RESISTANCE OF REINFORCED CONCRETE CANTILEVER SHEAR WALLS TO SEISMIC SHEAR DEMANDS by Mitchell P. Young B.A.Sc., University of New Brunswick, 2016 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) February 2019 © Mitchell P. Young, 2019 ii The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled: Resistance of reinforced concrete cantilever shear walls under seismic shear loading Submitted by Mitchell Young in partial fulfillment of the requirements for the degree of Master of Applied Science in Civil Engineering Examining Committee: Perry Adebar, Civil Engineering Supervisor Carlos Molina Hutt, Civil Engineering Additional Examiner iii Abstract Reinforced concrete shear wall core structures are very common among high-rise buildings in Vancouver, and increasingly so elsewhere. While the flexural behaviour of these structures is well understood, the shear behaviour is not. Much of the research regarding the shear behaviour is related to the amplifications of demands due to higher mode seismic shear, however, there has been little research regarding the resistance of these structures to higher mode seismic shear demands. It is theorized that due to the lack of experimental data on which more complex shear models could be based on, structural engineers have resorted to using building models with complex non-linear fibre section flexural stiffness, but a linear elastic shear stiffness. Which may have lead to higher mode shear demands to be overestimated. Therefore, the goal of this thesis is to complete an experimental program in which scaled shear wall core specimens are tested under higher mode demands in the cantilevered direction. Through the experimental program, topics that are investigated include: the effect of the rate of loading on the shear resistance, the effect of existing flexural base yielding on the shear resistance, and the presence of a plastic hinge at the base on the shear resistance. In addition to the experimental program, a series of dynamic analyses were completed on a simplified model, in order to better understand the behaviour of a high-rise reinforced concrete shear wall structure to higher mode effects, and the results of the experimental program are also compared to predictions made using common analytical tools. iv Lay Summary Reinforced concrete shear wall core structures are very common among high-rise structures in Vancouver, and increasingly so elsewhere. The resistance of these types of structures is well understood when being displaced in their first mode shape, but more research is required when they are displaced in their higher mode shapes. Therefore, the goal of this thesis is to complete an experimental program in which scaled shear wall core specimens are tested by forcing them into their second-mode shape in the cantilevered direction. Through the experimental program, topics that are investigated include: the effect of the rate of loading on the shear strength, the effect of existing flexural base yielding, and the presence of a plastic hinge at the base. v Preface This thesis is original, independent work by the author, Mitchell Young. vi Table of Contents Abstract .................................................................................................................................... iii Lay Summary ........................................................................................................................... iv Preface ....................................................................................................................................... v Table of Contents ..................................................................................................................... vi List of Tables ............................................................................................................................ x List of Figures .......................................................................................................................... xi Acknowledgements ............................................................................................................... xxii 1. Introduction ....................................................................................................................... 1 1.1 Background ..................................................................................................................... 1 1.2 Response Spectrum Analysis of 30-Story Core Wall Building ...................................... 3 1.3 Estimating Seismic Demands in Nonlinear Buildings .................................................... 7 1.4 Estimating Shear Force Demands on Nonlinear Concrete Walls ................................... 8 1.5 Shear Resistance of Concrete Walls ............................................................................. 10 1.6 Objective and Methodology of Current Study .............................................................. 11 1.7 Overview of Thesis ....................................................................................................... 12 2. Dynamic Analyses of Simplified Shear Wall Core Model ............................................. 14 2.1 Overview ....................................................................................................................... 14 2.2 Description of Analytical Model .................................................................................. 14 2.2.1 Overview of Analysis Approach ............................................................................ 14 2.2.2 Shear Wall Core Cross-Section ............................................................................. 17 2.2.3 Non-Linear Fibre Section Elements ....................................................................... 21 2.2.4 Model Parameters .................................................................................................. 22 2.2.5 Applied Pulse ......................................................................................................... 30 2.3 Results from Linear Model ........................................................................................... 31 2.3.1 Modal Decomposition ............................................................................................ 31 2.3.2 Linear Response when Excited at Period of First Mode ........................................ 35 2.3.3 Linear Response when Excited at Period of Second-mode ................................... 41 2.3.4 Summary of Results in Linear Elastic Range ........................................................ 47 2.4 Results of Non-Linear Model ....................................................................................... 50 vii 2.4.1 Curvature Distribution when Excited at Period of First Mode .............................. 51 2.4.2 Curvature Distribution when Excited at Period of Second-mode .......................... 56 2.4.3 Change in Modal Response Due to Base Yielding ................................................ 63 2.4.4 Non-Linear Response when Excited at Period of First Mode ............................... 74 2.4.5 Non-Linear Response when Excited at Period of Second-mode ........................... 80 2.4.6 Elongation of Period in Non-Linear Response ...................................................... 85 2.5 Summary of Dynamic Analyses ................................................................................... 90 2.6 Preliminary Design of Experiment Based on Analytical Data ..................................... 94 2.6.1 Preliminary Specimen Design ................................................................................ 94 2.6.2 Preliminary Displacement Protocol ....................................................................... 96 2.6.3 Preliminary Test Setup ......................................................................................... 100 3. Experimental Program .................................................................................................. 103 3.1 Overview ..................................................................................................................... 103 3.2 Testing Approach ........................................................................................................ 104 3.3 Details of Test Specimen ............................................................................................ 107 3.4 Material Properties ...................................................................................................... 113 3.4.1 Concrete Properties .............................................................................................. 113 3.4.2 Reinforcing Steel Properties ................................................................................ 115 3.5 Construction of Test Specimen ................................................................................... 121 3.5.1 Segmental Foundation ......................................................................................... 121 3.5.2 Precast Specimen Construction ............................................................................ 125 3.6 Test Setup Components .......................................................................................... 129 3.7 Instrumentation ........................................................................................................... 135 3.7.1 Displacement Coordinate Targets ........................................................................ 137 3.8 Testing Procedure ....................................................................................................... 140 4. Experimental Results .................................................................................................... 141 4.1 Overview ..................................................................................................................... 141 4.2 Summary of Results .................................................................................................... 141 4.2.1 Specimen 1 ........................................................................................................... 141 4.2.2 Specimen 2 ........................................................................................................... 147 4.2.3 Specimen 3 ........................................................................................................... 151 4.2.4 Specimen 4 ........................................................................................................... 164 viii 4.2.5 Specimen 5 ........................................................................................................... 176 4.3 Comparison of Test Results ........................................................................................ 188 5. Discussion of Experimental Results ............................................................................. 197 5.1 Overview ..................................................................................................................... 197 5.2 Analytical Methods ..................................................................................................... 197 5.2.1 Non-Linear Flexural Model ................................................................................. 198 5.2.2 Shear Model ......................................................................................................... 201 5.3 Discussion of Shear Strength ...................................................................................... 204 5.3.1 Shear Strength ...................................................................................................... 205 5.3.2 Bending Moment – Shear Force Interaction ........................................................ 212 5.4 Effect of Shear Stiffness Model on Prediction ........................................................... 214 5.5 Conclusion .............................................................................................................. 219 6. Conclusion and Recommendations ............................................................................... 220 6.1 Overview ..................................................................................................................... 220 6.2 Overall Summary ........................................................................................................ 220 6.3 Summary of Dynamic Analyses ................................................................................. 221 6.4 Summary of Experimental Results ............................................................................. 223 6.5 Summary of Analytical Predictions ............................................................................ 227 6.6 Recommendations for Future Research ...................................................................... 228 Bibliography ......................................................................................................................... 230 A. – Linear Elastic Response Spectrum Analysis Procedure ............................................ 233 B. – Data from Non-Linear Dynamic Analysis ................................................................. 238 B.1 Linear Elastic Model Excited by Sinusoidal Ground Acceleration ....................... 238 B.2 Non-Linear Model Excited by Sinusoidal Ground Acceleration ........................... 298 C. – Prediction Program .................................................................................................... 338 D. – Test Setup Design Drawings ...................................................................................... 345 D.1 Specimen and Foundation Formwork .................................................................... 345 D.2 Specimen and Foundation Reinforcement Detail ................................................... 347 D.3 Setup Components .................................................................................................. 351 E. – Summary of Experimental Tests ................................................................................ 352 E.1 Summary of Specimen 1 ........................................................................................ 352 E.1.1 Specimen 1 Crack Profile and Measurements ................................................ 354 ix E.1.2 Specimen 1 Load-Displacement Relationship ................................................ 362 E.1.3 Specimen 1 Additional Observations .............................................................. 370 E.2 Summary of Specimen 2 ........................................................................................ 371 E.2.1 Specimen 2 Crack Profile and Measurements ................................................ 373 E.2.2 Specimen 2 Load-Displacement Relationship ................................................ 380 E.2.3 Specimen 2 Additional Observations .............................................................. 387 E.3 Summary of Specimen 3 ........................................................................................ 388 E.3.1 Specimen 3 Crack Profile and Measurements ................................................ 390 E.3.2 Specimen 3 Load-Displacement Relationship ................................................ 399 E.3.3 Specimen 3 Displacement Coordinate Results ............................................... 407 E.3.4 Specimen 3 Additional Observations .............................................................. 418 E.4 Summary of Specimen 4 ........................................................................................ 420 E.4.1 Specimen 4 Crack Profile and Measurements ................................................ 422 E.4.2 Specimen 4 Load-Displacement Relationship ................................................ 430 E.4.3 Specimen 4 Displacement Coordinate Results ............................................... 438 E.4.4 Specimen 4 Additional Observations .............................................................. 449 E.5 Summary of Specimen 5 ........................................................................................ 450 E.5.1 Specimen 5 Crack Profile and Measurements ................................................ 453 E.5.2 Specimen 5 Load-Displacement Relationship ................................................ 460 E.5.3 Specimen 5 Displacement Coordinate Results ............................................... 465 E.5.4 Specimen 5 Additional Observations .............................................................. 476 F. – Analytical Prediction of Experimental Results .......................................................... 478 x List of Tables Table 2.1: Properties of prototype shear wall core and idealized I-beam cross-section. ........ 19 Table 2.2: Points of interest in moment-curvature relationship of 30-story building idealized cross-section. ........................................................................................................................... 21 Table 2.3: Modal participation factors and effective masses of model. ................................. 26 Table 2.4: Idealized “modes” of model for: (a) Displacement; (b) Forces; (c) Shear forces; (d) Bending moments. .................................................................................................................. 26 Table 2.5: Results of modal decomposition example. ............................................................ 33 Table 2.6: Intensity of ground acceleration pulse for each trial of analyses excited at period of first mode. ............................................................................................................................... 51 Table 2.7: Intensity of ground acceleration pulse for each trial for analyses excited at period of second-mode. .......................................................................................................................... 57 Table 2.8: Length of time required for each quarter cycle of top node first mode displacement response: (a) In seconds; (b) Normalized by the period of the first mode .............................. 87 Table 2.9: Length of time required for each cycle of top node second-mode displacement response: (a) In seconds; (b) Normalized by the period of the second-mode ......................... 88 Table 3.1: Location of cross-section regions along specimen height. .................................. 112 Table 3.2 Measured concrete compressive strengths from field and most cured cylinder samples. ................................................................................................................................. 114 Table 3.3: Measured concrete tensile strengths from field and moist cured cylinder samples. ............................................................................................................................................... 115 Table 5.1: Values used for tri-linear approximation of moment-curvature diagram for predictive model. ................................................................................................................... 200 Table 5.2: Summary of values used to determine shear strength of specimen using standard code procedures. ................................................................................................................... 211 Table 5.3: Comparison of maximum base shear strengths calculated using standard code procedures to observed peak base shear. .............................................................................. 211 Table A.1: Modal periods, participation factors, and mass ratios of 30 story building. ....... 233 Table A.2: Spectral acceleration and displacement values of first five modes of building in cantilevered direction. ........................................................................................................... 235 xi List of Figures Figure 1.1: Examples of shear wall core buildings in the City of Vancouver. ......................... 1 Figure 1.2: Total and individual modal displacement demands determined by RSA of a 30-story reinforced concrete core wall building in the cantilever wall direction. .......................... 5 Figure 1.3: Total and individual modal bending moment demands determined by RSA of a 30-story reinforced concrete core wall building in the cantilever wall direction. .......................... 5 Figure 1.4: Total and individual modal shear force demands determined by RSA of a 30-story reinforced concrete core wall building in the cantilever wall direction. ................................... 6 Figure 2.1: Prototype building modeled for dynamic analyses. ............................................. 15 Figure 2.2: Simplified model of cantilever walls used in the current study. .......................... 17 Figure 2.3: Cross-Section of prototype shear wall core used for non-linear model. .............. 18 Figure 2.4: Idealized cross-section of prototype shear wall core created in: (a) Response2000; (b) OpenSees (all units in mm). .............................................................................................. 19 Figure 2.5: Moment-Curvature relationship of idealized 30-story building cross-section: (a) Full moment-curvature response; (b) Linear and cracked phases of relationship. ................. 20 Figure 2.6: Material models used in model: (a) Concrete02; (b) Steel02 (UC Berkeley, 2017). ................................................................................................................................................. 23 Figure 2.7: Stress-strain relationship of shear stiffness model. .............................................. 24 Figure 2.8: Visualization of lumped masses and gravity load applied to model. ................... 25 Figure 2.9: Visualization of modes of wall for: (a) Displacement/forces; (b) Bending moment; (c) Shear force. ........................................................................................................................ 27 Figure 2.10: Relationship between period of response and damping ratio. ............................ 28 Figure 2.11: Effect of P-delta on the response of the model: (a) Linear; (b) Non-linear. ...... 29 Figure 2.12: Examples of ground accelerations: (a) Period of 0.62 s; (b) Period of 3.71 s. ... 30 Figure 2.13: Example of decomposing an arbitrary displacement vector into modal components at a given time step. ................................................................................................................ 33 Figure 2.14: Example of decomposition of total response into two modal components. ....... 34 Figure 2.15: Displacement response at top and mid-height nodes for structure excited at period of first mode in the linear elastic range. .................................................................................. 35 xii Figure 2.16: Modal displacement response for structure excited at period of first mode in the linear elastic range at: (a) top node; (b) mid-height node. ...................................................... 36 Figure 2.17: Bending moment response of structure excited at period of first mode in the linear elastic range. ........................................................................................................................... 36 Figure 2.18: Modal bending moment response for structure excited at period of first mode in the linear elastic range at: (a) Base node; (b) Mid-height node. ............................................. 37 Figure 2.19: Shear force response of structure excited at period of first mode in the linear elastic range. ....................................................................................................................................... 37 Figure 2.20: Modal shear force response for structure excited at period of first mode in the linear elastic range at: (a) Base element; (b) Top element. ..................................................... 38 Figure 2.21: Equivalent static force response of structure excited at period of first mode in the linear elastic range. ................................................................................................................. 39 Figure 2.22: Modal equivalent static force response for structure excited at period of first mode in the linear elastic range at: (a) Top node; (b) Mid-height node. .......................................... 39 Figure 2.23: Trend of modal responses for structure excited at period of first mode in the linear elastic range for: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static force. ....................................................................................................................................... 40 Figure 2.24: Displacement response at top and mid-height nodes for structure excited at period of second-mode in the linear elastic range. ............................................................................. 41 Figure 2.25: Modal displacement response for structure excited at period of second-mode in the linear elastic range at: (a) Top node; (b) Mid-height node. .............................................. 42 Figure 2.26: Bending moment response at base and mid-height nodes for structure excited at period of second-mode in the linear elastic range. ................................................................. 42 Figure 2.27: Modal bending moment response for structure excited at period of second-mode in the linear elastic range at: (a) Base node; (b) Mid-height node. ......................................... 43 Figure 2.28: Shear force response at top and base elements for structure excited at period of second-mode in the linear elastic range. ................................................................................. 44 Figure 2.29: Modal shear force response for structure excited at period of second-mode in the linear elastic range at: (a) Base element; (b) Top element. ..................................................... 44 Figure 2.30: Equivalent static force response at top and mid-height nodes for structure excited at period of second-mode in the linear elastic range. .............................................................. 45 xiii Figure 2.31: Modal equivalent static force response for structure excited at period of second-mode in the linear elastic range at: (a) Top node; (b) Mid-height node. ................................ 45 Figure 2.32: Trend of modal responses for structure excited at period of second-mode in the linear elastic range for: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static force. .............................................................................................................................. 46 Figure 2.33: Normalized maximum value of displacements, shear forces, and bending moments of model subjected to a sinusoidal ground acceleration pulse vs period of the applied pulse in the linear elastic range. .............................................................................................. 