in-plane forces in the wall and slab. Although one key difference is the slab is perpendicular to the wall in order to create the 3D space. Therefore, it is possible to model a 3D problem in a 2D space by modelling the wall and slab as 2D members subjected to in-plane forces, while accounting for the orientation of the slab (Figure 2.4). Figure 2.4: Orientations of the wall and slab as 2-dimensional planes perpendicular to one another. The reason for modelling in 2-dimensions is that VecTor2 can be used for the analysis. The VecTor suite does have a 3D version, VecTor3, however it has not been developed to the same extent as VecTor2 and there is no provided pre or postprocessor. As discussed previously, VecTor2 has been verified as a robust and accurate tool whereas the same level of verification is not available for VecTor3. However in order to be able to use VecTor2, a unique method is developed taking advantage of sectional wall behaviour and is described in the following subsections. Since a 2D analysis is being modified to emulate a 3D problem, the following model is labeled as a pseudo 3D slab model. The following subsections explore the development of the various pseudo slab models and selection of the best model for this study. Chapter 3 will study the slab influence on shear walls by using the optimal pseudo 3D model. 2.4.1 Original Pseudo Model Developing the pseudo 3D slab model relies on considering the wall and slab as two 2D plane members connected perpendicular to each other. In the wall-slab model, the slab surrounds the Y Z X Shear Wall (x-y plane) Floor Slab (x-z plane) 14 wall and is supported only by the wall; there are no outrigger effects. As such, loads can only be introduced into the slab from horizontal expansion of the wall and where the wall’s flexural rotation should not induce loads into the slab. These conditions can be satisfied by employing the sectional principle of plane sections remain plane. In sectional analysis, it is assumed that the vertical strains vary linearly along with length of the wall, defining curvature (Figure 2.5a). Although this principle does not hold completely true in regions near boundary conditions or in nonlinear shear critical sections (the current wall), it is a good starting point to develop the model (Figure 2.5b). a) Horizontal lines rotate while remaining straight b) Theory is adequate for even shear critical walls. Figure 2.5: Plane sections remain plane principle holds still holds approximately true. Using this theory, any component that is attached to the wall along a horizontal line will not add to the vertical stiffness of the wall since it will simply rotate with the wall. Considering this, if the slab is attached vertically along a horizontal line to the wall then it will not affect the vertical stiffness of the wall but it will provide horizontal restrain in the wall. This is the basis of the pseudo 3D slab model; see Figure 2.6 to visualize the steps involved. The first step is to directly connect the slab to wall along a single horizontal line by sharing the nodes at the wall-slab interface. Next, the slab is rotated from its original horizontal position to vertically in-line with the wall (Figure 2.6a & b). According to the plane sections principle, the slab will rotate along Rotating horizontal lines Linearly-varying vertical strains 15 with the wall without affecting the vertical wall curvature, but it will provide horizontal restraint (Figure 2.6c). In its final position, the slab affects the wall in the same manner as if it were in its original horizontal position because there are no out-of-plane forces in the wall, thus only horizontal restraint is required. Therefore, a slab model is developed where the slab can stiffen the wall’s horizontal (shear) stiffness but does not affect the flexural stiffness of the wall. This technique is able to emulate a 3D wall-slab system in a 2D computer program. a) Rotating slab plate vertically in line with wall b) 2D slab and wall planes can be modeled and analyzed together c) As the wall bends, the slab does not affect the flexural stiffness of the wall Figure 2.6: Steps to develop the pseudo slab model by rotating the slab vertically. To implement this model into VecTor2, the 3D slab and wall geometry is created with a single line of nodes shared between the slab and the wall where they intersect. Again, the slab is modeled vertically as described above. The slab and walls could be visualized as two separate models side by side where the nodal displacements are linked at the interface of the wall and slab. To reduce confusion, the slab dimensions will be defined in its original horizontal orientation even though it is modeled vertically. However, the computational requirement (bandwidth of the stiffness matrix) would increase significantly without considering the node numbering pattern. Increasing the bandwidth increases the computational time by approximately a power of two where doubling the bandwidth would quadruple the computational time (Wong & Vecchio, 2002). Reducing the bandwidth is achieved by creating elements with nodes numbers that are as close as possible. However, by having the slab and the wall sharing a line of nodes, the elements sharing nodes will naturally have a large node number difference, increasing the bandwidth (see Figure 2.7a). 16 Ignoring the node numbering pattern produces large bandwidth and as such results in unacceptable computational times. a) Without modifying numbering scheme results in large bandwidth b) Numbering the nodes in alternating layers towards the wall-slab interface results in much lower bandwidth Figure 2.7: Comparison of node numbering scheme and resulting bandwidth (indicator of computational time). 17 To reduce the bandwidth, the wall nodes are numbered sequentially in alternating layers from the top and bottom of the wall in towards the slab layer such that the nodes at the slab interface are the highest. Then the slab nodes are numbered sequentially in alternating layers outward from the wall interface such that the highest node number are at the extreme ends of the slab (see Figure 2.7b). The nodes are numbered in alternating layers where wall layer 1 has node numbers 1-19, layer 45 has numbers 20-38, layer 2 has numbers 39-57, and layer 44 has numbers 58-76 and so on. Using this method produces the node numbering scheme with the smallest possible bandwidth for the given slab configuration. Depending on the slab size and configuration, it is possible to achieve a bandwidth that is at least 6 times smaller, resulting in reducing the computational demand by 36 times. Even with this method and modern computer processors, a typical pushover analysis will take in excess of 30 minutes to complete while a single reverse cyclic loading may take several hours. Without this method a single analysis could take several days. This method was required in order to have reasonable computational times, although using this method still requires significant computational time. 2.4.2 Modified Pseudo Model The modified pseudo slab model is developed to address fundamental deficiencies with the 3D pseudo model previously described. The original pseudo model is based on the principle of plane sections remaining plane in order for the slab to not provide additional flexural stiffness. This assumption does not hold true in shear-critical sections, which is the basis of the current study. However, the original pseudo model can be modified such that vertical wall strains are not transferred to the slab. Through the use of truss bars, the connection between the slab and the wall is modified such that only horizontal displacements are transferred to the slab. Ideally, the perfect model would have slab nodes slaved to the wall nodes where only horizontal displacements are transferred. However, VecTor2 currently does not have that capability so the alternative is to use extremely stiff horizontal truss bars to transfer horizontal displacements between the wall and slab. Using truss bars to transfer only horizontal displacements depends on the mechanics of out of plane buckling of an axially-loaded member. Using axially rigid members (truss bars) connecting each slab node to adjacent wall nodes, the horizontal wall displacements is transferred through the truss bars to the slab (Figure 2.8a). However, once the wall nodes displace vertically then the slab nodes are not affected since the truss bar system 18 provides minimal out-of-plane (vertical) restraint (Figure 2.8b). In order to transfer all of the wall displacements to the slab a series of stiff truss bars connect the wall nodes to adjacent slab nodes. The truss bars must connect to adjacent slab nodes because connecting the truss bars to nodes in the same location would transfer both horizontal and vertical displacements. a) Horizontal displacements are transferred to slab b) Vertical wall displacements are not transferred to slab Figure 2.8: Illustration of negligible vertical truss stiffness that only allows horizontal wall displacements to be transferred to the slab. The configuration of the truss bar link between the slab and the wall can have an impact on the system behaviour. Since the wall node displacements are transferred to adjacent slab nodes, a different truss configuration can alter the slab loads. Figure 2.10 presents several different truss bar configurations that were developed in order to find the optimal truss bar configuration. Figure 2.9 illustrates how the exploded schematic of the truss bar configuration connects the wall and slab in the same coinciding line. It is important that the configuration is symmetrical in order for the response to be the same regardless of the loading direction. Typically, a node is shared between the wall and slab in order to maintain model stability and to fix the relative location of the slab. Figure 2.9: Exploded plan slab view of wall and slab nodes connected by truss bars. 19 a) Cross Truss with Center Point b) Outward Truss with Center Point c) Outward Truss with End Points d) Full Outward Truss not directly connected to slab Figure 2.10: Schematic of in-plane truss bar configurations (the wall and slab nodes coincide). Each truss bar configuration has advantages and disadvantages and the optimal model is selected from the best overall response in Section 2.4.5. The first bar configuration, Cross Truss, has each slab node connected to two adjacent wall nodes to better transfer the horizontal displacement. However, the bars may provide additional stiffness to the slab. The second configuration, Outward Truss, is similar to Cross Truss except that each slab node is influenced by only one adjacent wall node. This limits the influence of the bars but forces the slab to displace away from the wall center. The third configuration, End Outward Truss, modifies Outward truss such that there is no support at the center of the slab, which is where most of the deformation occurs. Finally, the Full Outward Truss configuration is a modification to the Outward Truss but it does not have a support at the center. This eliminates the support’s 20 influence on the slab loads but it also allows the slab to move globally away from the wall. Most of the truss bar configurations have an additional weak vertical truss bar which connects the end of the wall to the slab. This is required to prevent the slab from pivoting around the shared wall and slab node at the center of the wall. No force is transferred in this node; it is only mathematically required to keep the connected wall and slab nodes in line with each other. The primary objective for the truss bar configuration is to be able to transfer all of the horizontal nodal displacements from the wall to the slab. It is clear that this is directly related to the stiffness of the bars; if the truss bars are infinitely rigid, then all of the wall horizontal displacements must be transferred to the slab. If the truss configuration is perfect, the wall nodal displacements are perfectly transmitted to the slab where the slab’s stiffness can influence the wall. However, it is possible that the truss configuration could provide some additional axial stiffness, which will not reflect the true 3D slab response. This creates two conflicting conditions for the truss bars to pass: the configuration must be capable of transferring the nodal displacements but not be so stiff such that the truss bars themselves provide addition strength. These conditions will be analyzed when determining the optimal truss bar configuration in Section 2.4.4. 2.4.3 Pseudo Slab Model Issues The pseudo slab model emulates a 3D problem in a 2D space which inherently has issues preventing it from modeling the true slab response. One key issue is the 2D analysis cannot model the effect of wall flanges on the slab. In 2D analysis, element thickness determines the forces within the element but in reality the wall and slabs interact as 2D planes with no thickness. As such, the pseudo models cannot model the interaction the slab has with the wall flanges; all the walls are essentially rectangular. As can be seen in Figure 2.11, a full 3D model can capture the interaction between the flanges such as stress concentration at the corners whereas the pseudo model cannot. This will clearly alter how the internal slab forces are distributed and should be considered when the slab shear flow diagrams are presented Section 3.3. 21 3D Model Pseudo 3D Model 3D Rendering 3D Rendering 2D Slab Plane View a) 3D models would capture stress concentration around wall flanges 2D Slab Plane View b) Pseudo 3D models model wall web and flanges as a line of nodes Figure 2.11: Pseudo 3D model cannot capture the slab interaction with wall flanges. A deficiency with the original pseudo slab model is that the plane sections principle may not hold true and as such vertical strains may be introduced in the slab. The main purpose for developing the modified pseudo truss model is to prevent the slab from providing additional flexural stiffness and to prevent non-symmetrical loads in the slab. If only in-plane wall loading and displacement is considered, the load distribution in a 3D slab must be symmetrical on either side of the wall (provided the slab is symmetrical). Figure 2.12 compares the symmetry of the cracking patterns of 3 different pseudo models: original, outward truss and full outward truss. Note the slab cracking patterns presented is to compare the models; Chapter 3 will explain in detail the slab behaviour. 22 a) Original Pseudo Model with unsymmetrical cracking perpendicular to wall b) Full Outward Truss pseudo model with lopsided cracking parallel to wall c) Outward truss pseudo model with symmetrical and balanced cracking Figure 2.12: Plan view to compare the pseudo model effects on symmetrical slab cracking, SCW3 with 17 x 12 m slab at Δtop = 105 mm. 23 As can be seen in Figure 2.12a, the original model clearly has asymmetrical slab loading, where more cracking is on the top portion, incorrectly suggesting that one side of the slab will crack more. The full outward truss configuration in Figure 2.12b has symmetrical cracking perpendicular to the wall but it concentrates the loads towards the left end of the wall. This is the result of this configuration not having a shared slab and wall node which would fix the global position of the slab to the wall. As such, the entire slab shifts slightly to the left, which would not reflect the realistic 3D response. Finally, the outward truss pseudo model in Figure 2.12c is able to present symmetrical cracking perpendicular to the wall while not concentrating cracking towards an end of the wall. This last model will be shown in the next subsection to be the best 3D pseudo slab model. A deficiency with the modified pseudo slab model is the fact that the truss bars must be connected to adjacent nodes, thus shifting forces in the slab. The result is that forces can concentrate or disperse from parts of the slab, altering the slab capacity. In addition, the truss bars must have sufficient stiffness in order to act like a rigid link. The truss bars have a bar area of 500,000 mm2, a yield stress of 1000 MPa, an elastic modulus of 1,000,000 mm2 and are designed to remain elastic. The truss bar configuration can be tested by comparing the relative horizontal and vertical displacements of the wall nodes to the slab nodes (Figure 2.13 & Figure 2.14). If the truss bar configuration is perfect, the relative horizontal displacement would be zero while the relative vertical displacements would be non-zero. As a baseline, the original pseudo model (slab connected directly to wall) is presented to highlight zero relative node motion in Figure 2.13. Figure 2.13: Original pseudo model nodal displacements. -15 -10 -5 0 5 10 15 20 34 35 36 37 38 39 40 0 1500 3000 4500 6000 7500 9000 Y -a x is Di spl a ce m en t (m m ) X -a x is Di spl a ce m en t (m m ) Initial Node Horizontal Location (mm) Wall Δx Wall Δy 24 b) Cross Truss Model c) Outward Truss Model d) End Outward Truss Model Figure 2.14: Comparison of relative nodal displacements for various truss bar configurations, SCW3, Δtop = 90 mm. -10 -5 0 5 10 15 20 34 35 36 37 38 39 40 0 1500 3000 4500 6000 7500 9000 Y -A x is Di spl a ce m en t (m m ) X -a x is Di spl a ce m en t (m m ) Initial Node Horizontal Location (mm) Wall Δx Slab Δx Wall Δy Slab Δy -10 -5 0 5 10 15 20 34 35 36 37 38 39 40 0 1500 3000 4500 6000 7500 9000 Y -A x is Di spl a ce m en t (m m ) X -a x is Di sp la ce m ent ( m m ) Initial Node Horizontal Location (mm) Wall Δx Slab Δx Wall Δy Slab Δy -10 -5 0 5 10 15 20 34 35 36 37 38 39 40 0 1500 3000 4500 6000 7500 9000 Y -A x is Di spl a ce m en t (m m ) X -a x is Di spl a ce m en t (m m ) Initial Node Horizontal Location (mm) Wall Δx Slab Δx Wall Δy Slab Δy 25 As can be seen from the figure above, the truss models allow for relative vertical displacement satisfying their design principle. As expected, the cross truss is much better at transferring the horizontal wall displacements to the slab since there is little relative movement. Conversely, the outward truss has relative displacement between the wall and the slab of up to 0.7 mm for this load stage. The figure above also provides locations of stress concentration in the slab where the horizontal displacement changes significantly. For the cross truss, the stress is concentrated at the center of the wall and where there is virtually no concentration elsewhere. However, the outward truss has stress concentration on both sides of center which better matches the original model and the wall cracking pattern. However, the truss bars themselves are sufficiently rigid considering that the relative displacements between connected nodes are typically less than 0.001 mm. At the peak base shear, the maximum stress in a truss bar is 2.5 MPa and with a bar stiffness of 1,000,000 MPa that translates into an average strain of 0.0025 mm/m and relative displacement of 0.00125 mm. The end outward truss and the center outward truss have very similar responses in relative horizontal displacements. The only noticeable difference is the vertical displacements very slightly less with the end outward truss since the support is at the wall ends. Another issue with the pseudo slab model is that only one slab can be modelled at a time, which prevents the analysis of modelling a typical structure with multiple floor slabs. Although it is technically possible to model more than one slab, the size and number are severely limited due to the maximum number of elements and nodes allowed in VecTor2. In addition, high bandwidth is unavoidable, exponentially increasing the computational time. These issues are addressed in a simplified 2D slab layer model discussed in Chapter 4. 2.4.4 Shear Critical Walls for Model Comparison Assessing the slab effect on the shear capacity of walls requires walls that fail in shear. The shear critical walls must have a clear diagonal shear crack with horizontal yielding in the web. By achieving shear failure, it is easy to observe the wall response with and without slabs. The wall geometry is similar to aspect ratios from a wall test and typical wall sizes from Birely et al. (2010). The lightly-reinforced wall SCW1 is described in Figure 2.15. 26 a) Vertical wall geometry b) Section detail Figure 2.15: Wall properties of SCW1. The final wall properties may not represent a typical shear wall but they are necessary to have a clear shear failure. The wall is a flexural wall with a slenderness ratio of 3:1, with a 9 m wall length, based on a typical 30-storey wall, and a wall height of 27 m. Wall geometry from the study by Birely et al. (2010) was used as starting points to modify the wall until shear failure was achieved. The horizontal steel ratio is reduced to the minimum non-seismic ratio of 0.2% (A23.3 c.14) to achieve the worst-case scenario. SCW1 is loaded monotonically with displacement control at the top of the wall and has an axial load of 0.1 fc’Ag applied. The shear failure of the wall is shown in Figure 2.16 below. The wall has significant shear capacity beyond first yield at 6350 kN up to a peak load of 7430 kN, well beyond the predicted shear capacity of 5530 kN from the General Method of A23.3-04. However, the additional shear capacity is sufficient such that the peak load is governed by flexural hinging. Therefore, other shear critical walls, SCW3 and SCW4, are developed with greater vertical flange steel to ensure shear failure is achieved. The walls have light and moderate horizontal steel reinforcement at 0.2% and 0.5% for SCW3 and SCW4 respectively. These walls are discussed in greater detail in Chapter 3 and are presented in the following subsection to compare the different pseudo slab models over a range of wall parameters. Each 27 pseudo model is analyzed with a mid-height 200 mm thick slab with 0.4% steel each way. The slab size parallel and perpendicular is indicated by the dimensions AA x BB (in metres). a) Deformed shape with significant diagonal shear cracks b) Horizontal steel yielding in diagonal shear struts Figure 2.16: Shear failure in SCW1 with cracking and yielding in diagonal struts, Vbase = 7043 kN & Δtop = 120 mm. 2.4.5 Comparison of Pseudo 3D Slab Models The pseudo slab models create an artificial 3D space and as such are not perfect models. However, by comparing the advantages and disadvantages of the models the best pseudo model can be selected. The models are compared using the same load conditions and same slab parameters; only the modeling configurations discussed in Section 2.4.1 and 2.4.2 differ. The walls must fail in shear to assess the model effect on the shear capacity, therefore sufficient flexural steel is provided to prevent base hinging. The following discussion is only to select the best slab model; an in-depth analysis of the slab effect is explored in Chapter 3. The models are first compared by their load-deformation response, which is obtained from a pushover curve of the top wall displacement versus the base shear (Figure 2.17a). By preventing -110 fsy (MPa) +450 28 flexural hinging, the flexural component in the pushover is fairly linear; therefore any response change is due to shear. Assuming the total displacement consists of only the flexural and shear displacements, then the shear response is obtained by subtracting the flexural response, determined from the moment-area method and its moment-curvature response. This results in a derived base shear versus shear deformation pushover curve as provided Figure 2.17b. The shear pushover curve is very similar to the total pushover curve except that it highlights the dominant shear response. As such, the models are assessed using these curves. a) Total pushover curve b) Derived pushover curve from shear response Figure 2.17: Comparison of pseudo model response with total and shear pushover curve, SCW3 with slab: 17 x 12 m. 0 2000 4000 6000 8000 10000 12000 0 50 100 150 200 B a se S h ea r (k N ) Top Displacement (mm) Cross Truss with Center Point Outward Truss with End Points Outward Truss with Center Point Original Pseduo Slab Model 0 2000 4000 6000 8000 10000 12000 0 20 40 60 80 100 120 B a se S h ea r (k N ) Top Shear Displacement (mm) Cross Truss with Center Point Outward Truss with End Points Outward Truss with Center Point Original Pseduo Slab Model29 From Figure 2.17b, it is clear that the models predict fairly similar responses. The original model predicts the weakest response since the asymmetrical loading in the slab softens the model. The cross truss predicts the strongest response as a result of the truss bar configuration providing additional strength. This is a consistent observation for all slab conditions. The two outward truss configurations predict very similar responses, indicating that the location of the fixed node makes little difference. The shear response is also compared for each model against a very large, medium and small sized slab to capture model behaviour over a large of slab sizes. For the lightly-reinforced wall in Figure 2.18, the large slab is 17 m parallel and 12 m normal to the wall (6 m on either side), whereas the smallest slab has only 0.5 m surrounding the slab. For the moderately-reinforced wall in Figure 2.19, the large slab extends on both sides 3 m parallel and 5 m normal to the wall, whereas the smallest slab has only 0.2 m surrounding the slab. 30 a) 17 x 12 m Slab Size b) 10 x 6 m Slab Size c) 10 x 1 m Slab Size Figure 2.18: SCW3 comparison of derived shear pushover response for each model. 0 2000 4000 6000 8000 10000 12000 0 20 40 60 80 100 120 B a se S h ea r (k N ) Top Shear Displacement (mm) Cross Truss with Center Point Outward Truss with End Points Outward Truss with Center Point Original Pseduo Slab Model 0 2000 4000 6000 8000 10000 12000 0 20 40 60 80 100 120 B a se S h ea r (k N ) Top Shear Displacement (mm) Cross Truss with Center Point Outward Truss with End Points Outward Truss with Center Point Original Pseduo Slab Model 0 2000 4000 6000 8000 10000 12000 0 20 40 60 80 100 120 B a se S h ea r (k N ) Top Shear Displacement (mm) Cross Truss with Center Point Outward Truss with End Points Outward Truss with Center Point Original Pseduo Slab Model31 a) 9.4 x 10 m Slab Size b) 4 x 4 m Slab Size c) 4 x 0.4 m Slab Size Figure 2.19: SCW4 comparison of derived shear pushover response for each model. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 5 10 15 20 25 30 35 40 B a se S h ea r (k N ) Top Shear Displacement (mm) Cross Truss with Center Point Outward Truss with End Points Outward Truss with Center Point Original Pseduo Slab Model 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 5 10 15 20 25 30 35 40 B a se S h ea r (k N ) Top Shear Displacement (mm) Cross Truss with Center Point Outward Truss with End Points Outward Truss with Center Point Original Pseduo Slab Model 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 5 10 15 20 25 30 35 40 B a se S h ea r (k N ) Top Shear Displacement (mm) Cross Truss with Center Point Outward Truss with End Points Outward Truss with Center Point Original Pseduo Slab Model32 The pseudo models responses are similar for the largest slab but the response varies more with the smaller slabs. The small slab allows for a greater influence from the truss bar configuration, where if the slabs are very large most of the response is from the slab and the truss bar influence is negligible. For all the pseudo models, the small slab response is typically higher than expected although it is shown in Chapter 4 that the small slab size response is realistically closer to the original model. Although the truss bar models over predict the strength, most of the analysis is performed with the largest effective slab size, reducing the truss bar configuration as shown in Figure 2.18a and Figure 2.19a. Therefore, the effect of the truss bars influence on the wall strength is not a major concern. From the previous figures, the outward truss models are preferred as they typically predict an average response and not extremes of the others. The models are further compared by their internal horizontal strain around the slab interface in Figure 2.20 below for the same displacement. The figure highlights the element horizontal strain data for both the wall and the slab, where the small slab is the small strip of elements in front of the wall. At the current load stage, there are two large cracks; the primary crack extends down to the center of the slab and a secondary smaller crack extends up the left side of the web. The location where the crack meets the wall-slab interface is where horizontal strain should be concentrated. Note even though the outward truss appears to less strained, the base shear at 70 mm total displacement for the cross truss, center outward and end outward truss is 7775, 7754 and 7798 kN, respectively (original is 7159 kN). 33 a)Original Pseudo Model b)Cross Truss model c) Center Outward model d) End Outward model Figure 2.20: Horizontal strain with small slab, SCW3 Δtop = 70 mm. With no truss bars, the original pseudo model slab provides additional stiffness until the slab cracks and quickly yields, concentrating the horizontal strains at the cracked location. An element attracts significant straining once it cracks since there is not a second layer of elements to transfer the load. This effect is shown in Figure 2.20a where the slab is straining significantly at the crack locations but not elsewhere. In Figure 2.20b, the cross truss also concentrates horizontal strain at the wall crack but this is the result of the truss not allowing straining elsewhere. This does reflect realistic strain loading since cracks can develop away from the center of the wall. The center outward truss in Figure 2.20c provides a more accurate strain distribution by allowing straining due to the secondary crack but it disperses the strains from the large central crack. The end outward truss (Figure 2.20d) is similar to the center outward truss but it has a zone of relatively unstrained elements at the center. This is the result of the lack of a fixed node at the center and produces a less realistic response. From this analysis, the optimal -0.5 εx (mm/m) 12.0 Slab Layer Slab Layer 34 pseudo model is the center outward truss model because it has the most balanced response. The center outward truss model distributes the strains appropriately and strains in the correct locations near cracks, although the strains are shifted slightly since the truss bars connect to adjacent nodes. 2.4.6 Optimal Pseudo 3D Slab Model From the previous analysis, it is determined that the optimal pseudo 3D slab model is the center outward truss model. Through the use of the truss bars, the model transfers horizontal wall displacements to the slab while not providing additional flexural stiffness. The forces within the slab are balanced and symmetrical allowing for reasonable slab capacities over a wide range of slab sizes. Although the nodal displacements are shifted between the wall and slab, the consequences are negligible as shown by the horizontal strain distributions in the slab. Most importantly however, the center outward model provides reasonable shear stiffness throughout the load-deformation response, including a good estimate of the peak shear capacity. As the result of these many advantages and few disadvantages, this model is selected as the 3D pseudo slab model used for the rest of analysis. In the following chapters, the pseudo 3D slab model is based on the center outward slab model. 35 Influence of Floor Slabs on Shear Walls Chapter 3 - The following chapter explores in detail the influence of slabs on shear walls through the use of the pseudo 3D slab model. Modeling a wall with a slab affects the wall shear capacity and damage propagation. Correspondingly, the slab also experiences cracking and nonlinear behaviour as the wall is pushed further. The wall-slab system behaviour is complex and the objective of this chapter is to explain how the slab changes the shear wall response. 3.1 WALLS FOR ANALYSIS A set of three shear-critical shear walls are created to analyze the slab effect (Figure 3.1). The three walls developed are SCW3, SCW4 and TSE8 with horizontal steel ratios of 0.2, 0.5 and 1%, respectively. The 0.2% is selected as the minimum steel amount according to A23.3-04 and 1% was found to be the limit where slabs have an effect. SCW3 and TSE8 bound the typical values of steel ratios and SCW4 provides an intermediate response. a) SCW3: Lightly-Reinforced b) SCW4: Moderately- Reinforced c) TSE8: Heavily-Reinforced Figure 3.1: Wall geometry for analysis. 