47 Figure 2.34: Moment-to-Shear ratio vs period of the applied pulse in the linear elastic range. ................................................................................................................................................. 49 Figure 2.35: Maximum base moment vs maximum top displacement for each trial of dynamic non-linear analyses for structure excited at period of first mode. ........................................... 51 Figure 2.36: Results of dynamic non-linear analysis when excited at period of first mode, Trial 1: (a) Moment-curvature response at base; (b) Curvature envelope along height. ................. 52 Figure 2.37: Results of dynamic non-linear analysis when excited at period of first mode, Trial 2: (a) Moment-curvature response at base; (b) Curvature envelope along height. ................. 53 Figure 2.38: Results of dynamic non-linear analysis when excited at period of first mode, Trial 3: (a) Moment-curvature response at base; (b) Curvature envelope along height. ................. 54 Figure 2.39: Results of dynamic non-linear analysis when excited at period of first mode, Trial 4: (a) Moment-curvature response at base; (b) Curvature envelope along height. ................. 55 Figure 2.40: Maximum base moment vs maximum top displacement for each trial of dynamic non-linear analyses for structure excited at period of second-mode. ...................................... 57 Figure 2.41: Results of dynamic non-linear analysis when excited at period of second-mode, Trial 1: (a) Moment-curvature response at base; (b) Curvature envelope along height. ........ 58 Figure 2.42: Results of dynamic non-linear analysis when excited at period of second-mode, Trial 2: (a) Moment-curvature response at base; (b) Curvature envelope along height. ........ 59 Figure 2.43: Results of dynamic non-linear analysis when excited at period of second-mode, Trial 3: (a) Moment-curvature response at base; (b) Curvature envelope along height. ........ 60 Figure 2.44: Results of dynamic non-linear analysis when excited at period of second-mode, Trial 4: (a) Moment-curvature response at base; (b) Curvature envelope along height. ........ 61 xiv Figure 2.45: Comparison of curvature envelope for each trial for structure excited at period of: (a) First mode; (b) Second-mode. ........................................................................................... 62 Figure 2.46: Comparison between fixed, yielded, and pinned displacement profiles. ........... 63 Figure 2.47: Displacement response at top and mid-height nodes for structure excited at period of first mode, Trial 1: (a) Time history; (b) Modal trend. ....................................................... 64 Figure 2.48: Displacement response at top and mid-height nodes for structure excited at period of first mode, Trial 3: (a) Time history; (b) Modal trend. ....................................................... 65 Figure 2.49: Displacement response at top and mid-height nodes for structure excited at period of first mode, Trial 4: (a) Time history; (b) Modal trend. ....................................................... 65 Figure 2.50: Change in ratio of mid-height displacement to top displacement, based on slope of modal trendline of response in free vibration, as base curvature increases for structure excited at period of first mode. ............................................................................................... 66 Figure 2.51: Modal displacement response for structure excited at period of first mode, Trial 3: (a) Top node using linear mode shapes; (b) Top node using corrected mode shapes; (c) Mid-height node using linear mode shapes; (d) Mid-height node using corrected mode shapes. .. 67 Figure 2.52: Displacement response at top and mid-height nodes for structure excited at period of second-mode, Trial 1: (a) Time history; (b) Modal trend. .................................................. 68 Figure 2.53: Displacement response at top and mid-height nodes for structure excited at period of second-mode, Trial 3: (a) Time history; (b) Modal trend. .................................................. 69 Figure 2.54: Modal displacement response for structure excited at period of second-mode, Trial 3: (a) Top node using linear mode shapes; (b) Top node using corrected mode shapes; (c) Mid-height node using linear mode shapes; (d) Mid-height node using corrected mode shapes. .. 70 Figure 2.55: Displacement response at top and mid-height nodes for structure excited at period of second-mode, Trial 4: (a) Time history; (b) Modal trend. .................................................. 71 Figure 2.56: Modal displacement response for structure excited at period of second-mode, Trial 4: (a) Top node using linear mode shapes; (b) Top node using corrected mode shapes; (c) Mid-height node using linear mode shapes; (d) Mid-height node using corrected mode shapes. .. 72 Figure 2.57: Effect of effective shear stiffness on mode shape of: (a) First mode; (b) Second-mode. ....................................................................................................................................... 73 Figure 2.58: Displacement response at top and mid-height node for structure excited at period of first mode in the non-linear range. ...................................................................................... 74 xv Figure 2.59: Modal equivalent static force response for structure excited at period of first mode in the non-linear range at: (a) Top node; (b) Mid-height node. .............................................. 75 Figure 2.60: Equivalent static force response at top and mid-height nodes for structure excited at period of first mode in the non-linear range. ...................................................................... 75 Figure 2.61: Modal equivalent static force response for structure excited at period of first mode in the non-linear range at: (a) Top node; (b) Mid-height node. .............................................. 76 Figure 2.62: Trend of modal response for structure excited at period of first mode in the non-linear range at: (a) Bending moment response; (b) Shear force response. ............................. 77 Figure 2.63: Bending moment response of structure excited at period of first mode, Trial 1, with approximate cracking and yielding bending moments shown. ....................................... 78 Figure 2.64: Bending moment response of structure excited at period of first mode, Trial 3, with approximate cracking and yielding bending moments shown. ....................................... 78 Figure 2.65: Response of linear elastic structure with artificial plastic hinge excited at period of first mode for: (a) Displacement response; (b) Modal displacement trend; (c) Equivalent static force response; (d) Modal static force trend. ................................................................. 79 Figure 2.66: Displacement response at top and mid-height node for structure excited at period of second-mode in the non-linear range. ................................................................................. 80 Figure 2.67: Corrected modal displacement response for structure excited at period of second-mode in the non-linear range at: (a) Top node; (b) Mid-height node. .................................... 81 Figure 2.68: Equivalent static force response at top and base elements for structure excited at period of second-mode in the non-linear range. ..................................................................... 81 Figure 2.69: Modal equivalent static force response for structure excited at period of second-mode in the non-linear range at: (a) Top node; (b) Mid-height node. .................................... 82 Figure 2.70: Trend of modal responses for structure excited at period of second-mode in the non-linear range for: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static force. .............................................................................................................................. 83 Figure 2.71: Relationship between maximum base shear and mid-height displacement distinguished by period of excitation. ..................................................................................... 84 Figure 2.72: Normalized displacement histories of top node of structure excited at period of second-mode, with increasing intensity of each Trial 1 through 4. ........................................ 85 Figure 2.73: Example of a response cycle split into quarter cycles. ....................................... 86 xvi Figure 2.74: Normalized first mode displacement histories of top node of structure excited at period of second-mode, with increasing intensity of each Trial 1 through 4. ........................ 86 Figure 2.75: Normalized second-mode displacement histories of top node of structure excited at period of second-mode, with increasing intensity of each Trial 1 through 4. ..................... 87 Figure 2.76: Comparison of period elongation for modes 1 and 2. ........................................ 89 Figure 2.77: Resultant of positive and negative regions of first and second-mode force distribution along height of wall. ............................................................................................ 95 Figure 2.78: Summary of expected demands under second-mode where F is actuator force and L is total length of specimen for: (a) Actuator force; (b) Bending moment; (c) Shear force. 96 Figure 2.79: Analytical results of free vibration behaviour of structure excited at period of second-mode for: (a) displacement; (b) bending moments; (c) shear forces; (d) equivalent static forces. ...................................................................................................................................... 99 Figure 2.80: An example of a period of second-mode displacement loading. ..................... 100 Figure 2.81: Experimental setup of cantilever shear wall. .................................................... 101 Figure 3.1: Scale of actual high-rise building to test specimen of cantilevered core wall. .. 104 Figure 3.2: Variation in specimen cross-section. .................................................................. 107 Figure 3.3: Cross-section of test region of specimen including location of reinforcing steel. ............................................................................................................................................... 109 Figure 3.4: Change in cross-section along length of specimen: (a) Without reinforcement detail; (b) With reinforcement detail. .................................................................................... 111 Figure 3.5: Split cylinder tensile strength test of concrete cylinder. .................................... 114 Figure 3.6: Test setup of tensile stress-strain test on 6 mm reinforcing steel. ...................... 116 Figure 3.7: Stress-strain relationship of the 6 mm diamter reinforcing steel for: (a) Entire strain range; (b) Up to 2.0% strain; (c) Up to 0.5% strain. ............................................................. 118 Figure 3.8: Stress-strain relationship of 10M reinforcing bar samples: (a) Up to 0.5% strain; (b) Up to 2.0% strain. ............................................................................................................ 120 Figure 3.9: Profile of grouted specimen inside steel pocket of reusable foundation ............ 122 Figure 3.10: Layout of steel pipes in grouted steel pocket of foundation. ............................ 123 Figure 3.11: Layout of plastic wrapped steel pipes in grouted pocket of foundation: (a) Before removal of pipes; (b) After removal of pipes. ....................................................................... 124 Figure 3.12: Photos of specimen formwork prior to placement of reinforcing steel. ........... 126 xvii Figure 3.13: The 6 mm reinforcing steel shapes used for reinforcement cages: (a) Stirrups; (b) Containment steel in flanges. ................................................................................................ 127 Figure 3.14: Photos of specimen reinforcement cages placed in formwork. ........................ 128 Figure 3.15: Specimens after stripping of formwork. ........................................................... 128 Figure 3.16: Drawing of side-view of test setup. .................................................................. 129 Figure 3.17: Side-view of test setup. .................................................................................... 130 Figure 3.18: Reinforced concrete foundation that was used for specimen: (a) Formwork with steel cage; (b) Finished foundation. ...................................................................................... 131 Figure 3.19: Connection between actuators and specimen. .................................................. 132 Figure 3.20: Photograph of the actuators being used in the experiment. .............................. 133 Figure 3.21: Out-of-plane support for specimens: (a) Design of support; (b) Photograph of sliding bearing around specimen. ......................................................................................... 134 Figure 3.22: Instrumentation layout for test setup. ............................................................... 135 Figure 3.23: LVDT used to measure specimen displacement. ............................................. 136 Figure 3.24: Shape of target placed on specimen. ................................................................ 137 Figure 3.25: Target layout of displacement targets over the height of the structure: (a) Position of targets; (b) Geometry of displacements measured from the target coordinates.. ............. 138 Figure 3.26: Single cycle of the sinusoidal imposed displacement protocol. ....................... 140 Figure 4.1: Crack pattern of Specimen 1 test region for various stages of damage: (a) early cracking; (b) severe cracking; (c) crack pattern at failure. ................................................... 143 Figure 4.2: Approximate location and width of measured diagonal cracks during final load stage for Specimen 1. ............................................................................................................ 144 Figure 4.3: Specimen 1 force-displacement results for: (a) base shear (Vb) vs mid-height displacement (D1); (b) mid-height moment (Mm) vs mid-height displacement (D1). ........ 145 Figure 4.4: Crack pattern of Specimen 2 test region for various stages of damage: (a) early cracking; (b) severe cracking; (c) crack pattern at failure. ................................................... 148 Figure 4.5: Approximate location and width of measured residual diagonal cracks during final load stage for Specimen 2. .................................................................................................... 149 Figure 4.6: Specimen 2 force-displacement results for: (a) base shear (Vb) vs mid-height displacement (D1); (b) mid-height moment (Mm) vs mid-height displacement (D1). ........ 150 xviii Figure 4.7: Crack pattern of Specimen 3 test region for various stages of damage: (a) early cracking; (b) severe cracking; (c) crack pattern at failure. ................................................... 153 Figure 4.8: Approximate location and width of measured diagonal cracks during final load stage for Specimen 3. ............................................................................................................ 154 Figure 4.9: Specimen 3 Phase 1 force-displacement results for: (a) base shear (Vb) vs mid-height displacement (D1); (b) base moment (Mb) vs top displacement (D2). ..................... 155 Figure 4.10: Specimen 3 Phase 2 force-displacement results for: (a) base shear (Vb) vs mid-height displacement (D1); (b) mid-height moment (Mm) vs mid-height displacement (D1). ............................................................................................................................................... 156 Figure 4.11: Displaced shape determined from displacement coordinates in selected Load Stages for Specimen 3: (a) Phase 1; (b) Phase 2. .................................................................. 159 Figure 4.12: Flexural displacement calculated from displacement coordinates in selected Load Stages for Specimen 3: (a) Phase 1; (b) Phase 2. .................................................................. 160 Figure 4.13: Curvature profile calculated from displacement coordinates in selected Load Stages for Specimen 3: (a) Phase 1; (b) Phase 2. .................................................................. 161 Figure 4.14: Shear parameters calculated from displacement coordinates during selected Load Stages for Specimen 3: (a) Shear displacement; (b) Shear strain. ........................................ 162 Figure 4.15: Horizontal displacement calculated from displacement coordinates in selected Load Stages for Specimen 3: (a) Displacement; (b) Strain. .................................................. 163 Figure 4.16: Crack pattern of Specimen 4 test region for various stages of damage: (a) early cracking; (b) severe cracking; (c) crack pattern at failure. ................................................... 166 Figure 4.17: Approximate location and width of measured diagonal cracks during final load stage for Specimen 4. ............................................................................................................ 167 Figure 4.18: Specimen 4 Phase 1 force-displacement results for: (a) (a) base shear (Vb) vs mid-height displacement (D1); (b) base moment (Mb) vs top displacement (D2). ..................... 168 Figure 4.19: Specimen 4 Phase 2 force-displacement results for: (a) base shear (Vb) vs mid-height displacement (D1); (b) mid-height moment (Mm) vs mid-height displacement (D1). ............................................................................................................................................... 169 Figure 4.20: Displaced shape determined from displacement coordinates in selected Load Stages for Specimen 4: (a) Phase 1; (b) Phase 2. .................................................................. 171 xix Figure 4.21: Flexural displacement calculated from displacement coordinates in selected Load Stages for Specimen 4: (a) Phase 1; (b) Phase 2. .................................................................. 172 Figure 4.22: Curvature profile calculated from displacement coordinates in selected Load Stages for Specimen 4: (a) Phase 1; (b) Phase 2. .................................................................. 173 Figure 4.23: Shear parameters calculated from displacement coordinates during selected Load Stages for Specimen 4: (a) Shear displacement; (b) Shear strain. ........................................ 174 Figure 4.24: Horizontal displacement calculated from displacement coordinates in selected Load Stages for Specimen 4: (a) Displacement; (b) Strain. .................................................. 175 Figure 4.25: Crack pattern of Specimen 5 test region for various stages of damage: (a) early cracking; (b) severe cracking; (c) crack pattern at failure. ................................................... 178 Figure 4.26: Approximate location and width of measured diagonal cracks during final load stage for Specimen 5. ............................................................................................................ 179 Figure 4.27: Specimen 5 Phase 1 force-displacement results for: (a) (a) base shear (Vb) vs mid-height displacement (D1); (b) base moment (Mb) vs top displacement (D2). ..................... 180 Figure 4.28: Specimen 5 Phase 2 force-displacement results for: (a) base shear (Vb) vs mid-height displacement (D1); (b) mid-height moment (Mm) vs mid-height displacement (D1). ............................................................................................................................................... 181 Figure 4.29: Displaced shape determined from displacement coordinates in selected Load Stages for Specimen 5: (a) Phase 1; (b) Phase 2. .................................................................. 183 Figure 4.30: Flexural displacement calculated from displacement coordinates in selected Load Stages for Specimen 5: (a) Phase 1; (b) Phase 2. .................................................................. 184 Figure 4.31: Curvature profile calculated from displacement coordinates in selected Load Stages for Specimen 5: (a) Phase 1; (b) Phase 2. .................................................................. 185 Figure 4.32: Shear parameters calculated from displacement coordinates during selected Load Stages for Specimen 5: (a) Shear displacement; (b) Shear strain. ........................................ 186 Figure 4.33: Horizontal displacement calculated from displacement coordinates in selected Load Stages for Specimen 5: (a) Displacement; (b) Strain. .................................................. 187 Figure 4.34: Comparison of base shear (Vb) vs mid-height displacement (D1) backbone curves Specimens 1 through 3. ......................................................................................................... 189 Figure 4.35: Comparison of base shear (Vb) vs mid-height displacement (D1) backbone curves Specimens 1 through 5. ......................................................................................................... 190 xx Figure 4.36: Comparison of crack patterns in Specimen test region, diagonal crushing for: (a) Specimen 1; (b) Specimen 2; (c) Specimen 3; (d) Specimen 4; (e) Specimen 5. ............ 191 Figure 4.37: Comparison of total displacement profile for Specimens 3 through 5 for Load Stage that corresponds with maximum observed base shear force. ...................................... 193 Figure 4.38: Comparison of flexural ductility demands for Specimens 3 through 5 for Load Stage that corresponds with maximum observed base shear force: (a) Flexural displacement; (b) Curvature. ........................................................................................................................ 194 Figure 4.39: Comparison of shear ductility demands for Specimens 3 through 5 for Load Stage that corresponds with maximum observed base shear force: (a) Shear displacement; (b) Shear strain. ..................................................................................................................................... 195 Figure 4.40: Comparison of horizontal ductility demands for Specimens 3 through 5 for Load Stage that corresponds with maximum observed base shear force: (a) Displacement; (b) Strain. ............................................................................................................................................... 196 Figure 5.1: Comparison of Response2000 moment-curvature relationship and approximate tri-linear model up to maximum curvature of : (a) 200 rad/km; (b) 50 rad/km; (c) 12 rad/km. 200 Figure 5.2: Comparison of various shear models used to analyze specimen results. ........... 204 Figure 5.3: Change in principle tensile stress along depth of cross-section for Response2000 model subjected to pure shear. .............................................................................................. 207 Figure 5.4: Change in stirrup stress along depth of cross-section for Response2000 model subjected to pure shear. ......................................................................................................... 207 Figure 5.5: Change in stress angle along depth of cross-section for Response2000 model subjected to pure shear. ......................................................................................................... 208 Figure 5.6: Change in concrete shear stress along depth of cross-section for Response2000 model subjected to pure shear; flange width is 175 mm; web with is 50 mm. ..................... 209 Figure 5.7: Change in steel shear stress along depth of cross-section for Response2000 model subjected to pure shear; flange width is 175 mm; web with is 50 mm. ................................ 210 Figure 5.8: Change in total shear stress along depth of cross-section for Response2000 model subjected to pure shear; flange width is 175 mm; web with is 50 mm. ................................ 210 Figure 5.9: Comparison of moment-shear interaction diagram to base shear vs estimated moment at height of diagonal concrete crushing. ................................................................. 213 xxi Figure 5.10: Prediction of load-deformation backbone results of base shear (Vb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 1. ............................................................................................................................................... 214 Figure 5.11: Prediction of load-deformation backbone results of base shear (Vb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 2. ............................................................................................................................................... 215 Figure 5.12: Prediction of load-deformation backbone results of base shear (Vb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 3. ............................................................................................................................................... 216 Figure 5.13: Prediction of load-deformation backbone results of base shear (Vb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 4. ............................................................................................................................................... 217 Figure 5.14: Prediction of load-deformation backbone results of base shear (Vb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 5. ............................................................................................................................................... 218 xxii Acknowledgements First and foremost, I would like to acknowledge my supervisor, Dr. Perry Adebar, who provided me with guidance, spent countless hours meeting with me, and never hesitated to help me work through any obstacles that arose over the completion of this thesis. Thank you to Dr. Carlos Molina Hutt for being the second reviewer of this thesis, and providing me with helpful feedback. I would also like to thank the lab technicians of the UBC Structural Engineering Lab, who were always willing to teach me new skills, provide insight into the design, help build the specimens and test setup, and aid with data acquisition and controlling the actuators during the testing phase. A special thank you to Lixin Xu, who was instrumental in acquiring and shipping the 6 mm diameter deformed steel reinforcing bars from China that were used to construct the specimens used in the experimental study. Finally, I would like to thank all my friends and family, especially my parents, for providing me with support not only throughout this thesis, but throughout my entire academic career. 1 1. Introduction ___________________________________________________________________________ 1.1 Background In the City of Vancouver, and increasingly elsewhere, one of the most common structural forms of high-rise buildings is reinforced concrete shear wall cores. Commonly, the centre of a reinforced concrete building will have a series of reinforced concrete walls built around central elevator and stair shafts. These interconnected walls, which form the core of the building, work together to resist the lateral demands from wind or earthquakes, and are commonly referred to as the lateral force resisting system (LFRS) of the building. Examples of two shear wall core buildings are shown in Figure 1.1, where the portion of the core containing the elevators can be seen protruding through the roof of the buildings. Figure 1.1: Examples of shear wall core buildings in the City of Vancouver. 2 For typical Vancouver high-rise buildings, the concrete walls are solid in one of the principal directions of the core and act as cantilever walls. In the perpendicular direction, the walls have large openings so that the wall piers adjacent to the openings act as separate walls coupled by the wall segments (coupling beams) above and below the openings. The current study is focussed on the cantilever-wall direction of typical high-rise cores. In a typical 30-story high-rise building in Vancouver, the ratio of cantilever wall height above the grade level to the wall length (horizontal dimension) varies between 10 and 12. The seismic design of the above-grade portions of such cantilever shear walls includes three main aspects: 1. Ensuring that at every level, the flexural resistance of the wall is greater than the flexural (bending moment) demands due to the earthquake. 2. Ensuring the displacement capacity of the wall is greater than the displacement demands due to the earthquake. 3. Ensuring that at every level, the shear resistance of the wall is greater than the shear force demands due to the earthquake. Each of these is discussed briefly below. The flexural resistance of a reinforced concrete wall depends on the level of axial compression applied to the wall and the amount of vertical reinforcement. Typical cantilever core walls in 30-story buildings are subjected to significant axial compression from dead weight of the building (floor slabs and the walls themselves) and therefore very little vertical reinforcement is typically required to provide sufficient flexural resistance in the cantilever-wall direction of the core. The quantity of vertical reinforcement required in a core is usually controlled by the coupled-wall direction. 3 Ensuring that a cantilever core wall has adequate displacement capacity is achieved by ensuring the maximum compression strain demands in the concrete (maximum near the base of the wall) is less than the compression strain capacity of unconfined or confined concrete, as the case may be. For typical 30-story reinforced concrete core wall buildings in Vancouver, no confinement reinforcement is typically required to achieve adequate displacement capacity. This is because when the concrete walls are interconnected in a box shape (core), the transverse walls act as compression “flanges” and greatly reduce the compression strain demands on the concrete. The most challenging aspect of the design of a typical slender cantilever core wall is ensuring the shear resistance is greater than the shear demands from the earthquake. The shear force demands are typically maximum near the base of the wall. As there are two different types of shear failure modes in reinforced concrete members, the shear resistance involves two different aspects. A sufficient quantity of horizontal reinforcement in the wall is required to avoid diagonal tension failures in the wall, while sufficiently low or low enough concrete shear stresses (principal compression stresses) are required to avoid diagonal compression failure. 1.2 Response Spectrum Analysis of 30-Story Core Wall Building The earthquake demands (bending moments, displacements, shear forces) on high-rise concrete core walls in Canada are normally determined using linear dynamic – response spectrum analysis (RSA). To illustrate how these demands typically look, an example analysis was done on the 30-story Vancouver building example provided in Section 11.5 of the current edition of the CAC Concrete Design Handbook (Mitchell, Paultre and Adebar, 2016). The 4 building was subjected to the 2015 National Building Code of Canada design spectrum for Vancouver spectrum Site Class C. A summary of the analysis results are shown below, while further details are presented in Appendix A. Figure 1.2 presents the displacements from the first five modes of the building. The displacement demands from Mode 4 and 5 are so small they are not visible in the figure. As the displacement demands from the first mode are so much larger than the displacements from the second-mode, the combination of the displacements (SRSS) from all modes considered is indistinguishable from the displacements from the first mode. Figure 1.3 presents the bending moment demands from the first five modes of the building as well as the SRSS sum of all modes (in green) and the SRSS sum of the first two modes (in red). At the base of the wall, the total bending moment due to all modes is only 7.7% larger than the bending moment demand due to the first mode. From 15 m above the base to 25 m above the base (17% to 28% of the height), the total bending moment demand is equal to the first mode bending moment demand. Above that level, the second-mode makes a significant contribution to the total bending moment. For example, at 60 m above the base (67% of the height), the second-mode bending moment is 66% larger than the first mode bending moment. It is interesting to note that only over the top third of the wall, does the third mode make a noticeable contribution to the total bending moment; however, it is a small contribution. 5 Figure 1.2: Total and individual modal displacement demands determined by RSA of a 30-story reinforced concrete core wall building in the cantilever wall direction. Figure 1.3: Total and individual modal bending moment demands determined by RSA of a 30-story reinforced concrete core wall building in the cantilever wall direction. 01020304050607080901000 1 2 3 4 5 6Height Above Grade (m)Displacement (m)Mode 1 Mode 2Mode 3 Mode 4Mode 5 SRSSSRSS (1,2)01020304050607080901000 200000 400000 600000 800000 1000000 1200000 1400000Height Above Grade (m)Moment (kN-m)Mode 1 Mode 2Mode 3 Mode 4Mode 5 SRSSSRSS (1,2)6 Figure 1.4 presents the shear force demands from the first five modes of the building, as well as the SRSS sum of all modes (in green), and the SRSS sum of the first two modes (in red). This figure clearly demonstrates how the shear force demand has very significant contributions from the higher modes. At the base of the structure where the shear force demands are largest, the total shear force demand from all modes is 78% larger than the shear forced demand from only the first mode. In fact, the shear force demand from the second-mode is significantly larger than the shear force from the first mode, and the shear force demand from the third mode is about half as large as the shear force from the first mode. Figure 1.4: Total and individual modal shear force demands determined by RSA of a 30-story reinforced concrete core wall building in the cantilever wall direction. 01020304050607080901000 5000 10000 15000 20000 25000 30000Height Above Grade (m)Shear (kN)Mode 1 Mode 2Mode 3 Mode 4Mode 5 SRSSSRSS(1,2)7 1.3 Estimating Seismic Demands in Nonlinear Buildings The linear dynamic (response spectrum) analysis results are used to make an estimate of the earthquake demands on a nonlinear building. The displacement demands are assumed equal to the displacement demands determined in the linear analysis. Thus, the effective flexural rigidity of the wall used in the linear analysis must account for the reduction in stiffness due to flexural cracking (Adebar and Dezhdar, 2015). The bending moment demands, and shear force demands are determined by dividing the elastic force demands by the ductility-based force reduction factor Rd and the overstrength-based force reduction factor Ro defined in the National Building Code of Canada (NRC, 2015). For typical 30-story reinforced concrete core wall buildings in Vancouver, Rd = 3.5 and Ro = 1.6 in the cantilever axis. Thus, the design bending moment and design shear force is determined by dividing the elastic forces by the product Rd Ro = 5.6. As 1/5.6 = 0.18, a typical 30-story core wall building in Vancouver is designed for bending moment demands and shear force demands equal to 18% of the elastic force demands in the cantilever axis. Numerous nonlinear analyses have shown that after a high-rise cantilever wall reaches its flexural strength capacity – a plastic hinge forms at the base of the wall – the shear force demands continue to increase, these studies have been summarized by (Rutenberg, 2013). Thus, the approach of reducing the shear force demands in proportion to the reduction in bending moment demand is not correct, and the shear force demands must be increased again. This phenomenon is commonly referred to as ‘accounting for the inelastic effects of higher modes on the shear force demand,’ or simply as the ‘dynamic amplification of shear force.’ 8 1.4 Estimating Shear Force Demands on Nonlinear Concrete Walls The first prominent study which addressed the issue of higher mode shear amplification was completed by Blakely et al. (1975). From this study it was determined that the higher mode shear value can be amended by multiplying the forces found using the static procedure by a higher mode amplification factor which can be calculated based on Eqs. 1.1 and 1.2. 𝜔𝑣 = 0.9 + 𝑛10 𝑓𝑜𝑟 𝑛 < 6 (1.1) 𝜔𝑣 = 1.3 + 𝑛30 ≤ 1.8 𝑓𝑜𝑟 𝑛 ≥ 6 (1.2) Where 𝜔𝑣 is the higher mode shear amplification factor, and 𝑛 is the number of story’s in the structure. These factors were implemented by the 1982 New Zealand Design Standards (NZS, 1982), and were included as reference information in the commentary of the 1994 version of CSA A23.3-94 (CSA, 1994). The approach taken by Ghosh (1990) was based on the assumption that the maximum base shear was linked with the peak ground acceleration, as can be seen in Eq. 1.3. 𝑉𝑚𝑎𝑥 = 𝑉𝑠 + 𝐷𝑚𝑊(𝑃𝐺𝐴) (1.3) Where 𝑉𝑚𝑎𝑥 is the maximum shear at the base, 𝑉𝑠 is the maximum shear force calculated using a static procedure (essentially the shear caused by the first mode of response), 𝐷𝑚 is a factor typically taken as equal to approximately 0.3, 𝑊 is the weight of the structure, and PGA is the peak ground acceleration. Research completed by Keintzel (1992) investigated the possibility that the shear forces did not increase linearly with the yield moment, or linearly with the number of stories in the structure, but instead was based on the intensity of the seismic input of the analysis. 9 Keintzel (1992) also found that higher mode shears can be rationalized by considering a linear elastic cantilever wall with a rotational spring at the base. When the rotational spring is infinitely stiff, the model represents a fixed base, but as the stiffness of the spring is reduced it behaves more like a post-yielded wall. While in the post-yielded state, it was observed that the first mode of response effectively disappeared, however the higher modes were still present in the response and did not significantly change. Priestly and Amaris (2003) attempted to use the forces from the modal analysis direction, as shown in Eq. 1.4. 𝑉𝑎 = √𝑉𝑦12 + 𝑉𝑒22 + 𝑉𝑒32 +⋯ (1.4) Where 𝑉𝑎 is the amplified shear, 𝑉𝑒𝑖 is the elastic shear from the ith mode, and 𝑉𝑦1is the shear associated with yielding in a first mode distribution. Pugh (2012) proposed a modification to the modal method (which assumes that only the shear from the first mode is reduced by yielding), in which only the shear from the mode that contributes the most to the demand is reduced by the ductility factor and showed that this agreed well with non-linear analyses. A comprehensive study of other models, included those discussed above was presented by Rutenberg (2013). 10 1.5 Shear Resistance of Concrete Walls The shear resistance of concrete walls is typically defined in code-based approaches as being the sum of the individual concrete and steel contributions to the total strength. The shear resistance of reinforced concrete shear walls is defined by ACI 318-14 (ACI, 2014) in Clause 18.10.4.1, shown in Eq. 1.5. 𝑉𝑛 = 𝐴𝑐𝑣(𝛼𝑐√𝑓𝑐′ + 𝜌𝑡𝑓𝑦) (1.5) Where 𝑉𝑛 is the shear resistance, 𝐴𝑐𝑣 is the effective shear area of the cross-section, 𝛼𝑐 is a coefficient that defines the relative contribution from the concrete compressive strength to the shear strength, 𝑓𝑐′ is the concrete compressive strength, 𝜌𝑡 is the transverse steel reinforcing ratio, and 𝑓𝑦 is the transverse steel yield stress. The coefficient 𝛼𝑐 is based on the height to length ratio of the wall and is equal to 2 for walls with a ratio greater than 2, which is only applicable when the units for stress are in psi. The approach taken by CSA 23.3 is similar, where the total shear strength is a contribution of both the concrete and steel components, which are given in Eqs. 1.6 and 1.7 respectfully and are found in Clauses 11.3.4 and 11.3.5.1. 𝑉𝑐 = 𝛽√𝑓𝑐′𝑏𝑤𝑑𝑣 (1.6) 𝑉𝑠 = 𝐴𝑣𝑓𝑦𝑑𝑣𝑐𝑜𝑡𝜃𝑠 (1.7) Where 𝑉𝑐 is the concrete contribution to the shear strength, 𝑉𝑠 is the steel contribution to the shear strength, 𝛽 is a factor to account for the shear resistance of cracked concrete, 𝑏𝑤 is the web thickness, 𝑑𝑣 is the effective shear depth, which for a wall shall be taken as 80% of the 11 wall length, 𝐴𝑣 is the area of each stirrup, 𝑠 is the stirrup spacing, and 𝜃 is the angle of diagonal compressive stress, where all units are in metric. According to seismic design provisions for ductile shear walls given in Clause 21.5.9 of CSA A23.3-14, which accounts for the shear resistance of walls in the plastic hinge region, the 𝛽 coefficient must be assumed to be 0, unless the inelastic rotational demand on the wall is less than 0.015 radians. If the inelastic rotational demand is shown to be less than 0.005 radians, 𝛽 can be assumed to be 0.18, while for inelastic rotational demands between 0.015 and 0.005 the value for 𝛽 is determined by linear interpolation. This clause also specifies that the compression stress angle 𝜃 be taken as 45 degrees, unless the axial compression force acting on the wall is greater than 0.1𝑓𝑐′𝐴𝑔 (CSA, 2014). It should be noted however that all the code shear design approaches were developed based on experimental data obtained from specimens tested under pseudo-static loading, which is not reflective of seismic loading, where the shear force is applied very rapidly, and reverses immediately. 1.6 Objective and Methodology of Current Study The objective of the current thesis was to investigate the shear resistance of reinforced concrete cantilevered shear walls subjected to the typical shear demands from an earthquake. One approach that could be used to achieve this would be to test scaled models of concrete cantilever walls on a shake table that simulates the ground motion of an earthquake. The approach that was chosen for the current study was to use high-speed hydraulic actuators to 12 apply the earthquake demands on fixed-base cantilever wall models, and to use the results from nonlinear analysis to determine a suitable testing protocol for the cantilever wall models. The results from linear dynamic analysis in Figures 1.2 and 1.3 clearly indicate that the displacements of a cantilever wall are entirely first mode, and the bending moments are mostly first-mode. Thus, the flexural response of cantilever walls and the drift capacity of cantilever walls has been determined using fixed-based cantilever wall models subjected to earthquake demands using a single hydraulic actuator. See for example Orakcal and Wallace (2006), Birely (2012), and El-Azizy et al. (2015). Figure 1.4 indicates that the first and second-modes contribute the majority of the linear shear demands on a 30-story cantilever wall. The combination of shear demands from all modes (green line) exceeds the combination of shear demands from only the first and second-mode (red line) by only 7.7% at the base of the wall where the shear demands are maximum. Thus, the approach used in the current study to investigate the shear resistance of walls involved fixed-base cantilever wall models subjected to earthquake demands using two hydraulic actuators, one at the top of the wall and one at the mid-height. To determine the appropriate testing protocol for the cantilever wall models subjected to earthquake demands by two hydraulic actuators, a series of nonlinear analyses were conducted on a cantilever shear wall with two concentrated masses over the height. 1.7 Overview of Thesis Chapter 2 presents the results of a non-linear dynamic analysis of a high-rise concrete shear wall using the software OpenSees. In order to focus on the contributions from the first two 13 modes of vibration, the wall was modelled with two concentrated masses – one at the top of the wall and one at the mid-height of the wall. Results presented include displacements, shear forces, bending moments, as well as the curvature distribution along the height of the wall at various phases of non-linearity. The results of the study were used to determine the displacement protocol that was used in the experimental study. Chapter 3 presents a full description of the experimental program that was carried out. All details relating to the testing approach, specimen design, material details, test setup, instrumentation, and testing procedure can be found in this chapter. Chapter 4 presents the results of the experimental study and discusses the significant observations of the tests. While this chapter gives an overview of the results, the bulk of the data and figures are available in Appendix E. Chapter 5 contains discussions related to theories and predictive models relating to the specimen test results, using analytical tools that are common amongst current code and design guide standards. Chapter 6 presents conclusions related to all work presented throughout the thesis, including the non-linear dynamic analysis, the experimental program, and analysis of the experimental results. In addition, recommendations are made for future research related to this topic. 14 2. Dynamic Analyses of Simplified Shear Wall Core Model ___________________________________________________________________________ 2.1 Overview This chapter summarizes the dynamic analyses that were conducted on a simplified model of a high-rise cantilever shear wall, to gain insight into the behaviour of cantilever walls subjected to first and second-mode demands. Section 2.2 presents a description of the simplified cantilever shear wall model. Section 2.3 summarizes the results of the dynamic analyses on a linear-elastic model of the prototype wall, while Section 2.4 gives the results from the non-linear model. Section 2.5 presents a summary of the analytical results, as well as what conclusions were drawn from the analyses. Finally, Section 2.6 summarizes how the conclusions of the dynamic analyses influenced the design of the experimental program. 2.2 Description of Analytical Model 2.2.1 Overview of Analysis Approach A model of a high-rise concrete building may include many different members (shear walls, gravity columns, slabs, and substructure); with distributed mass along the height of the structure. The model is typically subjected to a suite of actual ground motions representative of the earthquake shaking intensity. For the current analytical study, a much simpler approach was taken. Firstly, only the concrete walls (core) resisting the lateral demands were included in the model. The gravity-load frame was not included in the model as is often done when investigating the collapse limit state of a core wall building. 15 The analytical study was meant to inform the experimental study that will be conducted on cantilever wall models subjected to lateral demands using only two hydraulic actuators. The results from the linear dynamic response spectrum analysis presented in Chapter 1 demonstrated that a large portion of the shear demands on a 30-story cantilever wall is due to the first and second-modes. Thus, the analyses conducted on the shear wall model was also done using only two concentrated masses in order to limit the number of modes to two for simplicity. Figure 2.1 shows the prototype shear wall core building that was modeled. Figure 2.1: Prototype building modeled for dynamic analyses. Height – 90 m Mass – 500 (103) kg/story Period – 3.71 s 16 There are two elements of equal height used to represent the shear wall core, where the height of each element was 45 m, giving the model a total height of 90 m. Based on the equivalent static force profile of the second-mode for a distributed mass model of only the core of the prototype building, it was determined that the resultant of the positive forces was located at 45% of the height, and the resultant of the negative forces was located at 91% of the height. However, to simplify the model, a total height of 90 m was used, with a concentrated mass located at the mid-height and top of the building, connected by the two elements. The total mass of the prototype building is given in the example as 500 x 103 kg/story, which gives a total mass of 15.5 x 106 kg. This mass was originally assumed to be divided evenly between the two nodes giving each concentrated mass a value of 7.75 x 106 kg. As the periods are based on the initial stiffness, the elastic modulus of the concrete was originally assumed to be 5000√𝑓𝑐′, which gave a reasonable estimate of the period. The values of initial stiffness, and distribution of the mass between the two nodes were then iterated until a first mode period of 3.71 s was obtained, and a second-mode period of 0.62 s, which is approximately 1/6th of the first mode period. The second-mode period is not given in the example, but the ratio between the first and second-mode periods for cantilever shear wall core buildings is commonly taken as 6:1. The heights of the elements were never iterated, and the final values used in the model were masses of 8.05 x 106 kg and 7.45 x 106 kg at the mid-height and top nodes respectively, and an initial elastic modulus of approximately 5230√𝑓𝑐′, which was equal in both elements. This is summarized in Figure 2.2, and further details are available throughout this section. 17 Figure 2.2: Simplified model of cantilever walls used in the current study. Earthquake ground motions are very complex as they include numerous wavelets with many different frequencies that can affect the behaviour of the structure. To further simplify the analyses, the simple model of the cantilever wall was subjected to sinusoidal ground acceleration and the response of the cantilever wall was studied during the free vibration. 2.2.2 Shear Wall Core Cross-Section The cross-section of the model was defined based on the shear wall core of the 30-story residential building in Vancouver example from section 11.5 of the current edition of the CAC Concrete Design Handbook (Mitchell, Paultre and Adebar, 2016) and can be seen in Figure 2.3. The shear-wall core is a generic example of a typical structure found in the city of Vancouver and therefore made for an ideal choice for this analysis. Only the first and second-mode demands in the cantilever direction will be tested experimentally, therefore the analysis 1218 was only completed in the cantilevered axis of this cross-section on a two-dimensional model, so the effects of the coupling beams are not considered. Figure 2.3: Cross-Section of prototype shear wall core used for non-linear model. Due to the fact that the prototype core is made up of three separate wall cross-sections, and that the forces will only be applied in the cantilevered direction, an idealized cross-section was derived to give very similar results, while significantly simplifying the analysis. The idealized cross-section has the same area, moment of inertia, and steel reinforcing ratio (in the flange and web respectively) but will allow the core to be analyzed as one cohesive cross-section. The properties that were used to create the idealized section, can be seen in Table 2.1, while the idealized I-beam core can be seen in Figure 2.4. 19 Table 2.1: Properties of prototype shear wall core and idealized I-beam cross-section. Ig (mm4) 1.65E+14 Ag (mm2) 1.55E+07 As,f (mm2) 5.40E+04 As,w (mm2) 2.92E+04 Reinforcing Ratio (%) 0.885 (a) (b) Figure 2.4: Idealized cross-section of prototype shear wall core created in: (a) Response2000; (b) OpenSees (all units in mm). The moment-curvature diagram of the cross-section, assuming an axial load of 10% of the capacity (0.1𝐴𝑔𝑓𝑐′) which is equal to roughly 6.2 (105) kN, can be seen in Figure 2.5, and was created using both Response2000 (Bentz, 2000), as well as by using OpenSees. While the analyses will be completed in OpenSees, the similarity of the two moment-curvature diagrams validates that the wall was modelled properly. The first diagram shows the entire curve up to ultimate capacity, while the other shows only up to the yielding, which begins at a curvature 20 of just above 0.33 rad/km. The values which differentiate each range of the phases of the response can be seen in Table 2.2. The ultimate moment-capacity is roughly 586,000 kN-m and a curvature of 6.15 rad/km, which is within 5% of the moment-capacity given in the prototype shear wall which is 560,000 kN-m, which validates that the assumptions made for the mass and idealization of the cross-section were sufficient. (a) (b) Figure 2.5: Moment-Curvature relationship of idealized 30-story building cross-section: (a) Full moment-curvature response; (b) Linear and cracked phases of relationship. 01000002000003000004000005000006000007000000 1 2 3 4 5 6 7Moment (kN-m)Curvature (rad/km)OpenSees CyclicResponse20000500001000001500002000002500003000003500004000004500005000000.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Moment (kN-m)Curvature (rad/km)OpenSees CyclicResponse200021 Table 2.2: Points of interest in moment-curvature relationship of 30-story building idealized cross-section. Moment (kN-m) Curvature (rad/km) Phase 250,000 0.043 Cracking 450,000 0.330 Initial Yielding 490,000 0.800 Fully Yielding 586,000 6.150 Ultimate The cross-section created in OpenSees was discretized using 20 area fibres for the web, and 10 area fibres in each flange. The steel reinforcement was discretized using point fibres with a pre-defined area. Information regarding the material models applied to the fibres will be presented further on within this subsection. 