36 All the walls have a wall height-to-length ratio of 3, which represents a flexural wall according to A23.3-04. A constant axial load of 0.1fc’Ag is applied to the top of the walls. SCW3 has the same geometry as SCW1 presented in Chapter 2 except the moment capacity is increased to prevent flexural hinging. TSE8 is developed from the geometry of a panel element test performed at the University of Toronto. The purpose of this wall is to compare the change in shear response of the same wall specimen as a membrane, in a wall and in a wall-slab system. This is explored in further detail in Chapter 5. SCW4 was created to represent typical shear wall geometry in a core wall with the same web and flange thickness and large flanges. All of the walls are analyzed with and without slabs, where a single slab is placed at mid-height of the wall. The slabs have a thickness of 200 mm with 0.4% steel each way, since it should represent a lower bound of typical floor slab parameters. Placing the slab at mid-height (or at 1.5lw) reduces the effect from the boundary conditions and is the closest to a sectional response. Only a single slab is modeled due limitations with the pseudo 3D analysis and to understand the simplest case before expanding to more complex configurations. To select the slab size parallel and normal to the wall, the slabs size is increased until the effect on the wall does not change; this is the largest effective slab size as shown in Figure 3.1. Throughout the analysis, flexural hinging is prevented resulting in large and impractical steel reinforcing ratios. These values are merely for academic purposes since the walls must be pushed to shear failure with slabs to assess the slab effect on its shear capacity. As will be discussed in the next section, the slabs can provide a significant increase in shear capacity which requires significant moment capacity. In some cases, the yield moment is required to be 2.5 times greater than the expected moment from the predicted shear capacity, as shown in Table 3.1. In the following table, Vn is the predicted shear capacity from the General Method in cl.11 of A23.3-04 and My is the nominal yield moment determined from Response-2000. Table 3.1: Elastic flexural capacity. SCW3 SCW4 TSE8 h 27 10.8 5.715 Vn (kN) 6380 3210 3808 My (kN) 330,000 86,500 25,000 ρfy 5.25% 12.0% 15.0% My/Vn*h 1.92 2.50 1.15 37 3.2 SLAB EFFECT ON SHEAR WALLS The shear-critical walls were analyzed with and without slabs using the pseudo 3D model to determine the slab effect on shear walls. In general, the floor slabs provide many benefits to the wall in terms of crack control, shear capacity, and reduction in damage and internal strains. These issues are explored further in the following subsections. 3.2.1 General Response The addition of the slab generally increases the wall stiffness and strength as indicated from the pushover curve, especially for lighter reinforcement (see Figure 3.2). The slabs have no effect prior to cracking, but the wall stiffness increases relatively linearly once cracking is initiated, specifically once the shear cracks reach the slab interface region. The slabs increase the peak base shear capacity for SCW3, SCW4 and TSE8 by 34, 22 and 1%, respectively. From these results, it can be inferred that the slabs provide a greater benefit to walls with less horizontal steel. This suggests that the slabs provide additional horizontal tension capacity, similar as if extra stirrups were provided. Supporting this theory is the fact that the heavily-reinforced wall, which is near compression failure, only has a minor increase in capacity from the slab. Since the influence of the slabs is minor for TSE8, further in-depth analysis is not required since the difference is negligible. 38 a) SCW3: Lightly-Reinforced b) SCW4: Moderately-Reinforced c) TSE8: Heavily-Reinforced Figure 3.2: Pushover curves without a slab and with a slab at mid-height. 0 2000 4000 6000 8000 10000 12000 0 25 50 75 100 125 150 B a se S h ea r (k N ) Top Total Displacement (mm) SCW3 - No Slab SCW3 - 17x12m Slab 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 10 20 30 40 50 60 70 B a se S h ea r (k N ) Top Total Displacement (mm) SCW4 - No Slab SCW4 - 9.6x10m Slab 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 10 20 30 40 50 B a se S h ea r (k N ) Top Total Displacement (mm) TSE8 - No Slab TSE8 - 3.8x2.6m Slab39 From the pushover curves, it is clear that the moment capacity must be increased with slabs in order to prevent flexural hinging. For instance, the first shear critical wall, SCW1, was designed with significant flange steel of 1.75% resulting in a moment capacity of 200,000 kNm determined from Response-2000. From the current loading conditions, a base shear of 7,400 kN is required to achieve flexural yielding, well above the expected nominal shear capacity of 5,500 kN from A23.3-04. Even though horizontal (shear) yielding does occur there is still sufficient shear capacity to increase the load until a flexural hinge is formed. The second shear critical wall, SCW2, the flange steel increased to 3.5% such that the moment capacity is 290,000 kNm (base shear of 10,700 kN). Shear failure does occur with this wall at 7,600 kN however adding a slab increases the shear capacity such that flexural hinge again forms even with the increased moment capacity. The third wall, SCW3, was designed with a moment capacity such that the wall would fail in shear with the slabs. To achieve this, the flange steel is increased to 5.25% for a moment capacity of 370,000 kNm (base shear 13,700 kN) since the base shear capacity with the slab is near 11,000 kN. From these first observations, achieving a shear failure for a typical wall is very difficult considering that the flange steel had to be increased beyond practical levels to prevent flexural hinging. The second observation is the floor slab helps prevent crack propagation, especially with the lightly-reinforced wall. As can be seen in Figure 3.3 below, both the walls with no slabs have large diagonal cracks through the web, and providing a mid-height slab reduces large cracks in the wall. Again, the slabs have the greatest benefit for the lightly-reinforced wall. SCW3 with no slab has a large diagonal crack surrounded by uncracked regions which propagates diagonally. However with a slab, the cracks are smaller and the slabs prevent propagation through the wall-slab interface. The slab increases crack control, by limiting large cracking from developing (darker crack lines), but it does not prevent uniform cracking from occurring in SCW4. This is due to moderately-reinforced walls naturally limit localized cracking and distribute the load though the width of the wall. The heavily-reinforced wall was not shown because the cracking pattern is essentially the same with and without a slab. 40 SCW3 SCW4 a) No Slabs b) Mid-Height Slab a) No Slabs b) Mid-Height Slab Figure 3.3: Wall crack pattern near failure with and without a mid-height slab, SCW3: Δtop = 100 mm, SCW4: Δtop = 42 mm. 3.2.2 Internal Shear Wall Damage The damage within the wall can provide insight to the mechanics of slab influence on the shear wall. The main signs of damage are tension failure and diagonal compression failure, indicated by horizontal and vertical steel yielding and localized concrete crushing. Figure 3.4 and Figure 3.5 on the following pages compare the internal damage for SCW3 and SCW4 with and without a slab. The cracking pattern is presented again for reference. Crushing failure is indicated by the ratio of the peak principle compression strain over the strain at peak concrete load, εc2/εcm. A value greater than 1 (red elements) indicates that the element is in a post-peak compression response, where there is limited strength remaining. Localized crushing is typically shortly followed by global system failure. The yielding failure is indicated by the ratio of steel strain value over the steel strain at yield, εs/εy. A value greater than 1 (red elements) indicates that the distributed steel in that element is yielding. 41 SCW3 – No Slab a) Cracking Pattern b) Concrete Crushing b) Horizontal Yielding d) Vertical Yielding SCW3 – With Mid-Height Slab a) Cracking Pattern b) Concrete Crushing b) Horizontal Yielding d) Vertical Yielding Figure 3.4: Internal damage indicating crushing and yielding failure, SCW3 Δtop = 100 mm. No damage Yielding/Crushing 42 SCW4 – No Slab a) Cracking Pattern b) Concrete Crushing b) Horizontal Yielding d) Vertical Yielding SCW4 – With Mid-Height Slab a) Cracking Pattern b) Concrete Crushing b) Horizontal Yielding d) Vertical Yielding Figure 3.5: Internal damage indicating crushing and yielding failure, SCW4 Δtop = 42 mm. No damage Yielding/Crushing 43 From Figure 3.4 and Figure 3.5 above, the slabs clearly reduce internal damage for the same top wall displacement. The slab splits the diagonal strut into 2 smaller struts above and below the slab. The slabs do not prevent horizontal yielding, although the extent of yielding and straining is reduced. However, the slabs do significantly reduce localized crushing, increasing the load capacity of the wall, particularly for SCW3. 3.2.3 Horizontal Restraint The slab can only apply horizontal strains and forces to the wall since the slab is connected along a single horizontal line and the pseudo 3D model prevents vertical displacements. To analyze the problem, the horizontal strain is averaged over each wall element layer to obtain the distribution over the height. The results are compared with and without a slab for SCW3 and SCW4 for the same applied base shear in Figure 3.6. In the case of SCW3, a base shear of around 7,750 kN results from a top displacement of 65 and 105 mm for with and without a slab respectively. For SCW4, a base shear of around 3,360 kN results from a top displacement of 28 and 48 mm for with and without a slab respectively. a) SCW3: Lightly-Reinforced b) SCW4: Moderately-Reinforced Figure 3.6: Average horizontal wall strain from mid-height slabs, SCW3: Vbase = 7750 kN, SCW4: Vbase = 3360 kN. Slab 0 4.5 9 13.5 18 22.5 27 0 1 2 3 H ei g ht ( m ) Horizontal Strain (mm/m) Slab at Mid-height No Slab Slab 0 1.8 3.6 5.4 7.2 9 10.8 0 2 4 6 H ei g ht ( m ) Horizontal Strain (mm/m) Slab at Mid-height No Slab44 It is clear that the slabs significantly reduce the horizontal strain at the wall-slab interface. This has the effect of pinching back all of the strains throughout the height of the wall. The slab reduces the strains to approximately 5% of the strain at mid-height without a slab. An approximate horizontal force-strain deformation plot is developed in Section 3.3.2. 3.2.4 Strain Profile Over Wall Height The main slab mechanism influencing the shear wall is the horizontal restraint at the wall-slab interface location. This horizontal restraint in turn also affects the shear and vertical strain distribution. The walls in this study have significant horizontal strains which contribute most significantly to the shear strains as shown in Figure 3.7. Whereas the vertical strains are limited as a result of preventing flexural hinging and do not contribute as much to the shear strains. As such, the slab significantly affects the shear strain profile over the wall by restraining εx over the wall height. Based on the MCFT compatibility equations, the horizontal and shear strains are related according to ( ) ⁄ . Figure 3.7: Biaxial strain state for elements with significant horizontal strains and limited vertical strains. The shear strain profile is presented in Figure 3.8 where there is clear similarity to horizontal strain profile presented in Section 3.2.3. The average layer shear strain is plotted in Figure 3.8a and selected layer profiles are presented in Figure 3.8b. By inspecting the shear strain profiles, it is clear that the slabs minimize extreme localized shear strains to reduce the average shear strain 45 in the layer. The slabs do not however reduce the shear strains of the entire layer; the strains are more evenly distributed over the wall length. L o ca l She a r St ra in (m m /m ) a) Shear strain profile over wall height b) Shear strain profile at select layers Figure 3.8: Vertical shear strain profile with select shear strain layer profiles, SCW3 Vbase = 7750 kN. One of the principle pseudo slab model assumptions is the slab does not provide additional flexural stiffness, however, the slabs indirectly affect the flexural response by reducing the shear strains. Without a slab, significant shear strains eventually induce localized vertical strains in the large cracks, modifying the vertical strain profile to non-planar. Subsequently, reducing extreme localized shear strains also reduces localized vertical strains which results in a more plane section. This is highlighted in Figure 3.9 where the curvature profile for the wall with and without a mid-height slab is not the same despite having the same applied moment demand. From the selected vertical strain profiles, it is clear that without a slab the vertical strains no longer remain plane whereas including a slab linearizes the profile. Although the moment Slab 0 4.5 9 13.5 18 22.5 27 0 1 2 3 H ei g ht ( m ) Average Layer Shear Strain (mm/m) Slab at Mid-height No Slab 0 2 4 6 8 10 0 4500 9000 0 5 10 15 20 0 4500 9000 0 5 10 15 20 0 4500 9000 Location Along Wall (mm) 46 capacity and stiffness are the same, the concept of a moment-curvature relationship does not apply to shear-critical members because the section no longer has a curvature from its theoretical definition. L o ca l V er tic a l S tr a in (m m /m ) a) Curvature profile over wall height b) Vertical strain profile at select layers Figure 3.9: Vertical curvature profile with select vertical strain layer profiles, SCW3 Vbas e= 7750 kN. By restraining the wall at mid-height, the slab should redistribute the flow of forces in the wall. As was shown in Section 3.2.2, the wall damage is split into struts above and below the slab. Analyzing the principle tension and compression stress field pattern highlights how the slab separates the diagonal strut and redistributes the forces. Figure 3.10 describes the flow of forces for the lightly-reinforced wall with and without a slab for the same displacement. Slab 0 4.5 9 13.5 18 22.5 27 0 0.1 0.2 0.3 0.4 H ei g ht ( m ) Curvature (rad/km) Slab at Mid-height No Slab -1 0 1 2 0 4500 9000 -1 0 1 2 0 4500 9000 -1 0 1 2 3 0 4500 9000 Location Along Wall (mm) 47 Principle Tension Stress Principle Compression Stress a) No Slab b) Slab at Mid-height c) No Slab d) Slab at Mid-height Figure 3.10: Principle stresses highlight regions of stress concentration for SCW3, Δtop = 100 mm. The main failure mechanism in SCW3 is diagonal tension failure as the result of a diagonal compression strut that extends the height of the wall (Figure 3.10c). The compression strut does turn slightly and merge with the vertical compression flange strut. In addition, the horizontal bars return some of the tension force back to the vertical tension flange, increasing the flange tension force towards the base (Figure 3.10a&b). Once the slab is provided, the diagonal strut splits into two struts above and below the slab and the strut is forced to intersect the vertical compression flange higher up the wall (Figure 3.10d). The slab acts like a large stirrup to tie back the diagonal strut to the vertical tension flange, where another compression strut starts. It is clear the slab transfers a significant amount of force since the vertical flange tension force increases significantly immediately below the slab. Although the slab provides substantial horizontal tension capacity, the distributed horizontal bars also continue to tie back the diagonal compression strut above and below the slab. Similar force flow patterns are observed in the moderately-reinforced wall, which is described in Figure 3.11. -2 f1 (MPa) +10 -9 f2 (MPa) +2 48 Principle Tension Stress Principle Compression Stress a) No Slab b) Slab at Mid-height c) No Slab c) Slab at Mid-height Figure 3.11: Principle stress regions with and without mid-height slab for SCW4, Δtop = 42 mm. The main diagonal strut in SCW4 without a slab is much wider than SCW3 and the strut distributes the forces more evenly over the wall. The horizontal bars tie back the diagonal strut throughout the height of the wall, where the tension stress in the flange increases consistently over the height. With the slab, the strut splits into regions above and below the slab while remaining wide. The diagonal compression strut is attracted to the vertical compression flange where most of the strut intersects the flange and not the slab. Again, the slab acts like a large tension tie, returning the force to the tension flange. Figure 3.12 below presents a simplified stress flow pattern as observed from SCW3 and SCW4, but it is not an equivalent strut and tie model. The line thickness indicates the stress magnitude and sign; tension and compression is represented by blue and red lines respectively. In both cases, the slab acts like a large external stirrup which ties back the diagonal compression strut to the vertical tension flange, splitting the strut. The diagonal compression strut naturally merges towards the vertical compression flange instead of the slab. However, if the slab spacing were -2 f1 (MPa) +10 -14 f2 (MPa) +2 49 reduced, increasing the strut angle, at some point the strut will be forced to intersect with the slab. This phenomenon is observed and discussed in Section 6.2.2. a)Wall with no slab b) Wall with Mid-height slab Figure 3.12: Simplified diagram of the stress field pattern in shear critical walls with and without mid- height floor slabs. 3.3 SLAB BEHAVIOUR The floor slabs have a significant effect on the shear behaviour of a shear wall but it is also important to study the behaviour of the slabs themselves. From the 3D pseudo analysis, information on the in-plane stresses and strains is easily obtained and analyzed. The behaviour of the slab is explored through the observations of slab cracking, force-deformation response, internal stresses and the flow of forces. Only the lightly and moderately-reinforced walls will be considered since the slab has a negligible effect on the heavily-reinforced wall. 3.3.1 Cracking The first observation from the slab response is the slab can experience cracking and nonlinear behaviour. This counters the standard practice assuming concrete floor slabs act as rigid plates. Cracking also provides an indication where the forces are concentrated within the slab. Figure 3.14 presents the progression of slab cracking for SCW4 along with its relative position on the pushover curve in Figure 3.13. SCW4 represents a typical cracking pattern, although SCW3 50 does experience relatively more cracking. It should be noted that the line angle indicating a crack is fixed and as such does not provide information on the crack angle, only that a crack exists. Figure 3.13: Stages of slab cracking along the wall pushover curve. a) Small Cracking b) Significant Cracking c) Peak Load 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 10 20 30 40 50 60 B a se S h ea r (k N ) Top Total Displacement (mm) SCW4 - No Slab SCW4 - 9.6x10m Slab51 a) Cracking pattern at first cracking, Δtop = 16 mm b) Crack pattern at significant cracking, Δtop = 28 mm c) Crack pattern at peak load, Δtop = 52 mm Figure 3.14: Progression of slab cracking patterns for SCW4. 52 From the figures above, the slab first cracks where the two pushover curves diverge, shortly after wall cracking has intersected the wall-slab interface. The increased wall stiffness after that point is clearly from engaging the slab as wall cracking places more load on the slab. For SCW4, the wall cracking is fairly uniform and as such the slab cracking is distributed as well. There are locations with significantly more cracking, such as at the end of the walls. As will be discussed in the following subsections, horizontal tension along the wall reverses to compression at the wall end, resulting in shear cracking extending diagonally from the wall ends. 3.3.2 Force-Deformation Response The floor slab restrains the wall like an external stirrup with significant stiffness and strength. The behaviour of the slab can be described using a force-deformation plot of the slab. The total horizontal force is calculated by summing the local slab force along the length of the wall. The local slab force is obtained by averaging the perpendicular slab horizontal stress, multiplied by the slab thickness and width. This procedure is repeated for the element steel stress and multiplied by the steel ratio to determine the slab’s steel contribution. This procedure calculates the equivalent total slab and steel force at the wall-slab interface. The equivalent slab strain at the wall-slab interface is calculated by averaging the slab element strain at the wall slab interface. The strain is easily converted to slab deformation by multiplying the strain by the wall length. Figure 3.15 below presents the slab force-top displacement plot and the slab force-strain for the lightly-reinforced wall. 53 a) Slab force-top displacement b) Slab force-Slab Strain Figure 3.15: Force-deformation plots for SCW3. From the figure above, the majority of the slab strength initially comes from the concrete, where the steel contributes more as the as the slab cracks. The concrete contributes a significant portion of the total strength because most of the slab remains uncracked; the slab only becomes significantly cracked near the peak load. Even when the slab is significantly cracked, approximately only 26% of the slab is cracked. From Figure 3.15a, the slab initially has only a small force up to 35 mm where wall cracking first intersects the wall-slab interface. The slab then enters a cracked region up to 80 mm (0.2 mm/m strain) where both the concrete and steel force increases linearly until significant cracking extends to the edges of the slabs. Once cracking extends to the edges, the remainder of the slab begins to crack more and the steel contribution increases significantly. The behavior is similar to a trilinear pushover curve with uncracked, cracked and yielding regions. 0 10000 20000 30000 40000 50000 60000 70000 0 40 80 120 160 Sl a b F o rc e (k N ) Top Displacement (mm) Total Slab Force Steel Slab Force 0 10000 20000 30000 40000 50000 60000 70000 0 0.4 0.8 1.2 1.6 Sl a b F o rc e (k N ) Average Horizontal Slab Strain (mm/m) Total Slab Force Steel Slab Force54 The same analysis is performed for SCW4 where the steel influence is less as seen in Figure 3.16. From the figures below, almost the entire force contribution comes from the concrete since most of the slab remains uncracked. The major difference between the SCW3 and SCW4 slab response is cracking remains relatively limited in SCW4 and does not extend to the slab ends. As such, the uncracked concrete continues to contribute the majority of the strength. Again, the steel contribution is linear with increasing strain because the steel does not yield. a) Slab force-top displacement b) Slab force-Slab Strain Figure 3.16: Force-deformation plot for SCW4. 3.3.3 Internal Stresses To understand the mechanics of the slab behaviour, it is important to develop an understanding of the stresses within the slab. Figure 3.17 below displays the slab shear stress and normal stresses parallel and perpendicular to the wall for SCW4. SCW4 represents the typical stress field distribution due to its good crack control. The stress scale is exaggerated in the following figure, where the max stress is 2 MPa or greater (green) and the minimum stress is -2 MPa or less (blue). The exaggerated stress gradient is required to better visualize the stress field. 0 10000 20000 30000 40000 50000 60000 0 6 12 18 24 30 36 42 Sl a b F o rc e (k N ) Top Displacement (mm) Total Slab Force Steel Slab Force 0 10000 20000 30000 40000 50000 60000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Sl a b F o rc e (k N ) Average Horizontal Slab Strain (mm/m) Total Slab Force Steel Slab Force55 a) Slab normal stress parallel to wall b) Slab normal stress perpendicular to wall c) Slab shear stress Figure 3.17: In-plane internal slab stress with the SCW4 slab, Δtop = 40 mm. <-2.0 Stress (MPa) >+2.0 56 The figure above depicts a typical stress pattern within a slab where the wall is heavily loaded and experiencing significant cracking (around 90% of peak load). The normal stress parallel to the wall is the main resisting mechanism where it resists the lateral expansion of the wall. As shown in Figure 3.17a, tension parallel to the wall is required to prevent expansion and a resulting compression force results at the end of the wall balancing the forces. These tension and compression regions are bounded by virtual lines that are approximately 45 degrees from the ends on the wall. In this diagonal region, perpendicular normal stress is required to balance the parallel stresses (see in Figure 3.17b). Shear stress is required as the result of the bi-axial stresses within this diagonal region (see in Figure 3.17c). The normal stress perpendicular to the wall and shear stress are secondary stresses caused by the normal parallel stress. In addition, a small amount of perpendicular normal compression stress forms at cracks perpendicular to the wall. Perpendicular tension forces also develop at the edges of the slab in line with the wall. A simplified diagram of the slab resisting mechanism is presented in Figure 3.18 below. In Figure 3.18a, the primary resisting mechanism is the tension parallel to the wall provided by the parallel bars and the uncracked concrete. Without the parallel bars perpendicular to the wall, the slab would crack brittlely and separate into two pieces, providing little resistance. The compression force results from balancing the horizontal tension forces within the slab ‘stirrup’. As the wall expands, the tension force holds the end regions in place and is compressed against the expanding wall. The line between the tension and compression region extends at 45 degrees from either side of the wall. In the diagram below, only regions near the wall are cracked, and the rest of the wall remains relatively uncracked which provides a very rigid perimeter. This rigid perimeter is required to prevent splitting at the slab edge as described in Figure 3.18b. 57 a) Simplified primary stress regions b) Simplified slab force mechanisms Figure 3.18: Simplified diagram of the resisting mechanism and forces within the slab. Figure 3.18b presents simplified force flow mechanisms within the slab, similar to a strut-and-tie model. The tension forces perpendicular to the wall can be replaced by an equivalent resultant tension force offset from the wall. Diagonal compression struts are required at the wall ends to prevent the wall from expanding. The diagonal strut creates the high compression zone at the 58 wall and along with the shear stress at 45o. Perpendicular tension at the slab edges, in line with the wall, results from the prying action due to the equivalent slab tension force offset from the wall. Although the uncracked slab region beyond the wall end does not directly affect the slab tension capacity, nominal capacity is required in this region to prevent the slab from splitting along the length of the wall. Figure 3.18b also highlights the local tension stress directly at the cracks. The concentration of local stress is mostly govern by the shear wall’s horizontal steel ratio; more steel provides better crack control in the wall and in turn distributes the forces more evenly within the slab. Based on the heavily-stressed regions in the diagram above, certain slab regions may be vulnerable to significant damage. The most likely damage is yielding in the bars parallel to the wall, especially the bars closest to the wall. Crushing occurring in the slab at the end of the wall may also occur. Although the wall is more likely to fail before the slab experiences significant damage, damage can be a concern for relatively small slab sizes. However, if the slab does fail, the wall strength simply reduces to the lower bound of the shear wall without slabs. One thing to note is although the slab behaviour is complex, it is primarily the result of the tension capacity of the slab parallel to the wall. Therefore, it would be possible to replicate the majority of the slab resistance by simply accounting for the slab tension region perpendicular to the wall. It is using this theory that we can create a simple 2D slab element layer discussed in Chapter 4. 59 Approximate 2D Model of Slabs Chapter 4 - The pseudo slab model is an effective tool to emulate the 3-dimensional slab effect in a 2- dimensional space. However, creating the pseudo slab model is both time and computationally intensive, as well as only one slab may be modelled at a time. Therefore, an approximate 2D slab model was developed to reproduce the 3D slab effects described in Chapter 3. Although the main purpose for the model is to replicate the pseudo 3D model behaviour, the simple procedure developed is relatively robust such that it could be applied to other situations. The benefit of the 2D slab model is that multiple slabs can be modeled on a wall while reducing computational demand. This is very useful for the analysis performed in Chapter 6 where walls are modeled with multiple slabs. 4.1 DEVELOPMENT OF 2D MODEL The 2D model is created by converting the 3D slab geometry to equivalent 2D geometry. The 2D slab model consists of a horizontal layer of elements with the same vertical thickness as the 3D slab thickness. In this 2D layer, the out-of-plane element thickness represents the 3D perpendicular slab length (Figure 4.1b). Due to the nature of 2D analysis, the 2D model can only capture the global restrain provided by the slab and not the local stress field within the slab. This is the result of the 2D analysis assuming the out of plane stresses are constant in a thick element (Figure 4.1a). Considering that slab damage is often minimal over typical shear loads, the main slab effect is the ability to clamp the wall, providing an external stirrup effect. Therefore, the slab layer must be able to capture the global slab behaviour by preventing crack propagation and providing restraint. 60 a) 3D rendering of slab layer b) Augustus model of 2D slab layer Figure 4.1: 3D visualization of 2D slab model. The most effective procedure to create the 2D model is to directly convert the 3D geometry to 2- dimensions. The slab thickness and the horizontal slab length remains the same and the slab length perpendicular to the wall converts directly to the element thickness. Although, this method is not as finely-tuned as it could be, it will be shown that this method is flexible and matches the 3D model very well. The steel quantity in the 3D model can be translated directly to the 2D layer as well. The parallel and perpendicular slab steel ratio becomes the horizontal and out-of-plane steel ratios for the slab layer respectively. Since the wall contains steel in the x-direction, the slab layer steel percent should increase as a ratio of [{ } { } ] but it is usually not necessary considering the slab out-of-plane depth is typically much larger than the wall thickness. However, the vertical steel ratio must be reduced such that the vertical steel amount remains the same as it passes through the slab layer. To obtain the slab vertical steel, the wall vertical steel ratio is reduced by the wall thickness over the slab out-of-plane length. The use of vertical slab steel can be avoided by providing vertical truss bars in the wall instead of distributed bars. To highlight the process to create a 2D slab element layer, an example for SCW3 is described. Using the methods from Chapter 2, a pseudo slab model is created with a slab 12 x 17 m by 200 mm thick with 0.4% steel each way (ρx and ρz). The out-of-plane slab steel, ρz, is required since X X Y Y B: Out-of-Plane Slab Length Z A: Horizontal Slab Length 61 VecTor2 considers its effect on the concrete strength. The geometry of the 2D layer becomes 17 m long, 200 mm high and an element thickness of 12 m. The wall and slab steel is combined in the slab layer and the appropriate steel ratio in the x, y, and z directions is obtained using the descriptions above (see Table 4.1). Since there is no out-of-plane wall steel, the ρz percent remains the same whereas the ρx percent should increase slightly but it is simpler and conservative to assume 0.4% in the x-direction. The slab contains no vertical steel thus the vertical steel is a ratio of the wall to slab thickness. The slab layer is created using the steel ratios below, and the pushover response of the slab layer is compared to the equivalent pseudo slab model in Figure 4.2. Table 4.1: Sample SCW3 conversions from 3D slab geometry to 2D slab layer (in mm). Wall Steel ratios Used Flang e b f: 90 0 ρx 0.2% (0.2·900+0.4·12000)/12000 0.415%→0.4% ρy 5.25% 5.25%· (900/12000) 0.394% ρz n/a from slab 0.4% 0.4% We b b w : 30 0 ρx 0.2% (0.2·300+0.4·12000)/12000 0.405%→0.4% ρy 0.375% 0.375· (300/12000) 0.094% ρz n/a from slab 0.4% 0.4% Figure 4.2: Comparison between 3D and 2D slab models for monotonic and reverse cyclic pushover response for SCW3. 