2.2.3 Non-Linear Fibre Section Elements The model consists of two non-linear fibre section force beam-column elements, with four integration points in each element. Forced-based derived elements were chosen, as they give a good estimate of the curvature along the length of the element, even when a low number of elements are used, as the curvature is calculated at each integration point. The other alternative would have been to use of displacement-based elements, where the curvature is assumed to be linear along the length of the element, meaning that many more elements are required in order to give a reasonable and accurate estimation of the curvature distribution (UC Berkeley, 2017). Displacement-based elements were not an option as it was required that the model will consist of only two elements in order to limit the number of mode shapes, therefore if they were used, the curvature distribution would not have been estimated accurately. As was determined in Chapter 1, only the first two modes need to be simulated to approximate most of the higher mode demands, and experimentally only two modes can be simulated, based on the number of 22 actuators that will be used, therefore a model was created where only the first two modes of vibration will contribute to the response. Four integration points were used as it was determined to give an accurate estimation of the curvature along the element, without causing localization of the curvature at the base. Use of too many integration points can lead to overestimates of the curvate, as the model will try to achieve the same rotation along a shorter length. This can lead to problems regarding estimating the yielding at the base of the structure. 2.2.4 Model Parameters The material models used were Concrete02 and Steel02 for the concrete and reinforcing steel respectively, and visualizations of these material models can be seen in Figure 2.6. The uniaxial material model relationships are applied at each fibre along the cross-section. For the concrete, the flanges were each discretized into 20 fibres, while the web was discretized into 10. The steel fibres are located at the location of the steel reinforcement, which were shown in the cross-section in Figure 2.3. For Concrete02, the values chosen were those that would be typical for a 30-story residential building, the compressive strength, 𝑓𝑐′, was set to 40 MPa, and the initial compressive modulus, 𝐸𝑐, was set equal to 5230√𝑓𝑐′. The reason the value used is so high is because the initial slope of the concrete stiffness was altered until the proper period was achieved, however the initial concrete stiffness is commonly taken somewhere between 4500√𝑓𝑐′ to 5500√𝑓𝑐′, therefore it is still a safe assumption. The peak concrete strain, 𝜀0, was determined using Eq. 2.1, and the 23 pre-peak compressive stress, 𝑓𝑐, is calculated using the Hognestad parabolic relationship, which can be seen in Eq. 2.2. 𝜀0 = 2𝑓𝑐′𝐸𝑐 (2.1) 𝑓𝑐 = 𝑓𝑐′ (2𝜀𝜀0− (𝜀𝜀0)2) (2.2) As has been mentioned, the purpose of the model was to remain simple, and gain a broad view of how the first and second-mode demands affect the response, which is the reason simple material models were used, as well as why no distinction was made between confined and unconfined concrete in the cross-section. For the Steel02 model, the yield stress 𝑓𝑦, was taken equal to 400 MPa, the elastic modulus, 𝐸𝑠, was taken equal to 200 GPa, and the strain hardening coefficient is set to 2%, which are all typical values assumed for reinforcing steel. (a) (b) Figure 2.6: Material models used in model: (a) Concrete02; (b) Steel02 (UC Berkeley, 2017). 24 The shear stiffness for the model was simulated using the multi-linear material model in OpenSees and was applied to the existing elements using the section aggregator function. The shear stiffness was developed using a tri-linear model based on recommendations by Oyen (2006), where the cracking stress is determined by Eq. 2.3 and the yield stress determined by Eq. 2.4. The shear stiffness of the uncracked section has a slope of the shear modulus of the concrete, G, while the fully cracked stiffness is equal to 10% of G. 𝑣𝑐𝑟 =16√𝑓𝑐′ (2.3) 𝑣𝑦 =23√𝑓𝑐′ (2.4) Figure 2.7: Stress-strain relationship of shear stiffness model. Lumped masses were placed at the middle and top nodes of the model; however, the gravity load was only applied at the top node so that the stiffness contribution from axial loads was did not vary in the two elements. OpenSees allows the user to define the mass and gravity load independently of each other so that this can be accomplished. The axial load was set equal to 10% of the axial resistance of the wall (0.1𝑓𝑐′𝐴𝑔), while the two masses were set equal to 8.05 (106) kg and 7.45 (106) kg at the mid-height and top node respectively. These two masses sum 00.511.522.533.544.50 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035Shear Stress (MPa)Shear Strain 10.1G25 to give a total of 15.5 (106) kg, which is the total mass of the structure given in the reference CAC example. The two masses were proportioned such that the period of the second-mode would be exactly 1/6th the period of the first mode, which is what would typically be expected for cantilevered shear wall core buildings. Figure 2.8: Visualization of lumped masses and gravity load applied to model. The modal participation factors and modal effective masses can be seen in Table 2.3. The second-mode accounts for 21% of the mass participation, highlighting its importance to the response of the structure. In order to determine how each mode contributes to the shear and moment force response of the structure, the shear and moment “mode shapes” were found from the forces at each node, while the forces were found by multiplying the mode shapes (displacements) by the stiffness matrix. The displacement, force, shear force, and bending moment modes can be seen in 𝑡 𝑡𝑎 𝑡 𝑡𝑎 𝑃 = 0.1𝑓𝑐′𝐴𝑔26 Table 2.4, where it should be noted that the displacement and force modes are roughly equal, because the stiffness of both elements is equal. A visualization of the displacement/force, shear force, and bending moment profiles can be seen in Figure 2.9, where each profile was normalized such that the largest value is 1. The displacement and force modes are shown in the same figure as they are approximately equal. Table 2.3: Modal participation factors and effective masses of model. Mode Participation Factor Effective Mass (kg 106) Mass Participation (%) 1 110.53 12.22 0.79 2 57.31 3.28 0.21 Table 2.4: Idealized “modes” of model for: (a) Displacement; (b) Forces; (c) Shear forces; (d) Bending moments. Displacement Forces Location Mode 1 Mode 2 Location Mode 1 Mode 2 Roof 1.00 -0.35 Roof 1.00 -0.33 Mid-Height 0.33 1.00 Mid-Height 0.35 1.00 (a) (b) Shear Force Bending Moment Location Mode 1 Mode 2 Location Mode 1 Mode 2 Element 2 0.74 -0.48 Mid-Height 0.43 -0.93 Element 1 1.00 1.00 Base 1.00 1.00 (c) (d) 27 (a) (b) (c) Figure 2.9: Visualization of modes of wall for: (a) Displacement/forces; (b) Bending moment; (c) Shear force. The damping in the model was defined using Rayleigh damping, as this is the only form of damping which can be implemented using OpenSees. Rayleigh damping can be used to set the damping ratio at two periods, which in the case of this model is not a problem as only the first two modes will be contributing to the response of the model. However, in the dynamic non-linear analysis the periods will elongate as the structure softens, and thus the damping ratios will slightly change. The damping ratio was set equal to 5%, at the point of the second modal period and at 1.5 times the first modal period as can be seen in Figure 2.10. This was done so that when the period elongates, the damping will not be unrealistically high or low. 12-1 -0.5 0 0.5 1Displacement/ForceMode 1 Mode 2Ele. 2Ele. 112-1 -0.5 0 0.5 1Bending MomentMode 1 Mode 2Ele. 1Ele. 212-1 -0.5 0 0.5 1Shear ForceMode 1 Mode 2Ele. 2Ele. 128 Figure 2.10: Relationship between period of response and damping ratio. In real structures, P-delta effects are always present and should be accounted for. However, the purpose of this analysis is not only to study how the first and second-mode demands affect the behaviour of the response, but to also guide the development of the experimental protocols. Due to the fact that P-delta effects will not be present in the experimental set up, they will not be used during the analysis either. In order to safely make this assumption, it is important to check if the P-delta effects would have a large impact on the response of the structure in both the linear and non-linear responses. 0123456789100 1 2 3 4 5 6Damping Ratio (%)Period (s)T2 1.5T129 (a) (b) Figure 2.11: Effect of P-delta on the response of the model: (a) Linear; (b) Non-linear. In Figure 2.11, the linear and non-linear responses are shown respectively. The P-delta effects do not affect the response of the structure significantly in either case, therefore the results of the analysis will not be significantly influenced by ignoring P-delta effects. -3000000-2000000-10000000100000020000003000000-1200 -900 -600 -300 0 300 600 900 1200Base Moment (kN-m)Top Displacement (mm)P-DeltaNo P-Delta-600000-400000-2000000200000400000600000-800 -600 -400 -200 0 200 400 600 800Base Moment (kN-m)Top Displacement (mm)P-DeltaNo P-Delta30 2.2.5 Applied Pulse To excite the model, a sinusoidal pulse ground acceleration was applied to the base. The period of the pulse varied such that the two modes would be excited in different ways. The periods of the pulses applied to the model ranged from 0.25 to 4 seconds, and the periods of the first and second-modes are within this range at approximately 3.71 and 0.62 seconds respectively. The pulse goes through three cycles before coming to a stop and allowing the model respond in free vibration. The choice to use a pulse as opposed to ground motions for the analysis was made to allow greater control over the period at which the model is excited could be had, in order to analyze the specific effect this had on the behaviour. Although it would be more realistic to use ground motions, they are made up of many different frequency contributions which can have effects on the behaviour that are difficult to predict. An example of a pulse intended to activate the second-mode, and one intended to trigger the first mode only, can be seen in Figure 2.12. Figure 2.12: Examples of ground accelerations: (a) Period of 0.62 s; (b) Period of 3.71 s. 0 1 2 3 4 5 6 7 8 9 10 11 12Acceleration (g)Time (s)Tp = T1Tp = T2Free VibrationFree VibrationT1 = 3.72T2 = 0.6231 2.3 Results from Linear Model The response of the structure was examined when subjected to a suite of sinusoidal ground acceleration pulses with varying periods. This was completed in order to determine how much influence the presence of the second-mode would have on the response of the structure. It is expected that if the period of the sinusoidal pulse is roughly equal to the period of the second-mode, then the second-mode of vibration will be excited, thus showing second-mode amplification of the resulting demands. It is also expected that when the structure is excited at a period close to that of the first mode, there will not be any influence on the response from the second-mode, however large demands will be observed to due resonance of the structure. In order to determine the influence that each mode has on the response of the structure, the response must be decomposed into its two modal contributions at each time step, which is explained in Subsection 2.3.1. 2.3.1 Modal Decomposition A structure in free vibration will respond in a way such that its response can be decomposed into its modal contributions, so long as the response is in the linear elastic range, this is a concept known as modal superposition. This decomposition process can be seen in Equations 2.5 through 2.8, where D1 and D2 are the displacements at the first node (mid-height) and second node (top) respectively, α and β are scaling factors for the first and second-mode respectively, and ϕij is the modal coordinate of the ith node in the jth mode shape. 𝐷1 = 𝛼𝜑11 + 𝛽𝜑12 (2.5) 𝐷2 = 𝛼𝜑21 + 𝛽𝜑22 (2.6) 32 [𝐷1𝐷2] = [𝜑11 𝜑12𝜑21 𝜑22] [𝛼𝛽] (2.7) [𝛼𝛽] = [𝜑11 𝜑12𝜑21 𝜑22]−1[𝐷1𝐷2] (2.8) This process can be explained easily by walking through a brief example, of which the results can be seen in Figure 2.13. In this example we can see the nodal displacements of the total response at a given time step are 0.6 and 0.8 at the mid-height (D1) and top nodes (D2) respectively. By multiplying this vector by the inverse of the modal matrix, as shown in Eq. 2.8, we are left with the following values for the alpha and beta scaling factors which can be seen in Eq. 2.9. [𝛼𝛽] = [0.33 11 −0.35]−1[0.970.34] = [0.610.77] (2.9) From this calculation we can see that the first mode must be scaled by a factor of 0.91, and the second-mode must be scaled by a factor of 0.30. Now that the values of alpha and beta are known, the modal components of the current time step can be found using the following equations, where Dij is the contribution to the displacement at node i from mode j. [𝐷11𝐷21] = 𝛼 [𝜑11𝜑21] (2.10) [𝐷12𝐷22] = 𝛽 [𝜑12𝜑22] (2.11) Once this calculation is completed, we are left with two vectors, which are the modal contributions of the first and second-mode and will combine to equal the exact displacement profile of the structure, which is 0.97 and 0.34. A simple visualization of the response, as well as the two modal components, can be seen in Figure 2.13. 33 Figure 2.13: Example of decomposing an arbitrary displacement vector into modal components at a given time step. Table 2.5: Results of modal decomposition example. Location Total Mode 1 Mode 2 Node 2 0.97 0.20 0.77 Node 1 0.34 0.61 -0.27 When this process is completed at each time step, we can plot the response of the two modal components over time and see that we are left with a very clear visualization of how the two modes of vibration contribute to the total response, as can be seen in Figure 2.14, where the displacement response versus time of the node at the mid-height of the model can be seen. Throughout this thesis, the actual combined response of the two modal components will be referred to as the “total” response, and the first and second-mode responses are referred to by their respective names. This figure leads into the next subsection of this chapter, which will examine in further detail how the two mode shapes contribute to the total response in the linear -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1Total Response-1 -0.5 0 0.5 1=+First Mode ResponseSecond Mode Response34 elastic range. In this figure it can be seen where the ground acceleration pulse ends at roughly 1.8 seconds, and how the second-mode contribution to the response becomes negligible at about 9 seconds, or 2 full cycles of first mode response. Figure 2.14: Example of decomposition of total response into two modal components. In addition to decomposition of the displacement response of the mode, the mode shear force, bending moment, and equivalent static force responses can be decomposed as well. The equivalent static forces can be calculated from the bending moment or shear responses of the structure at each time step. Equations 2.10 through 2.13 will show how they can be calculated based on the shear forces, and the bending moments, where F1 and F2 are the mid-height and top equivalent point load forces, Vb and Vt, which are the shear force in the base and top element, and Mb and Mm, which are the bending moments at the base and mid-height respectively. -15-10-5051015201.8 2.8 3.8 4.8 5.8 6.8 7.8D1(mm)Time (s)TotalMode 1Mode 235 𝐹1 = 𝑉𝑏 − 𝑉𝑡 (2.10) 𝐹2 = −𝑉𝑡 (2.11) 𝐹1 =2(𝑀𝑏 − 2 ∗ 𝑀𝑚) ℎ (2.12) 𝐹2 =2 ∗ 𝑀𝑚 ℎ (2.13) 2.3.2 Linear Response when Excited at Period of First Mode The first results shown are the linear responses of the structure when excited by a sinusoidal ground acceleration pulse with a period equal to that of the first mode. Figure 2.15 shows the displacement responses of both the top and mid-height nodes of the structure. Figure 2.15: Displacement response at top and mid-height nodes for structure excited at period of first mode in the linear elastic range. In Figure 2.16, the modal displacement response of each node can be seen, which shows that the entire response is composed entirely of the first mode. The first 11 seconds of the response are while the structure is being excited (the excitation is three periods long, at 3.71 seconds per -80-60-40-200204060801000 2 4 6 8 10 12 14 16 18 20Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)36 cycle), and after this the structure is in free vibration where it will continue to oscillate until it comes to rest. (a) (b) Figure 2.16: Modal displacement response for structure excited at period of first mode in the linear elastic range at: (a) top node; (b) mid-height node. In Figure 2.17, the bending moment response of the structure is shown, while Figure 2.18 shows the modal decomposition of the bending moment at the base, and at the mid-height respectively. Similarly to the displacement response, the bending moment response appears to be behave completely in the first mode of vibration. Figure 2.17: Bending moment response of structure excited at period of first mode in the linear elastic range. -80-60-40-2002040608010011 13 15 17 19D2(mm)Time (s)TotalMode 1Mode 2-30-20-10010203011 13 15 17 19D1(mm)Time (s)TotalMode 1Mode 2-200000-150000-100000-500000500001000001500002000000 2 4 6 8 10 12 14 16 18 20Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)37 (a) (b) Figure 2.18: Modal bending moment response for structure excited at period of first mode in the linear elastic range at: (a) Base node; (b) Mid-height node. In Figure 2.19, the shear force response of the structure is shown, while Figure 2.20 shows the modal decomposition of the shear force in the base and top elements respectively. During the excitation phase of the first 11 seconds, it can be seen that the shear force peaks in the two elements are slightly out of phase, but once the structure is in free vibration the peaks are in phase. Figure 2.19: Shear force response of structure excited at period of first mode in the linear elastic range. -200000-150000-100000-5000005000010000015000020000011 13 15 17 19Mb(kN-m)Time (s)TotalMode 1Mode 2-80000-60000-40000-2000002000040000600008000010000011 13 15 17 19Mm(kN-m)Time (s)TotalMode 1Mode 2-3000-2500-2000-1500-1000-500050010001500200025000 2 4 6 8 10 12 14 16 18 20Shear (kN)Time (s)Base (Vb)Top (Vt)38 While the response is again almost entirely in the first mode, the very beginning of the response in the free vibration phase shows an extremely small contribution from the second-mode, as the line is not completely flat, however this is damped out of the response after only a few seconds. (a) (b) Figure 2.20: Modal shear force response for structure excited at period of first mode in the linear elastic range at: (a) Base element; (b) Top element. In Figure 2.21, the equivalent static force response of the structure is shown, while Figure 2.22 shows the modal decomposition of the equivalent static force at the top and mid-height nodes respectively. Similar to the shear force response, the equivalent static force is also slightly out of phase during the excitation phase. In Figure 2.22 (b), the modal decomposition of the mid-height force shows a very clear contribution from the second-mode, between 11 and 15 seconds. This shows that even when the structure is excited at the exact period of the first mode, there is still a very slight influence from the second-mode, even in the linear elastic range. -3000-2500-2000-1500-1000-5000500100015002000250011 13 15 17 19Vb(kN)Time (s)TotalMode 1Mode 2-2000-1500-1000-500050010001500200011 13 15 17 19Vt(kN)Time (s)TotalMode 1Mode 239 Figure 2.21: Equivalent static force response of structure excited at period of first mode in the linear elastic range. (a) (b) Figure 2.22: Modal equivalent static force response for structure excited at period of first mode in the linear elastic range at: (a) Top node; (b) Mid-height node. Figure 2.23 shows the modal trend of each piece of the response of the structure. The slope of the line is equal the mode shape of the response. The “total” response is the output of the OpenSees analyses, and the slope represents a real modal trend. However, the first and second-mode response lines are calculated based on a pre-determined modal decomposition. Later it will be shown how using the slope of the real trend can aid in correcting the mode shape as the structure is pushed into the non-linear range. For the displacement, bending moment, and shear -2000-1500-1000-50005001000150020000 2 4 6 8 10 12 14 16 18 20Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-2000-1500-1000-500050010001500200011 13 15 17 19F2(kN)Time (s)TotalMode 1Mode 2-600-400-200020040060080011 13 15 17 19F1(kN)Time (s)TotalMode 1Mode 240 force response, the response is linear and follows the first mode trend line, which demonstrates that it is the dominant mode and there is little to no contribution from the second-mode, however the equivalent static force has a slight influence from the second-mode. (a) (b) (c) (d) Figure 2.23: Trend of modal responses for structure excited at period of first mode in the linear elastic range for: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static force. y = 0.3257x-30-20-100102030-100 -50 0 50 100D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.4252x-80000-60000-40000-20000020000400006000080000100000-200000 -100000 0 100000 200000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.74x-2000-1500-1000-5000500100015002000-3000 -2000 -1000 0 1000 2000 3000Vt(kN)Vb (kN)TotalMode 1Mode 2 y = 0.352x-600-400-2000200400600800-2000 -1000 0 1000 2000F1(kN)F2 (kN)TotalMode 1Mode 241 2.3.3 Linear Response when Excited at Period of Second-mode The structure was then excited at the period of the second-mode, where it is expected that the response of the structure will be influenced by the second-mode of vibration. However, it was observed that even when excited at the exact period of the second-mode, the structure responds in a combination of both the first and second-mode. In Figure 2.24, we can see the displacement response of the top and mid-height nodes of the structure. It should be noted that the influence of the second-mode is very clear in the first cycle of the response, however the structure begins to respond exclusively in the first mode after only a few cycles, when the second-mode damps out. This is because the second-mode is oscillating six times faster than the first mode, therefore it is expected that it will damp out six times faster. Figure 2.24: Displacement response at top and mid-height nodes for structure excited at period of second-mode in the linear elastic range. In Figure 2.25, we can see the modal response of the top and mid-height displacement of the structure. The response is only shown for the first full cycle of free vibration, as we are only interested in the second-mode contribution to the response, which damps out soon after. The top displacement is influenced only slightly by the second-mode, while the second-mode makes a much larger contribution to the mid-height displacement. The contribution from the -40-30-20-10010203040500 1 2 3 4 5 6 7 8 9 10Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)42 second-mode increases the maximum displacement at the mid-height node by approximately 50%. (a) (b) Figure 2.25: Modal displacement response for structure excited at period of second-mode in the linear elastic range at: (a) Top node; (b) Mid-height node. Figure 2.26 shows the bending moment response of the structure at the base and mid-height nodes. From this figure the second-mode plays a very large role in the force demands imposed on the structure. The influence is very evident in the first few second of free vibration, but the first moment again dominates the response once the second-mode damps out. Figure 2.26: Bending moment response at base and mid-height nodes for structure excited at period of second-mode in the linear elastic range. -40-30-20-10010203040501.8 2.8 3.8 4.8 5.8 6.8 7.8D2(mm)Time (s)TotalMode 1Mode 2-15-10-5051015201.8 2.8 3.8 4.8 5.8 6.8 7.8D1(mm)Time (s)TotalMode 1Mode 2-150000-100000-500000500001000001500002000000 1 2 3 4 5 6 7 8 9 10Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)43 Figure 2.27 shows that the base moment is roughly equally influenced by the first and second-mode, as the maximum bending moment is composed of approximately 50% of each mode. However, the mid-height moment is very strongly influenced by the second-mode, and the maximum peak is composed almost entirely of the second-mode. It should also be noted that the second-mode causes the moment to reverse between positive and negative directions very quickly. (a) (b) Figure 2.27: Modal bending moment response for structure excited at period of second-mode in the linear elastic range at: (a) Base node; (b) Mid-height node. -150000-100000-500000500001000001500002000001.8 2.8 3.8 4.8 5.8 6.8 7.8Mb(kN-m)Time (s)TotalMode 1Mode 2-150000-100000-500000500001000001500001.8 2.8 3.8 4.8 5.8 6.8 7.8Mm(kN-m)Time (s)TotalMode 1Mode 244 Figure 2.28 shows the shear force response of the structure in the base and top elements. From this figure the response is composed almost entirely of the second-mode during the first few seconds of free vibration. In Figure 2.29, the second-mode contribution is greater than the maximum shear force, as the two modal contributions are working against each other. Figure 2.28: Shear force response at top and base elements for structure excited at period of second-mode in the linear elastic range. (a) (b) Figure 2.29: Modal shear force response for structure excited at period of second-mode in the linear elastic range at: (a) Base element; (b) Top element. -8000-6000-4000-200002000400060000 1 2 3 4 5 6 7 8 9 10Shear (kN)Time (s)Base (Vb)Top (Vt)-8000-6000-4000-200002000400060001.8 2.8 3.8 4.8 5.8 6.8 7.8Vb(kN)Time (s)TotalMode 1Mode 2-4000-3000-2000-1000010002000300040001.8 2.8 3.8 4.8 5.8 6.8 7.8Vt(kN)Time (s)TotalMode 1Mode 245 Finally, in Figures 2.30 and 2.31, the equivalent static force response is shown, as well as the modal contributions of the force at the top and mid-height nodes respectively. The top force is strongly influenced by the second-mode; however, the mid-height force is composed entirely of the second-mode, and even after the second-mode contribution damps out, there is very little response left from the first mode. Figure 2.30: Equivalent static force response at top and mid-height nodes for structure excited at period of second-mode in the linear elastic range. (a) (b) Figure 2.31: Modal equivalent static force response for structure excited at period of second-mode in the linear elastic range at: (a) Top node; (b) Mid-height node. -10000-8000-6000-4000-200002000400060008000100000 2 4 6 8 10Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-4000-3000-2000-1000010002000300040001.8 2.8 3.8 4.8 5.8 6.8 7.8F2(kN)Time (s)TotalMode 1Mode 2-10000-8000-6000-4000-200002000400060008000100001.8 2.8 3.8 4.8 5.8 6.8 7.8F1(kN)Time (s)TotalMode 1Mode 246 Unlike the response of the structure excited at a period equal to the first mode, the modal trends shown in Figure 2.32, show a very strong influence from the second-mode. The increasing influence in the response from displacement to equivalent static force is shown very clearly. (a) (b) (c) (d) Figure 2.32: Trend of modal responses for structure excited at period of second-mode in the linear elastic range for: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static force. -15-10-505101520-40 -20 0 20 40 60D1(mm)D2 (mm)TotalMode 1Mode 2-150000-100000-50000050000100000150000-200000 -100000 0 100000 200000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2-4000-3000-2000-100001000200030004000-10000 -5000 0 5000 10000Vt(kN)Vb (kN)TotalMode 1Mode 2-10000-8000-6000-4000-20000200040006000800010000-4000 -2000 0 2000 4000F1(kN)F2 (kN)TotalMode 1Mode 247 2.3.4 Summary of Results in Linear Elastic Range In addition to the analyses that were completed in Subsections 2.3.2 and 2.3.3, a similar set of analyses was completed for the cases where the sinusoidal pulses were equal to a range of periods between 0.5 and 4 seconds. The time-history responses of each analysis can be found in Appendix B, but will not be discussed in detail, as they are similar to the results already shown. It should be noted that, while the period of the pulse was varied, the amplitude of the pulse was left constant, therefore the results are relatable to each other. After the model was excited with sinusoidal pulses of various periods, the maximum values of the displacements, shear forces, and bending moments were plotted. These values, which can be seen in Figure 2.33, have been normalized by dividing by the largest maximum response of the entire suite of pulses, which in every case appears to occur due to resonance when the structure was excited at the period of the first mode. This normalization allows us to examine the effect that the second-mode has on each parameter of the response on a relative scale. Figure 2.33: Normalized maximum value of displacements, shear forces, and bending moments of model subjected to a sinusoidal ground acceleration pulse vs period of the applied pulse in the linear elastic range. 