0 3000 6000 9000 12000 0 20 40 60 80 100 120 140 160 B a se S h ea r (k N ) Top Displacement (mm) 3D Pseudo - Monotonic 3D Pseudo - Cyclic 2D Layer - Monotonic 2D Layer - Cyclic62 As shown in Figure 4.2, the 2D slab layer matches very well to the 3D pseudo slab model in terms of stiffness and ultimate strength. However, the post-yield cyclic slab layer response deviates from the 3D model as a result of the layer of very thick elements emulating the varying out-of-plane slab stains. When the slab layer elements yield, it represents all of the out-of-plane steel yielding whereas the actual 3D slab has localized yielding near the wall and non-yielding elements away from the wall. Therefore, as the wall cycles again, the slab can engage more non- yielded elements while the slab layer cannot. A hybrid model is developed in an attempt to better capture the post-yield response is discussed in Section 4.4.3. There are several modifications to the 2D slab layer discussed in Section 4.4. 4.2 PARAMETRIC SLAB SIZE TEST A parametric study varying the dimensions of the slab was performed to assess the accuracy of the 2D model against the 3D model over a wide range of slab sizes. This was the second phase of the parametric study conducted in Chapter 2 to determine the optimal pseudo 3D slab model. The study is performed for both SCW3 and SCW4 and the slab lengths were varied both parallel and perpendicular to the wall. The 2D slab layer matched the 3D slab geometry except that the slab layer did not extend beyond the end of the wall. It was determined that these regions are ineffective for the 2D slab layer as will be discussed in Section 4.4.4. The objectives of the parametric study was to test that a) the 2D slab layer provides a good estimate of the 3D model and that b) the slab region extending beyond the end of the wall is relatively ineffective. Selected shear pushover curves are shown below as a characteristic cross section of the results. Note the 3D model is labeled Outward Truss AxB, which denotes the slab lengths in metres parallel and perpendicular to the wall with A and B respectively. The slab layer number defines its perpendicular length in metres. 63 a) SCW3 with 12 m slab normal to wall b) SCW3 with 6 m normal to wall c) SCW4 with 10 m slab normal to wall b) SCW4 with 4 m slab normal to wall Figure 4.3: Shear pushover curves for the same slab length perpendicular to wall. Considering that the 2D slab layer is a simple estimate of the 3D model, it matches the 3D model reasonably well for a wide range of slab sizes and wall conditions. The 2D layer provides a similar influence on the shear wall resulting in generally the same wall stiffness and peak shear capacity. However as shown in Figure 4.3b for slabs with a small out-of-plane length, the 2D slab layer predicts a lower peak shear capacity than the 3D models. This discrepancy is likely due to the truss bar configuration having a greater influence on the response for small slabs and as such they over-predict the peak shear capacity. This is considering the peak shear capacity 0 2000 4000 6000 8000 10000 12000 0 50 100 B a se S h ea r (k N ) Shear Displacement (mm) Outward Truss 17x12 Outward Truss 10x12 Slab Layer 12 0 2000 4000 6000 8000 10000 12000 0 50 100 B a se S h ea r (k N ) Shear Displacement (mm) Outward Truss 17x6 Outward Truss 10x6 Slab Layer 6 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 10 20 30 40 B a se S h ea r (k N ) Shear Displacement (mm) Outward Truss 9.6x10 Outward Truss 4x10 Slab Layer 10 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 10 20 30 40 B a se S h ea r (k N ) Shear Displacement (mm) Outward Truss 9.6x4 Outward Truss 4x4 Slab Layer 464 for the original pseudo 3D (no truss bars) and slab layer is 9,279 kN and 9,265 kN respectively while the 3D models with the truss bars predicts 10,257 kN and 10,126 kN respectively. As the slab narrows, the response should converge to a wall without a slab where the original pseudo model and 2D slab layer converge better than the 3D models with truss bars. In addition, there is a slight difference between the 3D models with different parallel length A. This indicates the slab region beyond the wall ends does influence the response, albeit a limited influence, as the result of 2 possible effects. First, the wide slab allows for the entire wall length to resist tension while the region beyond the end of the wall can undergo compression. When the slab is narrowed, the tension and compression must now resist within the wall length. This brings in the compression zone to the flange region, reducing the available tension length on the wall. Second, as discussed in Section 3.3.3, the resultant slab tension force is offset from the wall creating prying forces at the slab edges. A minimum amount of slab beyond the wall end is required to prevent splitting along the wall. A summary of the results is provided in Table 4.2 and Table 4.3 below. Table 4.2: SCW3 comparison of peak base shear from 3D model with 2D slab layer. Base Shear (kN) A-Length B-Length Average Slab Layer Difference 10m 17m 25m B -L eng th 1m 8728 n/a n/a 8728 8440 -3.3% 6m 10126 10257 n/a 10192 9265 -9.1% 12m 9900 10891 n/a 10395 10591 1.9% 15m n/a n/a 11462 11462 11459 0.0% Table 4.3: SCW4 comparison of peak base shear from 3D model with 2D slab layer. Base Shear (kN) A-length B-Length Average Slab Layer Difference 4m 9.6m B - L eng th 0.4m 3912 n/a 3912 3764 -3.8% 4m 4238 4398 4318 4247 -1.6% 10m 4259 4462 4360 4443 1.9% As can be seen from the tables above, the 2D slab layer predicts the peak base shear very well, typically within 4% of the 3D models. Therefore, the 2D slab layer is fairly robust over a wide range of slab sizes and could be used as an alternative to the 3D pseudo model. The 2D model does not always have the same stiffness as the pseudo model throughout the pushover but it is possible the 2D model is closer to the true response. This is likely due to modeling issues for 65 the 3D pseudo model discussed in Chapter 2. Another observation from the tables above is the slab zone beyond the end of the wall does not significantly influence the peak base shear capacity. In most cases, the wide and narrow slab widths achieve a base shear capacity within 5% of each other, where only 1 case had a 9% difference. Therefore, the slab region beyond the wall is relatively ineffective and is summarized in Section 4.4.4. 4.3 CAPTURING 3D SLAB EFFECTS The parametric study has shown the 2D slab layer is an effective tool at replicating the 3D slab effect on the wall’s pushover curve. To further evaluate the effectiveness of the 2D model, its effect on the wall is compared with the complete wall response. The slab effect on shear walls is compared by analyzing the cracking patterns, internal strains and forces, flow of forces, force- deformation plot 4.3.1 Wall Cracking and Damage The 2D slab layer accurately predicts the shear pushover curves but it must capture a similar cracking pattern and prevent crack propagation through the wall-slab interface. In addition, the wall also must achieve the same damage states as the 3D model. The following figures presents the cracking pattern and horizontal yielding state (red indicates element is yielding) for both models at the same load stage for SCW3 and SCW4. 66 3D Pseudo Model 2D Slab Layer Model a) Cracking Pattern b) fsx a) Cracking Pattern b) fsx Figure 4.4: Comparison of wall damage for the 3D and 2D model, SCW3 Δtop = 100 mm. 3D Pseudo Model 2D Slab Layer Model a) Cracking Pattern b) fsx a) Cracking Pattern b) fsx Figure 4.5: Comparison of wall damage for the 3D and 2D model, SCW4 Δtop = 42 mm. 67 The 2D slab layer predicts very similar cracking and damage patterns as the 3D pseudo model, especially for SCW3 (Figure 4.4). For SCW3, the slab limits crack propagation and damage is split into 2 struts above and below the wall. The horizontal yielding pattern is similar between the two models and the cracking patterns are relatively close. It is harder to predict the same cracking pattern for the moderately-reinforced SCW4 since the wall cracks over its entire length instead of a localized region (Figure 4.5). Finally, the slab layer accurately predicts the horizontal yielding in the wall. 4.3.2 Simplified Flow of Forces The 2D slab layer behaves differently than the 3D pseudo model and as such may alter the force flow in the wall. It is possible the slab layer may force the diagonal strut to intersect the vertical compression flange or the slab in another location, causing unexpected stress concentration. To test the 2D layer, the principle tension and compression stress fields are compared in Figure 4.6 below. Principle Tension Stress Principle Compression Stress a) 3D model b) 2D slab layer c) 3D model c) 2D slab layer Figure 4.6: Comparison of principle tension and compression stress for 3D and 2D layer model, SCW3 Δtop = 100 mm. -2 f1 (MPa) +10 -9 f2 (MPa) +2 68 As can be seen from the figure above, the stress field is very similar between the two models. The strut intersects the wall-slab interface at essentially the same location: the center of the vertical compression flange. The tension flange in both models also gradually increases in stress towards the base with a significant increase below the slab location. This indicates the slab in both models returns the large slab tension force to the tension flange. However, the models do differ significantly in how the slab tension forces are applied to the wall as shown in Figure 4.7. In the 3D model, a single node line transfers the slab stiffness to the wall affecting various wall elements above and below the node line. The slab tension force is distributed vertically within the wall whereas the 2D slab layer appropriately concentrates the slab tension force within the slab layer itself. This represents a more realistic response since the slab has a finite thickness when attached to the wall, not an infinitesimally thin line. This has a slight effect on the whole wall but has an overall negligible effect on the global wall response. a) 3D Pseudo Model b) 2D Slab Layer Model Figure 4.7: Comparison of the horizontal stress field for the 3D and 2D models, SCW3 Δtop = 100 mm. <-10 fx (MPa) >+2 69 4.3.3 Force-Deformation Plot The 2D layer has local stiffness and strength, much like the 3D pseudo model, where a load- deformation plot can describe its behaviour. Using the same techniques in Section 3.3.2, the total horizontal and steel force within the layer is obtained by summing the element stress in the layer. The layer strain is averaged over the layer for the respective load and this process is repeated until a full load deformation plot is obtained. The slab equivalent force-wall displacement plots are provided below for both the 3D pseudo and 2D slab layer model in Figure 4.8 a) SCW3 b) SCW4 Figure 4.8: Equivalent slab force-wall displacement plot confirms the slab force is similar between the models. 0 10000 20000 30000 40000 50000 60000 70000 0 25 50 75 100 125 150 Sl a b F o rc e (k N ) Top Displacement (mm) Total Slab Force Steel Slab Force 2D Layer - Total 2D layer - Steel 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 0 10 20 30 40 50 Sl a b F o rc e (k N ) Top Displacement (mm) Total Slab Force Steel Slab Force 2D Layer - Total 2D layer - Steel70 From the figure above, the 2D slab layer applies a similar force to the wall for the same top wall displacement. They both exhibit similar behaviour from cracking until yielding. The total force for SCW3 is slightly higher in the 2D layer than the 3D model but the steel contribution is relatively much higher (28% versus 13% for 100 mm top displacement). Conversely, the slab layer force response for SCW4 is less than the 3D model yet the steel force is essentially the same. This difference in steel contribution is likely due to the amount of slab cracking; SCW3 has much more slab cracking and as such the steel contributes more. The steel may contribute more for the 2D layer considering when the 2D slab layer cracks it represents the entire out-of- plane slab thickness cracking. However, the slab layer does not match the pseudo model as well when considering in the force- strain response in Figure 4.9. The slab layer is approximately twice as stiff as the pseudo model and has a linear cracked stiffness while the pseudo model gradually softens as more elements crack. This is the result of using the entire pseudo model slab size for the 2D layer model; although the entire slab area is required to achieve the required slab capacity, the large slab area increases the stiffness. The 2D layer physically cannot match the gradual curve of the pseudo slab model but the 2D layer can be modified such that its cracked stiffness is better (see Section 4.4.5). 71 a) SCW3 b) SCW4 Figure 4.9: Equivalent slab force-slab strain plots. Although, the slab layer does not fully replicate the force-strain response within the slab its main purpose is to replicate the 3D slab effect in the shear wall, which it has achieved. If analysis required in-depth knowledge of the slab behaviour than the complex 3D pseudo model is required. However, considering that the wall typically experiences most of the damage then a model is required to reproduce accurately the wall damage, and the 2D slab layer is quite capable. Therefore, the simple 2D model is used as the analysis tool for Chapter 5 and 6 because the behaviour of the shear wall is the main concern. 0 10000 20000 30000 40000 50000 60000 70000 0 0.1 0.2 0.3 0.4 Sl a b F o rc e (k N ) Average Horizontal Slab Strain (mm/m) Total Slab Force Steel Slab Force 2D Layer - Total 2D layer - Steel 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 0 0.05 0.1 0.15 0.2 Sl a b F o rc e (k N ) Average Horizontal Slab Strain (mm/m) Total Slab Force Steel Slab Force 2D Layer - Total 2D layer - Steel72 4.4 MODIFICATIONS The 2D slab layer model is a simplified model to emulate the 3D effects of the more complex pseudo model. Nevertheless, the model can be modified to better match the 3D response. However, the modifications can address a deficiency in the 2D layer model but may result in causing another issue that is worse than the initial deficiency. Therefore, the result is although the 2D layer model can be modified slightly, it is generally not worth the effort since the simplified 2D model works very well for a wide range of parameters. 4.4.1 Concrete Tension Strength A fundamental issue with the 2D layer is that the slab consists of very thick elements where if the element cracks it is equivalent to the entire slab cracking perpendicular to the wall. This does not reflect the true cracking pattern where the cracks begin at low loads near the wall and extend outward. The result is the 2D layer cracks at higher loads than the 3D model and has a sudden stiffness change whereas the 3D model gradually reduces its stiffness as more cracking occurs. Reducing the concrete tension strength to 50 or 70% of √ forces the slab to crack earlier but changes the strength of the layer. Figure 4.10 below compares the shear pushover response with 100% and 70% ft’ for SCW3 and SCW4. As can be seen, the responses are fairly similar but reducing the tension strength is unpredictable; it should decrease the stiffness but it can unexpectedly increase the stiffness as well. Reducing the tension strength also increases the steel contribution, which as discussed in Section 4.3.3 is already too large. Therefore, it was decided to keep the tension strength at 100% ft’ to stabilize the response. 73 a) SCW3 b) SCW4 Figure 4.10: Pushover curves for full and reduced concrete tension strength. 4.4.2 Steel Ratio Parallel to Wall From the force-deformation plot, the total slab force is a good approximation to the pseudo model but the steel contribution can be twice as much. Therefore, only the steel contribution should be reduced in order to better match the pseudo model. The steel amount parallel to the wall was reduced by half in an attempt to better match the steel contribution (Figure 4.11). Again, the modification has varying results between SCW3 and SCW4 where reducing the steel for SCW3 severely under predicted the wall capacity whereas the opposite effect occurred for SCW4. The likely reason for the different response is the steel contributes significantly more relative to the total resistance for SCW3 than SCW4. Therefore, reducing the steel amount clearly reduces the overall slab capacity for SCW3 but the effect is not the same for SCW4. Reducing the steel amount is unpredictable and is not an appropriate modifier. 0 2000 4000 6000 8000 10000 12000 0 20 40 60 80 100 B a se S h ea r (k N ) Top Displacement (mm) Slab Layer 12 - 1.0ft' Slab Layer 12 - 0.7ft' 0 1000 2000 3000 4000 5000 0 5 10 15 20 25 30 B a se S h ea r (k N ) Top Displacement (mm) Slab Layer 10 - 1.0ft' Slab Layer 10 - 0.7ft'74 a) SCW3 b) SCW4 Figure 4.11: Pushover curve for full and half slab steel ratio for SCW3 and SCW4. 4.4.3 Post-Yield Truss Bars for Stability The 2D slab layer matches the reverse cyclic response of the 3D model very well except the post-yield response varies since the 2D layer cannot engage more bars once the steel has yielded. In an attempt to improve the slab layer cyclic response, a hybrid of a regular slab layer and additional horizontal truss bars is developed. The slab layer was designed to capture the pre- yield response while the truss bars were designed to be engaged in the post-yield response. However, the result is that the truss bars simply add to the stiffness of the slab layer resulting in a similar response. Even when the bars are unbonded to the layer, it results in additional stiffness before yielding. The system could be modified to include additional bar-slip elements but the benefit would not outweigh the computational effort. 4.4.4 Ineffective Slab Region Beyond Wall End The principle slab resisting mechanism is the tension developed parallel to the wall whereas other forces developed in the 3D slab are secondary causes of the principle mechanism. As suggested in Section 3.3.3, virtually the entire lateral wall expansion due to cracking occurs in the web, therefore only the slab regions perpendicular to the web are effective (see Figure 4.12). This theory is tested with the parametric study in Section 4.2 where the shear response is similar for the pseudo slabs with and without this effective zone. This theory is also supported by the fact in 2D analysis the 2D slab layer elements beyond the end of the wall develop virtually no 0 2000 4000 6000 8000 10000 12000 0 50 100 B a se S h ea r (k N ) Top Displacement (mm) Slab Layer 12 - 1.0ρ Slab Layer 12 - 0.5ρ 0 1000 2000 3000 4000 5000 0 10 20 30 40 B a se S h ea r (k N ) Top Displacement (mm) Slab Layer 10 - 1.0ρ Slab Layer 10 - 0.5ρ 75 stress. Therefore, the 2D slab layer is modified such that the slab length parallel to the wall is equal to the wall length. Figure 4.12: Simplified diagram highlighting effective and ineffective slab regions to the slab resistance. 4.4.5 Reduced Stiffness From the force-deformation response in Section 4.3.3 it is clear that although the 2D layer achieves the same capacity as the 3D model it can have more than twice the stiffness. To reduce its stiffness, the slab area must be reduced but the strength must be increased proportionally to maintain the same capacity. The modified slab layer pushover curve is shown in Figure 4.13. a) SCW3 b) SCW4 Figure 4.13: Shear pushover curve for modified slab layer. 0 2000 4000 6000 8000 10000 12000 0 25 50 75 100 B a se S h ea r (k N ) Shear Displacement (mm) Outward Truss 17x12 Slab Layer 12 Modified Slab Layer 12 0 1000 2000 3000 4000 5000 0 10 20 30 40 B a se S h ea r (k N ) Shear Displacement (mm) Outward Truss 9.6x10 Slab Layer 10 Modified Slab Layer 1276 The 2D layer stiffness is reduced approximately by a half by reducing the slab length perpendicular to the wall by a half. In addition, the steel amount and the concrete tension strength are doubled (although the steel yield stress should instead be increased, in some cases the steel does not yield and the increased steel capacity would be nullified). The modifications to the slab do not affect the global response of the shear wall (Figure 4.13). However, the modified slab layer reduces the simple 2D layer model stiffness while maintaining the same capacity which better captures the 3D slab behaviour (Figure 4.14). a) SCW3 b) SCW4 Figure 4.14: Equivalent slab force-slab strain curve for modified 2D slab layer. 0 10000 20000 30000 40000 50000 60000 70000 0 0.1 0.2 0.3 0.4 Sl a b F o rc e (k N ) Average Horizontal Slab Strain (mm/m) Total Slab Force Steel Slab Force 2D Layer - Total 2D layer - Steel Modified 2D Layer - Total Modified 2D layer - Steel 0 10000 20000 30000 40000 50000 0 0.05 0.1 0.15 0.2 Sl a b F o rc e (k N ) Average Horizontal Slab Strain (mm/m) 77 The modified 2D slab layer also better represents the slab stress profile in the 3D slab by modelling an equivalent stress block as demonstrated in Figure 4.15. The figure plots the total and steel equivalent normal stress parallel to the wall for the 3D slab (black lines) which represents the tension force in the slab. The stress is highest near the wall and reduces away from the wall. As shown with the force-deformation plots, the simple 2D layer model achieves a similar slab capacity which is evident in the similar sizes stress block (blue lines). However, the stiffness is much larger because the area engaged is the entire slab width. The modified 2D layer (red lines) corrects this issue by reducing the slab area by half, while increasing the equivalent stress to maintain the same equivalent stress block. Figure 4.15: Equivalent total normal stress parallel to the wall, SCW3 Δtop = 100 mm. Although the modified 2D slab layer model better captures the slab force-deformation response, its cracking prediction is worse and the post-yield response can be less stable. Also, considering that the increased slab layer stiffness does not significantly affect the global wall response, the benefits may be insignificant. Therefore, in selecting an appropriate slab model the following guidelines should be followed: a) Simple 2D layer model: provides accurate wall behaviour response over a wide range of conditions. Wall 0 0.5 1 1.5 2 2.5 3 3.5 4 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 E q u iv a len t T o ta l St res s (M P a ) Out-of-Plane Distance from Wall (mm) 3D - Steel 3D - Total 2D Layer - Steel 2D Layer - Total Modified 2D Layer - Steel Modified 2D Layer - Total78 b) Modified 2D layer model: provides a more accurate slab behaviour mechanism, albeit still limited, may have some unexpected effects. c) Complex pseudo 3D model provides the best slab behaviour prediction, but with increased computational effort and a limitation of only one slab. As such, the simple 2D slab layer model is the selected tool to analyze the range of parameters studied in Chapter 5 and 6 for the analysis suite. 4.5 MATCHING 2D MODELS FOR WALL BEHAVIOUR STUDY The work in this chapter has shown how the simple 2D slab layer element can emulate the 3D slab effects from the complex pseudo model. Chapter 6 explores the behaviour of the shear walls with multiple slabs, therefore the 2D slab model is checked to ensure that it matches the response of pseudo walls. This check is performed because the walls are slightly different in Chapter 5 and 6 than in previous chapters. The largest effective slab size is first determined by increasing the slab size with the 3D pseudo model until there is no longer a change in effect. In most cases, the effective slab size was increased to ensure no failure in the slab. Once the largest effective slab size is determined, a 2D layer model was created to match the pseudo model. This 2D slab model is then used for all the slabs in the analysis suite. The pushover curves in Figure 4.16 highlight that the 2D models match the 3D response very well. From this analysis, the largest effective slab length perpendicular to the wall ranges between 1.6lw and 2.7lw. 79 a) SCW: Lightly-Reinforced b) SCW4: Moderately-Reinforced c) TSE8: Heavily-Reinforced Figure 4.16: Pushover curve matching the 2D model to the 3D model for use in the analysis suite. 0 2000 4000 6000 8000 10000 12000 0 25 50 75 100 125 150 175 B a se S h ea r (k N ) Top Displacement (mm) Outward Truss 25x15 Slab Layer 15 0 1000 2000 3000 4000 5000 0 10 20 30 40 50 60 B a se S h ea r (k N ) Top Displacement (mm) Outward Truss 9.6x10 Slab Layer 10 0 1000 2000 3000 4000 5000 0 10 20 30 40 50 B a se S h ea r (k N ) Top Displacement (mm) Outward Truss - 3.8x2.6 Slab Layer 2.680 Shear Behaviour of Membranes and Walls Chapter 5 - The previous chapters developed an analytical model which can emulate the 3D slab effect on walls with relative confidence in the results. The wall-slab system behaviour is complex; therefore the wall behaviour without slabs should be understood before the wall-slab system is considered. The flexural behaviour of walls is understood fairly well but the shear behaviour is complex and often gross simplifications are made. One simplification is to assume shear behaviour is linear which can overestimate the shear demands. Another simplification is to assume that the shear-resisting portion of the wall, usually assumed as the web, is a uniformly- loaded membrane element described by nonlinear shear models. Although, nonlinear models capture the response better than a linear model, the wall shear behaviour is not uniform; shear strains vary over the wall height and along the wall length. Therefore the study in this chapter compares the shear behaviour of walls with an equivalent nonlinear membrane. A small parametric study is performed by comparing walls with different axial loads and horizontal steel ratios. At the end of the chapter, a simple model is presented which better describes the shear behaviour of walls. 5.1 VECTOR2 PREDICTION OF SE8 MEMBRANE TEST VecTor2 must be able to predict the nonlinear behaviour of a membrane experiment to have confidence when comparing the shear response of a membrane against a wall. Many experiments have been performed at the University of Toronto on the nonlinear shear response of uniformly-loaded membrane elements. These experiments help lead to the shear design methods in A23.3. One of the membrane tests used to validate the nonlinear model developed by Gerin and Adebar (2009) is the test SE8 by Stevens (Stevens, Uzumeri, & Collins, 1991). Using this test, VecTor2 is validated and will provide some confidence when comparing the VecTor2 results with membranes, walls and wall-slab systems. The SE8 test is from a series of tests in the Shell Element Tester at the University of Toronto on membranes with varying shear conditions and reinforcement ratios. All the panels are 1524 x 1524 x 285 mm and have steel oriented at 45 degrees to horizontal as shown in Figure 5.1 (Stevens et al., 1991). By orienting the steel at 45o allows for pure shear to be applied to the 81 specimen by applying forces normal to the edges (Figure 5.1a). This in effect applies the principle compression and tension stress to the element, resulting in pure shear (Figure 5.1b). The panel SE8 has 0.98% and 2.94% steel in the y and x-directions respectively with a pure shear condition applied. a) Shell element tester setup b) Mohr’s circle of stress Figure 5.1: Shell element tester configuration (based on Stevens et. al., 1991). Several VecTor2 models were created in an attempt to match the test results but modelling the test is challenging due to the nature of the test setup. First, displacement control is preferred when capturing the post-yield response since the displacements increase significantly while the forces increase slightly. However, displacement control cannot be used this in this situation because the panel has different stiffness in the x and y direction. Second, the panel must be tested analytically as closely as possible to the test setup, that is, a single panel by itself. It would seem reasonable to model the panel as a small panel within a large member and apply shear to the larger component, then retrieve the response of the smaller panel. However, this would induce additional and non-uniform loads defeating the purpose of a membrane test. Third, rigid supports cannot be used because the panel edges length change as the membrane deforms Figure 5.2. This results in non-uniform edge deformation which induces local stress concentration. 82 a) Model A: Shear force applied at 90o b) Model B: Principle forces applied at 45o Figure 5.2: VecTor2 SE8 models with deformed shape. Several different VecTor2 models were created in an attempt to match SE8 and the best 2 models are shown in Figure 5.2 above. The first model does not replicate the test setup but it applies pure shear directly to each edge, with uniform shear forces applied to the edge nodes with steel oriented at 0 and 90o as shown in Figure 5.2a. The second model replicates the Stevens test by applying normal forces on each face with steel oriented at 45 and 135o (see Figure 5.2b). Both models produce a similar shear response except that model A produces slightly non-uniform loads within the panel as shown in Figure 5.3 below. Due to the non-uniform loading from model A, it was decided to use model B with steel oriented at 45 and 135o and by applying normal principle forces. The comparison between the experimental and VecTor2 shear strain- stress response is shown in Figure 5.4. The experimental data is obtained from Stevens (1987). a) Model A: Shear force applied at 90o b) Model B: Principle forces applied at 45o Figure 5.3: Comparison of Model A and B at reproducing uniform shear forces. 1.36 vyx (MPa) 7.46 83 a) SE8 Test Data b) VecTor2 prediction Figure 5.4: VecTor2 predicts the shear stress-strain behaviour of the SE8 test. Test data from Gerin (2003), © Marc Gerin, by permission. As can be seen in Figure 5.4, VecTor2 captures the behaviour of the experimental test including the complex change in stiffness as the load reverses. However, the VecTor2 model does predict slightly higher stiffness and shear capacity. Several different concrete material models were used in order to better capture the response but the difference was insignificant. This is most likely due to the fact that the computer program is able to model perfect uniform loading while the experimental test will have variations. This is a similar result to the study by Palermo and Vecchio (2007) where VecTor2 often slightly overestimated the cracked stiffness but it typically predicts the ultimate shear capacity reasonably well. 5.2 NONLINEAR MEMBRANE SHEAR MODEL Other studies have used a wide range of shear models for walls from simple linear models to more complex nonlinear membrane models. Oversimplifying the shear response as a linear model does not accurately reflect the true behaviour while nonlinear membrane models are an improvement but still may not reflect the true shear behavior. A handful of shear models from previous studies are explored to highlight their advantages and disadvantages. 