0.00.10.20.30.40.50.60.70.80.91.00 0.5 1 1.5 2 2.5 3 3.5 4Normalized Maximum ValuesPeriod of Exciation, Tp (s) Mid DisplacementTop DisplacementBase ShearTop ShearBase MomentMid MomentMode 1Mode 248 It can be observed that if the structure is in the linear range, each aspect of the response seems to increase along the same trend line. The exception to this is when the structure is excited around the period of the second-mode. The base shear is greatly influenced by the second-mode, while mid-height bending moment and top shear are somewhat influenced, the base moment and mid-height displacement are slightly influenced, and the top displacement seems to not be influenced at all by the second-mode. These findings relate to what was seen in the response spectrum analysis completed in Chapter 1, which found that while the shear is significantly influenced by the second-mode, the bending moment and displacement response are dominated by the first mode. It is also helpful to look at the moment-to-shear ratio for each analysis. This is another metric that is useful in determining how influential the shear force is to the overall response. The moment-to-shear ratio is calculated based on the maximum bending moment and shear at the base over the entire response, as well as the overall height of the structure. This simple calculation is shown in Equation 2.14. 𝑀𝐵𝑎𝑠𝑒𝑉𝐵𝑎𝑠𝑒 ∗ ℎ (2.14) In Figure 2.34, we can see that the results for the moment-to-shear ratio agree well with the results seen in Figure 2.33. The moment-to-shear ratio is very high, between 0.8 and 0.9, when excited near the period of the first mode, but decline to about 0.3 as the contribution of the second-mode becomes more influential. 49 Figure 2.34: Moment-to-Shear ratio vs period of the applied pulse in the linear elastic range. In summary, these results have been calculated using a linear elastic version of the model. The intensity of the pulse is the same for every sinusoidal ground acceleration pulse, however the period of excitation was varied between 0.5 and 4 seconds. Although the forces and displacement obtained from exciting the structure at the period of the first mode are much larger, the forces and displacements obtained from exciting the structure at the period of the second-mode contain a much higher contribution from the second-mode. The next section of this chapter presents the results observed when similar analyses were completed on a non-linear model. 00.10.20.30.40.50.60.70.80.910 0.5 1 1.5 2 2.5 3 3.5 4Mbmax/(Vbmax ·h w)Period of Excitation, Tp (s)Mode 1Mode 2hw50 2.4 Results of Non-Linear Model The model will be analyzed when subjected to an excitation equal to the period of the first and second-mode in order to easily see the contribution of the second-mode in the response. To understand how the model will respond at different levels of non-linearity, the ground excitation was scaled to four intensities, which resulted in four trials of response. These four trials will be where the structure is pushed to the linear range, cracked range, yielded range, and ultimate/significant yielding range. The envelope of the curvature distribution will be analyzed along with the moment-curvature response at the base of the structure to understand how the model responds to increasing non-linearity. First the curvature distribution of the structure excited at the period of the first mode will be presented, where it is not expected that there will be significant contribution, if any, from the second-mode. 51 2.4.1 Curvature Distribution when Excited at Period of First Mode The intensity of the pulse is measured by the maximum ground acceleration value, and the intensity of each pulse can be seen in Table 2.6. The maximum bending moment at the base, and maximum displacement at the top of the structure, for the linear and non-linear responses can be seen in Figure 2.35, where each of the trials of response is labelled 1 through 4. This figure clarifies how the structure is behaving in relation to a linear elastic model. Table 2.6: Intensity of ground acceleration pulse for each trial of analyses excited at period of first mode. Trial Intensity (g) Phase Max. Base Curvature (rad/km) 1 0.005 Linear 0.03 2 0.016 Cracked 0.2 3 0.050 Yielded 0.6 4 0.158 Ultimate 3.5 Figure 2.35: Maximum base moment vs maximum top displacement for each trial of dynamic non-linear analyses for structure excited at period of first mode. 12342 3 4010000002000000300000040000005000000600000070000000 500 1000 1500 2000 2500 3000Max. Base Moment (kNm)Max. Top Displacement (mm)LinearNon-Linear52 Figure 2.36 shows the results of the first trial that show clearly that the model is still in the linear elastic range. The moment-curvature response is perfectly linear, and the curvature profile resembles a triangle. (a) (b) Figure 2.36: Results of dynamic non-linear analysis when excited at period of first mode, Trial 1: (a) Moment-curvature response at base; (b) Curvature envelope along height. -200000-150000-100000-50000050000100000150000200000-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04Base Moment (kN-m)Base Curvature (rad/km)01020304050607080900.00 0.01 0.02 0.03 0.04Height (m)Curvature (Rad/km)53 Figure 2.37 shows the results of the second trial where cracking has begun to occur in each direction of the model based on the moment-curvature response at the base. The absolute curvature profile shows that cracking has begun to occur over the bottom 30 m of the model. (a) (b) Figure 2.37: Results of dynamic non-linear analysis when excited at period of first mode, Trial 2: (a) Moment-curvature response at base; (b) Curvature envelope along height. -400000-300000-200000-1000000100000200000300000400000-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30Base Moment (kN-m)Base Curvature (rad/km)01020304050607080900.00 0.05 0.10 0.15 0.20 0.25Height (m)Curvature (Rad/km)54 Figure 2.38 shows the results of the third trial where it is now seen that while the model has begun cracking in each direction, it has only begun yielding in the positive direction. The specimen was pushed to yielding once and cycled back demonstrating pinching behaviour which demonstrates the specimen has not yet experienced significant damage. The curvature profile shows that the entire bottom half of the structure should have cracking, and a long tail has begun to form due to the yielding at the base. (a) (b) Figure 2.38: Results of dynamic non-linear analysis when excited at period of first mode, Trial 3: (a) Moment-curvature response at base; (b) Curvature envelope along height. -600000-400000-2000000200000400000600000-0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80Base Moment (kN-m)Base Curvature (rad/km)01020304050607080900.00 0.20 0.40 0.60 0.80Height (m)Curvature (Rad/km)55 Finally, Figure 2.39 shows the results of the fourth trial where it is now seen that there is yielding in each direction, with significant yielding in the positive direction, although it was only one large cycle that caused the spike in curvature. The axial load in the model is large enough that it still exhibits pinching behaviour after the large amount of yielding. The curvature profile shows a very long tail at the base, with relatively much less curvature along the rest of the height, demonstrating formation of a plastic hinge at the base of the model. (a) (b) Figure 2.39: Results of dynamic non-linear analysis when excited at period of first mode, Trial 4: (a) Moment-curvature response at base; (b) Curvature envelope along height. -600000-400000-2000000200000400000600000-2.00 -1.00 0.00 1.00 2.00 3.00 4.00Base Moment (kN-m)Base Curvature (rad/km)01020304050607080900.00 1.00 2.00 3.00 4.00Height (m)Curvature (Rad/km)56 2.4.2 Curvature Distribution when Excited at Period of Second-mode Next, the curvature distribution of the structure excited at the period of the second-mode will be analyzed, where the results are slightly more complex compared to when the model was excited at the period of the first mode. In Table 2.7, the intensity of the pulse for each trial is shown, as well as the phase of the non-linear response that is observed. It should be noted that although the intensity of the pulse differs from those used in each trial in Subsection 2.4.2, the maximum base curvatures are approximately equal. This was completed so that each trial for a structure excited at the period of the first mode and one excited at the period of the second-mode would be comparable in the respective non-linear phases in which they were evaluate to. Figure 2.40, shows how the non-linear response compares with the linear response of each trial. 57 Table 2.7: Intensity of ground acceleration pulse for each trial for analyses excited at period of second-mode. Trial Intensity (g) Phase Max. Base Curvature (rad/km) 1 0.05 Linear 0.03 2 0.18 Cracked 0.2 3 0.40 Yielded 0.6 4 1.02 Ultimate 3.5 Figure 2.40: Maximum base moment vs maximum top displacement for each trial of dynamic non-linear analyses for structure excited at period of second-mode. 12342 34050000010000001500000200000025000003000000350000040000000 500 1000 1500 2000Max. Base Moment (kNm)Max. Top Displacement (mm)LinearNon-Linear58 Figure 2.41 shows the results of the first trial, the moment curvature response at the base shows that the model is still well within the linear elastic range of response, however the envelope of the curvature profile shows much more complex results compared to when the model was excited at the period of the first mode. While the largest curvature is still at the base, there is an a much larger amount of curvature concentrated at the mid-height of the structure. (a) (b) Figure 2.41: Results of dynamic non-linear analysis when excited at period of second-mode, Trial 1: (a) Moment-curvature response at base; (b) Curvature envelope along height. -150000-100000-50000050000100000150000200000-0.02 -0.01 0.00 0.01 0.02 0.03 0.04Base Moment (kN-m)Base Curvature (rad/km)01020304050607080900.00 0.01 0.02 0.03 0.04Height (m)Curvature (Rad/km)59 Figure 2.42 shows the results of the second trial, the moment curvature response at the base shows that the model has cracked in each direction, however the peak curvature in the positive direction is twice as large as the one reached in the negative direction. The envelope of the curvature profile shows again that there is additional curvature at the mid-height of the model, however the curvature at the base has increased by a greater relative amount. (a) (b) Figure 2.42: Results of dynamic non-linear analysis when excited at period of second-mode, Trial 2: (a) Moment-curvature response at base; (b) Curvature envelope along height. -400000-300000-200000-1000000100000200000300000400000-0.20 -0.10 0.00 0.10 0.20 0.30Base Moment (kN-m)Base Curvature (rad/km)01020304050607080900.00 0.05 0.10 0.15 0.20 0.25Height (m)Curvature (Rad/km)60 Figure 2.43 shows the results of the third trial, the moment curvature response at the base shows that the model is cracked in one direction and has reached yielding in the other. The envelope of the curvature profile is similar to that seen in Trial 2. (a) (b) Figure 2.43: Results of dynamic non-linear analysis when excited at period of second-mode, Trial 3: (a) Moment-curvature response at base; (b) Curvature envelope along height. -500000-400000-300000-200000-1000000100000200000300000400000500000600000-0.40 -0.20 0.00 0.20 0.40 0.60 0.80Base Moment (kN-m)Base Curvature (rad/km)01020304050607080900.00 0.20 0.40 0.60 0.80Height (m)Curvature (Rad/km)61 Finally, Figure 2.44 shows the results of the fourth trial, the specimen has yielded significantly in the positive direction, however in the negative direction has not accumulated much non-linearity. The envelope of the curvature profile shows that a plastic hinge has formed at the base of the structure, resulting in a large tail of curvature relative to any curvature along the rest of the height of the structure, including the curvature at the mid-height. (a) (b) Figure 2.44: Results of dynamic non-linear analysis when excited at period of second-mode, Trial 4: (a) Moment-curvature response at base; (b) Curvature envelope along height. -600000-400000-2000000200000400000600000-1.00 0.00 1.00 2.00 3.00 4.00Base Moment (kN-m)Base Curvature (rad/km)01020304050607080900.00 1.00 2.00 3.00 4.00Height (m)Curvature (Rad/km)62 In summary, when the structure is excited at the period of the first mode, all the non-linear behaviour occurs at the base of the structure. However, when the structure is excited at the period of the second-mode, we can see that there is much more non-linear behaviour occurring at the mid-height of the structure. As the structure is pushed further and further towards yielding, a plastic hinge forms at the base before any hinging occurs at the mid-height. Figure 2.45 shows a comparison of the curvature profiles of the model excited at periods equal to both the first and second-modes. The tail of the curvature from the plastic hinge is not shown to observe the non-linearity along the remaining height of the structure. From observation, the only significant difference between the two sets of analyses is the additional curvature at the mid-height. (a) (b) Figure 2.45: Comparison of curvature envelope for each trial for structure excited at period of: (a) First mode; (b) Second-mode. 01020304050607080900.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70Height (m)Curvature (rad/km)Tp = T101020304050607080900.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70Height (m)Curvature (rad/km)Tp = T263 2.4.3 Change in Modal Response Due to Base Yielding It was shown in the last section how the yielding at the base has caused a plastic hinge to form, which influences the response of the structure. The ratio between the top and mid-height displacement can be examined in order to determine how the behaviour of the response has changed. The ratio between the top and mid-height displacement is a representation of the first mode shape. A structure that is perfectly pinned at the base would behave such that the mid-height displacement would be exactly half of the top displacement. While a structure with a fixed base will have the mode shape we have observed throughout this thesis. A structure with a formed plastic hinge will take a mode shape somewhere in between these two extremes. A representation of this comparison can be seen in Figure 2.46, and will be presented further throughout this chapter. Figure 2.46: Comparison between fixed, yielded, and pinned displacement profiles. 120 0.5 1Normalized DisplacementMode 1Yielded BasePinned64 The formation of the plastic hinge means that the behaviour of the structure will be somewhere in between the two profiles shown above. In Figure 2.47 the time history response of a structure excited at a period equal to its first mode can be seen, and the trendline is in Figure 2.47(b) shows that the response is equal exactly to the first mode, with no contribution from the second-mode. The data shown is taken from Trial 1 of the set of analyses excited at the period of the first mode. (a) (b) Figure 2.47: Displacement response at top and mid-height nodes for structure excited at period of first mode, Trial 1: (a) Time history; (b) Modal trend. Figure 2.48 and 2.49 show similar plots to what was shown in Figure 2.47, however it is shown for Trials 3 and 4, where the structure is pushed into slight yielding and significant yielding. In both figures, as the amount of yielding increases, it can be seen that the slope of the modal ratio has increased to approximately 0.39 and 0.44, up from the linear elastic mode shape of 0.33. The change in slope is evidence of the altered mode shape as a plastic hinge forms at the base. -80-60-40-2002040608010011 13 15 17 19Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁) y = 0.3257x-30-20-100102030-100 -50 0 50 100D1(mm)D2 (mm)TotalMode 1Mode 265 (a) (b) Figure 2.48: Displacement response at top and mid-height nodes for structure excited at period of first mode, Trial 3: (a) Time history; (b) Modal trend. (a) (b) Figure 2.49: Displacement response at top and mid-height nodes for structure excited at period of first mode, Trial 4: (a) Time history; (b) Modal trend. -800-600-400-200020040060080011 13 15 17 19Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)y = 0.3921xy = 0.3257x-300-200-1000100200300-1000 -500 0 500 1000D1(mm)D2 (mm)TotalMode 1Mode 2-1500-1000-5000500100015002000250011 13 15 17 19Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)y = 0.4447xy = 0.3257x-600-400-20002004006008001000-1000 0 1000 2000 3000D1(mm)D2 (mm)TotalMode 1Mode 266 This change in mode shape was investigated further by running a series of analyses of increasing amplitude and plotting the change in ratio between the mid-height and top displacement as the maximum curvature at the base increases. In this plot we can see that the mode shape changes exponentially towards a perfectly pinned ratio of 0.5 as the curvature at the base increases. Figure 2.50: Change in ratio of mid-height displacement to top displacement, based on slope of modal trendline of response in free vibration, as base curvature increases for structure excited at period of first mode. Based on the observations presented thus far, we can correct the decomposition of the response into its two mode shapes. In Figure 2.51 (a) and (c), we can see the top and mid-height modal displacement responses, when decomposed using the linear elastic mode shape of 0.33, which are very noticeably incorrect. However, in (b) and (d), it can be seen that the when decomposed using a corrected mode shape, based on the slope of the data as observed above, a much-improved decomposition into the first and second-mode components is achieved. This serves as a confirmation that by using a corrected mode shape, we can properly decompose a 01234567890.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46ϕbase/ϕyieldD1/D267 displacement response into its modal components, even after it has been pushed past yielding. One assumption made in this process, is that the second-mode shape is not affected by yielding at the base. While this analysis was being completed, no noticeable observation was seen in the decomposition of the response into its mode shapes, when the second-mode was altered, which later proved as an important assumption in development of the experimental protocol. (a) (b) (c) (d) Figure 2.51: Modal displacement response for structure excited at period of first mode, Trial 3: (a) Top node using linear mode shapes; (b) Top node using corrected mode shapes; (c) Mid-height node using linear mode shapes; (d) Mid-height node using corrected mode shapes. -800-600-400-200020040060080011 13 15 17 19D2(mm)Time (s)TotalMode 1Mode 2-800-600-400-200020040060080011 13 15 17 19D2(mm)Time (s)TotalMode 1Mode 2-300-200-100010020030011 13 15 17 19D1(mm)Time (s)TotalMode 1Mode 2-300-200-100010020030011 13 15 17 19D1(mm)Time (s)TotalMode 1Mode 268 The next step is to apply this process to the displacement response of a structure that was excited at the period of the second-mode. Figure 2.52, shows the displacement response while in the linear elastic range. In this case it is not as simple, as there is a visible influence from the second-mode, however, the overall linear trendline of the data gives a slope very similar to the first mode of vibration (0.321 vs 0.325). (a) (b) Figure 2.52: Displacement response at top and mid-height nodes for structure excited at period of second-mode, Trial 1: (a) Time history; (b) Modal trend. -40-30-20-10010203040501.8 3.8 5.8 7.8 9.8Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)y = 0.3206x-15-10-505101520-40 -20 0 20 40 60D1(mm)D2 (mm)TotalMode 1Mode 269 In Figure 2.53, we can see the modal decomposition of the displacement response from Trial 3 of the structure excited at the period of the second-mode has a linear trendline with a slope very similar to the slope of the data from Trial 3 of the structure excited at the period of the first mode. This justifies using the slope of the data as a corrected non-linear mode shape regardless of the presence of the second-mode. (a) (b) Figure 2.53: Displacement response at top and mid-height nodes for structure excited at period of second-mode, Trial 3: (a) Time history; (b) Modal trend. -300-200-10001002003004005001.8 3.8 5.8 7.8 9.8Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)y = 0.3826x-150-100-50050100150200250-400 -200 0 200 400 600D1(mm)D2 (mm)TotalMode 1Mode 270 In Figure 2.54, we can see that using the slope of the trendline in Figure 2.53(b) as a corrected mode shape works very well in decomposing the displacement response into its first and second-mode shapes. Evidence of this is that the first and second-modes oscillate about the zero-displacement line in the decomposition using the corrected mode shape, while in the uncorrected decomposition the second-mode response is clearly not oscillating about this point. (a) (b) (c) (d) Figure 2.54: Modal displacement response for structure excited at period of second-mode, Trial 3: (a) Top node using linear mode shapes; (b) Top node using corrected mode shapes; (c) Mid-height node using linear mode shapes; (d) Mid-height node using corrected mode shapes. -300-200-10001002003004005001.8 3.8 5.8 7.8 9.8D2(mm)Time (s)TotalMode 1Mode 2-300-200-10001002003004005001.8 3.8 5.8 7.8 9.8D2(mm)Time (s)TotalMode 1Mode 2-150-100-500501001502002501.8 3.8 5.8 7.8 9.8D1(mm)Time (s)TotalMode 1Mode 2-150-100-500501001502002501.8 3.8 5.8 7.8 9.8D1(mm)Time (s)TotalMode 1Mode 271 A similar analysis for Trial 4 is shown in Figure 2.55, where again, regardless of the influence of the second-mode, a similar overall trendline of the displacement response gives a slope very similar to the one observed in Trial 4 of the structure excited at a period equal to the first mode. As a reminder, Trial 4 was when the structure is pushed to significant yielding, and as a result the mode shape has changed from roughly 0.33 to 0.45. This is further evidence of what has been asserted in this Section, in that the mode shape becomes more pin like as the plastic hinge forms at the base. (a) (b) Figure 2.55: Displacement response at top and mid-height nodes for structure excited at period of second-mode, Trial 4: (a) Time history; (b) Modal trend. Again, the corrected mode shape was taken as the slope of the linear trendline of the displacement response, which can be seen in Figure 2.56 to improve the decomposition into the first and second-mode shapes significantly. The inaccuracy of the uncorrected -1000-50005001000150020001.8 3.8 5.8 7.8 9.8 11.8Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)y = 0.4484x-400-20002004006008001000-1000 0 1000 2000D1(mm)D2 (mm)TotalMode 1Mode 272 decomposition is evidenced by of the large change in mode shape between the linear elastic and yielded structure. (a) (b) (c) (d) Figure 2.56: Modal displacement response for structure excited at period of second-mode, Trial 4: (a) Top node using linear mode shapes; (b) Top node using corrected mode shapes; (c) Mid-height node using linear mode shapes; (d) Mid-height node using corrected mode shapes. -1000-50005001000150020001.8 3.3 4.8 6.3 7.8 9.3 10.8D2(mm)Time (s)TotalMode 1Mode 2-1000-50005001000150020001.8 3.3 4.8 6.3 7.8 9.3 10.8D2(mm)Time (s)TotalMode 1Mode 2-400-200020040060080010001.8 3.3 4.8 6.3 7.8 9.3 10.8D1(mm)Time (s)TotalMode 1Mode 2-400-200020040060080010001.8 3.3 4.8 6.3 7.8 9.3 10.8D1(mm)Time (s)TotalMode 1Mode 273 In addition to a change in mode shape based on the flexural damage that occurs at the base of the structure, it was important to also investigate the influence of the reduction in shear stiffness on the mode shapes. This is important because the second-mode causes much higher shear demands on the structure which will affect the amount of shear damage. While there was not a significant amount of shear damage observed during these analyses, it is expected that the experimental tests will be strongly influenced by the shear stiffness. Figure 2.57 shows that as the shear stiffness decreases, the mode shape trends towards a perfectly pinned structure for the first mode, while for the second-mode the mode shape changes such that the difference between the top and mid-height nodes increases to -0.5. It should be noted that in this analysis the flexural stiffness was not changed from its gross value. (a) (b) Figure 2.57: Effect of effective shear stiffness on mode shape of: (a) First mode; (b) Second-mode. 00.10.20.30.40.50.60.70.80.910.300 0.350 0.400 0.450 0.500GAeff/GAgD1/D2Mode 100.10.20.30.40.50.60.70.80.91-0.500 -0.450 -0.400 -0.350 -0.300GAeff/GAgD2/D1Mode 274 2.4.4 Non-Linear Response when Excited at Period of First Mode In this subsection, the non-linear response histories of the model when excited at the period of the first mode will be presented. The displacement, bending moment, shear force, and equivalent static force responses that correspond with Trial 3 will be shown, while the other responses that correspond with the Trials 2 and 4 can be found in Appendix B, and Trial 1 can be found in Subsection 2.3.2. The displacement and equivalent static force response will be shown, as well as its decomposition into its two modal components. It was explored in the last subsection how a correction to this mode shapes can be rationally explained and applied for the displacement response of the structure, however there is no simple solution for decomposing the force response of the structure. The reason for completing this modal decomposition, although it is not valid, is to determine how well the response may react in its linear mode shapes, even though it is in the non-linear range. The displacement response is shown in Figure 2.58, while the modal decomposition is shown in Figure 2.59, using the corrected mode shape presented in the previous subsection. It can be seen at the mid-height displacement that there is a slight second-mode wave form. Figure 2.58: Displacement response at top and mid-height node for structure excited at period of first mode in the non-linear range. -800-600-400-20002004006008000 2 4 6 8 10 12 14 16 18 20Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)75 (a) (b) Figure 2.59: Modal equivalent static force response for structure excited at period of first mode in the non-linear range at: (a) Top node; (b) Mid-height node. In Figure 2.60, it is shown that the equivalent static force response has become very unpredictable and is no longer responding in the first mode as is seen with the displacement response. This shows that the force response cannot be reliably decomposed into the first and second-mode shapes, and therefore the bending moment and shear force responses will also not be able to be decomposed. There is also no simple solution that presents itself to be used as a simple correction to the mode shape, such as what was used for the displacement. Figure 2.60: Equivalent static force response at top and mid-height nodes for structure excited at period of first mode in the non-linear range. -800-600-400-200020040060080011 13 15 17 19D2(mm)Time (s)TotalMode 1Mode 2-300-200-100010020030011 13 15 17 19D1(mm)Time (s)TotalMode 1Mode 2-8000-6000-4000-2000020004000600080000 2 4 6 8 10 12 14 16 18 20Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)76 Although the modal decomposition is invalid, it was done anyways using the linear elastic mode shapes, which is shown in Figure 2.61. The results show that there is no well-defined wave that represents the first and second-mode contribution, however the overall influence of the second-mode is clear, regardless of the fact that the structure was excited at a period equal to its first mode. (a) (b) Figure 2.61: Modal equivalent static force response for structure excited at period of first mode in the non-linear range at: (a) Top node; (b) Mid-height node. The bending moment and shear force response are investigated by observing the slope of the trendline of the modal response. This is shown in Figure 2.62, where the mid-height moment is plotted against the base moment, and the top shear is plotted against the base shear. The slope of the trendline suggests that the overall ratio of the first mode contribution to the bending moment and shear force responses has not changed, but there is a second-mode presence, again regardless of the fact that the structure was excited at the period of its first mode. -8000-6000-4000-2000020004000600011 13 15 17 19F2(kN)Time (s)TotalMode 1Mode 2 -8000-6000-4000-20000200040006000800011 13 15 17 19F1(kN)Time (s)TotalMode 1Mode 277 (a) (b) Figure 2.62: Trend of modal response for structure excited at period of first mode in the non-linear range at: (a) Bending moment response; (b) Shear force response. It is suspected that the cause of this second-mode presence in the response is due to two reasons. First, as the structure cracks and yields the periods elongate, and thus both modes are being excited to some extent. Second, the base of the structure begins to behave more pin like at the base due to the formation of the plastic hinge. This was investigated by observing the bending moment response. Figure 2.63 shows that over time, neither the base or mid-height moments are greater than the cracking moment at any point in the response for Trial 1 in the linear elastic range, and thus there is no higher mode influence in the response. However, 2.64 shows the bending moment response for Trial 3, when the specimen is yielded at the base. In this case it is shown that the bending moment at the base is surpassing the estimated yield moment, and the mid-height moment is surpassing the cracking moment. In the first period of excitation, there is no evidence of any second-mode influence in the response, however as soon y = 0.4067x-250000-200000-150000-100000-50000050000100000150000200000250000300000-500000 -300000 -100000 100000 300000 500000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.6x-6000-4000-20000200040006000-10000 -5000 0 5000 10000Vt(kN)Vb (kN)TotalMode 1Mode 278 as the structure is pushing towards any non-linearity, the second-mode begins to influence the response. Figure 2.63: Bending moment response of structure excited at period of first mode, Trial 1, with approximate cracking and yielding bending moments shown. Figure 2.64: Bending moment response of structure excited at period of first mode, Trial 3, with approximate cracking and yielding bending moments shown. To investigate this further, an analysis was completed using a linear elastic model, however an artificial plastic hinge element was added to the base, by using a linear elastic beam-column with a length equal to 10% of the total height of the structure, and an effective flexural stiffness equal to 5% of the gross stiffness. The result of this analysis is shown in Figure 2.65, where it -500000-400000-300000-200000-10000001000002000003000004000005000000 2 4 6 8 10 12 14 16 18 20Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)CrackingYielding-500000-400000-300000-200000-10000001000002000003000004000005000000 2 4 6 8 10 12 14 16 18 20Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)CrackingYielding79 can be seen that the displacement response is not affected, however the mode shape has become more pin like, and the equivalent static force response is clearly influenced by the second-mode. This is evidence that pin like behaviour and period elongation lead to the response to be influenced by the second-mode, even when excited at the period of the first mode. (a) (b) (c) (d) Figure 2.65: Response of linear elastic structure with artificial plastic hinge excited at period of first mode for: (a) Displacement response; (b) Modal displacement trend; (c) Equivalent static force response; (d) Modal static force trend. -600-400-200020040060011 13 15 17 19Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)y = 0.4182x-250-200-150-100-50050100150200250-1000 -500 0 500 1000D1(mm)D2 (mm)TotalMode 1Mode 2-5000-4000-3000-2000-100001000200030004000500011 13 15 17 19Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)y = 0.4504x-2000-1500-1000-500050010001500200025003000-6000 -4000 -2000 0 2000 4000 6000F1(kN)F2 (kN)TotalMode 1Mode 280 2.4.5 Non-Linear Response when Excited at Period of Second-mode In this subsection, the non-linear response histories of the model when excited at the period of the second-mode will be presented. The displacement, bending moment, shear force, and equivalent static force responses that correspond with Trial 3 will be shown, while the other responses that correspond with the Trials 2 and 4 can be found in Appendix B, and Trial 1 was previously shown in Subsection 2.3.3. The displacement and equivalent static force response will be shown, as well as its decomposition into its two modal components. Similarly to the non-linear results for the structure excited at the first mode, a correction to the mode shape can be rationally explained and applied for the displacement response of the structure, however there is no simple solution for decomposing the force response of the structure. The reason for completing this modal decomposition, regardless of the fact that it is not valid, is to determine how well the response may react in its linear mode shapes, despite the fact that it is in the non-linear range. Figure 2.66 shows the displacement response for Trial 3 when excited at the period of the second-mode, and Figure 2.67 shows the decomposed response using the corrected mode shape shown in Subsection 2.4.3. Figure 2.66: Displacement response at top and mid-height node for structure excited at period of second-mode in the non-linear range. -300-200-10001002003004005001.8 2.8 3.8 4.8 5.8 6.8 7.8 8.8 9.8Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)81 (a) (b) Figure 2.67: Corrected modal displacement response for structure excited at period of second-mode in the non-linear range at: (a) Top node; (b) Mid-height node. Figure 2.68 shows the equivalent static force response, that demonstrates that similar to what was observed in the linear elastic case, the second-mode greatly influences the response of the structure. The modal decomposition in Figure 2.69 is technically invalid, however the influence that each mode has on the response is clearly shown, especially in the case of the second-mode influence on the mid-height force. Figure 2.68: Equivalent static force response at top and base elements for structure excited at period of second-mode in the non-linear range. -300-200-10001002003004005001.8 3.8 5.8 7.8 9.8D2(mm)Time (s)TotalMode 1Mode 2-150-100-500501001502002501.8 3.8 5.8 7.8 9.8D1(mm)Time (s)TotalMode 1Mode 2-30000-20000-1000001000020000300001.8 2.8 3.8 4.8 5.8 6.8 7.8 8.8 9.8Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)82 (a) (b) Figure 2.69: Modal equivalent static force response for structure excited at period of second-mode in the non-linear range at: (a) Top node; (b) Mid-height node. Figure 2.70 shows the response of the displacement, bending moment, shear force, and equivalent static force. When the structure was excited at the period of the first mode, it was very clear that the slope of the trendline of the response plotted in this way served as an effective tool to predict the dominant mode shape of the response, as well as a correction to the mode shape for the displacement. However, for the structure excited at the period of the second-mode, this trend becomes much more difficult to interpret. While this slope can be used as an effective correction to the displacement response, regardless of the influence of the second-mode, the second-mode plays a much larger role in the other areas of the response. -10000-8000-6000-4000-200002000400060008000100001.8 3.8 5.8 7.8 9.8F2(kN)Time (s)TotalMode 1Mode 2-30000-20000-1000001000020000300001.8 3.8 5.8 7.8 9.8F1(kN)Time (s)TotalMode 1Mode 283 (a) (b) (c) (d) Figure 2.70: Trend of modal responses for structure excited at period of second-mode in the non-linear range for: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static force. y = 0.3826x-150-100-50050100150200250-400 -200 0 200 400 600D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.2964x-500000-400000-300000-200000-1000000100000200000300000400000500000-600000-400000-200000 0 200000 400000 600000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = -0.0204x-10000-8000-6000-4000-20000200040006000800010000-20000 -10000 0 10000 20000Vt(kN)Vb (kN)TotalMode 1Mode 2y = -1.047x-20000-15000-10000-5000050001000015000200002500030000-10000 -5000 0 5000 10000F1(kN)F2 (kN)TotalMode 1Mode 284 In conclusion, the corrected mode shape can be applied to the modal decomposition of the displacement, but not to the force response. However, the first and second-mode can still be seen in the response, even if they cannot be predicted reliably. Figure 2.71 shows that the increase in force is linked with the increased displacement at mid-height, where the maximum mid-height displacement and base shear are plotted for each trial, for the case where the structure is excited at the period of the first and second-mode respectively. This figure shows that even though the two identical specimens were pushed to the exact same level of non-linearity in each trial, there is a significant difference in the observed shear demands due to the second-mode. It should also be noted that in each case, the increase in the shear force was limited by the flattening of the moment-curvature response. Figure 2.71: Relationship between maximum base shear and mid-height displacement distinguished by period of excitation. 05000100001500020000250000 200 400 600 800 1000Vbase(kN)D1 (mm)Tp = T1Tp = T285 2.4.6 Elongation of Period in Non-Linear Response As could be seen in the dynamic non-linear displacement response figures in Section 2.4.5, the time between peak displacements increases significantly when compared to the linear elastic displacement response due to elongation of the period. The period of the structure is calculated based on the mass and the stiffness, however as the stiffness degrades, the mass remains constant. This is what causes the period of each mode shape to elongate. In Figure 2.72, the displacement response of the top node can be seen for each trial and has been normalized such that each trial can be compared relatively. It can easily be seen that the amount of time required for the first cycle is increased as intensity of the demands in the response increase. Figure 2.72: Normalized displacement histories of top node of structure excited at period of second-mode, with increasing intensity of each Trial 1 through 4. In order to examine how much the period has elongated, the response of the top displacement was decomposed into its first mode response using the corrected mode shape procedure presented in Subsection 2.4.3, and can be seen in Figure 2.74. -1-0.8-0.6-0.4-0.200.20.40.60.811.8 4 6.2 8.4 10.6 12.8 15Normalized Top Displacement Time (s)123486 Figure 2.73: Example of a response cycle split into quarter cycles. The time required for each quarter cycle, as demonstrated in Figure 2.73, was multiplied by four, to represent the period of a full cycle, and these values can be found in Table 2.8. As expected for Trial 1, the period is equal to 3.68s, which is roughly equal to the actual period of 3.71. It should be noted that the first half of the cycle usually required more time, this is because the structure experienced more non-linearity in one direction than the other, as in the moment-curvature response was not symmetric as was shown in Subsection 2.4.2. Table 2.8(b) shows the level of elongation, by dividing the measured period of that trial by the measured period of the first trial. Figure 2.74: Normalized first mode displacement histories of top node of structure excited at period of second-mode, with increasing intensity of each Trial 1 through 4. -1-0.8-0.6-0.4-0.200.20.40.60.811.8 4 6.2 8.4 10.6 12.8 15Normalized Top Displacement due to Mode 1 Time (s)123487 Table 2.8: Length of time required for each quarter cycle of top node first mode displacement response: (a) In seconds; (b) Normalized by the period of the first mode Trial Intensity (g) 4t1 (s) 4t2 (s) 4t3 (s) 4t4 (s) Average 1 0.050 3.58 3.80 3.50 3.68 3.64 2 0.178 4.12 4.18 3.46 3.78 3.89 3 0.401 5.83 6.16 4.38 4.68 5.26 4 1.020 9.58 12.68 5.72 6.18 8.54 (a) Trial Intensity (g) 4t1/T1 4t2/T1 4t3/T1 4t4/T1 Average 1 0.050 0.98 1.04 0.96 1.01 1.00 2 0.178 1.13 1.15 0.95 1.04 1.07 3 0.401 1.60 1.69 1.20 1.29 1.45 4 1.020 2.63 3.48 1.57 1.70 2.35 (b) The contributions from the second-mode will be analyzed next, as can be seen in Figure 2.75. Since each period is far shorter, the first four periods of the response will be considered, as opposed to quarter cycles measured for the first period. It can be observed that even though the period elongates as non-linear behaviour increases, they all appear to damp out at relatively the same rate. Figure 2.75: Normalized second-mode displacement histories of top node of structure excited at period of second-mode, with increasing intensity of each Trial 1 through 4. -1-0.8-0.6-0.4-0.200.20.40.60.811.8 2.8 3.8 4.8 5.8 6.8 7.8 8.8 9.8Normalized Top Displacement due to Mode 2 Time (s)123488 From the results shown in Table 2.9, as expected, the average period remains almost unchanged for the response in the linear range, which was calculated to be 0.61 seconds compared to 0.62 seconds. In the far-right column, for the first 3 trials, the elongation of the two periods are very close, and with 4% difference. However, in Trial 4, where significant yielding occurred, the elongation of the first period was 35% larger than the elongation of the second period. This could be related to what was observed in Subsection 2.4.3, where it was noticed that for Trial 4 the second-mode contribution was much smaller. Table 2.9: Length of time required for each cycle of top node second-mode displacement response: (a) In seconds; (b) Normalized by the period of the second-mode (a) Trial Intensity (g) Cycle 1 (s) Cycle 2 (s) Cycle 3 (s) Cycle 4 (s) Average (s) 1 0.050 0.60 0.61 0.61 0.61 0.61 2 0.178 0.69 0.72 0.63 0.60 0.66 3 0.401 0.98 0.92 0.81 0.66 0.84 4 1.020 1.33 1.04 1.06 0.77 1.05 (b) Trial Intensity (g) Cycle 1 /T2 Cycle 2 /T2 Cycle 3 /T2 Cycle 4 /T2 Average Difference from Mode 1 (%) 1 0.050 0.99 1.00 1.01 1.00 1.00 0.00 2 0.178 1.14 1.19 1.04 0.99 1.09 2.18 3 0.401 1.62 1.52 1.34 1.08 1.39 3.91 4 1.020 2.19 1.72 1.74 1.26 1.73 35.68 In Figure 2.76, a comparison of the observed elongation of the periods is shown. The periods of both the first and second-mode elongate at approximately the same rate until Trial 4, which is where the structure was pushed far past yielding. In this case, the period of the first mode has elongated much more than that of mode 2, this is likely due to the fact that plastic hinging 89 has occurred at the base leading the structure to behave much more pin-like. This will lead to much larger displacements, and it has been observed throughout this chapter that the displacements are controlled by the first mode. Figure 2.76: Comparison of period elongation for modes 1 and 2. 0.500.751.001.251.501.752.002.252.500.3 0.4 0.5 0.6 0.7 0.8 0.9 1Original/Obererved PeriodNormalized Base MomentMode 1Mode 290 2.5 Summary of Dynamic Analyses In summary, a series of analyses were completed to study the influence of the first and second-mode on the response of a simplified analysis model devised to represent the behaviour of a high-rise shear wall core building. The purpose of this investigation was to answer the question of whether the second-mode demands act as an inherent displacement or force on the structure. In attempt to answer this question, the influence that the first and second-modes have on various aspects of the behaviour, such as the displacements, shear forces, bending moments, curvature distribution, and elongation of the period were studied. The structure analyzed was a two degree of freedom system modelled in OpenSees. The model was created using only two degrees of freedom so that the number of mode shapes present in the response of the model was limited to two. Two masses were used, one at each degree of freedom, because only two hydraulic actuators will be used for the experimental program. As has been previously stated, it was found that the influence of all higher modes can be reasonably approximated using only the second-mode shape for a 30-story structure. The cross-section of the model was discretized using material fibre section, in order to model the non-linear behaviour. The model was excited using a sinusoidal ground acceleration pulse, where the period of this sinusoidal pulse varied based on the periods of the model’s mode shapes. In order to study the influence of the higher modes, the model was first excited using a sinusoidal pulse with a period equal to that of the first mode, and then was excited using a sinusoidal pulse with a period equal to that of the second-mode. By comparing the results of these two sets of analyses, the influence of the second-mode on the response was evaluated. 91 First, the analyses were completed using a linear-elastic model, in order to observe the influence of the second-mode before any non-linear behaviour was considered. The time-history response of the model was decomposed into its first and second-mode components assuming modal superposition. This was not only completed for the displacement, but also for the shear, bending moment, and equivalent static force responses. When excited at the period of the first mode, the model responded purely in its first mode during free vibration. However, when excited at the period of the second-mode, the model responded in a combination of both the first and second-modes. From observing these results, the displacement response was influenced slightly at the mid-height, but not influenced at all at the top by the second-mode. The bending moment at the base is mostly controlled by the first mode, whereas the mid-height bending moments were influenced slightly by the second-mode. However, in the case of the shear forces, it was found that they were very strongly influenced by the presence of the second-mode, and that the time-history response is controlled mainly in the second-mode. Once the second-mode damps out, the shear force demands in the model decrease rapidly. It was also observed that the second-mode behaviour is present when the model is excited at the exact period of the second-mode, but if the period of excitation is not equal to the period of the second-mode, the model mainly responds in its first mode. Next, a non-linear model of the same structure was used. The model had the same initial stiffness as the linear-elastic one, so that the periods remained the same. In total 8 analyses were completed, 4 for each period of excitation, where in each trial the model was pushed to an increasing base curvature in order to observe how the model behaved when it was pushed to the linear, cracked, light yielding, and heavy yielding/ultimate strength range of the moment-curvature diagram. 92 By observing the curvature distribution of each trial, it was found that when the model was excited at the first mode, the curvature was located at the base in a triangular pattern, and the model ultimately yielded at the base. However, when the model was excited at the period of the second-mode, there was much more curvature at the mid-height. Flexural yielding was first observed at the mid-height, but ultimately the response of the model was governed by severe yielding at the base regardless of the period of excitation. Based on those results it was determined that the model would begin to exhibit more pin-like behaviour at the base as yielding began to occur, no matter the period of excitation. That assumption lead to the displacement response of the model being able to be decomposed into its modal components even though modal superposition is not technically valid for a non-linear structure, by modifying the first mode shape in order to account for pin like behaviour at the base. It was determined that although the response could be decomposed properly by modifying the first mode shape, modification of the second-mode shape has very little effect on the response. This suggests that the second-mode shape remains constant and is not affected by yielding at the base. It was also observed that while the period of the first and second-mode elongated at about the same rate for the first three trials, once yielding occurred at the base, the first mode period elongated at a larger rate. It was attempted to decompose the shear and bending moment time-history responses assuming the linear elastic “force mode shapes” as is shown in Figure 2.9, however the results clearly did not decompose properly into the modal components because the force mode shapes would change based on softening in the model, and the change in stiffness. The change in force mode shapes could not be rationalized, regardless, the influence of the second-mode could still be 93 inferred, and even when pushed to yielding at the base the second-mode was present in the shear response of the structure. Based on the results obtained, the non-linear analyses agreed with the findings of the linear elastic analyses, in that the shear force is very strongly influenced by the second-mode, no matter the level of cracking/yielding at the base. It was found that when the structure was excited at the period of the second-mode, the maximum shear force in both the top and base elements, was greater than the base shear when the structure is excited at the period of the first mode. In conclusion, it was determined that when the model was excited at its first mode, it will respond in its first mode during free vibration, however when excited at its second-mode, it will respond in a combination of both its first and second-modes during free vibration. When the presence of both the first and second-modes make up the response, the displacement and bending moment response of the structure are controlled by the first mode, however the shear force response is controlled by the second-mode. It was also found that as the structure is pushed to higher demands, yielding at the base leads to pin-like behaviour of the first-mode response, while the second-mode response is minimally effected. The next section will explain how the findings from this study and concluding theories helped influence the preliminary design of the experimental process, including the specimens, the test setup, and the demand protocol. . 94 2.6 Preliminary Design of Experiment Based on Analytical Data This section presents topics relevant to the preliminary development of the experimental testing program based on the findings of the analytical results presented in this chapter. The results of the analyses played an important role in influencing the design of the test specimens, the displacement test protocol, as well as the test setup. However, there are other constraints present that also played a role in its development that will be shown within this section. The analytical results were focused on understanding the behaviour of a structure when its first and second-modes of vibration are excited through sinusoidal ground accelerations, while the experiment will focus on how the seismic rate of loading of cyclical second-mode shear demands relates to the strength and stiffness of the wall specimens. The design process of the specimens, the protocol, and the test setup was an intrinsically iterative process; however, they will be presented independently in each of the following subsection. 