5.2.1 Other Nonlinear Models Many studies investigating the issue of dynamic shear magnification use a simple linear shear model (Boivin & Paultre, 2010; Panneton, Léger, & Tremblay, 2006; Rutenberg & Nsieri, 2006; Yathon, 2011). Often, the uncracked shear stiffness defines the model. Although these studies model the inelastic flexural response, assuming a linear shear model neglects the reduced shear -10 -8 -6 -4 -2 0 2 4 6 8 10 -12 -9 -6 -3 0 3 6 9 12 S h ea r St res s (M P a ) Shear Strain (mm/m) -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 shear strain, xy (m) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 sh ea r s tre ss , xy (M Pa ) G gr os s = 12 ,00 0 M Pa G cr ac ke d = 1, 20 0 M Pa G final = 6 00 MP a y-reinforcement yields -10 -8 -6 -4 -2 0 2 4 6 8 10 -12 -9 -6 -3 0 3 6 9 12 S h ea r St res s (M P a ) Shear Strain (mm/m) 84 stiffness due to cracking. With increased stiffness the shear force demand is greater, which can lead to their conclusions on the importance dynamic shear magnification. The Pacific Earthquake Engineering Research Center (PEER) has produced reports with guidelines to model and analyze tall buildings. As shown in Figure 5.5, common practice assumes linear uncracked shear rigidity up to the nominal shear strength. This crude model was adopted by FEMA 356 and is mostly due to the lack of test data (PEER, 2010). This models the shear behaviour as completely uncracked until the wall yields without accounting for softening due to cracking and other phenomena. However, this model does not reflect the shear response from actual test data (Stevens et al., 1991; F. Vecchio & Emara, 1992; F.J. Vecchio & Collins, 1986). A trilinear curve suggested by ASCE/SEI 41-06 in Figure 5.5 is a better representation of the actual shear behaviour. Figure 5.5: Different shear backbone curves (based on Peer, 2010). Another model to describe the shear behaviour is the Modified Compression Field Theory discussed in Section 2.2.1. The MCFT assumes the cracks are uniformly-distributed and rotating which does not accurately reflect the cracking for reverse cyclic shear (Gerin & Adebar, 2009). The rotating crack assumption is acceptable for monotonic loadings but the principle stress and strain angles can deviate substantially in reverse cyclic loading. 5.2.2 Nonlinear Membrane Shear Model Gerin and Adebar (2009) developed a simple rational nonlinear shear model which describes the complex shear behaviour with minimal empirical equations. This model assumes the shear- resisting portion of the wall is a membrane element described by uniform loads. The trilinear model developed is based on fundamental principles and is calibrated to the shear response of 85 several membrane element tests (Gerin, 2003). The model is described by a trilinear curve with uncracked, cracked and yielding regions as shown in Figure 5.6. The cracking stress vcr and nominal shear stress vn are calculated from code equations and other studies. The main contribution from Gerin and Adebar is developing equations to predict the shear strain at yield and ultimate. Figure 5.6: Simplified nonlinear trilinear regions for the membrane model. The stress and strain equations describing the various points along the curve are summarized in Table 5.1. There are several equations that can describe the cracking and nominal shear strength but only the equations in Table 5.1 are considered. The cracking stress is based on the ACI 318 cracking equation for a prestressed member but with a 0.6vn limit applied as suggested by PEER (2010). The yield stress (nominal stress) is defined by code equations by either the ACI upper and lower bound or the A23.3 General Method. The yield strain is defined by the strain of horizontal, vertical and shear components when the horizontal bars first yield (Gerin & Adebar, 2004). The ultimate shear strain is determined from the ductility which reduces as the element approaches crushing failure. Note the shear strain equations are for a simplified model with an assumed stress angle of 45o. 86 Table 5.1: Summary of equations to describe trilinear backbone membrane response. Stress (MPa) Strain (mm/mm) Cracking √ Yield a) √ b) √ c) √ Ultimate 5.2.3 Prediction of Experimental and VecTor2 SE8 Test The VecTor2 response is compared to the membrane backbone model developed by Gerin in Figure 5.7a. The nonlinear model predicts the SE8 experimental response very well as discussed by Gerin and Adebar (2004) when using the lower bound ACI 318 shear capacity of 5.73 MPa (dashed black line). However, the ACI equation assumes theta of 45o where theta is actually smaller seeing as ρy is 3ρx, increasing the steel contribution and the shear capacity. Instead of the ACI equation, we can modify Gerin’s model by using the shear stress at which steel yields in the VecTor2 model. The weak steel yields at 6.38 MPa and Gerin’s model matches the VecTor2 response very well when using this value (Figure 5.7b). As a result from this test, we are able to determine that a) VecTor2 predicts the shear behaviour very well, although slightly stronger and stiffer and b) Gerin’s nonlinear model can predict the shear response when an appropriate yield stress is used. Seeing as the nonlinear membrane model is accurate at predicting the experimental results, it will be used as the “equivalent membrane test” for walls that do not have a previous experimental test. 87 a) Prediction against experimental tests. b) Prediction against VecTor2 tests Figure 5.7: Nonlinear backbone model compared against the VecTor2 membrane element. Figure a) from Gerin (2003), © Marc Gerin, by permission. 5.3 PARAMETRIC WALL STUDY For the analysis suite, a parametric study is performed with the 3 walls discussed in Section 3.1. The properties include varying the horizontal web steel ratio, the applied axial load and the floor height between slabs. The horizontal steel ratio defines the 3 main walls with light, moderate and heavy horizontal reinforcement for walls SCW3, SCW4 and TSE8, respectively. For each of these walls, two levels of axial loads will be applied at 0 and 10% of fc’Ag, represented by the designation A and B respectively. In Chapter 6, the same walls are analyzed with a range of slab -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 shear strain, xy (m) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 sh ea r s tre ss, x y ( M Pa ) -6 -4 -2 0 2 4 6 shear stress, xy (MPa) -40 -30 -20 -10 0 10 20 30 40 str es s a ng le, c ( de g.) -12-10 -8 -6 -4 -2 0 2 4 6 8 10 12 shear strain, xy (m) -40 -30 -20 -10 0 10 20 30 40 str ain an gle , ( de g.) 0.5 0.0 -0.5 -1.0 -1.5 -2.0 min. principal strain, 2 (m) 0 -2 -4 -6 -8 -10 -12 mi n. pr inc ipa l s tre ss, c2 (M Pa ) 0 2 4 6 8 10 y-reinforcement strain, sy (m) -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 sh ea r s tra in, x y ( m ) a) b) c) d) e) Model Model, no shear failure check Experimental 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 0.0 2.0 4.0 6.0 8.0 10.0 S h ea r St res s (M P a ) Shear Strain (mm/m) VecTor2 Cyclic VecTor2 Monotonic Matched Gerin Model Gerin Model (ACI 318)88 spacing which forms a wall group with the same axial load and steel ratio (e.g. SCW4 B). Even though the walls are tested in a computer analysis, wall behaviour is assessed similarly to an experimental test. The advantage is a great deal more information is extracted and many more walls and more wall parameters are tested. All of the walls are pushed using displacement control in order to better capture yielding of the wall considering ultimate shear failures are relatively unstable. 5.3.1 Parameters Studied The 3 main walls SCW3, SCW4 and TSE8 have horizontal web steel ratios of 0.2, 0.5 and 1%, respectively. The lightly-reinforced SCW3 is designed to satisfy the minimum code requirement of 0.002Ag in cl.14.1.98.6 in A23.3 as the lower bound of steel ratios for non-seismic walls. Even though clause 14 walls are not analyzed with seismic demands, it is possible for these walls to be subjected to these demands nonetheless. The upper bound of 1% is selected because the walls become increasingly compression controlled where the increase in tension capacity from the slabs has little benefit. The walls are flanged and have a height-to-length ratio of 3. For high rise core walls, it is possible for hw/lw to be approximately 10 for a typical shear wall (Yathon, 2011). Applying a moment to shear ratio of 3 to a typical shear wall is equivalent of a force applied at 0.3hw. From RSA, a moment to shear ratio of 3 represents a higher mode response; approximately 2nd mode. The axial load of 10% fc’Ag is reasonable for a typical shear wall, as opposed to columns which are more heavily loaded. Each wall is also analyzed with no axial load to assess its influence on the shear response. The two levels of axial load are important in determining the vertical flange steel since the walls are designed not to form a flexural hinge, which limits the base shear. The vertical flange steel is kept constant for the various slabs spacing in each wall group so the walls all have the same geometry and flexural capacity. The walls with the closest slab spacing have the highest base shear capacity and govern the flange steel for the rest of the wall group. The flexural capacity is similar for the wall groups A and B for the same steel ratio, which allows for comparison of results with different axial load. 5.3.2 Development of Global Shear Stress-Strain Response The shear strain-stress response of a wall and wall-slab system is important in developing a model to describe shear behaviour regardless of geometry. The shear stress is converted to the 89 shear force by multiplying the stress by the shear area and the shear deformation is calculated by multiplying the shear strain by the height. The shear stress is obtained from the applied shear force in VecTor2 and dividing by an effective shear area. However, the shear strain must be measured from the VecTor2 results. The shear stress is assumed to the uniform over the height of the wall and across the wall length. Although this assumption does not hold completely true, especially in regions near boundaries (Figure 5.8c); it is still a fairly reasonable assumption (Figure 5.8b). The desired shear stress value is the average stress over the web region where the flanges have minimal shear stress and have minimal effect on the shear response (Figure 5.8a). The average web shear stress is predicted by dividing the applied shear force by an effective shear area. In the Canadian codes, the shear area is bw by dv where bw is the web width and dv is the lever arm between the tension and compression forces (CAC, 2004). The value for dv in this analysis for flanged walls is assumed to be the center-to-center flange distance, as it is the approximate lever arm. This is a good approximation as shown in Figure 5.8b where the dashed lines is the predicted shear value using dv against the measured average shear stress from VecTor2. a) Shear stress in mid-height layer b) Shear stress profile over wall height c) Shear stress in lower wall layer Figure 5.8: Shear stress profiles in wall. The average web stress is predicted using dv from flange to flange, SCW3 B – Wall Only, Δtop = 60 mm. 0 4500 9000 13500 18000 22500 27000 0 1 2 3 4 H ei g ht ( m m ) Shear Stress (MPa) υxy -web υxy - wall dv - 0.83lw dv - 1.11lw90 The shear strain can be measured several ways by averaging VecTor2 strains, using nodal displacements or deriving it from the flexural deformations. Deriving the shear strain from flexural displacements begins by measuring the curvature distribution to calculate the flexural displacement using the moment-area method. The shear displacement is calculated by subtracting the flexural component from the total displacement and the shear strain is the shear deformation divided by the height. However, two issues may result in inaccuracies: 1) plane sections do not remain plane in shear critical sections, hence, incorrect curvatures and 2) the total displacement consists of more than just a shear and flexural component. The shear strain can also be measured from the nodal displacements and length change between the 4 nodes in a rectangular area. Flexural and shear displacements are determined from the angle and length changes (Beyer, Dazio, & Priestley, 2011). The issue with this method is the flexural curvatures change over the height of the rectangle and can overestimate the shear deformation. The third method to measure shear strain is to obtain the element shear strain values directly from VecTor2 and average over some height. This method is computationally intensive due to the amount of VecTor2 data, but it is straight forward and provides consistent results. All of the methods are compared in Figure 5.9 against the assumed true response of VecTor2 (black line). Figure 5.9: Comparison of shear strain measurement methods, SCW3 B - Wall Only Δtop = 60 mm. 0 4500 9000 13500 18000 22500 27000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 H ei g ht ( m m ) Average Shear Strain (mm) γxy - VT2 measured 5 layer average 10 layer average Derived from curvatures Measured from nodal displacements91 Both the nodal displacement and curvature method loosely capture the variation of shear strain over the height but can severely over and under predict the shear strains. Averaging the shear strains over a couple element layers provides a reasonable depiction of the shear strain profile; there is enough detail to capture the variation but not too much detail such that the curve becomes confusing. Therefore for the analysis suite, the global shear strain is measured from the VecTor2 shear strain data, averaged over 8 to 10 element layers. This simplifies the shear strain profile but still provides sufficient detail. The shear stress-strain plot measures the response from the middle third of the wall in order to avoid the effects from boundary conditions. As can be seen from the Figure 5.10, the averaged shear stress-strain plot is able to predict the shear deformation of the wall very well. The predicted flexural component is determined from a tri-linear moment-curvature model and the moment-area theorem. Combining the predicted shear and flexural deformations provides a very good estimate of the total wall displacement. The small discrepancy is due to averaging errors, shear and dowel slippage. However, the right and left side wall displacements can differ significantly due to local cracking and damage. Therefore, the global response from the shear stress-strain response is not enough to describe the shear behaviour; some indicator of localized damage is necessary. The following subsection develops localized damage indices. Figure 5.10: Predicted versus measured wall displacements over the height, SCW4 Δtop = 38 mm. 0 1.8 3.6 5.4 7.2 9 10.8 0 10 20 30 40 H ei g ht ( m m ) Horzontal Deformation (mm) VecTor Δx Left VecTor Δx Right Predicted Flexural Component Predicted Total Deformation92 5.3.3 Development of Local Damage Indicators Indicators of local damage are helpful in determining the mechanics of the wall behaviour and provide insight into how close the wall is to failure. The damage indicators are split into 2 groups: shear strain profiles and component damage ratio. The component damage ratio provides the damage level of the horizontal steel and concrete where a value greater than one indicates the element is yielding or crushing. The horizontal bar damage ratio, Yld%, is the ratio of the steel strain over the steel strain at yield and the concrete crushing ratio, εcm%, is the concrete principle compression strain over the principle compression strain at peak stress. By comparing the relative level of damage between the horizontal bars and the concrete, we can infer whether the wall is experiencing a tension or compression failure. Figure 5.11 below compares the damage of local crushing and yielding at first yield, where it is clear that the lightly-reinforced wall is experiencing diagonal tension failure while the heavily-reinforced wall is experiencing local crushing as well as tension yielding. Local crushing is a good indicator whether the wall is near its ultimate shear capacity since significant local crushing is typically quickly followed by ultimate failure. SCW3 – Lightly-Reinforced TSE8 – Heavily Reinforced a) Steel Yielding b) Crushing c) Steel Yielding d) Crushing Figure 5.11: Local yielding and crushing damage values indicate if wall is under diagonal tension or compression failure. Damage state at first yield. No damage Yielding/Crushing 93 The shear strain profile indicator provides insight into the behaviour and load state of the wall. As discussed in the previous section, the average shear strain in a layer relates to the global response of the structure. To complement the average strain, the maximum shear strain within the layer is also provided since it typically drives local damage. A third shear strain indicator provides the strain in a local region, where it provides the third highest strain value in the layer. This local shear strain indicates how concentrated the peak strain is compared to a small local region. The shear strain values are compared for a number of strain profiles in Figure 5.12. When the maximum value is significantly higher than the average, that indicates a highly localized failure, such as in a lightly-reinforced wall in Figure 5.12a & b, and the opposite is true for heavily-reinforced walls in Figure 5.12c and d. The local shear strain provides an indication of how concentrated is the failure, where it is very localized in the case of (a), and not as localized in (d). Also, it is possible to determine the shear behaviour of the wall by comparing the relative values between the maximum, local and average shear strain. a) Extreme local strain above low average b) Wide localized strain above low average c) Local strain with high average d) Wide local strain with high average Figure 5.12: Relation between average, maximum and local shear strain indicates wall behaviour. 0 0.3 0.6 0.9 1.2 1.5 1.8 0 1500 3000 4500 6000 7500 9000 S h ea r St ra in (m m /m ) Wall Length (mm) Layer Profile Average Local 0 1 2 3 4 5 6 0 1500 3000 4500 6000 7500 9000 S h ea r St ra in (m m /m ) Wall Length (mm) Layer Profile Average Local 0 2 4 6 8 10 0 300 600 900 1200 1500 1800 S h ea r St ra in (m m /m ) Wall Length (mm) Layer Profile Average Local 0 2 4 6 8 10 0 300 600 900 1200 1500 1800 S h ea r St ra in (m m /m ) Wall Length (mm) Layer Profile Average Local94 5.4 BEHAVIOUR OF MEMBRANES AND WALLS The shear behaviour of walls and membranes are significantly different because the members are loaded differently. Membranes are applied with uniform shear and axial load whereas walls have shear, axial and flexural loads which vary over the wall height and along its length. The behaviour of walls is analyzed with its general response, local damage indices and global shear stress-strain response. Walls can only be compared to the membrane through the shear stress- strain response since the membrane cracks and damages uniformly. Table 5.2 summarizes the capacity and drift for the walls in this parametric study. The following subsection highlights noteworthy observations; see Appendix A.3 for the complete set of results. Table 5.2: Summary of parametric wall results. Name Axial Load ρx (%) Mn (kNm) Vn (kN) Peak Drift SCW3 A - Wall Only 0% 0.20 708,000 7,140 0.33% SCW3 B - Wall Only 10% 0.20 641,000 8,150 0.28% SCW4 A - Wall Only 0% 0.50 120,000 3,600 0.59% SCW4 B - Wall Only 10% 0.50 104,000 3,720 0.61% TSE8 A - Wall Only 0% 0.98 29,200 4,350 0.80% TSE8 B - Wall Only 10% 0.98 29,000 4,500 0.80% 5.4.1 General Response The general response provides quick insight to the behaviour of the wall. One issue encountered during the analysis suite is shear-critical components can become unstable near the peak load. Unlike with flexural hinging where the curves behave well, shear-critical failures often result in a localized crushing and can lead to sudden brittle failure or unstable yielding. For the lightly- reinforced members, local crushing failure does not necessarily result in global failure. Global failure is defined as the peak base shear prior to negative wall stiffness or a sudden drop in base shear. The wall can often redistribute the loads to other portions of the wall where the load capacity can still increase, even though failure has occurred. As shown in Figure 5.13a, the monotonic pushover appears to have a stable yielding plateau, however the wall quickly fails when a reverse cyclic load is applied; therefore this wall has unstable yielding. A reverse cyclic analysis is performed on each wall to determine the actual failure point during unstable shear failure; if the cyclic analysis achieves the same strength as the monotonic then ultimate shear failure has not occurred. Reverse cyclic analysis is performed for all the walls as a check to 95 determine the true failure point, where any response after the failure point is ignored. Figure 5.13b highlights how the cyclic response achieves the same peak base shear, indicating the monotonic response is stable until sudden failure around 60 mm. a) SCW3 B – Cyclic analysis reveals wall failed before unstable plateau b) SCW4 B – Cyclic Analysis reveals a stable yielding plateau Figure 5.13: Reverse cyclic analysis to determine the actual failure point on the monotonic curve. The monotonic pushover curves for all the walls are provided in Figure 5.14 below comparing the response with and without axial load. In every case, the wall with 10% axial load has higher strength for the same displacement but each wall pair achieves the same peak base shear at approximately the same top displacement regardless of the axial load. Each wall pair has approximately the same flexural stiffness therefore virtually all the change in stiffness is due to the shear response. The different behaviour of between SCW3 and TSE8 highlights the limited ductility in the heavily-reinforced TSE8. -10000 -5000 0 5000 10000 -200 -100 0 100 200 B a se S h ea r (k N ) Top Displacement (mm) Cyclic Monotonic -5000 -2500 0 2500 5000 -100 -50 0 50 100 B a se S h ea r (k N ) Top Displacement (mm) Cyclic Monotonic96 a) SCW3 – Lightly-Reinforced Wall b) SCW4 – Moderately-Reinforced Wall c) TSE8 – Heavily-Reinforced Wall Figure 5.14: Pushover curves for the walls with 0 and 10% axial load. 0 2000 4000 6000 8000 10000 0 25 50 75 100 B a se S h ea r (k N ) Top Displacement (mm) 10% axial load No axial load Failure Load 0 1000 2000 3000 4000 0 20 40 60 80 B a se S h ea r (k N ) Top Displacement (mm) No axial load 10% axial load Failure Load 0 1500 3000 4500 6000 0 15 30 45 60 B a se S h ea r (k N ) Top Displacement (mm) No axial load 10% axial load Failure Load97 The cracking patterns are compared for the walls with 0% axial load in Figure 5.15 and walls with 10% axial load in Figure 5.16. The axial load has little effect on the cracking pattern; perhaps the crack angle is slightly more vertical. As shown previously, the lightly-reinforced wall develops a large crack surrounded by uncracked zones whereas TSE8 is uniformly cracked with no large cracks until just prior to failure. The crack patterns provide an indication of the loading and behaviour of the wall: the lightly-reinforced wall is also lightly-loaded with severe local failure, the heavily-reinforced wall is heavily-loaded with wide-spread failure, and the moderately-reinforced wall behaves between the two extremes. SCW3 SCW4 TSE8 a) 0.2% Steel Ratio b) 0.5% Steel Ratio c) 1.0% Steel Ratio Figure 5.15: Cracking pattern at failure load for walls with 0% axial load. 98 SCW3 SCW4 TSE8 a) 0.2% Steel Ratio b) 0.5% Steel Ratio c) 1.0% Steel Ratio Figure 5.16: Cracking pattern at failure load for walls with 10% axial load. 5.4.2 Local Damage Indicator The damage indicators are provide for a select few load stages which describe the typical wall behaviour in Figure 5.17. Each load stage provides the applied shear stress vn (MPa) and shear stress ratio (vn/fc’). The shear strain values over the height of the structure are plotted in black on the top axis (average: solid line, maximum: thick dashed line, local: small dashed line). The steel yielding and crushing, blue and red lines respectively, are plotted on the bottom axis as damage ratio; if the value crosses the thick grey line at 1.0, then the region is crushing or yielding. SCW3 has large localized shear strains where the max value drives the horizontal steel yielding (Figure 5.17a) and has significant ductility remaining because the wall is far from crushing. A similar response occurs in SCW4 except that the shear strains are distributed more evenly over the wall (Figure 5.17b). SCW4 also experiences relatively more crushing due to its heavier loading. 99 a) SCW3 A – Horizontal Yielding – 2.32 MPa (0.05) b) SCW4 A – Horizontal yielding, more distributed strains and crushing damage – 4.39 MPa (0.15) c) TSE8 A – Heavily loaded steel and concrete – 8.01 MPa (0.22) d) TSE8 B – Axial load increases the crushing damage – 8.48 MPa (0.23) Figure 5.17: Select damage indices for all walls. Shear stress for each figure is labelled as vn (vn/fc’). 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 0.5 1.0 1.5 2.0 Shear Strain (mm/m) h/H w Damage Ratio -1.0 1.0 3.0 5.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 0.5 1.0 1.5 2.0 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 0.5 1.0 1.5 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 0.5 1.0 1.5 Shear Strain (mm/m) h/H w Damage Ratio Yld% εcm% γmax γlocal Damage Limit γmean 100 TSE8 is clearly more heavily loaded than the other walls because the average and maximum shear strains are relatively close (Figure 5.17c&d). In addition, the higher load increases concrete compression to near crushing once the steel yields. By comparing Figure 5.17c & d we can observe the internal effect the axial load has on the wall where the load states have approximately the same shear strain profile and similar shear stress. The concrete compression strain is higher with the 10% axial load wall, which would be expected because the section is experiencing shear and axial load. However, the axial load is not significant enough that it has a large effect on the wall. 5.4.3 Global Shear Stress-Strain Response The global shear strain-stress is the one parameter that allows us to compare the response of both the walls and membranes. The wall shear response is obtained from the VecTor2 results as described in Section 5.3.2. The equivalent membrane has the same web vertical and horizontal steel and a thickness and length equal to the wall bw and dv respectively. The membrane shear stress-strain response is determined from the equations in Section 5.2.2. The results comparing the wall and membrane response for all the walls are presented in Figure 5.18. The first observation is the VecTor2 shear strain curves have a similar shape to the pushover curves in previous sections. The ductility of the wall is evident in SCW4 and TSE8 but SCW3 appears to have limited ductility due the scale. Conversely, SCW3 has significant ductility because horizontal yielding occurs shortly after cracking near a shear strain of 0.5mm/m but continues to gain strength beyond 2.0 mm/m. The most significant observation is the membrane behaviour does not predict the wall shear behaviour very well, especially for SCW3. The General Method from CSA A23.3-04 can reasonably predict the nominal shear capacity but the nonlinear membrane shear model significantly overestimates the shear yield strain and ductility. The cracking stress equation is fairly accurate except that it underestimates for SCW3 B. The results from the ACI 318 equations are not shown but consistently underestimate the shear capacity since they do not account of applied moment and axial loads. 101 a) SCW3 – 0.2% Steel Ratio b) SCW4 – 0.5% Steel Ratio c) TSE8 – 1.0% Steel Ratio Figure 5.18: Shear stress-strain response of wall and an equivalent membrane element. Solid lines are VecTor2 results and dashed lines are the membrane backbone model. 0 1 2 3 4 0 4 8 12 16 S h ea r St res s (M P a ) Shear Strain (mm/m) SCW3 A - Wall Only SCW3 A - Backbone SCW3 B - Wall Only SCW3 B - Backbone 0 1 2 3 4 5 6 0 3 6 9 12 S h ea r St res s (M P a ) Shear Strain (mm/m) SCW4 A - Wall Only SCW4 A - Backbone SCW4 B - Wall Only SCW4 B - Backbone 0 2 4 6 8 10 0 2 4 6 8 S h ea r St res s (M P a ) Shear Strain (mm/m) TSE8 B - Wall Only TSE8 A - Wall Only TSE8 A - Backbone TSE8 B - Backbone102 From the above figures, it is clear that the wall behaves stiffer than the membrane element. Except for SCW3, the membrane element reasonably predicts the absolute value of plastic shear strain. The backbone trilinear curve from the nonlinear membrane model reasonably reproduces the wall shear response; however the yield shear strain estimate must be reduced. A possible reason for this discrepancy is the wall has non-uniform loads therefore some sections of the wall will experience higher strains than the average. As such, some regions will yield first, long before the whole wall yields, reducing the global yield strain. This is a reasonable hypothesis since the more heavily-reinforced walls are more uniformly loaded where its membrane yield strain estimate is not as poor (see Figure 5.18c). 5.5 SHEAR RESPONSE MODEL FOR SHEAR WALLS The nonlinear membrane shear stress-strain response does not accurately predict the actual shear response of walls; it severely under predicts the cracked shear rigidity. However, the membrane response does capture the general behaviour and as such it can be modified to better fit the wall shear response. The following subsections discuss how the nonlinear shear model for membranes can be altered. The model is modified based on phenomena observed from the analysis and is not necessarily a generalized model. As shown in Figure 5.19, the modified model predicts a reasonable response but SCW3 is a unique case because the section appears to yield long after it actually does. A summary of the equations for the nonlinear wall shear model is provided in Table 5.3. Note the limits to the equations in Table 5.1 still apply. Table 5.3: Summary of equations for modified nonlinear wall shear model. Parameter Membrane Model Modified Wall Model Cracking √ same ⁄ same Yield √ same ⁄ Ultimate same ⁄ same 103 a) SCW3 – Lightly-Reinforced b) SCW4 - Moderately-Reinforced c) TSE8 - Heavily-Reinforced Figure 5.19: Shear stress-strain response for wall and the modified membrane nonlinear model. Solid lines are VecTor2 results and dashed lines are the membrane backbone model. 0 1 2 3 4 0 2 4 6 S h ea r St res s (M P a ) Shear Strain (mm/m) SCW3 B - Matched Backbone SCW3 B - Wall Only SCW3 A - Matched Backbone SCW3 A - Wall Only 0 1 2 3 4 5 6 0 3 6 9 S h ea r St res s (M P a ) Shear Strain (mm/m) SCW4 B - Matched Backbone SCW4 B - Wall Only SCW4 A - Matched Backbone SCW4 A - Wall Only 0 2 4 6 8 10 0 3 6 9 S h ea r St res s (M P a ) Shear Strain (mm/m) TSE8 B - Matched Backbone TSE8 B - Wall Only TSE8 A - Matched Backbone TSE8 A - Wall Only104 5.5.1 Shear Strength The shear strength of the walls can be determined from the ACI and CSA A23.3 code equations. The best matching equation is the General Method from clause 11 from A23.3-04 since it accounts for variation in theta and beta based on the moment, shear and axial load demand (Equation 4). A variable theta is crucial since theta can be significantly different between two walls, modifying the contribution from the concrete and steel. Conversely, a fixed stress angle adopted in code equation is for simplified, conservative design, not necessarily for accurate analysis. The ACI equations assume an angle of 45o and the simplified A23.3 method assumes an angle of 35o. The General Method is flexible and has been calibrated to many tests (Bentz & Collins, 2006), yet the steps are considerably more complex than other code equations. However, the added accuracy from the General Method outweighs the added complexity to the shear stress-strain model considering it provides fairly reasonable predictions. The equations required to estimate the nominal shear strength are outlined in Equations 4 to 7 and are used following the methods in Clause 11 of A23.3-04. ⁄ (4) (5) (6) √ (7) 5.5.2 Yield Strain The shear yield strain equation provided by Gerin and Adebar (2004) reasonably predicts the yield strain of membrane elements but overestimates the wall shear strain. It is crucial to accurately predict the yield strain since the slope between the cracking and yielding point defines the cracked rigidity. This is likely due to the fact membrane elements experience uniform stress and strain fields while walls experience localized strain concentration. As such, it is the yielding of a small region within the wall which reduces the stiffness of the entire wall (Figure 5.20). However, the global shear strain-stress response is developed from the average shear strain, which is clearly much less than the yielding of the local elements. Therefore, the wall appears to globally have a lower yield strain while the local elements are yielding at a much higher strain. 105 SCW3 SCW4 Figure 5.20: Localized yielding of a few elements cause a reduction in stiffness for the entire wall. To account for the global reduction in shear strain, the membrane yield strain is reduced by a ratio of the local over the average wall shear strain. The only unknown is the local shear strain value seeing as the average wall strain and membrane yield strain are known from previous analysis. The local shear strain value must be over a large enough region that yielding there will affect the global wall stiffness; likely the average strain over a couple elements. One measure of the localized shear strain is the local shear strain indicator which measures the 3rd highest element shear strain in a layer as discussed in Section 5.3.3. The ratio of the local over the average strain is measured from the middle third of the wall, as this region typically fails and it is the farthest from the boundary conditions. The ratio is averaged over 4 different load stages to obtain the average local to average strain ratio (Table 5.4). This ratio varies little over the pushover curve since it is related to how well the wall can distribute the load. In order to better match the response, the ratio is increased by a factor of 1.1. This is similar to measuring the strain of the 2.7th element or the local strain in 15% of the wall Localized Yielding 106 length. The modified yield strain value is obtained by dividing the membrane yield strain by the average values for each wall in Table 5.4. Table 5.4: Measured local over average shear strain ratios. SCW3 SCW4 TSE8 Load A B A B A B D 2.01 2.02 1.62 1.52 1.81 1.79 C 2.85 1.95 2.31 1.65 1.51 1.48 B 2.30 2.10 1.35 1.36 1.46 1.45 A 2.76 1.14 1.93 1.30 1.46 1.42 Average 2.48 1.80 1.80 1.46 1.56 1.53 5.5.3 Shear Strain at Ultimate and Ductility The nonlinear membrane shear model predicts the ultimate shear strain by estimating the element ductility from the shear stress ratio: ductility defines the ultimate shear strain by . As the nominal shear stress approaches the assumed crushing stress of 0.25 fc’, the ductility reduces to 1. With low shear stress, the ductility increases to a maximum value of 4. It is reasonable to assume a plastic yielding plateau at the nominal shear strength; however, this does not necessarily apply to SCW3. SCW3 appears to have no ductility because the predicted yield point actually occurs when the wall crushes, however the actual yield point occurs much earlier. For the other walls however, using the predicted ductility from Gerin and Adebar (2004) provides a reasonable estimate of the ductility and a conservative estimate of the ultimate shear strain. Using the absolute value of plastic shear strain could provide a more accurate estimate for some walls but may be unconservative for others. 107 Behaviour of Wall-Slab Systems Chapter 6 - In Chapter 5, the shear behaviour of walls was found to be significantly different than an idealized equivalent membrane. Expanding on that analysis, the shear behaviour of walls as the more complex wall-slab systems is explored in the following chapter. All of the previous work it utilized to assess the behaviour of shear-critical walls accounting for the 3D effects from concrete floor slabs spaced at regular intervals. Slabs are modeled over the wall height using multiple 2D slab layers developed in Chapter 4. A study is performed using the same walls and techniques from Chapter 5 but the walls now have slabs spaced at 0.5, 1.0 and 1.5 the length of the wall. 6.1 SLAB SPACING PARAMETERS The slab spacing intervals are designed to capture a wide range of possible floor heights and wall sizes. The slabs are spaced at 0.5lw, 1.0lw and 1.5lw and for a wall height to length ratio of 3:1 translate into 6, 3 and 2 slabs over the height of the wall, respectively (Figure 6.1). Note the top boundary condition has a slab in place to prevent local failure; therefore the number of slabs between the boundary conditions is 5, 2 and 1 slab, respectively. The different slab intervals are labeled 1.5:1, 1.0:1 and 0.5:1 Ratio. Each wall has 4 slab intervals (including without a slab) for each axial load, resulting in a total of 24 walls in the analysis. a) 1.5:1 Ratio b) 1.0:1 Ratio c) 0.5:1 Ratio Figure 6.1: Slab spacing intervals for parametric study. 108 The various slab spacing intervals capture a wide range of floor heights for various wall sizes (Table 6.1). The closely-spaced slabs can represent a tall first floor height for a typical high rise structure (wall length of 9m). In contrast, the wide slab spacing (1.5:1 Ratio) can represent a typical floor height for a low rise building (wall length of 1.9m). Although some floor heights are not realistic, the range in slab intervals is designed to capture the different wall failures that occur by varying spacing. Based on the typical floor heights, typical high rise structures stand to gain the greatest benefit from slabs because floors are spaced relatively close together. For these walls, the typical slab spacing could be as low as 0.25 or 0.3lw. Table 6.1: Equivalent floor height relative to wall length. Wall Length (m) Slab spacing (lw) 1.5 1 0.5 9.0 13.50m (44.3ft) 9.00m (29.5ft) 4.50m (14.8ft) 3.6 5.40m (17.7ft) 3.60m (11.8ft) 1.80m (5.9ft) 1.9 2.86m (9.4ft) 1.91m (6.3ft) 0.95m (3.1ft) To maintain model consistency, a single VecTor2 model mesh is created for each of the 3 walls. Therefore all the walls have 6 element layers with 200 mm height which can be assigned either wall or slab layer properties. Walls are modeled with different slab spacing by assigning the 2D slab layer elements at the desired slab location. If a separate model was created for each slab spacing, the flow of forces within the wall could be altered by the different mesh configuration and not by just the slabs themselves. As a result, the mesh of SCW3 appears to always have 6 slabs, even though they may not be assigned as slabs. The slab location will be indicated on the figures to aid with distinguishing between slab spacing. 6.2 GENERAL WALL-SLAB BEHAVIOUR The shear behaviour of the wall and wall-slab system differs due to the additional tension capacity from the floor slabs. The following section explores the change in internal mechanics due to slabs for the lightly, moderately and heavily-reinforced walls. Only a select few interesting observations are discussed below; see Appendix A.4 for the complete data set. 6.2.1 General Response The effect of slabs and their spacing has a significant influence on the shear response of walls. This is evident in the increase in base shear capacity for walls with slabs, where more slabs 109 increase the shear capacity (see Table 6.2). The greatest increase in shear capacity is the lightly- reinforced wall with up to 193% additional shear capacity, almost 3 times the initial wall capacity. As expected, the heavily-reinforced wall has the smallest increase in shear capacity with up to a 22% increase for the tightest slab spacing. The walls also fail at similar levels of top wall drift, typically between 0.5-0.7%. However, there is little correlation between an increase or decrease in drift with an increasing number of slabs. The moment capacity is the same for each wall group, determined by the 0.5:1 ratio since it produces the largest base shear capacity. The nominal moment capacity is determined from Response-2000. Table 6.2: Summary of results from the full analysis suite. Name Axial Load Slab Spacing ρx Mn (kNm) Vn (kN) Vn Increase Drift SCW3 SCW3 A - 0.5:1 Ratio 0% 0.5lw 0.2% 708,000 20,900 193% 0.65% SCW3 A - 1.0:1 Ratio 1.0lw 14,850 108% 0.50% SCW3 A - 1.5:1 Ratio 1.5lw 12,200 71% 0.48% SCW3 A - Wall Only n/a 7,140 0% 0.33% SCW3 B - 0.5:1 Ratio 10% 0.5lw 0.2% 641,000 21,600 165% 0.70% SCW3 B - 1.0:1 Ratio 1.0lw 14,970 84% 0.50% SCW3 B - 1.5:1 Ratio 1.5lw 12,910 58% 0.43% SCW3 B - Wall Only n/a 8,150 0% 0.28% SCW4 SCW4 A - 0.5 Ratio 0% 0.5lw 0.50% 120,000 7,250 101% 0.54% SCW4 A - 1.0 Ratio 1.0lw 6,100 69% 0.61% SCW4 A - 1.5 Ratio 1.5lw 4,850 35% 0.61% SCW4 A - Wall Only n/a 3,600 0% 0.59% SCW4 B - 0.5 Ratio 10% 0.5lw 0.50% 104,000 7,620 105% 0.63% SCW4 B - 1.0 Ratio 1.0lw 5,730 54% 0.52% SCW4 B - 1.5 Ratio 1.5lw 4,470 20% 0.48% SCW4 B - Wall Only n/a 3,720 0% 0.61% TSE8 TSE8 A - 0.5 Ratio 0% 0.5lw 0.98% 29,200 5,090 17% 0.70% TSE8 A - 1.0 Ratio 1.0lw 4,710 8% 0.70% TSE8 A - 1.5 Ratio 1.5lw 4,630 6% 0.73% TSE8 A - Wall Only n/a 4,350 0% 0.80% TSE8 B - 0.5 Ratio 10% 0.5lw 0.98% 29,000 5,490 22% 0.73% TSE8 B - 1.0 Ratio 1.0lw 4,930 10% 0.70% TSE8 B - 1.5 Ratio 1.5lw 4,720 5% 0.70% TSE8 B - Wall Only n/a 4,500 0% 0.80% 110 Another observation is the change in stiffness and strength from the pushover curves (see Figure 6.2). Only the 10% axially-loaded wall response is presented for clarity since their response is similar. As in Chapter 5, a reverse cyclic analysis is performed to determine the failure point of the walls with slabs. The slabs have virtually no effect on the initial uncracked region but all walls experienced an increase in wall stiffness after cracking. Once again, the figures highlight the minimal effect slabs have on the heavily-reinforced wall as opposed to the lightly-reinforced wall. In addition, the walls with slabs appear to have minimal plastic ductility even though in some cases they achieve similar or higher drifts. This is expected since the slabs force the walls to be more concrete controlled, resulting in sudden failures. As such, the potential exists that although the slabs provide additional shear capacity, they may consequentially limit plastic shear yielding. However, the flexural component is included in the pushover curves; therefore the actual shear ductility is explored through the shear strain-stress responses discussed in Section 6.2.4 111 a) SCW3 B b) SCW4 B c) TSE8 B Figure 6.2: Pushover curves highlighting slab effect on global wall capacity. Walls have 10% axial load. 0 5000 10000 15000 20000 25000 0 50 100 150 200 B a se S h ea r (k N ) Top Displacement (mm) 0.5:1 Ratio 1.0:1 Ratio 1.5:1 Ratio Wall Only Failure Load 0 2000 4000 6000 8000 10000 0 20 40 60 80 B a se S h ea r (k N ) Top Displacement (mm) 0.5:1 Ratio 1.0:1 Ratio 1.5:1 Ratio Wall Only Failure Load 0 1000 2000 3000 4000 5000 6000 0 15 30 45 60 B a se S h ea r (k N ) Top Displacement (mm) 0.5:1 Ratio 1.0:1 Ratio 1.5:1 Ratio Wall Only Failure Load112 Generally, slabs limits crack propagation and distribute localized cracking over the width of the wall. As shown in Figure 6.3, the slabs split the cracking struts into regions between the slabs. By splitting the cracking struts, the slabs distribute the cracking over the wall length and reduce severe local cracking. This effect is only shown for SCW3 since the other walls with heavier reinforcement already have good crack control and the difference is less pronounced. Near peak load, the tightly-spaced slabs force the lightly-reinforced wall to effectively behave as a heavily- reinforced wall with distributed and uniform cracking. Note each figure appears to have the same slab configuration but only the layers indicated are assigned slab layer properties as discussed in Section 6.1. a) No Slab b) 1.5:1 Ratio c) 1.0:1 Ratio d) 0.5:1 Ratio Figure 6.3: Crack pattern of SCW3 B at 70 mm top displacement (slab layers in blue). 6.2.2 Flow of Forces Floor slabs spaced at regular intervals clearly modify the flow of forces within the wall. The slab spacing affects the strut angle where the angle flattens with tighter slab spacing (theta increases). This results from the slab forcing the compression strut towards the vertical compression flange and limiting strut action through the slab. The increasing strut angle is apparent in Figure 6.4 which presents the principle compression stress for SCW4 A at a top displacement of 50 mm. 113 This wall series is selected because the struts are very clear and it represents typical behaviour experienced by the other walls. a) No Slab b) 1.5:1 Ratio c) 1.0:1 Ratio d) 0.5:1 Ratio Figure 6.4: Principle compression stress highlights shallow strut angle with closer slab spacing, SCW4 A Δtop = 50 mm. The principle compression stress highlights the diagonal strut regions and provides insights into how the forces flow through the wall. Similarly to the force flow diagrams in Chapter 3 and 4, the slabs are in tension while the web regions between the slabs are in compression. The slabs split the large diagonal strut into smaller struts between each slab, which increases the strut angle, but only to an upper limit. For the 1.5 and 1.0:1 ratio (Figure 6.4b & c), the compression strut is redirected into the flange whereas with the 0.5:1 ratio the strut angle would need be very large to intersect the flange. In the latter case, the strut instead intersects with the slab which distributes the compression strut over the web width. This creates a wide strut and more uniform loading over the web, reducing the potential for localized failure. These two cases are highlighted in Figure 6.5. -32 .0 f2 (MPa) 1.3 114 Figure 6.5: Diagram of flow of forces in wall with wide and tight slab spacing. Although the slabs increase the strut angle, it can only increase to an upper limit. The minimum strut angle required to intersect the compression flange for a slab spacing of 1.5lw, 1.0lw and 0.5lw is 33, 45, and 63o, respectively. It is fairly clear that 63o is too large of an angle for the strut to intersect with the wall, therefore it must intersect with the flange. The ACI 318-08 Commentary on strut and tie models suggests limiting the stress to 25o between the compression strut and the tension ties, equivalent the 65o in our case (ACI, 2008). Figure 6.6 below presents the change in average total element stress angle in the middle third height of each wall in the analysis suite. The total element stress angle is selected because it correlates to the applied stress on the wall and accounts for the combined steel and concrete response. However, the difference between the total and net stress angle is minimal. As expected, the axial load reduces the stress angle (steeper strut) as well as closer slabs increase the stress angle (flatter strut). The stress angle can change significantly during the pushover, but generally trends towards smaller angles with increasing displacement (shear force). In addition, increasing the wall steel generally reduces angle variability where the wall steel can control the strut angle more. The stress angle helps determine the shear capacity of the section where √ according to the Canadian code. Although the theta from this analysis is not the same type of theta used in code equations, the effect is still the same: increasing theta reduces the steel contribution and increases the concrete demand. Therefore the slabs provide additional tension capacity while increasing theta, which effectively places more demand on the concrete and ultimately results in concrete failure. This is a similar effect as increasing shear reinforcement is only effective at increasing the shear capacity up to a maximum concrete shear stress of 0.25 fc’. 115 a) SCW3 b) SCW4 c) TSE8 Figure 6.6: Variation of theta with increasing displacement for all wall pairs. 20 25 30 35 40 45 0 50 100 150 200 St res s An g le ( ra d ) Top Wall Displacement (mm) 10% Axial Load No Axial Load 20 25 30 35 40 45 0 20 40 60 80 St res s An g le ( ra d ) Top Wall Displacement (mm) 20 25 30 35 40 45 0 15 30 45 60 St res s An g le ( ra d ) Top Wall Displacement (mm) 0% - Wall Only 10% - Wall Only 0% - 1.5:1 Ratio 10% - 1.5:1 Ratio 0% - 1.0:1 Ratio 10% - 1.0:1 Ratio 0% - 0.5:1 Ratio 10% - 0.5:1 Ratio 116 6.2.3 Local Damage Local damage indices are provided for select walls representing typical behaviour. The internal damage of SCW3 A is presented in in Figure 6.7 when the average shear strain at mid-height is approximately 0.5-0.6mm/m (black line). Comparing the damage at the same average shear strain highlights the effect resulting from the slab spacing. Again, the shear strain profiles plot the average, maximum and local strain along the top axis and the yielding and crushing indices are on the bottom axis. Although the plots are for 0% axial load, similar damage trends are observed with 10% axial load, as well as for the walls SCW4 and TSE8. In Figure 6.7 it is clear the maximum shear strain varies substantially even though the walls have an average shear strain around 0.5 mm/m. The ratio of the maximum shear strain over the average shear strain varies from 6.5 for no slabs to as small as 2.5 for 0.5:1 Ratio. This indicates the tighter slab spacing can distribute the shear strains over the wall length instead of being concentrated in a large crack. Supporting this observation is the local shear strain indicator is closer to the maximum shear strain for 0.5:1 Ratio than it is for wider slab spacing. The yield strain indicator highlights that the slabs provide addition horizontal tension capacity since the steel demand is reduced with an increasing number of slabs. At the load stage in Figure 6.7, the wall without slabs has significant yielding and is being driven by the maximum shear strain. Conversely, the 0.5:1 Ratio wall has significantly less steel demand such that the concrete compression and steel tension damage ratios are very similar. The crushing indicator is approximately 0.25 for all the walls suggesting that the walls are relatively undamaged and should have significant strain capacity remaining. This is true considering the shear strain in Figure 6.7 is 0.5 mm/m and failure for the walls do not start until 2.7 mm/m. 117 a) Wall Only – 2.32 MPa (0.05) b) 1.5:1 Ratio – 3.07 MPa (0.06) c) 1.0:1 Ratio – 3.55 MPa (0.07) d) 0.5:1 Ratio – 5.08 MPa (0.10) Figure 6.7: Comparison of internal damage for SCW3 A with same average shear strain near 0.5 mm/m. Shear stress for each figure is labelled as vn MPa (vn/fc’). 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 0.5 1.0 1.5 2.0 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 0.5 1.0 1.5 2.0 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 0.5 1.0 1.5 2.0 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 0.5 1.0 1.5 2.0 Shear Strain (mm/m) h/H w Damage Ratio Yld% εcm% γmax γlocal Damage Limit γmean 118 A summary of the shear strain profiles for the analysis suite is provided in Figure 6.8, where the maximum to average and local to average shear strain ratios are compared. For SCW3, the maximum/average ratio (left side) can be significant for the wall with no slabs and the ratio generally reduces with more slabs. Although a pattern is not obvious, tighter slab spacing generally reduces and stabilizes the maximum to average ratio. Conversely, the local to average ratio remains relatively constant during the pushover, suggesting the shear strain distribution, and hence load distribution, is fairly constant. The figure below again reiterates the general wall behaviour: SCW3 has extreme local failure and has the greatest slab benefit, TSE8 has relatively uniform loading and minor slab benefits while SCW4 is blend of the other walls. 119 Maximum/Average Ratio Local/Average Ratio a) SCW3 – Maximum b) SCW3 – Local c) SCW4 – Maximum d) SCW4 – Local e) TSE8 – Maximum f) TSE8 – Local Figure 6.8: Maximum and local shear over average shear strain ratios for the analysis suite. 0 1 2 3 4 5 6 7 8 0 50 100 150 200 M a x /A v er a g e R a ti o Top Wall Displacement (mm) 0 1 2 3 4 5 6 7 8 0 50 100 150 200 L o ca l/A v er a g e St ra in Top Wall Displacement (mm) 0 1 2 3 4 5 6 7 8 0 20 40 60 80 M a x /A v er a g e St ra in Top Wall Displacement (mm) 0 1 2 3 4 5 6 7 8 0 20 40 60 80 L o ca l/A v er a g e St ra in Top Wall Displacement (mm) 0 1 2 3 4 5 6 7 8 0 15 30 45 60 M a x /A v er a g e St ra in Top Wall Displacement (mm) 0 1 2 3 4 5 6 7 8 0 15 30 45 60 L o ca l/A v er a g e St ra in Top Wall Displacement (mm) 120 6.2.4 Shear Stress-Strain Response The shear stress-strain response directly describes the shear behaviour and is useful in determining the mechanics of the wall system. Describing the shear behaviour is complex since the wall has many possible failure mechanisms; understanding how each wall fails is crucial. Labelling failure points such as first yield and crushing provides insight into the mechanisms which cause a change in shear stiffness. A summary of the labels used is provided in Table 6.3. The following figures also plot a matching quadrilinear curve (dashed lines) along with the VecTor2 results (solid lines). The quadrilinear curve simplifies the wall-slab shear response and is discussed in Section 6.3. Table 6.3: Definitions of chart labels describing failure mechanisms. Chart Label Definition X-Yield First yield of horizontal web steel Strut Y-Yield Yielding of vertical web steel in diagonal strut Crushing Concrete crushing in web region (εcm%> 1) Slab Cracking First cracking appearing in slab layer Base Y-Yield (-) Compression yielding of vertical flange steel Localized Strain Concentration Small region suddenly experiences extreme local shear strains Cracks Combine Two cracked regions expand and merge creating one large cracked region, reducing stiffness Sig. Crushing Significant concrete crushing Strain Expansion & Struts Merge Shear strain is distributed over the wall length and 2 separate diagonal struts merge into one 121 a) SCW3 A (0% axial load) b) SCW3 B (10% axial load) Figure 6.9: Shear stress-strain response for SCW3 with failure mechanisms labelled. The shear strain plot is similar to the pushover curve but the shear ductility is clearer. The shear strain at ultimate is fairly constant for SCW3 A and somewhat constant for SCW3 B, where the X-Yield Slab Cracking X-Yield Base Y-Yield then Crushing Crushing 0 2 4 6 8 10 12 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.5-1 Ratio - Group 2/3 1.0-1 Ratio - Group 2/3 1.5-1 Ratio - Group 2/3 Wall Only - Group 2/3 Wall Only 1.5-1 Ratio 1.0-1 Ratio 0.5-1 Ratio Failure Load X-Yield Slab Cracking X-Yield Strain Expansion & Struts Merge Crushing Crushing 0 2 4 6 8 10 12 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 S h ea r St res s (M P a ) Shear Strain (mm/m) 122 pushover curves do not suggest this. The shear response follows a similar pattern for most of the lightly-reinforced walls: first, horizontal web yielding softens the wall, then slab cracking softens the wall further, eventually followed by concrete crushing prior to the ultimate load. Typically ultimate failure results from significant localized concrete crushing, often from vertical sliding shear between the slabs; not a typical diagonal or horizontal sliding failure. Floor slabs do not appear to change this pattern significantly except the 0.5lw slab spacing has slab cracking prior to horizontal steel yielding. This highlights the fact the slabs modify the shear response where slab cracking also reduces the system’s stiffness, which must be accounted for when creating a shear model. Axial load does not appear to have an effect on the ultimate shear capacity considering the peak load is approximately the same for equal slab spacing with 0 and 10% axial load. Axial load does however increase the wall cracking point from around 1.5 MPa to 2.7 MPa for 0% and 10% axial load respectively. However, slabs appear to have little effect on the cracking stress. Even though axial load does not affect the ultimate shear capacity it does stiffen the response. Axial load stiffens the initial cracked response and then softens significantly achieving the same failure shear. Creating a model matching this response must incorporate axial load throughout the response. Slabs affect the moderately-reinforced SCW4 where more slabs increases the shear stiffness and capacity but decreases the ultimate strain capacity (Figure 6.10). Similarly to SCW3, the slabs do not affect the cracking stress but unlike SCW3, slab cracking is not prominent in this wall system. Except for 0.5:1 Ratio walls, the general failure sequence is horizontal steel yielding, followed by concrete crushing which typically softens the response further. Ultimate failure results from significant compression failure, usually horizontal shear sliding failure between the slabs. Again, the axial load increases the cracking stress and has a slightly stronger shear response, although relatively much less than the axial load effect from SCW3. One noteworthy observation unique to SCW4 is vertical steel yielding in a diagonal strut in a couple cases, even before horizontal yielding (0.5:1 Ratio, Figure 6.10a & b). SCW4 is the only wall with equal vertical and horizontal steel ratios where the additional flexural demands on the vertical bars exceeded the large horizontal shear demand. A check for this failure should be incorporated into the nonlinear wall shear model. 123 a) SCW4 A (0% axial load) b) SCW4 B (10% axial load) Figure 6.10: Shear stress-strain response for SCW4 with failure mechanisms labelled. Strut Y-Yield X-Yield Crushing Slab Cracking Crushing 0 2 4 6 8 10 12 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.5-1 Ratio - Group 2/3 1.0-1 Ratio - Group 2/3 1.5-1 Ratio - Group 2/3 Wall Only - Group 2/3 Wall Only 1.5-1 Ratio 1.0-1 Ratio 0.5-1 Ratio Failure Load Crushing X-Yield Crushing X-Yield Strut Y-Yield Cracks Combine Strut Y-Yield 0 2 4 6 8 10 12 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 S h ea r St res s (M P a ) Shear Strain (mm/m) 124 The shear stress-strain response in Figure 6.11 for TSE8 highlights that the slabs do have an effect on the shear response, albeit very limited, which was not as evident in the pushover curves. Primarily, the slabs do slightly increase the shear stiffness and capacity and but also reduce the ultimate shear strain capacity to approximately the same value. The slabs force TSE8 to be compression controlled where concrete crushing precedes horizontal yielding, if any. Since horizontal yielding occurs late in the pushover, slab cracking provides a significant reduction in stiffness, such that it resembles horizontal yielding. It is interesting to note the 1.5lw and 1.0lw slab spacing shear response is virtually identical. This is likely the closer slab spacing does not fully intersect the diagonal compression strut, which does not directly fully engage the slab. The shear stress angle between the 1.5:1 and 1.0:1 Ratio remains very similar, unlike with SCW3 or SCW4, suggesting the closer slabs are not altering the diagonal strut path. 125 a) SCW4 A (0% axial load) b) TSE8 B (10% axial load) Figure 6.11: Shear stress-strain response of TSE8 with failure mechanisms labeled. Crushing Slab Cracking X-Yield & Crushing Sig. Crushing X-Yield Localized Strain Concentration 0 2 4 6 8 10 12 0.0 1.0 2.0 3.0 4.0 5.0 6.0 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.5-1 Ratio - Group 2/3 1.0-1 Ratio - Group 2/3 1.5-1 Ratio - Group 2/3 Wall Only - Group 2/3 Wall Only 1.5-1 Ratio 1.0-1 Ratio 0.5-1 Ratio Failure Load Crushing X-Yield Base Y-Yield (-) Localized Strain Concentration Slab Cracking Crushing 0 2 4 6 8 10 12 0.0 1.0 2.0 3.0 4.0 5.0 6.0 S h ea r St res s (M P a ) Shear Strain (mm/m) 126 6.3 SHEAR RESPONSE MODEL FOR WALL-SLAB SYSTEMS Creating a model to predict the wall-slab shear response is complex and developing the full shear stress-strain curve is beyond the scope of this study. However, a prediction of the wall-slab ultimate shear capacity is developed in the following sections. In addition, the immediate section describes the general behaviour of the wall-slab stress-strain response. This amalgamates the behaviour observed in Section 6.2.4. 6.3.1 Simplified Shear Stress-Strain Response A starting point to create a model for the wall system response could be to use the nonlinear shear model developed by Gerin & Adebar (2004) for membrane elements. Their model is trilinear with an uncracked region, softened cracked region and a plastically yielding region. The lightly-reinforced walls tend to follow a similar trend where there is the uncracked region then horizontal yielding followed by local crushing prior to failure. However, slab cracking is another failure mechanism unique to wall systems which tends to soften the wall response. Since additional shear strength is being provided by the stiff slabs, the slab layer will certainly be softer once the slab cracks; therefore it softens the response of the entire wall. This is most obvious for the SCW3 walls with 0.5:1 slab spacing: slab cracking prior to yielding significantly softens the response where it appears the slab cracking point is actually the yield point on a trilinear curve. As the result of this, the trilinear curve is inadequate at describing the wall system shear behaviour since it cannot account for the additional mechanism of slab cracking. Therefore we must produce a quadrilinear shear model to describe the shear behaviour of a wall system which defines the points at which the wall cracks, the wall yields, wall crushes and when the slab cracks. A quadrilinear curve is able to capture the general behaviour of the walls as shown in Section 6.2.4 and the response for the walls with 10% axial load is shown in Figure 6.12. The response is not shown for 0% axial load since they are very similar. 127 a) Lightly-Reinforced SCW3 b) Moderately-Reinforced SCW4 c) Heavily-Reinforced TSE8 Figure 6.12: General behaviour and mechanisms of the wall-slab shear stress-strain response. 0 2 4 6 8 10 12 0 0.5 1 1.5 2 2.5 3 3.5 4 S h ea r S tres s (M P a ) Shear Strain (mm/m) 10% - Wall Only 10% - 1.5-1 Ratio 10% - 1.0-1 Ratio 10% - 0.5-1 RatioWall Cracking Yielding Slab Cracking Crushing 0 2 4 6 8 10 12 0 0.5 1 1.5 2 2.5 3 3.5 4 S h ea r S tres s (M P a ) Shear Strain (mm/m) 10% - Wall Only 10% - 1.5-1 Ratio 10% - 1.0-1 Ratio 10% - 0.5-1 RatioWall Cracking Yielding Crushing 0 2 4 6 8 10 12 0 0.5 1 1.5 2 2.5 3 3.5 4 S h ea r S tres s (M P a ) Shear Strain (mm/m) 10% - Wall Only 10% - 1.5-1 Ratio 10% - 1.0-1 Ratio 10% - 0.5-1 Ratio Wall Cracking Slab Cracking Crushing then Yielding 128 From Figure 6.12, all the walls have an initial uncracked region with a sudden change in stiffness when the wall cracks. The walls also generally have concrete crushing just prior to ultimate failure. After crushing, the wall behaves similarly to plastic yielding but with some nominal amount of strain hardening. However, the intermediate stiffness changes result from horizontal steel yielding and slab cracking and the failure progression differs for each wall. Generally each wall set experiences the same failure progression (i.e. yielding then slab cracking or vice versa) except tight slab spacing can reverse this progression. For the lightly-reinforced walls and walls with wide slab spacing, often horizontal yielding occurs first and is followed by slab cracking. Generally, the reverse progression occurs with heavily-reinforced walls or walls with tight slab spacing. Whichever is first of these two failure mechanisms often produces the greatest change in stiffness. As a result of this variation in failure mechanisms, a quadrilinear model would have to be versatile where it would be able to capture the variation in failure mechanisms. That is, not all mechanisms occur for each wall and not always in the same order. 