2.6.1 Preliminary Specimen Design The goal of the experimental program is to replicate the cyclical second-mode shear demands that are applied at a high rate of loading that are observed during higher mode excitation of tall slender shear wall core structures. Therefore, the ideal specimen will be one that represents a scaled structure such as this. While it is common in experimental tests of walls to take an effective specimen height, based on the location of the resultant loading force applied by a single actuator, many of these tests share a common denominator in that they are testing resistance of the specimen to first mode demands, which are idealized by an inverted triangular load. However, the tests completed in this thesis are different in that the goal is to test the second-mode demands. 95 To determine the appropriate effective height of the specimen, modal analysis was completed on the same 30-story example structure. As can be seen in Figure 2.77, the forces of the first mode are completely in one direction, while the forces of the second-mode have a positive and negative region. The negative resultant occurs at roughly 92% of the height, while the positive resultant is taken at approximately 45% of the height. While the overall second-mode resultant (not shown) occurs even lower at roughly 33%, however a specimen of this height would ignore what is happening in the rest of the specimen subjected to second-mode demands. Therefore, it was decided that the specimen would be a full-height scaled wall, and for simplification the forces would be applied at the top and mid-height portions of the specimen, which is approximately in accordance with the locations of the two resultants. Figure 2.77: Resultant of positive and negative regions of first and second-mode force distribution along height of wall. As the imposed displacement protocol will be the second-mode shape, it is possible to try and predict a rough value for the maximum shear and bending moment forces that will be imposed upon the specimen, based on the shear and moment “mode shapes” presented in Section 2.3, 0.000.100.200.300.400.500.600.700.800.901.00-1.00 -0.50 0.00 0.50 1.00Normalized HeightNormalized Modal ForceMode 1Mode 2Mode 2 Negaitve ResultantMode 1 ResultantMode 2 Positive Resultant96 which are shown again in Figure 2.78, except this time with generalized expressions that predict the magnitude of the demands based on the mode shape. (a) (b) (c) Figure 2.78: Summary of expected demands under second-mode where F is actuator force and L is total length of specimen for: (a) Actuator force; (b) Bending moment; (c) Shear force. As the moment arm between the top actuator and base is exactly twice as long as the moment arm between the mid-height actuator and base, then the base moment will be equal to the maximum base shear multiplied by ¼ of the length of the specimen, which is equal to FL/6, where F is the actuator force, and L is the total length of the specimen. 2.6.2 Preliminary Displacement Protocol The ideal protocol would be to replicate the time-history response observed from the analytical data, with a reference example shown in Figure 2.79. As a reminder, the model was excited with a sinusoidal ground acceleration pulse where the period of the pulse was varied in order to observe its influence on the behaviour of the model. It was observed that even when the ForceEle. 2Ele. 1FF/3Bending MomentEle. 1Ele. 2FL/6FL/6Shear ForceEle. 2Ele. 1F/32F/397 model was excited at the exact period of the second-mode, the response of the wall contained components of both the first and second-modes, in its displacement, shear force, and bending moment response. While the displacement and bending moment behaviour is controlled by the first mode, the shear forces are heavily influenced by the presence of the second-mode, resulting in a high rate of loading and cyclic demands where the peak force is reached only for a fraction of a second before reversing. This seismic shear force behaviour is the desired demand pattern to be imposed upon the specimen. In Figure 2.79, the excitation phase is cut-off, and only the free vibration phase starting at 1.8 seconds is shown. However, there are a few problems with using the exact output of the analytical model as the displacement protocol. One of these problems is that the maximum observed shear forces are occurring almost instantly, as soon as the wall enters the free vibration phase, and then decreasing as the response damps out. This is due to the fact that the shear force response is controlled by the second-mode, and therefore reaches its first peak much faster than the first peak of the first mode component (recall that the second-mode period is roughly six times shorter than the fundamental period for a 10:1 structure), which governs the displacement and bending moment response. In the experimental tests it is desired that the shear forces will ramp up over time, in order to observe the effects of increasing demands before failure of the specimen. This will also allow the change in stiffness of the specimen to be measured as it becomes increasingly damaged. 98 Another potential problem, is that although the combination of both modes contributed to the response during the analytical study, the dominant failure mode was still always flexural yielding at the base, despite the significantly increased shear demands caused by the second-mode. The purpose of these experiments is to examine the shear strength, therefore a protocol and specimen must be developed that can allow the specimen to fail in shear prior to flexural yielding, while still applying the cyclic, high rate of loading shear demands observed in the analytical results. In summary, the behaviour observed from the analyses will not be used as the displacement protocol for the experiment because: 1) The maximum shear demands occur instantaneously, and do not ramp up over one period of response. 2) The presence of the first mode leads to flexural failures, while shear failures are desired in the test. (a) (b) -40-30-20-10010203040501.8 2.8 3.8 4.8 5.8Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-150000-100000-500000500001000001500002000001.8 2.8 3.8 4.8 5.8Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)99 (c) (d) Figure 2.79: Analytical results of free vibration behaviour of structure excited at period of second-mode for: (a) displacement; (b) bending moments; (c) shear forces; (d) equivalent static forces. The proposed solution to these issues is for the imposed displacement protocol to be only the second-mode contribution of the response. While this deviates from the behaviour observed in in the analytical results, it will allow the experimental portion to focus more on the shear strength of the walls and the seismic rate of loading. As was shown in Subsection 2.6.1, the purely second-mode loading is the optimal protocol that will both minimize flexural demands, and maximize shear demands, while still maintaining the cyclic, high rate of loading. An example of one period of second-mode displacement can be seen in Figure 2.80. -8000-6000-4000-200002000400060001.8 2.8 3.8 4.8 5.8Shear (kN)Time (s)Base (Vb)Top (Vt)-10000-8000-6000-4000-200002000400060008000100001.8 2.8 3.8 4.8 5.8Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)100 Figure 2.80: An example of a period of second-mode displacement loading. Recall that while the second-mode shape remained consistent even as the base began yielding, but the force response behaved in a way that made prediction very difficult. For this reason it was decided that the protocol of the test would be an imposed displacement, rather than an imposed force. This will allow the protocol to be applied in a manner that is relatively predictable, even as the demands and non-linear behaviour increase. In summary, each test will impose a displacement-controlled protocol on the specimen using two actuators, which will force the specimen to deform in its second-mode shape, which will result in cyclic reversing shear demands. In total there will be five tests completed on five identical specimens, where in each test the imposed displacement protocol will be the variable. 2.6.3 Preliminary Test Setup The desired test setup is one that it will closely resemble the model used to generate the analytical results. This would be a slender cantilever shear wall, with two free nodes (one at the top, and one at mid-height) that would be present to simulate the first and second-mode -0.15-0.1-0.0500.050.10.150 0.1 0.2 0.3 0.4 0.5Displacement (mm)Time (s)TopDisplacementMid-HeightDisplacement101 shapes of the specimen. In the experimental setup these nodes exist in the form of hydraulic actuators which are connected to the specimen at the top, and mid-height. It has been shown in detail that at least two nodes are required to simulate the second-mode, and that the second-mode is sufficient to predict the behaviour of a slender cantilever wall structure that is about 20 to 30 stories in height. While the analytical model was simulated using a sinusoidal ground acceleration pulse, the experimental test will be completed by imposing displacement demands using two hydraulic actuators. The specimens will be fixed at one end, and free at the other, as can be seen in Figure 2.81. Figure 2.81: Experimental setup of cantilever shear wall. Mid-Height Actuator, D1Top Actuator, D2102 An important part of this study, is that the imposed displacements on the specimen must be applied at the rate expected during an actual seismic event, and not in a pseudo-static way, to demonstrate if there is a correlation between the seismic rate and the shear strength/stiffness of the wall. In order to accomplish this, specific dynamic actuators had to be used. While these actuators provided the desired rate, they come with the constraint of a maximum applicable load that had to be accounted for, as it was desired for the specimens to be pushed to failure during the tests. This constraint will be presented in Chapter 3, along with more details related to the test specimen and setup. 103 3. Experimental Program ___________________________________________________________________________ 3.1 Overview This chapter summarizes the experimental program. The testing approach is presented in Section 3.2, while the design of the test specimens is shown in Section 3.3. The measured material properties of concrete and reinforcing steel used to construct the specimens is presented in Section 3.4. Construction of the specimens and connection between the specimens and the foundation is given in Section 3.5. The test setup, which includes the hydraulic actuators used, connections between actuators and the specimen, the foundation, and the out-of-plane support is in Section 3.6, while the instrumentation and data acquisition setup is presented in Section 3.7. Finally, the specimen testing procedures are summarized in Section 3.8. The theoretical relationship between the dynamic analyses and the experimental program was presented in Section 2.6. 104 3.2 Testing Approach The purpose of the experimental tests is to measure the resistance of a high-rise reinforced concrete core wall to second-mode shear demands. Much of the preliminary test design, and how it relates to theoretical analyses, was presented in Section 2.6. As has been mentioned, the test specimen will be a scaled version of an full-height reinforced concrete shear wall core, and the dynamic movement at the top and mid-height of the structure will be simulated using two hydraulic actuators. As the goal of the test was to fail the specimens in shear, the limitations of these two hydraulic actuators played a large role in governing the design of the specimen. An example of the scale of the test specimen to an actual 30-story building is shown in Figure 3.1. Figure 3.1: Scale of actual high-rise building to test specimen of cantilevered core wall. 105 The three important properties of the actuators were the maximum applicable force, displacement, and velocity. The goal of the tests was to impose displacements on the specimen at a high velocity, and thus a pair of actuators with these capabilities were chosen. It is specified that the maximum dynamic load that the actuators are each able to impose is 142 kN (32 kips), however, the two actuators will be working against each other. Based on the second “shear force mode” shown in Figure 2.78(c), the top force is equal to approximately 1/3rd of the mid-height force for the linear elastic case. Therefore, the maximum applicable base shear will be equal to 2/3 of the maximum actuator load, which is equal to roughly 95 kN. The desired failure mode of the test is diagonal concrete crushing due to shear; therefore, it is important that the specimen is designed such that it can fail in shear before any yielding begins to occur in the flexural steel. However, it is also expected that under high rates of loading the shear strength could be higher than the typical strength that would be observed under static loading. Therefore, the specimen was designed such that its flexural resistance will be high enough that the steel will not yield when the maximum base shear of 95 kN is applied under the dynamic second-mode protocol. This resulted in a design yield moment of roughly 40 kN-m. The ratio of shear span to effective depth of the specimen test region (the bottom web) is equal to approximately 2.7, which according to Figure N11.1.2(b) of the CAC handbook (CAC, 2014) classifies it near the intersection of elements that should be analyzed by using a strut-and-tie model and those that should be analyzed using a sectional model. This suggests that there was some strut action in the specimen, but it was not the dominant contributor to the shear strength. 106 While typical high-rise shear wall core buildings have a height to length ratio of 10:1, the specimen had a ratio of slightly over 5:1. The specimens have no axial compression and have a much smaller flange area than a realistic shear wall core, which limited the amount of flexural reinforcement, resulting in a weaker flexural resistance. This change in height to length ratio significantly reduced the moment arms from the actuators to the base of the specimen, reducing the flexural demands, but had no effect on the shear strength. It should be noted that a 5:1 ratio shear wall is still considered a slender wall according to ACI 318-14 (ACI, 2014). The cross-section of the specimen was chosen as an I-shape, so that the flexural strength could be maximized while having a shear critical web. The regions near the actuators were left as rectangular to ensure there would be no failure there, as is shown in Figure 3.2. As the goal of the tests was to investigate the effects of the protocol on the specimen, all the specimens were constructed identically. Section 3.3 presents more details related to the final design of the test specimens. 107 Figure 3.2: Variation in specimen cross-section. 3.3 Details of Test Specimen After several iterations, the specimen geometry and reinforcement were determined. The longitudinal reinforcing steel consists of four 10M bars in each flange, giving roughly 20 mm of concrete cover, as well as two 6 mm diameter bars along the web which are spaced 75 mm centre-to-centre. The 6 mm bars are slightly offset from the centre of the cross-section, so that the transverse reinforcement can be placed directly at the centre. This resulted in a total vertical reinforcing ratio of 3.12%, with localized reinforcing ratios of 4.57% in the flange, and 0.6% 108 in the web. While the amount of longitudinal reinforcement is quite high, it is still within the maximum limit for compression members stated in CSA A23.3-14, which is a ratio of 8% (CSA, 2014). The transverse reinforcement consisted of 6 mm diameter stirrups spaced every 125 mm in the webbed area, giving a horizontal reinforcing ratio of roughly 0.45%. The stirrups were hooked at each end at approximately a 115-degree angle, and the hooks at each end pointed in opposite directions. The orientation of the stirrups was switched for every second bar, to ensure that the overall configuration of the reinforcing cage was balanced. This is presented in Figure 3.3, which shows the cross-section including the transverse reinforcement. The solid line represents the hooked transverse reinforcing bar, while the dashed line represents every second bar with the reversed orientation. In addition to the stirrups in the web, there are also hooped bars in the flanges, to contain the longitudinal reinforcement as well as the hooks of the stirrups. Containment of all the reinforcement in the flanges will prevent the edges of the flanges from breaking off during the tests. The bars were hooped on each side and overlap each other over the width of the web. These bars are also fabricated from the 6 mm reinforcing steel, and are located at every stirrup location, as well as additional 6 mm diameter transverse reinforcement at the base and at the location of the connections to the actuators. The concrete cover from the outside edge of the hooped containment bars to the edge of the flange is 10 mm. A visualization of the reinforcement in the cross-section can be seen in Figure 3.3. 109 Figure 3.3: Cross-section of test region of specimen including location of reinforcing steel. The total height of the specimen is 2300 mm, while the cross-section varies between two separate geometries along the height. In addition to the flanged cross-section, there is also a rectangular cross-section that is present in specific regions. The purpose of expanding some regions along the height to a rectangular cross-section was to ensure that the specimen would have increased strength in these regions, to minimalize the risk of failure. The regions with rectangular cross-sections include the length that will be embedded into the foundation, the bearing zones where the actuators will connect, as well as an extension above the bearing zone of the top actuator. This extension of the specimen above the bearing zone allows the longitudinal bars to have enough length to develop hooks and is also used to connect the specimen to the out-of-plane support. The rectangular region is 175 mm by 300 mm, as the entire rectangle that bounds the “I” shape of the shear critical regions is filled in. 110 A summary of the cross-section regions along the specimen can be seen in Table 3.1, and a visualization of how the cross-section changed along the height is shown in Figure 3.4. The transverse reinforcement is spaced at 125 mm in the shear critical regions of the specimen, the spacing is cut in half to 62.5 mm in the foundation region of the specimen to ensure that no unwanted shear failures occur there. There is also extra transverse reinforcement in the bearing zones, with a stirrup on each side of the holes at approximately 67 mm centre-to-centre spacing, as well as an addition reinforcing bar in the extension at the top of the specimen. A visualization of where the reinforcement is located along the length of the specimen is presented in Figure 3.4(b). This figure also shows the hooks of the longitudinal 10M bars at each end of the specimen, as well as the length of the two 6 mm diameter longitudinal bars in the web along the specimen. 111 (a) (b) Figure 3.4: Change in cross-section along length of specimen: (a) Without reinforcement detail; (b) With reinforcement detail. A E D C B 112 Table 3.1: Location of cross-section regions along specimen height. Region Length (from base) (mm) Cross-Section Type A 2000 – 2300 Rectangular B 1300 – 2000 I – Shape C 1200 – 1300 Rectangular D 450 – 1200 I – Shape E 0 – 450 Rectangular The length of the specimen determined to be embedded into the foundation was determined based on the development length of the 10M reinforcement bars that are used as the main longitudinal reinforcement of the specimen. The required development length was determined based on the equation given in Clause 12.2.3 of CSA A23.3-14 (CSA, 2014). Where 𝑘 is taken as 0.8 for bars smaller than 20M, 𝑑𝑏 is the diameter of the reinforcing steel which is equal to approximately 11.7 mm for 10M bars, and the values for steel yielding and concrete strength are assumed as 400 and 30 MPa, respectively. This results in a minimum required development length of 410 mm, therefore to be conservative, a length of 450 mm was chosen as the length that would be embedded into the foundation. The size of the bearing zones where the actuators connect to the specimen were determined based on the allowable compressive stress in the concrete. It was desired that the compressive stress in the bearing zone would not exceed 10 MPa. While it is known that the maximum applicable force from the actuator is approximately 150 kN, and the width of the specimen is 175 mm wide, the height of the bearing zone was determined to be at least 85.7 mm. Therefore, it was decided that the bearing zone would be 100 mm in height. 113 In each bearing zone there are two holes going through the specimen, which are used for the connection to the actuators. These holes are roughly 30 mm in diameter and are each spaced 38.1 mm (1.5 inches) from the center. The location and size of the holes was determined based on how they fit in the specimen along with the reinforcement cage, and more details will be given in Section 3.5 3.4 Material Properties This section provides details on all measured material properties used in construction of the specimens. The materials were all tested at the UBC Civil Engineering Materials Lab. The compressive and tensile strength of the concrete was determined from cylinders poured on the same day as the specimens, that are both field and moist cured, with more details in Section 3.4.1. The steel properties were measured based on tensile tests of 8-inch long samples, and details are provided in Section 3.4.2. 3.4.1 Concrete Properties The concrete cylinders were poured into plastic moulds with a height of 200 mm and a diameter of 100 mm. All concrete cylinder samples were removed from the moulds 24 hours after being poured. Half of the cylinders were wrapped in plastic and left next to the specimen to be field cured, while the other half were moved into a moist room in the UBC Civil Engineering Materials Lab, to be moist cured. Compression tests on both the field and moist cured cylinder samples were completed at 28 days, as well as the day before testing began (92 days), and the day after testing was finished (178 days). Split cylinder tension tests were completed on both the field and moist cured cylinder samples the day before the testing began, and the day after 114 testing was finished. A photo of the split cylinder test is shown in Figure 3.5. A summary of the results is available in Tables 3.2 and 3.3. Table 3.2 Measured concrete compressive strengths from field and most cured cylinder samples. Curing Cylinder Age (Days) Height (mm) Weight (g) Density (kg/m³) Compressive Strength (MPa) Average (MPa) Field 1 28 202 4000 2452 30.7 29.9 Field 2 28 199 3950 2458 29.0 Field 3 92 204 4049 2458 37.0 36.6 Field 4 92 201 3955 2436 36.2 Field 5 178 201 3956 2437 41.8 41.0 Field 6 178 199 3952 2459 40.2 Moist 1 28 200 4000 2477 34.0 33.2 Moist 2 28 200 4000 2477 32.4 Moist 3 92 203 4030 2458 38.1 37.5 Moist 4 92 201 3948 2432 37.0 Moist 5 178 199 3973 2472 40.4 39.7 Moist 6 178 199 3956 2461 39.1 Figure 3.5: Split cylinder tensile strength test of concrete cylinder. 115 Table 3.3: Measured concrete tensile strengths from field and moist cured cylinder samples. Curing Cylinder Age (Days) Height (mm) Weight (g) Density (kg/m³) Tensile Strength (MPa) Average (MPa) Field 1 92 206 4050 2435 3.62 3.68 Field 2 92 206 4000 2405 3.75 Field 3 178 203 3978 2427 4.22 4.34 Field 4 178 202 3991 2447 4.45 Moist 1 92 207 4050 2423 3.12 3.18 Moist 2 92 207 4100 2453 3.23 Moist 3 178 203 4064 2479 3.83 3.43 Moist 4 178 202 4068 2494 3.03 3.4.2 Reinforcing Steel Properties Two types of steel were used for longitudinal and transverse reinforcement respectively. The transverse reinforcement has a nominal diameter of 6 mm and is grade HRB400 based on the Chinese standard GB1499.2-2007. Due to the size of the specimens, it was required to use small diameter reinforcing steel that is not obtainable in North America, thus, the steel was acquired from China. The tensile stress-strain relationship of the steel was obtained based on tests completed at UBC, where the test setup can be seen in Figure 3.6. Each specimen was clamped in place giving a free length of roughly 150 mm. The strain was measured by attaching an extensometer directly to the specimen which measured the elongation over a gauge length of 50 mm. The load was measured via a load cell connected to the testing machine. The specimens were tested under load control, as displacement control was not possible with the available equipment. The output of the test was given in load and displacement, which were then converted into stress and strain by dividing the load by an area of 28.3 mm2, and dividing the displacement by the gauge length of 50 mm. 116 Figure 3.6: Test setup of tensile stress-strain test on 6 mm reinforcing steel. In total, three samples were tested, and the results of the tests can be seen in Figure 3.7. Based on the likeness of the three results it was decided that no further testing would be necessary. While the stress-strain relationship does not have a clearly defined yield plateau, it can be observed that all three samples have begun yielding at approximately 400 MPa. The ultimate tensile strength of all three specimens was roughly 610 MPa. The maximum elongation of Samples 2 and 3 was around 22- 23%, whereas the elongation of Sample 1 was roughly 15%. The reason that the elongation of Sample 1 is lower is due to the fact the necking occurred outside of the region measured by the extensometer. A summary of the results measured of each specimen is available in Table 3.4. As the stress-strain results of the 6 mm diameter steel bars did not have a well-defined yield plateau, the results were compared at several levels of strain along the curve. 