6.3.2 Wall-Slab Shear Capacity Prediction Two possible methods were explored to predict the wall-slab system shear capacity: a sectional model for flexural regions and a strut and tie model. Although the strut and tie method is versatile it can be computationally intense and it is a lower-bound approach. As a lower-bound model, any load path which satisfies the stress limits is acceptable even if that load path does not represent the actual load path. Therefore, the model may not accurately account for the full contribution from the slabs, steel and concrete. In addition, a great deal of time can be spent creating a model that satisfies the requirements where it may be simpler to create a finite element model in the first place. The second method to predict the wall-slab shear capacity is sectional method based on the equations from clause 11.3 in A23.3-04. This method assumes the wall is in a B-region of relatively uniform stresses and strains which would be true of the walls with tight slab spacing. Clause 11.3 predicts the total section capacity by summing the contribution from the concrete and steel, Vc, and Vs respectively. This method is modified by adding a third contribution from the slab, Vslab, as shown in Figure 6.13. In order for shear failure to occur a diagonal crack must intercept multiple closely-spaced slabs; the same mechanism which engages stirrups. The only difference is the slabs are spaced farther apart a have much more capacity than a single steel tie. 129 Figure 6.13: For tall walls, Vslab is analogous to Vs. The slab contribution is similar to the steel contribution since there are a finite number of slabs at constant slab spacing. If the spacing or cracking angle increases, less slabs are engaged reducing the slab contribution. The Vs equation in clause 11.3 can be direct converted to an equation for the slab shear capacity contribution as shown Equation (8). The slab spacing, sslab, and lever arm, dv, are known so θ and the equivalent slab force, Aslabfslab, are the only unknowns for the slab equation. The following sections explore determining an appropriate value for these unknown, as well as β for the concrete contribution (8) Where the complete wall-slab shear capacity equation is: (9) √ (10) (11) (12) 130 6.3.3 Prediction Using VecTor2 Data Results Before a general model is created to predict the wall-slab system shear capacity, the method is first verified with the measured analysis data. Throughout this analysis, parameters which remain constant are shown in Table 6.4. The β value is assumed to remain constant regardless of the number of slabs since it is not directly measurable; β is often back-calculated from the total shear capacity minus the steel contribution. The β is selected as the value for the walls without slabs from the General Method, one value for each wall and axial load (6 total). It is possible to back-calculate β using ( )⁄ where εx is determined from . However, this method is very sensitive to variations in θ. For instance, θ for SCW3 A with no slabs fluctuates from 31.6 to 27.5o between nearby load stages resulting in β of 0.25 and 0.57 respectively. Therefore, β is assumed constant considering the concrete strength should not vary significantly with the addition of slabs. As for the other variables, dv is assumed to be the center to center flange length as well as Av and s values remain constant for each wall as per equation ( )⁄ . Table 6.4: Constant variables for wall-slab shear model. SCW3 SCW4 TSE8 ρx 0.2% 0.5% 0.98% Axial Load 0% 10% 0% 10% 0% 10% Vc Calculation β 0.255 0.280 0.256 0.265 0.165 0.176 fc’ (MPa) 50 30 37 bw (mm) 300 200 285 dv (mm) 7500 3400 1715 Vs Calculation Av (mm 2) 200 200 200 s (mm) 333 200 71.5 fy (MPa) 450 400 479 Sslab (mm) 0.5:1 Ratio 4500 1800 952.5 1.0:1 Ratio 9000 3600 1905 1.5:1 Ratio 13500 5400 2857.5 Wall Only ∞ ∞ ∞ The remaining parameters, θ and Aslabfslab, are calculated from the VecTor2 data. The sectional shear model is compared against the peak base shear capacity from the pushover analysis. The Aslab area is taken as the cross-sectional area of the slab layer perpendicular to the walls, which are 0.2 x 15 m, 0.2 x 10 m and 0.19 x 2.85 m for SCW3, SCW4 and TSE8, respectively. The 131 slab stress, fslab, is as the average horizontal stress in the slab layer at peak load. For multiple slabs, the slab stress is averaged over all the slab layers. Finally, θ is measured as the total stress element theta at peak load as shown in Figure 6.6 in Section 6.2.2. The results are shown below in Table 6.5 to Table 6.7. Table 6.5: SCW3 wall-slab system shear capacity using VecTor2 data. Axial Load 0% 10% Wall 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only M eas u re d θ (deg) 28.8 26.6 25.3 31.7 27.4 28.3 23.6 23.8 Aslab (mm 2) 3.0E+6 3.0E+6 3.0E+6 0 3.0E+6 3.0E+6 3.0E+6 0 Fslab (MPa) 1.348 1.409 1.487 0.000 1.310 1.336 1.463 0.000 Vc kN 4065 4065 4065 4065 4457 4457 4457 4457 % 20% 27% 30% 55% 21% 31% 30% 49% Vs kN 3691 4049 4279 3286 3907 3767 4631 4586 % 18% 27% 32% 45% 19% 26% 32% 51% Vslab kN 12271 7039 5231 0 12620 6208 5570 0 % 61% 46% 39% 0% 60% 43% 38% 0% Vr kN 20027 15153 13574 7350 20984 14432 14658 9043 Vbase kN 20900 14850 12200 7140 21600 14970 12910 8150 Difference (%) -4.2% 2.0% 11.3% 2.9% -2.9% -3.6% 13.5% 11.0% Table 6.6: SCW4 wall-slab system shear capacity using VecTor2 data. Axial Load 0% 10% Wall 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only M eas u re d θ (deg) 40.3 33.3 31.1 22.9 39.1 33.1 29.3 21.9 Aslab (mm 2) 2.0E+6 2.0E+6 2.0E+6 0 2.0E+6 2.0E+6 2.0E+6 0 Fslab (MPa) 1.319 1.487 1.456 0.000 1.385 1.403 1.219 0.000 Vc kN 955 955 955 955 987 987 987 987 % 11% 13% 15% 23% 11% 14% 16% 23% Vs kN 1606 2067 2252 3219 1675 2085 2426 3381 % 19% 28% 36% 77% 18% 29% 39% 77% Vslab kN 5885 4269 3036 0 6446 4061 2738 0 % 70% 59% 49% 0% 71% 57% 45% 0% Vr kN 8445 7290 6242 4174 9109 7133 6151 4368 Vbase kN 7250 6100 4850 3600 7620 5730 4470 3720 Difference (%) 16.5% 19.5% 28.7% 15.9% 19.5% 24.5% 37.6% 17.4% 132 Table 6.7: TSE8 wall-slab system shear capacity using VecTor2 data. Axial Load 0% 10% Wall 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only M eas u re d θ (deg) 34.1 33.1 32.4 30.1 32.2 31.1 30.4 28.0 Aslab (mm 2) 541E+3 541E+3 541E+3 0 541E+3 541E+3 541E+3 0 Fslab (MPa) 1.197 1.498 1.640 0.000 1.264 1.528 1.580 0.000 Vc kN 492 492 492 492 522 522 522 522 % 9% 10% 10% 11% 9% 9% 10% 11% Vs kN 3394 3519 3615 3961 3654 3809 3924 4323 % 61% 69% 73% 89% 60% 68% 74% 89% Vslab kN 1724 1118 838 0 1959 1235 877 0 % 31% 22% 17% 0% 32% 22% 16% 0% Vr kN 5609 5129 4945 4453 6136 5567 5323 4845 Vbase kN 5090 4710 4630 4350 5490 4930 4720 4500 Difference (%) 10.2% 8.9% 6.8% 2.4% 11.8% 12.9% 12.8% 7.7% From the tables above, the sectional method with Vslab can reasonably predict the shear capacity using the measured slab forces and θ. The best strength prediction is for SCW3, on average within 7% of the VecTor2 peak base shear value. The prediction for TSE8 is slightly worse with an average less than 10% of the VecTor2 value. However, this model is not as accurate for the moderately-reinforced SCW4, which predicts on average 23% over the VecTor2 base shear. This poor prediction is likely due to the assumption that the slab area is the total cross sectional area; the effective area is likely smaller as will be shown in Section 6.4.2. It is simple to determine the relative contribution each component contributes since the model calculates the shear capacity for concrete, steel and slabs separately. For the lightly-reinforced wall with no slabs, the concrete contributes approximately half to the shear capacity (β = 0.26- 0.28). Conversely, the heavily-reinforced wall with no slabs has only 11% of its shear capacity from the concrete (β = 0.16-0.17). With the addition of slabs spaced at 0.5lw to SCW3, the concrete and steel now both contribute around 20% with the slabs contributing the remaining 60%. For TSE8, the concrete contributes virtually the same regardless of the number of slabs, and the wall steel and slab trade off the remaining 90% capacity (30% maximum slab contribution). 133 As expected, the steel contributes relatively more for the more heavily-reinforced walls and the slabs contribute less to the total shear capacity. The exception to this observation is SCW4 where the relative slab contribution is higher compared to the lightly-reinforced SCW3. This is counter-intuitive considering as the steel ratio increases the walls become more concrete- controlled, thus the slabs should contribute less as is observed with TSE8. Again, these observations are likely due to the slab area used is larger than the effective slab size. In addition, θ varies significantly more for SCW4 between 22.9 to 40.3o while θ only varies by 4o to 6o for the other walls. 6.4 WALL-SLAB SYSTEM SHEAR CAPACTIY The previous section demonstrated the wall-slab system shear capacity can be predicted by modifying the code-defined Vc and Vs equations to include a Vslab component. This method requires a value for θ and Aslabfslab to calculate the shear capacity. To extend this model to general walls, a procedure is developed to determine a reasonable estimate of these unknown values. 6.4.1 Estimate of β and θ for Wall-Slab System Section 6.2.2 revealed that θ changes over the course of a pushover and with various the slab spacing. The measured value of θ varies significantly which affects the steel and slab demand considerably. Based on the General Method, increasing θ also correspondingly reduces β resulting in a reduction in Vs and Vc contribution. Figure 6.6 in section 6.2.2 highlights that θ generally increased with an increased number of slabs. Therefore, using the wall-slab shear capacity equations from Section 6.3 would result in lower concrete, steel and slab capacity with more slabs. As such, it would be unconservative to have a constant β and θ with changing the number of slabs. However, θ calculated from the General Method for the wall without slabs generally is higher than the measured θ from VecTor2. As a result, using a constant β from the General Method would result in a conservative estimate of Vr for design. Therefore, the wall- slab shear capacity model is calculated using a constant β and θ determined for each wall and axial load from the General Method ignoring slabs. Table 6.8 to Table 6.10 below summarized the results. 134 Table 6.8: SCW3 wall-slab system shear capacity using β and θ from the General Method. Axial Load 0% 10% Wall 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only M eas u re d θ (deg) 31.6 31.6 31.6 31.6 31.0 31.0 31.0 31.0 Aslab (mm 2) 3.0E+6 3.0E+6 3.0E+6 0 3.0E+6 3.0E+6 3.0E+6 0 Fslab (MPa) 1.348 1.409 1.487 0.000 1.310 1.336 1.463 0.000 Vc kN 4065 4065 4065 4065 4457 4457 4457 4457 Vs kN 3290 3290 3290 3290 3374 3374 3374 3374 Vslab kN 10937 5719 4021 0 10899 5560 4058 0 Vr kN 18292 13073 11376 7354 18730 13391 11890 7831 Vbase kN 20900 14850 12200 7140 21600 14970 12910 8150 Difference (%) -12.5% -12.0% -6.8% 3.0% -13.3% -10.5% -7.9% -3.9% Table 6.9: SCW4 wall-slab system shear capacity using β and θ from the General Method. Axial Load 0% 10% Wall 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only M eas u re d θ (deg) 31.6 31.6 31.6 31.6 31.4 31.4 31.4 31.4 Aslab (mm 2) 2.0E+6 2.0E+6 2.0E+6 0 2.0E+6 2.0E+6 2.0E+6 0 Fslab (MPa) 1.319 1.487 1.456 0.000 1.385 1.403 1.219 0.000 Vc kN 955 955 955 955 987 987 987 987 Vs kN 2209 2209 2209 2209 2230 2230 2230 2230 Vslab kN 8097 4563 2979 0 8582 4345 2517 0 Vr kN 11261 7726 6143 3164 11799 7562 5734 3217 Vbase kN 7250 6100 4850 3600 7620 5730 4470 3720 Difference (%) 55.3% 26.7% 26.7% -12.1% 54.8% 32.0% 28.3% -13.5% Table 6.10: TSE8 wall-slab system shear capacity using β and θ from the General Method. Axial Load 0% 10% Wall 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only M eas u re d θ (deg) 35.6 35.6 35.6 35.6 35.0 35.0 35.0 35.0 Aslab (mm 2) 542E+3 542E+3 542E+3 0 542E+3 542E+3 542E+3 0 Fslab (MPa) 1.197 1.498 1.640 0.000 1.264 1.528 1.580 0.000 Vc kN 492 492 492 492 522 522 522 522 Vs kN 3207 3207 3207 3207 3287 3287 3287 3287 Vslab kN 1629 1019 744 0 1762 1066 735 0 Vr kN 5328 4718 4442 3699 5572 4875 4544 3809 Vbase kN 5090 4710 4630 4350 5490 4930 4720 4500 Difference (%) 4.7% 0.2% -4.1% -15% 1.5% -1.1% -3.7% -15% 135 Comparing the Vr to the peak Vbase from VecTor2, Vr from the wall-slab shear model is generally lower than Vbase, especially for the tighter slab spacing. Again, SCW3 and TSE8 are predicted reasonably well where Vr is on average less than Vbase by 9 and 6% respectively. However, SCW4 is significantly over-estimated on average 31% compared to the 23% when using the measured values. This is partly the result of the measured θ measured being higher than θ from the General Method, where θ measured for the 0.5lw slab spacing is 40.3 o as opposed to 31.6o. This is the equivalent to reducing Vs and Vslab by 27%. As mentioned previously, the increased SCW4 Vr prediction is partly due to an effective slab area that is too large as will be addressed in the following section. 6.4.2 Effective Slab Force The second unknown variable in the wall-slab shear capacity model is the effective slab force Aslabfslab. The steel stirrup contribution to the shear capacity depends on the stirrup cross- sectional area multiplied by its stress which is fy assuming the bars are yielding. Similarly for slabs, the slab force is the cross-sectional slab area perpendicular to the wall multiplied by the slab force. However, two issues arise: first, walls are often surrounded by large slab area where only a portion near the wall is effective and second, the slab is a composite of concrete and steel each with different yield stresses. Therefore, a conservative estimate of these values must be determined. The effective slab force can be easily measured from the horizontal slab forces using the pseudo slab model from Chapter 2 and 3. By averaging the total and steel normal stress parallel to the wall, we can observe the change in slab stress perpendicular to the wall, including the relative contribution from the steel and concrete. The variation in horizontal stress is shown for a series of different slab widths at the same load in Figure 6.14 for both SCW3 and SCW4. Averaging the horizontal stress in the slab is required to get the global slab response since the slabs have non-uniform cracking and yielding. The effective slab force is the combined effect from the entire slab and not just the local regions of yielding and cracking. 136 a) SCW3, Vbase at 9950 kN b) SCW4, Vbase at 3870 kN Figure 6.14: Variation in normal stress parallel to the wall in the slabs. From the figures above, the total stress within the slab remains relatively constant near 1.5 MPa for large slabs where the stress exceeds that only for very small slabs. Since the slabs have the same thickness and steel ratio, only the length parallel to the wall changes, 9.0 m and 3.6 m for Wall 0 0.5 1 1.5 2 2.5 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 N o rm a l St res s P a ra llel t o W a ll ( M p a ) Distance Perpendicular from Wall (mm) Total - 6m Steel - 6m Total - 12m Steel - 12m Total - 15m Steel - 15m Wall -0.5 0 0.5 1 1.5 2 2.5 3 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 N o rm a l St res s P a ra llel t o W a ll ( M p a ) Distance Perpendicular from Wall (mm) Total - 0.4m Steel - 0.4m Total - 4m Steel - 4m Total - 10m Steel - 10m137 SCW3 and SCW4 respectively. Although the two walls clearly have different shear demands, the slab demand at the wall-slab interface is very similar. Note that the drop in stress at the wall for SCW3 is largely due to the significant compression at the end of the wall reducing the average. The figures above also reveal the effective slab area; SCW3 has virtually the entire slab at 1.5 MPa while only a port of the slab for SCW4 is in tension. Therefore, the effective slab area for SCW3 and SCW4 is approximately 200 x 1500 mm and 200 x 4800 mm. The limit of the effective area is indicated by the steel force in the slab where the extent of the effective region is where the steel is in compression. Contrary to the total slab stress, the steel stress varies approximately linearly from a peak at the wall to zero at the edge of the effective area. Although the slab width was not large enough for SCW3 to test the extent of the effective area, the steel stress is very small suggesting a 15m width is almost at the limit of the effective slab area. The effective slab area is limited by the ability for wall stresses to extend outward from the wall ends at some angle. This limit is defined by a ratio of the slab width to wall length in Figure 6.15. For the three walls, the ratio of the effective slab width to the wall length is 1.67, 1.33 and 1.50. Therefore, the maximum effective slab cross-sectional area is assumed to be 1.5lw multiplied by the slab thickness. Figure 6.15: Plan of effective slab area. 138 The peak stress of 1.5 MPa is directly attributed to the tension capacity of the slab; slab cracking stress and steel yielding. For low strains, the slab tension force is governed by the concrete cracking stress fcr while at high strains the stress is limited by the steel yielding. The cracking stress of concrete is similar to the tensile strength of concrete ft’ which is assumed to be √ . With a slab strength of 25 MPa, the cracking stress is approximately 1.6 MPa. Once the element yields at high strain, the stress is governed by the steel fy where the total stress is ρfy. For a yield stress of 400 MPa and 0.4% steel, the maximum slab tension stress is 1.6 MPa. However, not every element in the slab is experiencing cracking and yielding, which reduces the average total stress to 1.5 MPa. In Table 6.11, this peak slab stress is also reflected in the peak slab stress observed in the analysis suite (note: the steel yield stress is higher for TSE8 resulting in a peak stress greater than 1.6 MPa). Therefore, it is reasonable to assume the slab stress is fairly constant at 1.5 MPa which is used for the value of fslab. However, this value would change for higher-strength concrete or slabs with a significant amount of steel. Table 6.11: Summary of average slab horizontal stress. Axial Load Slab Spacing SCW3 SCW4 TSE8 Average 0% 0.5 Ratio 1.348 1.319 1.197 1.288 1.0 Ratio 1.409 1.487 1.498 1.465 1.5 Ratio 1.487 1.456 1.640 1.528 10% 0.5 Ratio 1.310 1.385 1.264 1.320 1.0 Ratio 1.336 1.403 1.528 1.422 1.5 Ratio 1.463 1.219 1.580 1.421 6.4.3 Simple Wall-Slab Shear Model A simple wall-slab shear capacity model is created using an effective slab area of tslab by l.5lw, a slab force of 1.5 MPa and using β and θ determined from the General Method (see Table 6.12 to Table 6.14). This method would not be appropriate for high-level analytical analysis since it ignores the variation in β, θ and slab stress with different slab spacing. However, it is shown to be an effective tool for a fairly accurate prediction of the shear capacity, which tends to slightly conservative. 139 Table 6.12: SCW3 wall-slab system shear capacity from simplified model. Axial Load 0% 10% Wall 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only M eas u re d θ (deg) 31.6 31.6 31.6 31.6 31.0 31.0 31.0 31.0 Aslab (mm 2) 2.7E+6 2.7E+6 2.7E+6 0 2.7E+6 2.7E+6 2.7E+6 0 Fslab (MPa) 1.500 1.500 1.500 0.000 1.500 1.500 1.500 0.000 Vc kN 4065 4065 4065 4065 4457 4457 4457 4457 % 22% 32% 37% 55% 23% 33% 39% 57% Vs kN 3290 3290 3290 3290 3374 3374 3374 3374 % 18% 26% 30% 45% 18% 25% 29% 43% Vslab kN 10955 5477 3652 0 11235 5618 3745 0 % 60% 43% 33% 0% 59% 42% 32% 0% Vr kN 18309 12832 11006 7354 19067 13449 11576 7831 Vbase kN 20900 14850 12200 7140 21600 14970 12910 8150 Difference (%) -12.4% -13.6% -9.8% 3.0% -11.7% -10.2% -10.3% -3.9% Table 6.13: SCW4 wall-slab system shear capacity from simplified model. Axial Load 0% 10% Wall 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only M eas u re d θ (deg) 31.6 31.6 31.6 31.6 31.4 31.4 31.4 31.4 Aslab (mm 2) 1.1E+6 1.1E+6 1.1E+6 0 1.1E+6 1.1E+6 1.1E+6 0 Fslab (MPa) 1.500 1.500 1.500 0.000 1.500 1.500 1.500 0.000 Vc kN 955 955 955 955 987 987 987 987 % 12% 17% 20% 30% 12% 17% 20% 31% Vs kN 2209 2209 2209 2209 2230 2230 2230 2230 % 27% 39% 46% 70% 27% 39% 46% 69% Vslab kN 4971 2485 1657 0 5018 2509 1673 0 % 61% 44% 34% 0% 61% 44% 34% 0% Vr kN 8134 5649 4821 3164 8235 5726 4890 3217 Vbase kN 7250 6100 4850 3600 7620 5730 4470 3720 Difference (%) 12.2% -7.4% -0.6% -12.1% 8.1% -0.1% 9.4% -13.5% 140 Table 6.14: TSE8 wall-slab shear capacity from simplified model. Axial Load 0% 10% Wall 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only 0.5 Ratio 1.0 Ratio 1.5 Ratio Wall Only M eas u re d θ (deg) 35.6 35.6 35.6 35.6 35.0 35.0 35.0 35.0 Aslab (mm 2) 543E+3 543E+3 543E+3 0 543E+3 543E+3 543E+3 0 Fslab (MPa) 1.500 1.500 1.500 0.000 1.500 1.500 1.500 0.000 Vc kN 492 492 492 492 522 522 522 522 % 9% 10% 11% 13% 9% 11% 12% 14% Vs kN 3207 3207 3207 3207 3287 3287 3287 3287 % 56% 68% 73% 87% 56% 68% 73% 86% Vslab kN 2046 1023 682 0 2097 1049 699 0 % 36% 22% 16% 0% 36% 22% 16% 0% Vr kN 5745 4722 4381 3699 5907 4858 4509 3809 Vbase kN 5090 4710 4630 4350 5490 4930 4720 4500 Difference (%) 12.9% 0.3% -5.4% -15.0% 7.6% -1.5% -4.5% -15.3% Even though the simplified wall-slab shear model has significant assumptions, it generally predicts within 10% of the peak base shear from VecTor2. The most significant benefit from the simplified model is the effective slab size limit where the SCW4 prediction is better using a smaller slab size. The results are also very similar when using measured VecTor2 data since the simplified model has fairly balanced assumptions. Considering the General Method θ is constant and typically higher than the measured values, it reduces the steel and slab contribution. However, keeping the slab force constant at 1.5 MPa is generally slightly higher than the measured slab force. These two counteracting assumptions result in a total wall-slab system shear capacity which is very similar when using measured data and both methods are close the VecTor2 shear capacity. Throughout this analysis, the shear limit on the code-based Vr equations has been neglected where the code requires Vr,max to be limited to . This limit is to ensure the transverse reinforcement will yield prior to diagonal crushing (CAC, 2004). However, this simple limit is ignored since VecTor2 naturally considers diagonal crushing where the crushing stress can vary significantly. Although diagonal crushing does lead to wall failure, the wall stress at diagonal crushing varies between 0.06 to .18 fc’ for SCW3, 0.17 to 0.33 fc’ for SCW4 and 0.25 to 0.27 fc’ for TSE8. In most cases, the walls with tight slab spacing allows the walls to experience higher 141 stress than 0.25 fc’ before diagonal crushing. However, even though the simplified model can reasonably predict the capacity beyond this stress limit, the limit should still be applied for design since the wall-behaviour becomes less stable at these high stresses. Summarizing the equations and assumptions from previous sections, the final wall-slab system shear model to predict the peak shear stress at failure is presented below in Equations (13) to (17) and in Table 6.15: (13) Where: √ (14) (15) (16) And Vr is limited to: (17) Table 6.15: Summary of variables and values for simplified wall-slab shear model. Variable Units Value fc’ MPa Wall concrete compressive strength bw mm Wall web thickness dv mm Center-to-center flange distance β - General Method for walls with no slabs (eq. 11-11 A23.3-04) θ deg. General Method for walls with no slabs (eq. 11-12 A23.3-04) fy MPa Wall horizontal web steel yield stress Av mm 2 Wall horizontal web steel area s mm Wall horizontal web steel spacing fs,slab MPa Slab peak tension stress: 1.5 MPa Aslab mm 2 Cross-sectional area of slab perpendicular to wall: tslab x 1.5lw sslab mm Floor-to-floor slab height 142 6.5 FUTURE DEVELOPMENT OF SHEAR STRESS-STRAIN MODEL Due to the limited scope of this study, a complete model describing the stress-strain shear response of the wall-slab system cannot be defined. However, from this analysis we can determine how one would create such a model. The model would have to be piece-wise quadrilinear with stress-strain defined at the 4 important points: wall cracking, horizontal wall steel yield, slab cracking and ultimate crushing failure. As shown from the stress-strain curves in Section 6.2.4, the cracking point is does not change regardless of the number of slabs. However, the current cracking models can vary significantly and determining the actual cracking point can be difficult. Defining the point where the wall is “cracked” is challenging since most walls have slow cracking propagation. Yet once a good estimate of the cracking stress is provided, the strain at crack is easily determined from the shear modulus. The point where horizontal wall steel yields depends on many parameters including, but not limited to, the amount of web steel, axial load and the influence of slabs. Adebar and Gerin (2009) were able to create a rational stress-strain model for membrane elements which could be extended wall systems. As shown in Chapter 5, their model could be modified to account for the additional stiffness walls have due to non-uniform cracking and yielding. A modified model should also account for the amount of crack control. For instance, lightly-reinforced walls have poor crack control and hence have more concentrated local yield, increasing the global stiffness relative to a membrane analysis. Accounting for the slab effects on the wall further complicates the model. First, the slabs can reduce the loads on the wall steel, where the size, spacing and geometry of the slabs affect the steel loading. Second, the slabs also increase the system stiffness where cracking of the slab itself can significantly reduce the wall stiffness. Slab cracking can also have a greater influence than wall steel yielding. Due to these variables, a single membrane shear model may be too simple to account for the variation in strains and cracking within the wall. A strut-and-tie model could be effective at modelling the wall behaviour, with discrete slab layers defining the local slab stress-strain response. The model could push the wall in increments checking for slab cracking or steel yielding to reduce the stiffness. However, the strut and tie model must be able 143 to account for the variation of the strut angle during a pushover as observed in Section 6.2.2. The strut does not necessarily intercept convenient strut and tie nodes and intercept the flanges as the wall-slab interface. The simple model developed to predict the peak shear capacity is a good foundation for a more robust model. A model must be able to capture the variation in theta, beta and the effective slab force. Much like Gerin and Adebar’s shear model, the strain at peak stress could combine the strain states separately of the horizontal, vertical and shear components. These components would combine the effect of moment and axial load, shear demand with additional horizontal stiffness from slabs. However, the assumption that the peak shear is limited by steel yielding does not always hold true and other failure mechanisms would have to be considered. Finally the model would have to account for the decrease in wall ductility with more slabs. As the shear capacity increases, so does the concrete contribution which pushes the wall towards a more brittle response. 144 Conclusion Chapter 7 - The design of reinforced concrete shear walls for shear is complex and challenging. Designers often oversimplify the wall system as an isolated wall in order to efficiently model the wall. However, isolated walls do not reflect the realistic response of a wall in a building. Furthermore, the current modelling techniques of the isolated wall models do not necessarily represent the realistic response of an isolated wall. Many topics were discussed within this study, yet four main results standout. The first few chapters focused on the development of finite element slab models which reasonably capture the 3D slab effects in a 2D analysis. The more powerful pseudo 3D slab model can capture the complex interaction between the wall and slab, yet it can also capture the variations in stress and strains within the slab. Although there are limitations to the model, it is fairly robust and should be easily adapted to other finite element programs. A less powerful but easier to implement 2D slab layer model is also developed to extend this analysis to walls with multiple slabs. Considering most analysis is primarily concerned with just the wall response, the 2D slab layer model is appropriate for most applications. Even though these models were extensively tested analytically they still require experimental verification. However, experimental verification would be challenging since the tests must be specifically designed fail in shear with floor slabs. Considering that much higher shear capacities are possible, it is more likely that some other non-shear failure mechanism will occur before the slabs fail. Of course these models still require more parametric testing to refine the models and to determine the model limits. The second key result of this study is demonstrating that isolated walls behave differently than some typical shear models, including some nonlinear shear models. The assumption that walls have brittle shear failure is not necessarily true, especially for lightly-reinforced walls. This study found that walls are stiffer than membrane specimens seeing as walls are not uniformly- loaded and have strain variation. Increased shear stiffness does affect the shear demand, although flexure almost entirely governs the displacement demand for typical walls. Nevertheless, the study also found the code equations from the General Method from A23.3-04 are fairly accurate in predicting the wall shear capacity. 145 The third key result is quantifying the additional wall shear capacity from slabs; which was shown to be up to 3 times the nominal wall capacity for lightly-reinforced walls. However, the slab benefits are not as significant for heavily-reinforced walls. In addition to increased system shear capacity, the slabs help reduce extreme local cracking and the wall shear behaviour becomes more well-behaved. With increased diagonal tension capacity, the wall-slab system is pushed to a concrete-controlled mechanism which can lead to brittle failure. Further analysis is required to determine if the increased stiffness and shear capacity also increases the shear demand in NLTHA. Although the walls can gain significant strength they also lose some energy-dissipating mechanisms. The final key result is the development of simple shear models to predict the shear capacity of wall-slab systems as well as the shear stress-strain response of isolated wall systems. The wall- slab system shear model is developed by adding a Vslab component to the code sectional shear models. This simple model can reasonably predict the peak shear capacity of the system, typically within 10% of the finite element response. Modelling the stress-strain response of isolated wall systems was based on modifying the trilinear model developed by Gerin and Adebar (2009). Their model was modified by reducing the yield strain with a local-to-average strain factor, increasing the cracked stiffness. This factor accounted for the non-uniform strain within a wall and was shown to reasonably match the VecTor2 wall shear stress-strain response. Finally some guidelines were provided for potential future development of a quadrilinear wall- slab system shear model. In design, simplifications are often made in order to analyze complex systems more efficiently such as modelling a wall as an isolated member. Even though this simplifies analysis, it does not necessarily reflect the behaviour of the wall system. This study has shown there can be added benefits to considering the system as a whole. The models and methods developed could be used as an extra tool to potentially help designers satisfy their design requirements. If the shear demand is too great for a given wall, possibly from seismic upgrading or dynamic shear magnification, the designer could use these tools to analyze their wall-slab system. Analyzing the wall as a wall-slab system could increase the wall capacity to meet the demands. In conclusion, the overall result from this thesis is a better understanding of the nonlinear shear response of concrete shear walls. 146 References ACI. (2008). Building Code Requirements for Structural Concrete (ACI 318-08) and Commentary. Farmington Hills, Mich.: American Concrete Institute (ACI). Bentz, E. C., & Collins, M. P. (2006). Development of the 2004 Canadian Standards Association (CSA) A23.3 Shear Provisions for Reinforced Concrete. Canadian Journal of Civil Engineering, 33(5), 521-534. doi:10.1139/L06-005 Beyer, K., Dazio, A., & Priestley, M. J. N. (2011). Shear Deformations of Slender Reinforced Concrete Walls under Seismic Loading. ACI Structural Journal, 108(2), 167-177. Birely, A., Lowes, L. N., Lehman, D. E., Marley, K., Hart, C., & Kuchma, D. (2010). Investigation of the seismic response of slender planar concrete walls. Proceedings of the 9th U.S. National and 10th Canadian Conference on Earthquake Engineering. Toronto, ON. Bohl, A. G. (2006). Plastic hinge length in high-rise concrete shear walls. University of British Columbia. Boivin, Y., & Paultre, P. (2010). Seismic performance of a 12-storey ductile concrete shear wall system designed according to the 2005 National building code of Canada and the 2004 Canadian Standard Association standard A23.3. Canadian Journal of Civil Engineering, 37(1), 1-16. doi:10.1139/L09-115 CAC. (2004). Explanatory Notes on CSA Standard A23.3-04. Ottawa, Ont.: Cement Association of Canada (CAC). CSA. (2004). Design of Concrete Structures (A23.3-04). Rexdal, Ont.: Canadian Standard Association (CSA). Fischnger, M., Isaković, T., & Kante, P. (2006). Shaking table response of a thin h-shaped coupled wall. Proceedings of the 8th U.S. National Conference on Earthquake Engineering, (890). Gerin, M. (2003). Reverse-Cyclic Shear In Reinforced Concrete Elements. University of British Columbia. Gerin, M., & Adebar, P. (2004). Accounting for shear in seismic analysis of concrete structures. 13th World Conference on Earthquake Engineering., Vancouver, BC, CD Rom Paper (Vol. 1747, p. 13). 147 Gerin, M., & Adebar, P. (2009). Simple Rational Model for Reinforced Concrete Subjected to Seismic Shear. Journal of Structural Engineering, 135(7), 753. Retrieved from http://link.aip.org/link/?JSENDH/135/753/1 Ghorbani-Renani, I., Velev, N., Tremblay, R., Palermo, D., Massicotte, B., & Leger, P. (2009). Modeling and Testing Influence of Scaling Effects on Inelastic Response of Shear Walls. ACI Structural Journal, 106(3), 358-367. Retrieved from http://www.concrete.org/PUBS/JOURNALS/OLJDetails.asp?Home=SJ&ID=56500 Ghorbanirenani, I., Tremblay, R., El-Sokkary, H., Galal, K., Léger, P., & Leclerc, M. (2010). Shake table tests and repair of ductile slender reinforced concrete shear walls. Proceedings of the 9th U.S. National and 10th Canadian Conference on Earthquake Engineering. Toronto, ON. PEER. (2010). Modeling and Acceptance Criteria for Seismic Design and Analysis of Tall Buildings: Task 7 (PEER/ATC-72-1). October (Vol. 111). Redwood City, CA. Palermo, D., & Vecchio, F. (2002). Behavior of three-dimensional reinforced concrete shear walls. ACI Structural Journal, 99(1), 81–89. American Concrete Institute, 38800 International Way P. O. Box 9094, Farmington Hills, MI, 48333, USA. Palermo, D., & Vecchio, F. J. (2007). Simulation of Cyclically Loaded Concrete Structures Based on the Finite-Element Method. Journal of Structural Engineering, 133(5), 728. doi:10.1061/(ASCE)0733-9445(2007)133:5(728) Panagiotou, Marios. (2008). Seismic Design, Testing And Analysis Of Reinforced Concrete Wall Buildings. University of California, San Diego. Retrieved from http://gradworks.umi.com/33/07/3307372.html Panneton, M., Léger, P., & Tremblay, R. (2006). Inelastic analysis of a reinforced concrete shear wall building according to the National Building Code of Canada 2005. Canadian Journal of Civil Engineering, 33(7), 854-871. doi:10.1139/L06-026 Restrepo, J. I., Conte, J. P., & Panagiotou, M. (2010). The Design and Shake Table Testing of a Full-scale 7-storey Reinforced Concrete Cantilever Wall. 2010 NZSEE Conference. Retrieved from http://db.nzsee.org.nz/2010/Paper03.pdf Rutenberg, A., & Nsieri, E. (2006). The Seismic Shear Demand in Ductile Cantilever Wall Systems and the EC8 Provisions. Bulletin of Earthquake Engineering, 4(1), 1-21. doi:10.1007/s10518-005-5407-9 Stevens, N. J. (1987). Analytical Modelling of Reinforced Concrete Subjected To Monotonic and Reversed Loadings. University of Toronto. 148 Stevens, N. J., Uzumeri, S. M., & Collins, M. P. (1991). Reinforced Concrete Subjected to Reverse Cyclic Shear - Experiments and Constitutive Model. ACI Structural Journal, 88(2), 135-146. Thomsen, J. H., & Wallace, J. W. (2004). Displacement-Based Design of Slender Reinforced Concrete Structural Walls—Experimental Verification. Journal of Structural Engineering, 130(4), 618. doi:10.1061/(ASCE)0733-9445(2004)130:4(618) Tremblay, R., Ghorbanirenani, I., Velev, N., Léger, P., Leclerc, M., Koboevic, S., Bouaanani, N., et al. (2008). Seismic response of multi-storey reinforced concrete walls subjected to eastern north america high frequency ground motions. 14 th World Conference on Earthquake Engineering: Innovation Practice Safety. Beijing, China. Retrieved from http://www.iitk.ac.in/nicee/wcee/article/14_05-01-0526.PDF Vecchio, F., & Emara, M. B. (1992). Shear deformations in reinforced concrete frames. ACI Structural Journal, 89(1), 46–56. ACI. Retrieved from http://www.civ.utoronto.ca/vector/journal_publications/jp16.pdf Vecchio, F.J., & Collins, M. P. (1986). The Modified Compression-Field Theory For Reinforced Concrete Elements Subjected To Shear. ACI Journal, 83(2), 219–231. Retrieved from http://www.civ.utoronto.ca/vector/journal_publications/jp2.pdf Vecchio, F.J., Lai, D., Shim, W., & Ng, J. (2001). Disturbed stress field model for reinforced concrete: validation. Journal of Structural Engineering, 127(4), 350. Retrieved from http://www.civ.utoronto.ca/vector/journal_publications/jp36.pdf Wong, P. S., & Vecchio, F. J. (2002). VecTor2 & Formworks User’s Manual. Yathon, J. S. (2011). Seismic shear demand in reinforced concrete cantilever walls. University of British Columbia. 149 Appendices List of appendices: A.1 Single-Stage Custom Postprocessors A.2 Multi-Stage Custom Postprocessors A.3 Results from Wall Only Test A.4 Results from the Complete Analysis Suite 150 A.1 SINGLE-STAGE CUSTOM POSTPROCESSORS The single stage processor performs and in-depth analysis of a single stage of a VecTor2 pushover test. The program is developed for Excel® 2010. By providing the excel program wall geometry including node numbering and element location (see attached), the excel program performs a variety analysis. The single-stage processor was primarily used to explore the complex wall-slab behaviour in the initial chapters. A full list of features is provided below: Deflection Summary: Compares wall displacement over wall height directly from node deflections from VecTor2 versus calculated deflection from strains (see attached) Layer Strains: Obtains the VecTor2 element strain data and arranges data in horizontal element layers according to the input geometry. Averages for horizontal, vertical and shear strains as well as curvature are provided for each layer. Layer Stresses: Same analysis as layer strains except for element stresses. Horizontal Deformation: Using the layer curvature, the program calculates the wall flexural deformation based on the moment-area method. Using the average shear strains, the program calculates the shear deformation based on the shear strain multiplied by the layer height. Layer Summary: Summarizes the strain and stress distribution profiles over the height of the wall (see attached). Moment Curvature: From the calculated curvatures and based on the input moment demand profile, a simple moment-curvature response is calculated. Wall Damage Pattern: Based on the provided wall geometry, the program obtains the crack information, yield and crushing state for each element and displays the results graphically (see attached). Slab Damage Pattern: Similar to wall damage pattern, except the damage information is obtained for the slab. 151 A.1.1 Model Geometry Input Page (Raw data and calculation cells are omitted due to size of spreadsheet) Figure A1.1: Representative image of model geometry and input page. Name: SCW3 - Wall Only Wall Element Pattern Reinforcement #1 (deg) : 0 Reinforcement #2 (deg) : 90 No. Nodes: 874 Point Load: 27000 Point1 1 27000 kN 27000 mm Column No. Elements: 810 Left Flange 1500 Point2 0 0 kN 80100 mm Dist to Node 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000 8500 No. Layers 45 Web 6000 Total 27000 kN Layer 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 No. Supports 19 Right Flange 1500 Control: Displacement 45 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 Flange Width 900 Web Width 300 44 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 Pushover Analysis 43 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 42 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 File Name: SCW3 - Wall Only 41 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 Folder Name: 40 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 LoadID: Mono 39 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 Load Steps 26 38 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 37 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 Layer height Floor Height Layer Node 36 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 left Web Left Web Right Right mm Left Right 35 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 0 1 4 16 19 0 0 0 0 0 1 19 34 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 1 39 42 54 57 600 1 1 4200 7 267 285 33 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 2 77 80 92 95 1200 2 2 8400 14 533 551 32 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 3 115 118 130 133 1800 3 3 13800 23 856 874 31 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 4 153 156 168 171 2400 4 4 18000 30 590 608 30 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 5 191 194 206 209 3000 5 5 22200 37 324 342 29 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 6 229 232 244 247 3600 6 6 26400 44 58 76 28 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 7 267 270 282 285 4200 7 7 13800 23 856 874 27 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 8 305 308 320 323 4800 8 8 13800 23 856 874 26 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 9 343 346 358 361 5400 9 9 13800 23 856 874 25 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 10 381 384 396 399 6000 10 10 13800 23 856 874 24 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 11 419 422 434 437 6600 11 11 13800 23 856 874 23 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 12 457 460 472 475 7200 12 12 13800 23 856 874 22 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 13 495 498 510 513 7800 13 13 13800 23 856 874 21 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 14 533 536 548 551 8400 14 14 13800 23 856 874 20 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 15 571 574 586 589 9000 15 15 13800 23 856 874 19 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 16 609 612 624 627 9600 16 18 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 17 647 650 662 665 10200 17 17 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 18 685 688 700 703 10800 18 16 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 19 723 726 738 741 11400 19 15 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 20 761 764 776 779 12000 20 14 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 21 799 802 814 817 12600 21 13 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 22 837 840 852 855 13200 22 12 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 23 856 859 871 874 13800 23 11 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 24 818 821 833 836 14400 24 10 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 25 780 783 795 798 15000 25 9 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 26 742 745 757 760 15600 26 8 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 27 704 707 719 722 16200 27 7 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 28 666 669 681 684 16800 28 6 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 29 628 631 643 646 17400 29 5 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 30 590 593 605 608 18000 30 4 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 31 552 555 567 570 18600 31 3 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 32 514 517 529 532 19200 32 2 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 33 476 479 491 494 19800 33 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Node LOAD CASE FOR ANALYSIS 21 C:\Users\Steph n\Word Documents\MASc\Thesis\Vector\Model Verification\SCW3 - Wall Only\Load Full Data Set Update Load Case152 A.1.2 Deflection Summary (Raw data and calculation cells are omitted due to size of spreadsheet) Figure A1.2: Representative image of deflection summary page. Date Modified: 13-Jun-11 Title: SCW3 - Wall Only Summary Load Case Factor Displacement Check: OK Δx top (mm) Lateral 100 Strain Check: OK Vector 98.37 Axial 1 Stress Check: OK Calculated 84.94 Displacements Offset: 1761 Calculations Floor Height Node Node Δx Δy Node Node Δx Δy Δx total Δx flexural Δx shear shear/total mm (check) mm mm (check) mm mm mm mm mm % Left Right 0 0 1 1 0 0 19 19 0 0 0.00 0.00 0.00 0.0% 0.0% 0.0% 1 4200 267 267 2.427 4.236 285 285 8.889 -4.959 3.57 1.49 2.08 58.3% -47.0% 59.9% 2 8400 533 533 10.307 8.538 551 551 26.794 -7.866 14.74 5.31 9.44 64.0% -43.0% 45.0% 3 13800 856 856 31.645 13.436 874 874 52.189 -9.614 36.33 13.24 23.09 63.6% -14.8% 30.4% 4 18000 590 590 53.889 15.239 608 608 69.568 -10.124 53.60 21.60 32.00 59.7% 0.5% 23.0% 5 22200 324 324 75.583 15.445 342 342 84.257 -10.353 69.64 31.53 38.11 54.7% 7.9% 17.4% 6 26400 58 58 96.097 15.729 76 76 96.765 -10.452 83.30 42.48 40.82 49.0% 13.3% 13.9% 7 13800 856 856 31.645 13.436 874 874 52.189 -9.614 36.33 13.24 23.09 63.6% -14.8% 30.4% 8 13800 856 856 31.645 13.436 874 874 52.189 -9.614 36.33 13.24 23.09 63.6% -14.8% 30.4% 9 13800 856 856 31.645 13.436 874 874 52.189 -9.614 36.33 13.24 23.09 63.6% -14.8% 30.4% 10 13800 856 856 31.645 13.436 874 874 52.189 -9.614 36.33 13.24 23.09 63.6% -14.8% 30.4% 11 13800 856 856 31.645 13.436 874 874 52.189 -9.614 36.33 13.24 23.09 63.6% -14.8% 30.4% 12 13800 856 856 31.645 13.436 874 874 52.189 -9.614 36.33 13.24 23.09 63.6% -14.8% 30.4% 13 13800 856 856 31.645 13.436 874 874 52.189 -9.614 36.33 13.24 23.09 63.6% -14.8% 30.4% 8621 8621 Layer Node Height Δx Left Graph Node Height Δx Right Graph 0 1 0 0 100 19 0 0 98.371 1 39 600 0.075 96.097 57 600 0.47 96.765 2 77 1200 0.235 93.213 95 1200 1.202 95.111 3 115 1800 0.483 90.289 133 1800 2.242 93.401 4 153 2400 0.821 87.369 171 2400 3.551 91.647 5 191 3000 1.255 84.448 209 3000 5.108 89.856 6 229 3600 1.789 81.514 247 3600 6.893 88.028 7 267 4200 2.427 78.561 285 4200 8.889 86.163 8 305 4800 3.176 75.583 323 4800 11.076 84.257 9 343 5400 4.042 72.578 361 5400 13.429 82.308 10 381 6000 5.029 69.546 399 6000 15.92 80.313 11 419 6600 6.144 66.481 437 6600 18.525 78.269 12 457 7200 7.39 63.379 475 7200 21.219 76.174 13 495 7800 8.776 60.24 513 7800 23.981 74.026 14 533 8400 10.307 57.075 551 8400 26.794 71.824 15 571 9000 11.999 53.889 589 9000 29.64 69.568 16 609 9600 13.864 50.674 627 9600 32.506 67.256 17 647 10200 15.904 47.44 665 10200 35.378 64.889 18 685 10800 18.123 44.212 703 10800 38.245 62.465 19 723 11400 20.509 41.005 741 11400 41.097 59.983 20 761 12000 23.074 37.832 779 12000 43.926 57.443 21 799 12600 25.809 34.705 817 12600 46.723 54.844 22 837 13200 28.676 31.645 855 13200 49.48 52.189 23 856 13800 31.645 28.676 874 13800 52.189 49.48 Left Side Right Side Left Side Data Right Side Data % Error 24000 27000 30000 Comparison of Deformations Δx Left Δxflexural Δxtotal Δx Right 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 He ig ht (m m) Horzontal Deformation (mm) Comparison of Deformations Δx Left Δxflexural Δxtotal Δx Right153 A.1.3 Wall Damage Sheet Input – Part A (Raw data and calculation cells are omitted due to size of spreadsheet) Figure A1.3: Representative image of input element numbering and location for damage calculations. Wall Element Pattern VITAL SIGNS Offset: 470 CRACK CONDITIONSOffset: 1288 Wall Element Pattern Vital Signs Strains Crack Conditions Strains Element Cracked? Crack Ratio of Ratio of Element Crack Angle Layer Column Width f'c εcu Reinf 1 Reinf 2 No. (deg) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 1 1 0.09 0 0 0 0.42 1 178.7 0 -0.06808 45 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 2 2 1 0.07 0 0 0 0.36 2 178.4 0 -0.0838 44 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 3 3 1 0.06 0 0 0 0.3 3 178.1 0 -0.09952 43 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 4 4 1 0.2 0 0 0 0.26 4 175.7 0 -0.22557 42 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 5 5 1 0.17 0 0 0 0.22 5 175.5 0 -0.23611 41 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 6 6 1 0.14 0 0 0 0.18 6 175.1 0 -0.25719 40 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 7 7 1 0.11 0.006 0.006 0 0.14 7 173.9 0 -0.32061 39 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 8 8 1 0.08 0.004 0.011 0 0.1 8 169.2 0 -0.57228 38 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 9 9 1 0.06 0.012 0.013 0.01 0.06 9 161.5 0 -1.00379 37 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 10 10 1 0.03 0.02 0.021 0.01 0.03 10 148.4 0 -1.84561 36 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 11 11 1 0.02 0.052 0.043 0.01 -0.01 11 129.4 0 -3.65226 35 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 12 12 1 0.02 0.099 0.076 0.01 -0.04 12 120.5 0 -5.09299 34 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 13 13 1 0.03 0.138 0.104 0.02 -0.06 13 120.6 0 -5.07272 33 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 14 14 1 0.02 0.156 0.118 0.01 -0.07 14 123.2 0 -4.58448 32 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 15 15 1 0.15 0.201 0.152 0.07 0.01 15 131.6 0 -3.37898 31 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 16 16 1 0.06 0.101 0.083 -0.01 -0.04 16 135 0 -3 30 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 17 17 0 0 0.395 0.275 0.01 -0.28 19 179.4 0 -0.03142 29 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 18 18 0 0 0.689 0.47 0.04 -0.55 20 178.9 0 -0.0576 28 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 19 19 1 0.09 0.001 0.001 0 0.42 21 178.5 0 -0.07856 27 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 20 20 1 0.07 0 0 0 0.36 22 173.3 0 -0.35242 26 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 21 21 1 0.06 0 0 0 0.3 23 173.6 0 -0.3365 25 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 22 22 1 0.19 0 0 0 0.25 24 174.1 0 -0.31002 24 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 23 23 1 0.16 0 0 0 0.21 25 170.4 0 -0.50741 23 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 24 24 1 0.13 0 0 0 0.17 26 162.2 0 -0.96319 22 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 25 25 1 0.1 0 0 0 0.12 27 150.7 0 -1.68352 21 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 26 26 1 0.07 0.012 0.013 0.01 0.09 28 133.5 0 -3.16134 20 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 27 27 1 0.05 0.016 0.018 0.01 0.05 29 120.9 0 -5.01263 19 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 28 28 1 0.04 0.034 0.031 0.02 0.01 30 118.7 0 -5.47961 18 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 29 29 1 0.03 0.077 0.061 0.02 -0.03 31 120 0 -5.19615 17 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 30 30 1 0.04 0.133 0.101 0.04 -0.05 32 124.9 0 -4.3004 16 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 31 31 1 0.05 0.176 0.133 0.04 -0.07 33 124.4 0 -4.38139 15 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 32 32 1 0.08 0.207 0.157 0.05 -0.06 37 179.8 0 -0.01047 14 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 33 33 1 0.39 0.368 0.279 0.31 0 38 179.4 0 -0.03142 13 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 34 34 0 0 0.112 0.093 -0.01 -0.06 39 178.8 0 -0.06284 12 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 35 35 0 0 0.392 0.296 0.03 -0.28 40 172.2 0 -0.41095 11 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 36 36 0 0 0.66 0.51 0.07 -0.49 41 172.3 0 -0.40562 10 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 37 37 1 0.09 0 0 0 0.42 42 172.3 0 -0.40562 9 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 38 38 1 0.07 0 0 0 0.36 43 165.5 0 -0.77585 8 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 39 39 1 0.06 0 0 0 0.3 44 155.3 0 -1.37985 7 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 40 40 1 0.19 0 0 0.01 0.25 45 142.3 0 -2.31866 6 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 41 41 1 0.16 0 0 0 0.21 46 125.7 0 -4.17494 5 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 42 42 1 0.12 0.001 0.001 0 0.16 47 117.6 0 -5.73847 4 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 43 43 1 0.09 0.006 0.006 0.01 0.12 48 117.5 0 -5.76295 3 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 44 44 1 0.08 0.016 0.017 0.02 0.08 49 120.1 0 -5.17527 2 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 45 45 1 0.06 0.025 0.025 0.03 0.04 50 123.3 0 -4.56706 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 % of yield154 A.1.4 Wall Damage Output – Part B (Raw data and calculation cells are omitted due to size of spreadsheet) Figure A1.4: Wall damage output indicating cracking and values of crushing/yielding. Wall Crack Pattern Compression Strain Horizontal Steel Yielding Wall Crack Pattern Critical Crack Width: 1 mm Compression Strain Horizontal Steel Yielding Cracked:: \ Thick Crack: Ԇ Note: Compression Strains are very small and treated as small positive values Layer Column Layer Column Layer Column 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 45 \ \ \ 45 0.23 0.19 0.18 0.32 0.13 0.05 0.01 0.00 0.01 0.00 0.00 0.00 0.01 0.03 0.19 0.19 0.02 0.00 45 0.21 0.22 0.18 0.10 0.08 0.02 0.01 0.03 0.05 0.30 0.03 0.02 0.01 0.01 0.03 0.03 0.01 0.00 44 \ \ \ \ \ 44 0.05 0.04 0.07 0.30 0.17 0.11 0.07 0.04 0.02 0.00 0.01 0.02 0.03 0.06 0.09 0.10 0.06 0.01 44 0.05 0.01 0.16 0.93 0.00 0.04 0.04 0.03 0.01 0.02 0.00 0.00 0.00 0.00 0.06 0.03 0.00 0.00 43 \ Ԇ \ \ \ 43 0.00 0.02 0.07 0.55 0.51 0.35 0.26 0.19 0.13 0.07 0.05 0.09 0.12 0.14 0.14 0.10 0.06 0.02 43 0.00 0.01 0.10 1.77 0.52 0.03 0.06 0.06 0.06 0.06 0.03 0.02 0.00 0.03 0.07 0.03 0.01 0.00 42 \ \ \ 42 0.00 0.00 0.01 0.57 0.52 0.37 0.28 0.22 0.15 0.09 0.06 0.08 0.11 0.12 0.11 0.09 0.06 0.03 42 0.00 0.00 0.00 1.90 1.52 0.04 0.05 0.06 0.06 0.05 0.04 0.02 0.00 0.03 0.04 0.02 0.01 0.00 41 \ Ԇ \ 41 0.00 0.00 0.01 0.43 0.65 0.40 0.29 0.24 0.18 0.13 0.10 0.09 0.10 0.11 0.10 0.08 0.06 0.03 41 0.00 0.00 0.00 1.63 2.44 0.38 0.03 0.04 0.04 0.04 0.03 0.02 0.00 0.01 0.02 0.02 0.01 0.00 40 \ \ \ 40 0.00 0.00 0.00 0.33 0.68 0.53 0.28 0.25 0.21 0.16 0.13 0.11 0.11 0.10 0.10 0.08 0.06 0.03 40 0.00 0.00 0.00 1.27 3.00 1.10 0.01 0.02 0.03 0.03 0.02 0.01 0.00 0.01 0.01 0.01 0.01 0.00 39 \ Ԇ \ \ \ 39 0.00 0.00 0.00 0.26 0.49 0.64 0.32 0.27 0.23 0.19 0.16 0.14 0.12 0.11 0.10 0.08 0.06 0.03 39 0.00 0.00 0.00 1.13 3.01 2.01 0.16 0.01 0.02 0.02 0.01 0.01 0.00 0.00 0.01 0.01 0.01 0.00 38 \ Ԇ \ Ԇ \ 38 0.00 0.00 0.00 0.19 0.43 0.82 0.43 0.28 0.25 0.21 0.18 0.16 0.14 0.12 0.11 0.08 0.06 0.03 38 0.00 0.00 0.00 1.00 2.59 2.98 0.66 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.01 0.01 0.01 0.00 37 \ \ \ \ \ 37 0.00 0.00 0.00 0.15 0.31 0.93 0.58 0.29 0.27 0.24 0.20 0.18 0.15 0.13 0.12 0.09 0.06 0.03 37 0.00 0.00 0.00 0.80 2.10 3.62 1.63 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.00 36 \ \ \ \ \ \ 36 0.00 0.00 0.01 0.17 0.25 0.71 0.82 0.30 0.28 0.26 0.22 0.19 0.17 0.15 0.12 0.09 0.06 0.04 36 0.00 0.00 0.00 0.70 1.60 3.79 2.94 0.02 0.01 0.01 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.00 35 \ \ \ Ԇ \ \ \ 35 0.00 0.00 0.00 0.19 0.19 0.52 1.48 0.47 0.32 0.27 0.23 0.21 0.18 0.16 0.14 0.10 0.07 0.04 35 0.00 0.00 0.00 0.76 0.97 3.71 3.90 0.62 0.01 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.01 34 \ \ Ԇ \ Ԇ Ԇ Ԇ 34 0.00 0.01 0.01 0.23 0.13 0.43 1.79 0.52 0.34 0.28 0.25 0.22 0.20 0.17 0.14 0.10 0.07 0.04 34 0.00 0.00 0.01 1.04 0.34 3.30 4.20 1.97 0.01 0.00 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 33 \ \ \ Ԇ Ԇ \ Ԇ 33 0.00 0.01 0.01 0.23 0.07 0.35 1.66 0.61 0.35 0.31 0.28 0.24 0.21 0.18 0.15 0.11 0.08 0.04 33 0.00 0.00 0.00 1.43 0.02 2.46 4.34 3.42 0.03 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 32 \ \ \ Ԇ \ \ Ԇ \ 32 0.01 0.01 0.01 0.24 0.09 0.26 0.99 1.19 0.45 0.32 0.29 0.25 0.22 0.19 0.16 0.12 0.08 0.05 32 0.00 0.00 0.01 1.78 0.01 1.55 4.40 4.23 0.57 0.02 0.02 0.02 0.02 0.01 0.01 0.02 0.01 0.01 31 \ \ \ \ \ Ԇ \ \ 31 0.00 0.01 0.02 0.32 0.09 0.23 0.51 1.91 0.50 0.34 0.29 0.26 0.23 0.20 0.17 0.12 0.09 0.05 31 0.00 0.01 0.01 2.10 0.01 0.85 4.08 4.48 1.82 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.01 30 \ \ \ Ԇ \ Ԇ \ \ 30 0.00 0.01 0.02 0.43 0.06 0.14 0.43 2.22 0.65 0.33 0.30 0.28 0.24 0.21 0.18 0.13 0.09 0.06 30 0.00 0.01 0.02 2.48 0.03 0.28 3.11 4.88 3.30 0.05 0.04 0.03 0.02 0.02 0.02 0.02 0.01 0.01 29 \ \ \ \ \ \ \ \ \ 29 0.01 0.02 0.02 0.44 0.15 0.10 0.28 1.58 1.12 0.47 0.31 0.28 0.25 0.22 0.18 0.13 0.10 0.06 29 0.00 0.01 0.02 2.64 0.37 0.03 1.70 5.44 4.00 0.74 0.04 0.03 0.02 0.02 0.02 0.02 0.01 0.01 28 \ \ \ \ Ԇ \ \ \ Ԇ 28 0.00 0.02 0.02 0.41 0.24 0.12 0.17 1.36 2.05 0.47 0.35 0.29 0.26 0.23 0.19 0.14 0.10 0.07 28 0.00 0.01 0.02 2.40 1.12 0.02 0.53 5.39 4.39 1.83 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.01 27 \ \ \ \ Ԇ \ \ Ԇ 27 0.00 0.02 0.03 0.41 0.27 0.13 0.13 0.00 2.43 0.53 0.34 0.30 0.27 0.23 0.20 0.15 0.11 0.07 27 0.00 0.02 0.03 2.15 1.86 0.02 0.03 4.53 4.69 3.02 0.05 0.03 0.03 0.02 0.02 0.02 0.01 0.01 26 \ \ \ \ Ԇ \ Ԇ \ \ 26 0.00 0.02 0.03 0.41 0.34 0.11 0.12 0.50 2.44 0.82 0.44 0.31 0.28 0.24 0.21 0.15 0.12 0.08 26 0.00 0.02 0.03 2.04 2.40 0.02 0.01 3.41 4.40 4.01 0.67 0.04 0.03 0.02 0.02 0.02 0.02 0.01 25 \ \ \ \ \ \ Ԇ \ Ԇ 25 0.00 0.02 0.03 0.40 0.40 0.11 0.13 0.35 1.53 1.17 0.47 0.35 0.28 0.24 0.21 0.16 0.12 0.09 25 0.00 0.02 0.03 2.07 2.78 0.01 0.02 2.30 4.44 4.31 1.56 0.03 0.02 0.02 0.01 0.02 0.02 0.01 24 \ \ \ \ \ Ԇ Ԇ \ Ԇ 24 0.01 0.03 0.03 0.40 0.52 0.11 0.14 0.28 0.55 2.31 0.42 0.35 0.28 0.25 0.22 0.16 0.13 0.09 24 0.00 0.02 0.03 2.15 3.08 0.02 0.02 1.26 4.71 4.39 2.27 0.03 0.02 0.01 0.01 0.02 0.02 0.01 23 \ \ \ Ԇ \ \ \ Ԇ Ԇ Ԇ 23 0.00 0.02 0.03 0.38 0.58 0.18 0.14 0.21 0.45 3.38 0.48 0.32 0.28 0.25 0.22 0.17 0.14 0.10 23 0.01 0.02 0.03 1.94 3.32 0.38 0.02 0.40 4.13 4.52 3.48 0.03 0.02 0.02 0.02 0.02 0.02 0.01 22 \ \ \ Ԇ Ԇ \ \ \ \ 22 0.00 0.02 0.02 0.37 0.48 0.26 0.14 0.13 0.36 2.66 0.59 0.33 0.27 0.25 0.22 0.17 0.14 0.12 22 0.01 0.02 0.02 1.78 3.14 1.04 0.02 0.03 2.78 4.93 4.56 0.06 0.04 0.03 0.02 0.02 0.02 0.02 21 \ \ \ Ԇ \ \ \ \ \ \ 21 0.01 0.03 0.03 0.26 0.52 0.30 0.17 0.16 0.28 1.95 1.84 0.33 0.26 0.24 0.22 0.18 0.15 0.13 21 0.01 0.02 0.02 1.73 2.66 1.74 0.01 0.01 1.53 5.20 4.97 0.50 0.04 0.03 0.03 0.02 0.02 0.02 20 \ \ \ Ԇ \ Ԇ \ \ \ Ԇ 20 0.00 0.02 0.03 0.24 0.44 0.40 0.13 0.17 0.24 1.08 2.74 0.35 0.26 0.23 0.22 0.18 0.16 0.14 20 0.00 0.02 0.02 1.43 2.27 2.44 0.03 0.03 0.54 5.06 5.16 1.27 0.03 0.03 0.03 0.02 0.02 0.02 19 \ \ \ Ԇ \ Ԇ \ \ \ Ԇ Ԇ 19 0.00 0.01 0.03 0.24 0.40 0.45 0.19 0.16 0.19 0.59 3.69 0.37 0.23 0.23 0.21 0.18 0.17 0.15 19 0.00 0.01 0.02 1.02 2.03 2.77 0.25 0.02 0.05 4.27 5.36 2.13 0.03 0.03 0.03 0.02 0.02 0.02 18 \ \ \ \ Ԇ \ Ԇ \ \ Ԇ 18 0.00 0.01 0.02 0.19 0.36 0.45 0.28 0.18 0.18 0.41 4.12 0.35 0.23 0.21 0.21 0.18 0.17 0.17 18 0.00 0.01 0.02 0.64 1.69 2.49 1.00 0.01 0.01 2.80 5.54 3.21 0.04 0.03 0.03 0.02 0.02 0.02 17 \ \ \ \ Ԇ \ Ԇ \ \ \ \ 17 0.00 0.01 0.02 0.14 0.29 0.43 0.31 0.19 0.21 0.33 3.63 0.60 0.25 0.20 0.20 0.18 0.18 0.18 17 0.00 0.01 0.01 0.41 1.13 2.07 1.80 0.02 0.02 1.47 5.38 4.25 0.26 0.02 0.02 0.02 0.02 0.03 16 \ \ \ \ \ \ \ \ \ \ Ԇ \ 16 0.00 0.00 0.02 0.12 0.21 0.37 0.43 0.19 0.22 0.30 2.55 1.17 0.27 0.20 0.19 0.18 0.19 0.19 16 0.00 0.00 0.01 0.29 0.64 1.65 2.20 0.11 0.03 0.51 5.03 4.72 0.82 0.02 0.02 0.02 0.03 0.03 15 \ \ \ \ \ \ Ԇ \ Ԇ Ԇ Ԇ 15 0.00 0.00 0.01 0.09 0.15 0.29 0.41 0.22 0.22 0.29 1.15 2.02 0.28 0.17 0.18 0.18 0.19 0.21 15 0.00 0.00 0.01 0.22 0.37 1.07 2.28 0.29 0.02 0.05 4.82 4.49 1.42 0.02 0.02 0.02 0.03 0.03 14 \ \ \ \ \ \ Ԇ \ Ԇ Ԇ \ 14 0.00 0.00 0.01 0.08 0.11 0.22 0.37 0.25 0.21 0.32 0.79 3.25 0.30 0.13 0.17 0.18 0.20 0.23 14 0.00 0.00 0.00 0.19 0.25 0.58 1.97 0.50 0.00 0.02 4.03 4.37 2.02 0.04 0.03 0.02 0.03 0.03 13 \ \ \ \ \ \ Ԇ \ \ Ԇ \ \ \ 13 0.00 0.00 0.00 0.06 0.09 0.16 0.30 0.27 0.21 0.32 0.53 3.44 0.33 0.15 0.16 0.17 0.21 0.24 13 0.00 0.00 0.00 0.16 0.20 0.34 1.24 0.71 0.09 0.02 2.58 4.68 2.49 0.28 0.02 0.02 0.03 0.03 12 \ \ \ \ \ \ \ \ \ \ \ \ \ 12 0.00 0.00 0.00 0.05 0.07 0.12 0.20 0.25 0.24 0.32 0.49 3.30 0.35 0.19 0.14 0.17 0.21 0.26 12 0.00 0.00 0.00 0.13 0.17 0.26 0.57 0.67 0.27 0.04 1.06 4.87 2.77 0.75 0.01 0.02 0.03 0.04 11 \ \ \ \ \ \ \ \ \ \ \ \ \ Ԇ 11 0.00 0.00 0.00 0.04 0.06 0.09 0.14 0.19 0.23 0.28 0.48 2.57 0.37 0.20 0.11 0.16 0.22 0.28 11 0.00 0.00 0.00 0.09 0.13 0.18 0.27 0.38 0.30 0.04 0.13 4.48 2.93 1.29 0.02 0.02 0.03 0.04 10 \ \ \ \ \ \ \ \ \ \ \ \ \ 10 0.00 0.00 0.00 0.04 0.05 0.07 0.10 0.13 0.17 0.22 0.37 1.55 0.44 0.27 0.07 0.15 0.23 0.30 10 0.00 0.00 0.00 0.06 0.09 0.13 0.16 0.19 0.17 0.00 0.03 3.55 2.81 1.76 0.02 0.02 0.03 0.04 9 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9 0.00 0.00 0.00 0.03 0.04 0.05 0.07 0.10 0.12 0.16 0.27 0.80 0.70 0.34 0.09 0.14 0.23 0.33 9 0.00 0.00 0.00 0.04 0.06 0.09 0.11 0.12 0.07 0.01 0.02 2.13 2.89 1.95 0.15 0.02 0.03 0.05 8 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8 0.00 0.00 0.00 0.02 0.03 0.04 0.05 0.07 0.09 0.13 0.26 0.47 0.79 0.41 0.15 0.12 0.23 0.35 8 0.00 0.00 0.00 0.03 0.04 0.06 0.08 0.09 0.06 0.01 0.02 0.71 2.98 1.81 0.41 0.02 0.03 0.05 7 \ \ \ \ \ \ \ \ \ \ \ \ Ԇ \ \ 7 0.00 0.00 0.00 0.01 0.02 0.03 0.04 0.06 0.08 0.11 0.21 0.30 0.60 0.48 0.24 0.11 0.24 0.37 7 0.00 0.00 0.00 0.02 0.02 0.04 0.06 0.07 0.07 0.00 0.06 0.14 2.32 1.62 0.62 0.01 0.03 0.05 6 \ \ \ \ \ \ \ \ \ \ \ \ Ԇ \ \ 6 0.00 0.00 0.00 0.01 0.01 0.02 0.03 0.04 0.06 0.09 0.17 0.25 0.43 0.51 0.34 0.10 0.25 0.39 6 0.00 0.00 0.00 0.01 0.01 0.02 0.04 0.06 0.06 0.02 0.05 0.08 1.19 1.52 0.82 0.01 0.03 0.06 5 \ \ \ \ \ \ \ \ \ \ \ \ \ Ԇ \ 5 0.00 0.00 0.00 0.00 0.01 0.01 0.02 0.03 0.05 0.08 0.13 0.21 0.30 0.43 0.39 0.10 0.25 0.41 5 0.00 0.00 0.00 0.01 0.01 0.01 0.03 0.04 0.05 0.04 0.04 0.10 0.38 1.24 0.88 0.00 0.03 0.06 4 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.02 0.04 0.06 0.11 0.17 0.21 0.35 0.39 0.10 0.26 0.43 4 0.00 0.00 0.00 0.01 0.01 0.01 0.02 0.03 0.04 0.04 0.05 0.08 0.13 0.68 0.82 0.00 0.03 0.06 3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.03 0.04 0.08 0.13 0.17 0.23 0.36 0.10 0.28 0.46 3 0.00 0.00 0.00 0.01 0.00 0.00 0.01 0.02 0.03 0.03 0.04 0.06 0.06 0.21 0.64 0.00 0.03 0.06 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.03 0.06 0.10 0.13 0.16 0.28 0.09 0.30 0.51 2 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.02 0.02 0.04 0.04 0.05 0.31 0.01 0.03 0.07 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.02 0.04 0.08 0.10 0.12 0.15 0.08 0.28 0.47 1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.02 0.01 0.07 0.01 0.01 0.04155 A.1.5 Strain and Stress Distribution Profiles Figure A1.5: Stress and strain profiles output. Title: SCW3 - Wall Only 0 5 10 15 20 25 30 35 40 45 0.0000 0.0500 0.1000 0.1500 0.2000 600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 7200 7800 8400 9000 9600 10200 10800 11400 12000 12600 13200 13800 14400 15000 15600 16200 16800 17400 18000 18600 19200 19800 20400 21000 21600 22200 22800 23400 24000 24600 25200 25800 26400 27000 La ye r Curvature in Layer (rad/km) He ig ht (m m) Curvature Distribution Over Height 0 5 10 15 20 25 30 35 40 45 -8.0000 -6.0000 -4.0000 -2.0000 0.0000 2.0000 4.0000 600 1800 3000 4200 5400 6600 7800 9000 10200 11400 12600 13800 15000 16200 17400 18600 19800 21000 22200 23400 24600 25800 27000 La ye r Average Vertical Stress in Layer (MPa) He ig ht (m m) Vertical Stress Distribution Over Height Without Axial Load 0 5 10 15 20 25 30 35 40 45 0.0000 0.5000 1.0000 1.5000 2.0000 2.5000 3.0000 600 1800 3000 4200 5400 6600 7800 9000 10200 11400 12600 13800 15000 16200 17400 18600 19800 21000 22200 23400 24600 25800 27000 La ye r Average Shear Strain (mm/m) He ig ht (m m) Average Shear Strain Distribution Over Height 0 5 10 15 20 25 30 35 40 45 0.0000 0.5000 1.0000 1.5000 2.0000 2.5000 3.0000 600 1800 3000 4200 5400 6600 7800 9000 10200 11400 12600 13800 15000 16200 17400 18600 19800 21000 22200 23400 24600 25800 27000 La ye r Average Shear Stress in Layer (MPa) He ig ht (m m) Shear Stress Distribution Over Height156 A.2 MULTI-STAGE CUSTOM POSTPROCESSOR The multi-stage processor is similar to the single stage program described previously except that it analyzes specific values over the entire pushover. The program is developed for Excel® 2010. By providing the program with the wall geometry and key load stages (see attached), it performs a series of analysis for each load stage and combines the results graphically. The multi-stage processor is the main tool to compare the wall responses in the analysis suite. A full list of features is provided below: Shear Calculations: For each load stage, the program obtains the shear strain for each element and arranges the data in the appropriate element pattern. The wall is divided into equal regions over its height and the average, peak and distributed strain values are obtained for each region. The total horizontal reaction is also obtained and converted to shear stress which corresponds with the measured shear strain. This process is repeated for each load stage and the data is compiled into a seamless shear stress-strain plot. Damage Calculations: The same analysis is performed as the shear calculations except for the measured concrete crushing, steel yielding and theta. Pushover curves: Plots the base shear-horizontal displacement curves to verify the data matches with VecTor2’s Augustus output (see attached). Damage Summary: From the input summary, 4 key load stages are selected which generally corresponds to when the wall yields, significant damage, concrete crushing and peak base shear. The measured data for these 4 load stages (shear strain, shear stress, crushing, yielding and theta) are combined into single graphs which provide a snapshot of the wall state at the 4 key load stages (see attached). Selected results from this analysis are provided in Section 5.4.2 and 6.3.2 (see Appendix A.3 and A.4) Summary Shear Stress-Strain: The shear strain distribution over the height and the shear stress-strain response for each region determined from the shear calculations is plotted together. The shear strain distribution for the 4 key load stages are plotted together to observe the change in distribution during the pushover (see attached). For shear stress- strain response, the middle third region is selected as the characteristic wall stress-strain response. These results are provided in Appendix A.3 and A.4 157 A.2.1 Geometry and Load Input Sheet Figure A2.1: Model geometry and input page. Version: 23/07/2011 Pushover Analysis Support Nodes Loads To Plot SCW3 A - Wall Only Loadstage 19 12 8 7 File Name: SCW3 A - Wall Only No. Nodes: 1273 Node X- RestraintY-Restraint Colour Black Blue Red Green Folder Name:C:\Users\Stephen\Word Documents\MASc\Thesis\Vector\Analysis Suite\SCW3 A - Wall Only\ No. Elements: 1188 1 1 1 LoadID: Mono No. Layers 66 2 1 1 Layer 1 2 3 4 Load Steps: 20 No. Supports 19 3 1 1 Colour Black Blue Red Green Data Section: Total bw (mm) 300 4 1 1 Section Check: ELEMENT TOTAL STRAINS dv(mm) 7500 5 1 1 Groups Section Offset: 83 Wall Height (mm) 27000 6 1 1 Small 11 2nd Section: Vital Wall Length (mm) 9000 7 1 1 Med 22 Section 2 Check: VITAL SIGNS f'c 50 8 1 1 Large 33 Section 2 Offset: 1279 Elements/Layer 18 9 1 1 3nd Section: Stresses Layer Height 430 10 1 1 Wall Web Section 3 Check: ELEMENT STRESSES 11 1 1 Length 9000 6000 Section 3 Offset: 2473 12 1 1 1st Element Offset 0 3 13 1 1 No. Elements 18 12 14 1 1 15 1 1 16 1 1 17 1 1 18 1 1 19 1 1 Import Data158 A.2.2 Pushover Curves Figure A2.2: Pushover curves from the VecTor2 data to verify with Augustus postprocessor. Pushover Curve Control: Displacement Title: Pushover Curve LoadSteps 20 Xtitle: Top Displacement (mm)Ytitl : Base Shear (kN) Load Offset 20 Loadstep Load FactorTop Displ. Base Shear 1 0 0 0 2 5 5 1408.8 3 10 10 2618.8 4 15 15 3445.9 5 20 20 4030.9 6 25 25 4510.6 7 30 30 4920.5 8 35 35 5218.2 9 40 40 5579.7 10 45 45 5919.7 11 50 50 6267.2 12 55 55 6543.5 13 60 60 6748 14 65 65 7005.1 15 70 70 7140 16 75 75 7442.8 17 80 80 7636.1 18 85 85 7750.4 19 90 90 7874.5 20 95 95 7074.3 21 0 0 0 22 0 0 0 23 0 0 0 24 0 0 0 25 0 0 0 26 0 0 0 27 0 0 0 28 0 0 0 29 0 0 0 30 0 0 0 31 0 0 0 32 0 0 0 33 0 0 0 34 0 0 0 35 0 0 0 36 0 0 0 37 0 0 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0 20 40 60 80 100 B as e Sh ea r (k N ) Top Displacement (mm) Pushover Curve 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0 20 40 60 80 100 B as e Sh ea r (k N ) Top Displacement (mm) Pushover Curve159 A.2.3 Damage Summary (Raw data and calculation cells are omitted due to size of spreadsheet) Figure A2.3: Representative image of damage summary sheets for selected load stages. Indication of Wall Damage Title Wall Damage Indicator Damage Limit 0 1 Primary X axis Damage RatioY Axis h/Hw 1 1 SecondaryX axis Shear Strain (mm/m)Y Axis h/Hw Yield Ratio Compression Ratio Shear Mean Shear Max Yld% εcm γmean γmax Loadstage 19 12 8 7 Loadstage 19 12 8 7 Loadstage 19 12 8 7 V (MPa) 3.50 2.91 2.32 2.19 V (MPa) 3.50 2.91 2.32 2.19 V (MPa) 3.50 2.91 2.32 2.19 Vratio 0.07 0.06 0.05 0.04 Vratio 0.07 0.06 0.05 0.04 Vratio 0.07 0.06 0.05 0.04 GRAPH Yld% - 3.50 MPa (0.07)Yld% - 2.91 MPa (0.06)Yld% - 2.32 MPa (0.05)Yld% - 2.19 MPa (0.04) GRAPH εcm - 3.50 MPa (0.07)εcm - 2.91 MPa (0.06)εcm - 2.32 MPa (0.05)εcm - 2.19 MPa (0.04) GRAPH γmean - 3.50 MPa (0.07)γmean - 2.91 MPa (0.06)γmean - 2.32 MPa (0.05)γmean - 2.19 MPa (0.04) Name Load Stage 19Load Stage 12Load Stage 8Load Stage 7 Name Load Stage 19Load Stage 12Load Stage 8Load Stage 7 Name Load Stage 19Load Stage 12Load Stage 8Load Stage 7 Height\LoadBlack Blue Red Green Height\LoadBlack Blue Red Green Height\LoadBlack Blue Red Green Rows 13 Rows 13 Rows 13 0 0 0 0 0 Rows 1 0 0 1.2373 0.7845 0.3055 0.1645 1 0 0 0.4268 0.2895 0.1760 0.1456 1 0 0 0.7509 0.4946 0.3131 0.2632 1 2 4500 0.16667 1.2373 0.7845 0.3055 0.1645 2 4500 0.16667 0.4268 0.2895 0.1760 0.1456 2 4500 0.16667 0.7509 0.4946 0.3131 0.2632 2 3 4500 0.16667 4.1073 2.9809 1.4564 0.7482 3 4500 0.16667 1.5128 0.5895 0.2388 0.1963 3 4500 0.16667 2.1862 1.2247 0.6580 0.4842 3 4 9000 0.33333 4.1073 2.9809 1.4564 0.7482 4 9000 0.33333 1.5128 0.5895 0.2388 0.1963 4 9000 0.33333 2.1862 1.2247 0.6580 0.4842 4 5 9000 0.33333 4.9055 4.2718 1.8945 0.8009 5 9000 0.33333 2.7452 1.1403 0.2753 0.2015 5 9000 0.33333 2.9611 1.4048 0.6663 0.4736 5 6 13500 0.5 4.9055 4.2718 1.8945 0.8009 6 13500 0.5 2.7452 1.1403 0.2753 0.2015 6 13500 0.5 2.9611 1.4048 0.6663 0.4736 6 7 13500 0.5 4.5945 3.1855 1.7509 0.7155 7 13500 0.5 3.4487 0.6403 0.2732 0.1960 7 13500 0.5 2.5897 0.9943 0.4620 0.3262 7 8 18000 0.66667 4.5945 3.1855 1.7509 0.7155 8 18000 0.66667 3.4487 0.6403 0.2732 0.1960 8 18000 0.66667 2.5897 0.9943 0.4620 0.3262 8 9 18000 0.66667 3.4827 2.3564 0.8609 0.5809 9 18000 0.66667 1.4788 0.4818 0.2326 0.1836 9 18000 0.66667 1.2844 0.5982 0.2839 0.2050 9 10 22500 0.83333 3.4827 2.3564 0.8609 0.5809 10 22500 0.83333 1.4788 0.4818 0.2326 0.1836 10 22500 0.83333 1.2844 0.5982 0.2839 0.2050 10 11 22500 0.83333 1.4827 1.1527 0.2700 0.1509 11 22500 0.83333 0.4338 0.2617 0.1143 0.0805 11 22500 0.83333 0.4511 0.2432 0.1247 0.1048 11 12 27000 1 1.4827 1.1527 0.2700 0.1509 12 27000 1 0.4338 0.2617 0.1143 0.0805 12 27000 1 0.4511 0.2432 0.1247 0.1048 12 13 27000 1 0.0000 0.0000 0.0000 0.0000 13 27000 1 0.0000 0.0000 0.0000 0.0000 13 27000 1 0.0000 0.0000 0.0000 0.0000 13 14 0 0 0.0000 0.0000 0.0000 0.0000 14 0 0 0.0000 0.0000 0.0000 0.0000 14 15 0 0 0.0000 0.0000 0.0000 0.0000 15 0 0 0.0000 0.0000 0.0000 0.0000 15 16 0 0 0.0000 0.0000 0.0000 0.0000 16 0 0 0.0000 0.0000 0.0000 0.0000 16 17 0 0 0.0000 0.0000 0.0000 0.0000 17 0 0 0.0000 0.0000 0.0000 0.0000 17 18 0 0 0.0000 0.0000 0.0000 0.0000 18 0 0 0.0000 0.0000 0.0000 0.0000 18 19 0 0 0.0000 0.0000 0.0000 0.0000 19 0 0 0.0000 0.0000 0.0000 0.0000 19 20 0 0 0.0000 0.0000 0.0000 0.0000 20 0 0 0.0000 0.0000 0.0000 0.0000 20 21 0 0 0.0000 0.0000 0.0000 0.0000 21 0 0 0.0000 0.0000 0.0000 0.0000 21 22 0 0 0.0000 0.0000 0.0000 0.0000 22 0 0 0.0000 0.0000 0.0000 0.0000 22 23 0 0 0.0000 0.0000 0.0000 0.0000 23 0 0 0.0000 0.0000 0.0000 0.0000 23 24 0 0 0.0000 0.0000 0.0000 0.0000 24 0 0 0.0000 0.0000 0.0000 0.0000 24 25 0 0 0.0000 0.0000 0.0000 0.0000 25 0 0 0.0000 0.0000 0.0000 0.0000 25 26 0 0 0.0000 0.0000 0.0000 0.0000 26 0 0 0.0000 0.0000 0.0000 0.0000 26 27 0 0 0.0000 0.0000 0.0000 0.0000 27 0 0 0.0000 0.0000 0.0000 0.0000 27 28 0 0 0.0000 0.0000 0.0000 0.0000 28 0 0 0.0000 0.0000 0.0000 0.0000 28 29 0 0 0.0000 0.0000 0.0000 0.0000 29 0 0 0.0000 0.0000 0.0000 0.0000 29 30 0 0 0.0000 0.0000 0.0000 0.0000 30 0 0 0.0000 0.0000 0.0000 0.0000 30 31 0 0 0.0000 0.0000 0.0000 0.0000 31 0 0 0.0000 0.0000 0.0000 0.0000 31 32 0 0 0.0000 0.0000 0.0000 0.0000 32 0 0 0.0000 0.0000 0.0000 0.0000 32 33 0 0 0.0000 0.0000 0.0000 0.0000 33 0 0 0.0000 0.0000 0.0000 0.0000 33 34 0 0 0.0000 0.0000 0.0000 0.0000 34 0 0 0.0000 0.0000 0.0000 0.0000 34 35 0 0 0.0000 0.0000 0.0000 0.0000 35 0 0 0.0000 0.0000 0.0000 0.0000 35 36 0 0 0.0000 0.0000 0.0000 0.0000 36 0 0 0.0000 0.0000 0.0000 0.0000 36 .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 2.000 Shear Strain (mm/m) h/ Hw Damage Ratio Wall Damage Indicator Damage Limit Yld% - 2.32 MPa (0.05) εcm - 2.32 MPa (0.05) γmean - 2.32 MPa (0.05) γmax - 2.32 MPa (0.05) γother - 2.32 MPa (0.05) 0.0 2.0 4.0 6 0 8.0 10.0 12.0 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.5 0 1.000 1.500 2.000 2.5 0 3. 3.50 4.000 4.500 Shear Strain (mm/m) h/ Hw Damage Ratio Wall Damage Indicator Damage Limit Yld% - .91 MPa (0. 6) εcm - 2.91 MPa (0.06) γmean - 2.91 MPa (0.06) γmax - 2.91 MPa (0.06) γother - 2.91 MPa (0.06) 0.0 5.0 10.0 15.0 20.0 25.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 4.000 5.000 6.000 Shear Strain (mm/m) h/ Hw Damage Ratio Wall Damage Indicator Damage Limit Yld% - 3.50 MPa (0. 7) εcm - 3.50 MPa (0.07) γmean - 3.50 MPa (0.07) γmax - 3.50 MPa (0.07) γother - 3.50 MPa (0.07)160 A.2.4 Shear Stress & Strains (Raw data and calculation cells are omitted due to size of spreadsheet) Figure A2.4: Sample of averaged shear strain profiles over the wall height. Collation of Group Data GAM-XY Layers/Group 11 Group offset 8 Load Step Offset 78 Layer Height 430 Data Set 1 Height 27000 Set Check: Mean Graph 1 Title Shear Strain Distribution Over Height Graph 2 Title Shear Stress-Strain Response Graph 3 X axis Shear Strain (mm/m)Y Axis h/Hw X axis Shear Strain (mm/m)Y Axis Shear Stress (MPa) Plot Series 19 12 8 7 Plot Series 1 2 3 4 Name Load Stage 19Load Stage 12Load Stage 8Load Stage 7 Name Group 1 - G11roup 2 - G11roup 3 - G11roup 4 - G11 Height\LoadBlack Blue Red Green Force (kN)Stress\StrainBlack Blue Red Green Rows Rows 13 Rows 20 7 1 0 0 0.7509 0.4946 0.3131 0.2632 1 0 0 0.0000 0.0000 0.0000 0.0000 1 4500 2 4500 0.16667 0.7509 0.4946 0.3131 0.2632 2 1408.8 0.626133 0.0315 0.0343 0.0343 0.0343 2 9000 3 4500 0.16667 2.1862 1.2247 0.6580 0.4842 3 2618.8 1.163911 0.0638 0.0645 0.0642 0.0639 3 13500 4 9000 0.33333 2.1862 1.2247 0.6580 0.4842 4 3445.9 1.531511 0.1129 0.1227 0.0925 0.0844 4 18000 5 9000 0.33333 2.9611 1.4048 0.6663 0.4736 5 4030.9 1.791511 0.1711 0.2201 0.1861 0.1315 5 22500 6 13500 0.5 2.9611 1.4048 0.6663 0.4736 6 4510.6 2.004711 0.2198 0.3374 0.3161 0.2179 6 27000 7 13500 0.5 2.5897 0.9943 0.4620 0.3262 7 4920.5 2.186889 0.2632 0.4842 0.4736 0.3262 7 27000 8 18000 0.66667 2.5897 0.9943 0.4620 0.3262 8 5218.2 2.3192 0.3131 0.6580 0.6663 0.4620 9 18000 0.66667 1.2844 0.5982 0.2839 0.2050 9 5579.7 2.479867 0.3667 0.7954 0.8362 0.5805 10 22500 0.83333 1.2844 0.5982 0.2839 0.2050 10 5919.7 2.630978 0.4151 0.9308 0.9932 0.7261 11 22500 0.83333 0.4511 0.2432 0.1247 0.1048 11 6267.2 2.785422 0.4549 1.0833 1.1810 0.8675 12 27000 1 0.4511 0.2432 0.1247 0.1048 12 6543.5 2.908222 0.4946 1.2247 1.4048 0.9943 13 27000 1 0.0000 0.0000 0.0000 0.0000 13 6748 2.999111 0.5300 1.3974 1.6656 1.1791 14 7005.1 3.113378 0.5768 1.5543 1.8703 1.3654 15 7140 3.173333 0.6181 1.7192 2.1159 1.5765 16 7442.8 3.307911 0.6576 1.8907 2.3039 1.7379 17 7636.1 3.393822 0.7097 2.0593 2.5105 1.9319 18 7750.4 3.444622 0.7346 2.1733 2.7558 2.2382 19 7874.5 3.499778 0.7509 2.1862 2.9611 2.5897 20 7074.3 3.144133 0.5699 2.4588 3.7676 3.0519 21 0 0 0.0000 0.0000 0.0000 0.0000 22 0 0 0.0000 0.0000 0.0000 0.0000 23 0 0 0.0000 0.0000 0.0000 0.0000 24 0 0 0.0000 0.0000 0.0000 0.0000 25 0 0 0.0000 0.0000 0.0000 0.0000 26 0 0 0.0000 0.0000 0.0000 0.0000 27 0 0 0.0000 0.0000 0.0000 0.0000 28 0 0 0.0000 0.0000 0.0000 0.0000 29 0 0 0.0000 0.0000 0.0000 0.0000 30 0 0 0.0000 0.0000 0.0000 0.0000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0000 0.5000 1.0000 1.5000 2.0000 2.5000 3.0000 3.5000 h/ H w Shear Strain (mm/m) Shear Strain Distribution Over Height Load Stage 7 Load Stage 8 Load Stage 12 Load Stage 19 0 0.5 1 1.5 2 2.5 3 3.5 4 0.0000 0.5000 1.0000 1.5000 2.0000 2.5000 3.0000 3.5000 4.0000 Sh ea r St re ss (M Pa ) Shear Strain (mm/m) Shear Stress-Strain Response Group 4 - G11 Group 3 - G11 Group 2 - G11 Group 1 - G11161 A.3 RESULTS FROM WALL ONLY TEST The following sheets summarize the VecTor2 wall data for the walls without slabs as discussed in Chapter 5 (total of 8 walls). Each sheet summarizes the key data for one wall including the name, axial load (in percent of fc’Ag), peak base shear and drift at failure. Each sheet contains 5 figures: Figure A: Shear stress-strain response with the 4 key load stages labeled A through D. In addition, the mechanism forming at that load stage is indicated (if applicable). Figures B to E: A snapshot of the damage state at the current load stage. Some results presented in 5.4.2. See Section 5.3.3 for a description of each damage parameter. 162 Wall: SCW3 A – Wall Only a) Shear Stress-Strain Response ρx: 0.2% Axial Load: 0% Total Slabs: 0 Peak Base Shear: 7,140kN Peak Drift: 0.33% Legend for Figures b) to d) b) Damage at Load Stage A: 2.19 MPa c) Damage at Load Stage B: 2.32 MPa d) Damage at Load Stage C: 2.91 MPa e) Damage at Load Stage D: 3.50 MPa Figure A3.1: Summary of data results for SCW3 A - Wall Only. X-Yield Crushing D C B A 0 1 2 3 4 0 1 2 3 4 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 1.0 2.0 3.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 2.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 15.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 2.000 4.000 6.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 10.0 20.0 30.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 2.000 4.000 6.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 163 Wall: SCW3 B – Wall Only a) Shear Stress-Strain Response ρx: 0.2% Axial Load: 10% Total Slabs: 0 Peak Base Shear: 8,150kN Peak Drift: 0.28% Legend for Figures b) to d) b) Damage at Load Stage A: 2.61 MPa c) Damage at Load Stage B: 2.80 MPa d) Damage at Load Stage C: 3.10 MPa e) Damage at Load Stage D: 3.48 MPa Figure A3.2: Summary of data results for SCW3 B - Wall Only. X-Yield Crushing D C B A 0 1 2 3 4 0 1 2 3 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 0.5 1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 0.5 1.0 1.5 2.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 4.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 15.0 20.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 2.000 4.000 6.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 164 Wall: SCW4 A – Wall Only a) Shear Stress-Strain Response ρx: 0.5% Axial Load: 0% Total Slabs: 0 Peak Base Shear: 3,600 kN Peak Drift: 0.59% Legend for Figures b) to d) b) Damage at Load Stage A: 2.22 MPa c) Damage at Load Stage B: 4.25 MPa d) Damage at Load Stage C: 4.59 MPa e) Damage at Load Stage D: 5.29 MPa Figure A3.3: Summary of data results for SCW4 A - Wall Only. X-Yield Crushing D C B A 0 1 2 3 4 5 6 0 2 4 6 8 10 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 0.5 1.0 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 2.000 4.000 6.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 15.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 2.0 4.0 6.0 8.0 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 165 Wall: SCW4 B – Wall Only a) Shear Stress-Strain Response ρx: 0.5% Axial Load: 10% Total Slabs: 0 Peak Base Shear: 3,720 kN Peak Drift: 0.61% Legend for Figures b) to d) b) Damage at Load Stage A: 3.72 MPa c) Damage at Load Stage B: 4.68 MPa d) Damage at Load Stage C: 4.78 MPa e) Damage at Load Stage D: 5.48 MPa Figure A3.4: Summary of data results for SCW4 B - Wall Only. X-Yield Crushing Cracks Combine D C B A 0 1 2 3 4 5 6 0 2 4 6 8 10 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 0.5 1.0 1.5 2.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 2.000 4.000 6.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 15.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 2.000 4.000 6.000 8.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 166 Wall: TSE8 A – Wall Only a) Shear Stress-Strain Response ρx: 0.98% Axial Load: 0% Total Slabs: 0 Peak Base Shear: 4,350 kN Peak Drift: 0.80% Legend for Figures b) to d) b) Damage at Load Stage A: 6.03 MPa c) Damage at Load Stage B: 8.01 MPa d) Damage at Load Stage C: 8.73 MPa e) Damage at Load Stage D: 8.92 MPa Figure A3.5: Summary of data results for TSE8 A - Wall Only. X-Yield & Crushing Localized Strain D C B A 0 2 4 6 8 10 12 0 2 4 6 8 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 15.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 2.000 4.000 6.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 167 Wall: TSE8 B – Wall Only a) Shear Stress-Strain Response ρx: 0.98% Axial Load: 10% Total Slabs: 0 Peak Base Shear: 4,500 kN Peak Drift: 0.80% Legend for Figures b) to d) b) Damage at Load Stage A: 8.13 MPa c) Damage at Load Stage B: 8.48 MPa d) Damage at Load Stage C: 9.08 MPa e) Damage at Load Stage D: 9.20 MPa Figure A3.6: Summary of data results for TSE8 B - Wall Only. X-Yield Localized Strain Crushing D C B A 0 2 4 6 8 10 12 0 2 4 6 8 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 15.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 2.000 4.000 6.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 168 A.4 RESULTS FROM THE COMPLETE ANALYSIS SUITE The following sheets summarize the VecTor2 wall data for the walls with slabs as discussed in Chapter 6 (total of 18 walls). Each sheet summarizes the key data for one wall including the name, axial load (in percent of fc’Ag), number of slabs, peak base shear and drift at failure. Each sheet contains 5 figures. Figure A: Shear stress-strain response with the 4 key load stages labeled A through D. In addition, the mechanism forming at that load stage is indicated (if applicable). Figures B to E: A snapshot of the damage state at the current load stage. Some results presented in 6.2.3. See Section 5.3.3 for a description of each damage parameter. 169 Wall: SCW3 A – 1.5:1 Ratio a) Shear Stress-Strain Response ρx: 0.2% Axial Load: 0% Total Slabs: 2 Peak Base Shear: 9,500kN Peak Drift: 0.48% Legend for Figures b) to d) b) Damage at Load Stage A: 3.07 MPa c) Damage at Load Stage B: 3.66 MPa d) Damage at Load Stage C: 5.20 MPa e) Damage at Load Stage D: 5.45 MPa Figure A4.1: Summary of data results for SCW3 A - 1.5:1 Ratio. Slab Cracking Crushing X-Yield D C B A 0 1 2 3 4 5 6 7 0 1 2 3 4 5 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 2.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 4.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 10.0 20.0 30.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 2.000 4.000 6.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 170 Wall: SCW3 A – 1.0:1 Ratio a) Shear Stress-Strain Response ρx: 0.2% Axial Load: 0% Total Slabs: 3 Peak Base Shear: 14,850kN Peak Drift: 0.50% Legend for Figures b) to d) b) Damage at Load Stage A: 3.82 MPa c) Damage at Load Stage B: 4.01 MPa d) Damage at Load Stage C: 6.48 MPa e) Damage at Load Stage D: 6.60 MPa Figure A4.2: Summary of data results for SCW3 A - 1.0:1 Ratio. Slab Cracking Crushing X-Yield D C B A 0 1 2 3 4 5 6 7 8 0 2 4 6 8 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 1.0 2.0 3.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 4.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 171 Wall: SCW3 A – 0.5:1 Ratio a) Shear Stress-Strain Response ρx: 0.2% Axial Load: 0% Total Slabs: 6 Peak Base Shear: 20,900 kN Peak Drift: 0.65% Legend for Figures b) to d) b) Damage at Load Stage A: 5.08 MPa c) Damage at Load Stage B: 6.68 MPa d) Damage at Load Stage C: 9.11 MPa e) Damage at Load Stage D: 9.29 MPa Figure A4.3: Summary of data results for SCW3 A - 0.5:1 Ratio. X-Yield Base Y- Yield then Crushing Slab Cracking D C B A 0 2 4 6 8 10 12 0 1 2 3 4 5 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 0.5 1.0 1.5 2.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 172 Wall: SCW3 B – 1.5:1 Ratio a) Shear Stress-Strain Response ρx: 0.2% Axial Load: 10% Total Slabs: 2 Peak Base Shear: 12,910 kN Peak Drift: 0.43% Legend for Figures b) to d) b) Damage at Load Stage A: 3.14 MPa c) Damage at Load Stage B: 4.24 MPa d) Damage at Load Stage C: 5.25 MPa e) Damage at Load Stage D: 5.74 MPa Figure A4.4: Summary of data results for SCW3 B - 1.5:1 Ratio. Slab Cracking Crushing X-Yield D C B A 0 1 2 3 4 5 6 7 0 1 2 3 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 1.0 2.0 3.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 4.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 15.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 2.000 4.000 6.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 173 Wall: SCW3 B – 1.0:1 Ratio a) Shear Stress-Strain Response ρx: 0.2% Axial Load: 10% Total Slabs: 3 Peak Base Shear: 14,970 kN Peak Drift: 0.50% Legend for Figures b) to d) b) Damage at Load Stage A: 4.34 MPa c) Damage at Load Stage B: 4.77 MPa d) Damage at Load Stage C: 6.51 MPa e) Damage at Load Stage D: 6.65 MPa Figure A4.5: Summary of data results for SCW3 B - 1.0:1 Ratio. Slab Cracking Crushing X-Yield D C B A 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 1.0 2.0 3.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 4.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 15.0 20.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 2.000 4.000 6.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 174 Wall: SCW3 B – 0.5:1 Ratio a) Shear Stress-Strain Response ρx: 0.2% Axial Load: 10% Total Slabs: 6 Peak Base Shear: 21,600 kN Peak Drift: 0.70% Legend for Figures b) to d) b) Damage at Load Stage A: 5.75 MPa c) Damage at Load Stage B: 7.02 MPa d) Damage at Load Stage C: 8.14 MPa e) Damage at Load Stage D: 9.60 MPa Figure A4.6: Summary of data results for SCW3 B - 0.5:1 Ratio. X-Yield Strain Expansion & Struts Merge Slab Cracking D C B A 0 2 4 6 8 10 12 0 1 2 3 4 5 6 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 0.5 1.0 1.5 2.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 2.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 15.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 4.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 175 Wall: SCW4 A – 1.5:1 Ratio a) Shear Stress-Strain Response ρx: 0.5% Axial Load: 0% Total Slabs: 2 Peak Base Shear: 4,850 kN Peak Drift: 0.61% Legend for Figures b) to d) b) Damage at Load Stage A: 3.78 MPa c) Damage at Load Stage B: 5.12 MPa d) Damage at Load Stage C: 5.96 MPa e) Damage at Load Stage D: 7.13 MPa Figure A4.7: Summary of data results for SCW4 A - 1.5:1 Ratio. X-Yield Crushing D C B A 0 1 2 3 4 5 6 7 8 0 1 2 3 4 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 1.0 2.0 3.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 15.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 2.000 4.000 6.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 176 Wall: SCW4 A – 1.0:1 Ratio a) Shear Stress-Strain Response ρx: 0.5% Axial Load: 0% Total Slabs: 3 Peak Base Shear: 6,100 kN Peak Drift: 0.61% Legend for Figures b) to d) b) Damage at Load Stage A: 6.89 MPa c) Damage at Load Stage B: 7.61 MPa d) Damage at Load Stage C: 8.15 MPa e) Damage at Load Stage D: 8.97 MPa Figure A4.8: Summary of data results for SCW4 A - 1.0:1 Ratio. Crushing Slab Cracking X-Yield D C B A 0 2 4 6 8 10 12 0 1 2 3 4 5 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 15.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 4.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 177 Wall: SCW4 A – 0.5:1 Ratio a) Shear Stress-Strain Response ρx: 0.5% Axial Load: 0% Total Slabs: 6 Peak Base Shear: 7,250 kN Peak Drift: 0.54% Legend for Figures b) to d) b) Damage at Load Stage A: 6.34 MPa c) Damage at Load Stage B: 8.82 MPa d) Damage at Load Stage C: 9.69 MPa e) Damage at Load Stage D: 10.66 MPa Figure A4.9: Summary of data results for SCW4 A - 0.5:1 Ratio. Strut Y- Yield Crushing D C B A 0 2 4 6 8 10 12 0 1 2 3 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 1.0 2.0 3.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 178 Wall: SCW4 B – 1.5:1 Ratio a) Shear Stress-Strain Response ρx: 0.5% Axial Load: 10% Total Slabs: 2 Peak Base Shear: 4,470 kN Peak Drift: 0.48% Legend for Figures b) to d) b) Damage at Load Stage A: 5.41 MPa c) Damage at Load Stage B: 6.26 MPa d) Damage at Load Stage C: 6.51 MPa e) Damage at Load Stage D: 6.76 MPa Figure A4.10: Summary of data results for SCW4 B - 1.5:1 Ratio. Crushing Strut Y- Yield Strut Y- Yield X-Yield D C B A 0 2 4 6 8 10 0 1 2 3 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 4.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 15.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 4.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 179 Wall: SCW4 B – 1.0:1 Ratio a) Shear Stress-Strain Response ρx: 0.5% Axial Load: 10% Total Slabs: 3 Peak Base Shear: 5,730 kN Peak Drift: 0.52% Legend for Figures b) to d) b) Damage at Load Stage A: 6.82 MPa c) Damage at Load Stage B: 7.66 MPa d) Damage at Load Stage C: 8.14 MPa e) Damage at Load Stage D: 8.44 MPa Figure A4.11: Summary of data results for SCW4 B - 1.0:1 Ratio. Crushing Strut Y- Yield X-Yield D C B A 0 2 4 6 8 10 12 0 1 2 3 4 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 2.000 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 1.000 2.000 3.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 180 Wall: SCW4 B – 0.5:1 Ratio a) Shear Stress-Strain Response ρx: 0.5% Axial Load: 10% Total Slabs: 6 Peak Base Shear: 7,620 kN Peak Drift: 0.63% Legend for Figures b) to d) b) Damage at Load Stage A: 8.56 MPa b) Damage at Load Stage A: 10.05 MPa b) Damage at Load Stage A: 11.17 MPa e) Damage at Load Stage D: 11.21 MPa Figure A4.12: Summary of data results for SCW4 B - 0.5:1 Ratio. Crushing X-Yield Strut Y- Yield D C B A 0 2 4 6 8 10 12 14 0 2 4 6 8 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 181 Wall: TSE8 A – 1.5:1 Ratio a) Shear Stress-Strain Response ρx: 0.98% Axial Load: 0% Total Slabs: 2 Peak Base Shear: 4,630 kN Peak Drift: 0.73% Legend for Figures b) to d) b) Damage at Load Stage A: 3.17 MPa b) Damage at Load Stage A: 4.53 MPa b) Damage at Load Stage A: 8.29 MPa e) Damage at Load Stage D: 9.49 MPa Figure A4.13: Summary of data results for TSE8 A - 1.5:1 Ratio. Slab Cracking X-Yield & Crushing D C B A 0 2 4 6 8 10 12 0 1 2 3 4 5 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 0.5 1.0 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 2.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 182 Wall: TSE8 A – 1.0:1 Ratio a) Shear Stress-Strain Response ρx: 0.98% Axial Load: 0% Total Slabs: 3 Peak Base Shear: 4,710 kN Peak Drift: 0.70% Legend for Figures b) to d) b) Damage at Load Stage A: 4.74 MPa b) Damage at Load Stage A: 8.16 MPa b) Damage at Load Stage A: 8.94 MPa e) Damage at Load Stage D: 9.65MPa Figure A4.14: Summary of data results for TSE8 A - 1.0:1 Ratio. Crushing X-Yield Slab Cracking D C B A 0 2 4 6 8 10 12 0 1 2 3 4 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 1.0 2.0 3.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 183 Wall: TSE8 A – 0.5:1 Ratio a) Shear Stress-Strain Response ρx: 0.98% Axial Load: 0% Total Slabs: 6 Peak Base Shear: 5,090 kN Peak Drift: 0.70% Legend for Figures b) to d) b) Damage at Load Stage A: 5.26 MPa b) Damage at Load Stage A: 9.25 MPa b) Damage at Load Stage A: 10.42 MPa e) Damage at Load Stage D: 10.42 MPa Figure A4.15: Summary of data results for TSE8 A - 0.5:1 Ratio. Crushing Sig. Crushing Slab Cracking D C B A 0 2 4 6 8 10 12 0 1 2 3 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 0.5 1.0 1.5 2.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 184 Wall: TSE8 B – 1.5:1 Ratio a) Shear Stress-Strain Response ρx: 0.98% Axial Load: 10% Total Slabs: 2 Peak Base Shear: 4,720 kN Peak Drift: 0.70% Legend for Figures b) to d) b) Damage at Load Stage A: 5.07 MPa b) Damage at Load Stage A: 8.40 MPa b) Damage at Load Stage A: 8.77 MPa e) Damage at Load Stage D: 9.67 MPa Figure A4.16: Summary of data results for TSE8 B - 1.5:1 Ratio. Crushing X-Yield Slab Cracking D C B A 0 2 4 6 8 10 12 0 1 2 3 4 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 0.5 1.0 1.5 2.0 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 4.0 5.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 5.0 10.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 2.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 185 Wall: TSE8 B – 1.0:1 Ratio a) Shear Stress-Strain Response ρx: 0.98% Axial Load: 10% Total Slabs: 3 Peak Base Shear: 4,930 kN Peak Drift: 0.70% Legend for Figures b) to d) b) Damage at Load Stage A: 4.83 MPa b) Damage at Load Stage A: 8.66 MPa b) Damage at Load Stage A: 9.03 MPa e) Damage at Load Stage D: 10.09 MPa Figure A4.17: Summary of data results for TSE8 B - 1.0:1 Ratio. Crushing X-Yield Slab Cracking D C B A 0 2 4 6 8 10 12 0 1 2 3 4 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 0.5 1.0 1.5 2.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 8.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 2.000 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax 186 Wall: TSE8 B – 0.5:1 Ratio a) Shear Stress-Strain Response ρx: 0.98% Axial Load: 10% Total Slabs: 6 Peak Base Shear: 5,490 kN Peak Drift: 0.73% Legend for Figures b) to d) b) Damage at Load Stage A: 5.82 MPa b) Damage at Load Stage A: 9.74 MPa b) Damage at Load Stage A: 10.60 MPa e) Damage at Load Stage D: 11.24 MPa Figure A4.18: Summary of data results for TSE8 B - 0.5:1 Ratio. Crushing Base Y- Yield (-) Slab Cracking D C B A 0 2 4 6 8 10 12 0 1 2 3 S h ea r St res s (M P a ) Shear Strain (mm/m) 0.0 0.5 1.0 1.5 2.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio 0.0 2.0 4.0 6.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.000 0.500 1.000 1.500 Shear Strain (mm/m) h/H w Damage Ratio Yld% γmean γlocal Damage Limit εcm% γmax
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Nonlinear shear response of cantilever reinforced concrete...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Nonlinear shear response of cantilever reinforced concrete shear walls with floor slabs Mercer, Stephen Sterling 2012-04-20
pdf
Page Metadata
Item Metadata
Title | Nonlinear shear response of cantilever reinforced concrete shear walls with floor slabs |
Creator |
Mercer, Stephen Sterling |
Publisher | University of British Columbia |
Date Issued | 2012 |
Description | The nonlinear shear behaviour of cantilever reinforced concrete shear walls is complex and not fully understood. Design assumptions often oversimplify the wall response and can yield results which do not reflect the true response of the shear wall. One such assumption is analyzing the wall ignoring the effects from multiple floor slabs connected to the wall over its height. Floor slabs can provide a significant increase in wall shear capacity. This thesis examines the nonlinear shear response of walls, including the effect of floor slabs as a wall-slab system, through state-of-the-art nonlinear finite element analysis. Finite element slab models were developed to emulate the 3D slab effect within a 2D analysis environment: a high-end pseudo 3D model for in-depth slab analysis and a simple 2D slab layer model for typical wall analysis. The slab effects are explored through a parametric study varying the wall size, concrete strength, axial load, horizontal steel ratio and the slab dimensions parallel and perpendicular to the wall. The slabs were found to act like large external stirrups which provide additional tension capacity for the slab and limit shear cracking and failure. The slabs can significantly increase the shear capacity of lightly-reinforced walls. Using the developed slab models, the bounds of the slab effect were investigated by a parametric study with lightly to heavily-reinforced walls, with and without axial load, as well as varying the slab spacing. Within this study, the nonlinear response of isolated walls is compared to nonlinear uniformly-loaded membrane models. It is determined that although code-based shear capacity equations are fairly accurate, the membrane models can underestimate the shear stiffness and over predict the ductility. This study also reveals that tightly spaced slabs can increase up to 3 times the isolated wall capacity for walls with minimum horizontal steel, whereas there is little effect for walls with horizontal steel above 1%. Finally, methods were developed to predict the nonlinear shear stress-strain response of isolated walls and the peak shear capacity of wall-slab systems. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-04-20 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivs 3.0 Unported |
DOI | 10.14288/1.0050939 |
URI | http://hdl.handle.net/2429/42164 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2012-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/3.0/ |
Aggregated Source Repository | DSpace |
Download
- Media
- 24-ubc_2012_spring_mercer_stephen.pdf [ 6.56MB ]
- Metadata
- JSON: 24-1.0050939.json
- JSON-LD: 24-1.0050939-ld.json
- RDF/XML (Pretty): 24-1.0050939-rdf.xml
- RDF/JSON: 24-1.0050939-rdf.json
- Turtle: 24-1.0050939-turtle.txt
- N-Triples: 24-1.0050939-rdf-ntriples.txt
- Original Record: 24-1.0050939-source.json
- Full Text
- 24-1.0050939-fulltext.txt
- Citation
- 24-1.0050939.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0050939/manifest