117 (a) (b) 01002003004005006007000 5 10 15 20 25Axial Stress (MPa)Axial Strain (%)Sample 1 Sample 2 Sample 301002003004005006000 0.5 1 1.5 2Axial Stress (MPa)Axial Strain (%)Sample 1 Sample 2 Sample 3Ab = 28.3 mm2 Ab = 28.3 mm2 118 (c) Figure 3.7: Stress-strain relationship of the 6 mm diamter reinforcing steel for: (a) Entire strain range; (b) Up to 2.0% strain; (c) Up to 0.5% strain. Table 3.4: Measured tensile stress-strain values for 6 mm diameter reinforcing bar samples. Specimen Stress at 0.2% (MPa) Stress at 0.4% (MPa) Stress at 1.0% (MPa) Stress at 2.0% (MPa) Peak Stress (MPa) Maximum Strain (%) 1 321.5 391.4 448.2 498.7 614.7 15.3* 2 333.3 377.0 442.4 491.2 611.9 22.5 3 337.0 381.1 438.3 483.1 604.9 21.8 Average 330.6 383.1 443.0 491.0 610.5 22.1 *Necking fracture occurred outside of gauge length, not included in average. The longitudinal reinforcing steel in the specimens consisted of 10M Grade W400 reinforcing bars obtained from Harris Rebar in Richmond, BC. The exact same setup that was used to test the 6 mm diameter reinforcing bars was used, however due to limitations in the testing equipment, the 10M reinforcing bars were only able to be tested up to 500 MPa. This was determined to not be a issue, as it is desired that during the actual testing of the shear wall specimens, the flexural steel in the specimens should not yield, therefore it was only necessary 01002003004005000 0.1 0.2 0.3 0.4 0.5Axial Stress (MPa)Axial Strain (%)Sample 1 Sample 2 Sample 3200 000 MPa1Ab = 28.3 mm2 119 to know the yield strength of the 10M reinforcing steel, and not the ultimate strength. In total three samples were tested and the results can be observed in Figure 3.8. All three samples have a clearly defined yield plateau at a strength of roughly 425 MPa, because the yield plateau of all three samples were similar, no further tests were completed. The yield strain was not clearly observed in the stress-strain curve, but there was a flat yield plateau in the stress which was compared at a strain of 0.4% and is shown in Table 3.5. 120 (a) (b) Figure 3.8: Stress-strain relationship of 10M reinforcing bar samples: (a) Up to 0.5% strain; (b) Up to 2.0% strain. Table 3.5: Measured tensile stress-strain values for 10M reinforcing bar samples. Specimen Stress at 0.4% (Mpa) 1 419.6 2 425.4 3 420.0 Average 421.7 01002003004005000 0.1 0.2 0.3 0.4 0.5Stress (MPa)Strain (%)Sample 1 Sample 2 Sample 3200 000 MPa101002003004005000 0.5 1 1.5 2Stress (MPa)Strain (%)Sample 1 Sample 2 Sample 3Ab = 100 mm2 Ab = 100 mm2 121 3.5 Construction of Test Specimen This section presents the construction of the test setup and the specimen. The test setup and specimen were constructed in the UBC Structural Engineering Lab and were setup on a rigid concrete strong wall/floor. The most prominent components of the experiment are the precast specimens, as well as the reusable specimen concrete foundation block. The construction of the foundation and precast specimens will be presented in detail, while other components of the experiment, such as the out-of-plane support, actuators, and actuator connections, will also be shown in Section 3.6 3.5.1 Segmental Foundation As there were five specimens, it was deemed excessive for each specimen to be cast with its own foundation. For this reason, a single foundation was constructed that would be able to be used for each specimen, while the five specimens could be precast in advance of the actual testing. This was accomplished by placing a pocket made from steel plates at the center of a reinforced concrete foundation block. In addition to the construction of the box, a method had to be developed that would allow for the specimens to be inserted and not only have a reliably strong fixed connection, but also be able to be efficiently removed. Positioning the specimen in the pocket and completely filling it with grout would certainly be the strongest solution, and the ideal choice if the specimen would be permanently embedded in the foundation, however, would be far too difficult to remove. After several iterations the chosen solution that both 122 provided sufficient strength, as well as ease of removal was composed of several components that will be summarized, with a profile of the design shown in Figure 3.9. Figure 3.9: Profile of grouted specimen inside steel pocket of reusable foundation First the bottom face of the specimen was covered in plastic, to ensure that there would be no concrete bond, which would be difficult to break when the specimen was to be removed. The specimen was then placed on a precast concrete pedestal that was plastered in place at the center of the base of the pocket. The specimen was plastered to the pedestal so that would be able to sufficiently transfer compression stress into the pedestal, as well as remain in place during construction of the remaining components of the base. Once the specimen was in place, a layer of Styrofoam was placed 300 mm deep into the pocket, the Styrofoam pieces were cut slightly larger than the area, such that they would be in compression and be held in place by the friction from the specimen and sides of the box. Duct tape was placed around the edges of the Styrofoam to ensure there would be no leakages. Then a thin half inch layer of concrete Steel Pipes Specimen Sytrofoam Void Area Concrete Pedestal Thin Grout Layer Grout 123 was poured on top of the Styrofoam, giving a strength to hold up the steel pipes and the rest of the grout. Before the thin grout layer had cured, a pattern of steel pipes wrapped in plastic were placed in the pocket around the specimen. The pipes were wrapped in plastic so that they would not bond to the grout and could be pulled easily after the grout had hardened. The steel pipes provided a solution that gave sufficient strength and reduce the amount of grout. The layout of the steel pipes is shown in Figure 3.10. Figure 3.10: Layout of steel pipes in grouted steel pocket of foundation. Once the steel pipes were in place, the grout was poured around the specimen, and held up by the Styrofoam layer. The mix design of the grout was a ratio of 3:2:1 for sand, cement, and water, respectively. The grout was left to harden for at least 24 hours before testing began. Upon completion of each test, the first step in removing the specimen was to pull out the steel pipes. Due to the plastic wrapping, there was not a bond to the concrete and the pipes were able to be pulled out relatively easily. There was a 9.5 mm diameter hole drilled in the region 124 of the pipe sticking out of the grout, so that a bolt could be inserted to aid in its removal. Figure 3.11 shows the steel pipes embedded in the grout before and after they were removed. Once the pipes were removed, there was a thin section of grout between each hole that could be drilled away, which can be seen in Figure 3.11(b). The layer of Styrofoam was able to be broken as there was access to it given through the pipe holes, and a concrete drill was then used to remove the grout. This resulted in a lot of silica dust, however the void area beneath the Styrofoam shown in Figure 3.9 served as a space for this dust to go. Early iterations of this solution did not have the void space, and dust removal served as a difficult challenge when trying to remove the specimen. Once the concrete was sufficiently drilled away, the specimen was easily pulled out of the steel pocket using the crane in the lab. Further details regarding the construction of the foundation will be given in subsection 3.6. (a) (b) Figure 3.11: Layout of plastic wrapped steel pipes in grouted pocket of foundation: (a) Before removal of pipes; (b) After removal of pipes. 125 3.5.2 Precast Specimen Construction The specimens were constructed in the UBC Structural Engineering Lab. There were three phases of construction for the specimen; 1) Construction of formwork 2) Fabrication of reinforcing bars cages 3) Pouring of concrete The formwork was constructed such that the specimens were poured on their side, with the face of the web parallel to the ground. This was done due to reduce the difficulty of ensuring that there would be full consolidation of the concrete throughout the formwork. The specimens were all poured side by side, with boxes connected to the plywood base to form one side of the web and hanging boxes that formed the other side of the web. Holes were cut in the formwork such that PVC pipes could be inserted. These pipes will allow a hole to be through the specimen at the locations of the bearing areas for the actuator connection. A photo of the formwork before the reinforcing bars cages were put in place can be seen in Figure 3.12. 126 Figure 3.12: Photos of specimen formwork prior to placement of reinforcing steel. The steel reinforcement cages were constructed using 10M bars for the flexural steel and 6 mm diameter bars used for the shear reinforcement, as well as for containment in the flanges. The 10M reinforcing bars were ordered cut and bent in the desired shape, however the 6 mm diameter reinforcing bars were cut and bent into shape at the UBC Structural Engineering Lab. The shear reinforcement was hooked on each end, with the hooks bent in opposite directions as seen in Figure 3.13(a), while containment hooped bars are shown in Figure 3.13(b). 127 (a) (b) Figure 3.13: The 6 mm reinforcing steel shapes used for reinforcement cages: (a) Stirrups; (b) Containment steel in flanges. The reinforcement cages were fabricated and put into the form work, each cage is held in place using the PVC pipes, which go in between the pieces of longitudinal steel. The locations of the flexural steel are what determined the locations of the holes, as the holes would have to go through the specimen where no reinforcement was located. The PVC pipes helped hold the cages up off of the base of the formwork, to ensure that the desired concrete cover was achieved. Once the cages were all put in place, the hanging forms were placed on top and the specimens were ready to be poured. Figure 3.14 shows the finished reinforcement cages in the formwork, the locatons of the PVC pipe holding the formwork up in the bearing area can be seen, as well as the base box that forms the bottom face of the webs, and the hanging box that forms the top face of the webs. 128 Figure 3.14: Photos of specimen reinforcement cages placed in formwork. The formwork was stripped 28 days after the concrete was poured, and the PVC pipe was cut to be flush with the specimen surface which left the five specimens finalized as can be seen in Figure 3.15. Figure 3.15: Specimens after stripping of formwork. 129 3.6 Test Setup Components All of the components of the test setup are shown in Figure 3.16, some of the components have already been presented, such as the precast specimens, and the segmental foundation, however the remaining components will be briefly explained in this section. Figure 3.16: Drawing of side-view of test setup. Strong Floor Strong Wall C H G F E D A B I A: Reinforced concrete segmental foundation to ensure specimens will have fixed base. B: Steel pocket constructed at center of foundation so that specimens can be input/removed. C: Precast concrete test specimen. D: Steel bolts that will be used to connect the actuators to the specimen. E: Steel bearing plates used to impose the load on the specimens from the actuators. F: Steel channels used to connect bearing plates to actuators. G: Steel mounting plates for specimen connection to actuators. H: Actuators used for loading the specimen. I: Steel mounting plates to ensure rigid connection between actuators and strong wall. 130 The previous figure is complemented by Figure 3.17 which shows a photograph of the completed test setup, which includes the LVDT’s that measure the specimen displacement on the left-hand side of the specimen, as well as the out-of-plane support at the top of the specimen. Figure 3.17: Side-view of test setup. 131 Figure 3.18 shows the construction of the foundation, before and after the concrete is poured. The dimensions of the foundation are 550 mm high, with base dimensions of approximately 810 by 1630 mm, and constructed from reinforced concrete. A welded steel box was placed at the center, with an interion dimension of roughly 400 by 375 mm, a thickness of 9.5 mm, a 25 mm thick steel plate at the base, and the concrete was poured around it. To ensure that the steel box bonded properly with the concrete poured around it, several one-inch length bolts were inserted such that they protruded from the box into the concrete. (a) (b) Figure 3.18: Reinforced concrete foundation that was used for specimen: (a) Formwork with steel cage; (b) Finished foundation. The connection between the actuator and the specimen was completed in a way that would allow the area of the applied demand to be reduced to the size of the bearing zone in the specimen which is approximately 100 by 175 mm as mentioned previously, compared to the face of the actuator which has a diameter of approximately 280 mm. It was also necessary that the actuator be connected to the specimen in a way that would allow it to not only push, but also pull the specimen, to properly apply a cyclic loading protocol. This was accomplished by putting a steel bearing plate on each face of the specimen with steel bolts running through, 132 channels welded to one of the bearing plates would connect it to another plate connected to the actuator face via another weld. Photos of this connection can be seen in Figure 3.19. Figure 3.19: Connection between actuators and specimen. The two actuators used in the experiment are identical and specially manufactured for the UBC Structural Engineering Lab by Norcan. The actuators have a maximum applicable static force of 38 kips (169 kN), a maximum stroke length of +/- 5 inches (125 mm), and a maximum velocity of 32 inches/second (812.8 mm/s). The maximum velocity of these actuators was critical in order to impose a displacement on the specimens that would result in a realistic rate of loading for a 30-story structure being excited in its second-mode shape, however they came with the disadvantage of having a relatively low maximum applicable force. There was also a hanging support in order to assist in carrying the gravity load of the specimens. This helped reduce the bending moment applied to the specimen from the weight of the actuators. A photo of the actuators can be seen in Figure 3.20. 133 Figure 3.20: Photograph of the actuators being used in the experiment. In addition to all the components previously mentioned, there was also an out-of-plane support that ensured the actuators imposed the displacement on the specimens in the desired in-plane axis and prevent out-of-plane displacement. The support was connected to a steel brace and was tightened on the specimens. The sliding bearing was constructed from two steel channels with Teflon on the surface to reduce friction with the specimen. A photo of the out-of-plane support is shown in Figure 3.21. 134 (a) (b) Figure 3.21: Out-of-plane support for specimens: (a) Design of support; (b) Photograph of sliding bearing around specimen. Steel Brace Specimen Foundation Sliding Bearing 135 3.7 Instrumentation The instrumentation used during the testing to record data includes load cells attached to the actuators used to measure the load, LVDT’s built into the actuators that are used to measure the actuator displacements (which are also used as the control displacement during the tests), and LVDT’s placed on the other side of the specimen used to measure the actual displacement of the specimen. In addition, there were targets placed in the test region, which is the bottom half of the specimen, that were used to measure the displaced shape which will be presented in subsection 3.7.1. The instrumentation layout can be seen in Figure 3.22 Figure 3.22: Instrumentation layout for test setup. 136 Figure 3.23 shows one of the LVDT’s used to measure the specimen displacement. The displacement was measured at the top and mid-height of the structure, roughly 75 mm above the centre of the bearing plate. During testing of Specimen 1, data acquisition was sampled at a rate of 10 Hz, however all subsequent specimens were sampled at a rate of 25 Hz due to the higher rate of loading. Figure 3.23: LVDT used to measure specimen displacement. 137 3.7.1 Displacement Coordinate Targets In addition to the recorded data, a set of targets was also setup in the test region on each specimen. The targets were placed in such a way that the specimen could be divided into a series of shear panels, with vertical, horizonal, and diagonal components. These targets were photographed using a high definition camera on a tripod, so that they were always taken from the same reference point, and a photo was taken of the test region for every load stage, at the unloaded and peak load times of the protocol. The photos of the test region were uploaded into AutoCAD and assigned coordinates such that the displaced shape could be measured and visualized geometrically. An example of the target shape is shown in Figure 3.24. Figure 3.24: Shape of target placed on specimen. The line crossing in the target shape allows for an accurate coordinate to be pin pointed from specimen photographs. The targets were placed on the specimen in a specific pattern, spaced at 125 mm, except for the bottom set of targets which are placed at 50 mm from the base, as the targets would not be blocked in photographs by the steel pipes used in the segmental foundation protruding from the grout. Figure 3.25 shows the layout of the targets in the test 138 region of the specimen, as well as the geometry of the displacements measured from the target coordinates. (a) (b) Figure 3.25: Target layout of displacement targets over the height of the structure: (a) Position of targets; (b) Geometry of displacements measured from the target coordinates.. The coordinates were plotted using excel and resulted in 7 shear panels in which the lengths of the two vertical lines, two horizontal lines, and two diagonal lines of each panel are known for the unloaded and loaded specimen, as is shown in Figure 3.25(b). 139 From these coordinates, the rotation at the centre of each panel can be estimated, by dividing the change in height on each side of the panel by the width of the panel. These rotations can be used to estimate the flexural displacement and curvature along the height of the specimen. It should be noted that the real curvature profile would be sloped, while the known rotations only allow for it to be estimated at the centre of each panel. The horizontal compressive and tensile strain can also be estimated by measuring the change in length of the horizontal components of each panel. In addition to the flexural displacement, the shear displacement is also able to be calculated, and from these shear displacements the shear strain can be determined. The shear displacements (∆𝑠) were measured using the change in the diagonal components of each panel, using a method developed by Hiraishi (1984), that accounts for the change in curvature over the height of the shear panel, which is shown in Eq. 3.1. ∆𝑠=14𝑏((𝑑 + 𝛿2)2 − (𝑑 + 𝛿1)2) − (𝛼 − 0.5)𝜃(ℎ𝑠ℎ)ℎ𝑠ℎ (3.1) Where b is the width of the shear panel, d is the length of the diagonal of the shear panel, 𝛿𝑖 is the change in length of each diagonal, ℎ𝑠ℎ is the height of the shear panel, 𝜃(ℎ𝑠ℎ) is the difference in rotation at the top and bottom of the panel, and 𝛼 is a coefficient to account for the variation of curvature over the height of the panel. The first term of Eq. 3.1 accounts for the shear displacement based on the change in length of the diagonals in the panel, while the second term is a correction factor based on the change in curvature. While the value of 𝛼 can be calculated theoretically, that was not possible as the curvature distribution of the specimen could only be estimated, Beyer et al. (2011) recommend using a value of 2/3 for a triangular curvature distribution, which was the value that was used for this thesis. 140 3.8 Testing Procedure As has been previously explained, each specimen is identical, and the parameter that varied between the specimens was the imposed displacement protocols. The protocol is applied by an imposed displacement at the top and mid-height of the specimen. As can be seen in Figure 3.26, the displacement is applied in a sinusoidal pattern, with the top displacement being approximately 35% of the mid-height displacement, but in the opposite direction. The protocol was only applied in one direction at a time, in half cycles, and a unique data set was recorded for each half cycle. Each load stage consisted of one to three full cycles (two half cycles) of equal target actuator displacements. The amplitude of the displacements will increase after each cycle in order to push the specimens to higher levels of damage, and testing was finished once the strength of the specimen began to degrade and became unstable. A full summary on the specific testing procedure of each specimen is given in Chapter 4. Figure 3.26: Single cycle of the sinusoidal imposed displacement protocol. -0.15-0.1-0.0500.050.10.150 0.1 0.2 0.3 0.4 0.5Displacement (mm)Time (s)TopDisplacementMid-HeightDisplacement141 4. Experimental Results ___________________________________________________________________________ 4.1 Overview In this chapter the results of the five specimens will be briefly presented. The key results that are presented include the strength, crack pattern, failure mode, stiffness, ductility, and the shear strain. The results within this chapter focus mainly on the test region of the specimen, which is the lower web of the specimen. The lower web was the critical element of the specimen and the desired location of failure for each test. While a brief overview and comparison of the key results is presented within this chapter, a full detailed overview of all results and observations of each respective specimen is available in Appendix E. A more comprehensive discussion of the results, including models and theories is presented in Chapter 5. 4.2 Summary of Results In the following results, positive displacement is when the actuator is pushing (Cycle “A”), while the negative displacement is when the controlling actuator is pulling (Cycle “B”). 4.2.1 Specimen 1 Specimen 1 was tested by applying the imposed displacement from the mid-height and top actuators at a very slow rate of loading (between 0.01 mm/s and 0.03 mm/s), in order to determine baseline results on the shear strength and ductility of the specimen subjected to the second-mode displacement protocol under minimal interfering conditions. The two actuators were brought to contact the specimen with a force of less than 1 kN, and were then tightened 142 in place, to ensure a firm connection. Before imposing any displacement using the actuators, the data acquisition system was set to zero displacement and then recording started. The mid-height displacement was applied slowly over a course of 600 seconds (10 minutes), the actuators then held the specimen in place for a period of 300 to 600 seconds (5 to 10 minutes), so that the displaced shape could be studied. While the specimen was held in place, the cracks were measured, and photographs of crack patterns were taken, there were no displaced shape targets placed on the first specimen. The specimen was then unloaded over the same length of time as it was applied (10 minutes). Once the protocol had finished and the displacement returned to zero, the data recording was stopped. The displacement recording was then set to zero before the protocol was applied in the other direction. A full table of all the displacement and force demands can be found in Appendix E. The specimen was loaded to two load cycles while remaining uncracked and in the linear elastic range. The magnitudes of the first two load stages were between mid-height displacements of -0.40 and +0.35 mm, and base shear force demands of approximately -18 to +15 kN. The first cracks were observed during Load Stage 3, which were diagonal cracks in the upper portion of the lower web, diagonal cracks in the top web, flexural cracks in the top web, as well as several flexural cracks along the flanges on each side of the specimen. Photos of the crack profiles in the lower web are shown in Figure 4.1(a), while full specimen photos are available in Figure E.1. The mid-height displacements of Load Stage 3 were between approximately -1.0 mm and + 0.8 mm, and base shear force demands of approximately +/- 30 kN, which is shown in the hysteretic results availabile in Figure E.9. 143 (a) (b) (c) Figure 4.1: Crack pattern of Specimen 1 test region for various stages of damage: (a) early cracking; (b) severe cracking; (c) crack pattern at failure. The approximate location and width of measured diagonal cracks are shown in Figure 4.2. Cracks denoted “A” are when the mid-height actuator is pushing, while those denoted “B” are when the mid-height actuator is pulling. A full summary of measured crack widths, including a photo of the measured cracks is available in Appendix E. 144 Figure 4.2: Approximate location and width of measured diagonal cracks during final load stage for Specimen 1. In Load Stages 4 through 6, cracks continued forming throughout both webs, with significant diagonal cracking in the lower web. The largest crack width was measured as approximately 0.95 mm at a height of 500 mm from the base of the specimen in the centre of the web. The crack profile in the lower web after Load Stage 6 is shown in Figure 4.1(b), while the full crack profiles of Load Stages 4 through 6 are shown in Figures E.2 through E.4. The magnitude of the specimen demands were mid-height displacements of +/- 3.75 mm, and peak base shear force demands of +/- 60 kN. The hystertic data of Load Stages 4 through 6 is shown in Figures E.10 to E.12. 020040060080010001200140016000.00 0.30 0.60 0.90 1.20 1.50Height of Crack Measurement (mm)Measured Diagonal Crack Width (mm)0 0.3 0.6 0.9 1.2 1.5Measured Diagonal Crack Width (mm)LS8ALS8B145 (a) (b) Figure 4.3: Specimen 1 force-displacement results for: (a) base shear (Vb) vs mid-height displacement (D1); (b) mid-height moment (Mm) vs mid-height displacement (D1). 146 The peak base shear strength of approximately +/- 68 kN was observed during Load Stage 7, which occurred during the first cycle of this load stage and coincided with mid-height displacements of +/- 5.2 mm. During the second cycle of Load Stage 7, degradation was observed in the positive force direction (mid-height actuator pushing) which resulted in a drop in shear strength to 50 kN, and a slight increase in mid-height displacement, and minor degradation was observed in the negative direction (mid-height actuator pulling). While some cracks began opening significantly during this load stage, being measured as high 1.25 mm, many smaller diagonal cracks developed as the stress redistributed through the web. The full crack profile of Load Stage 7 is shown in Figure E.5, while the hysteretic data is shown in Figure E.13. During Load Stage 8 the magnitude of the mid-height displacement was between -7.3 and +7.5 mm, and the corresponding shear strength at these ductility demands was between -40 and +30 kN, indicating a drop in strength of approximately 40 to 55%. Due to the large drop in strength the specimen was considered failed, with the failure mode being ductile shear as a result of diagonal concrete crushing in the bottom web after yielding in the stirrups occurred. The concrete crushing was observed along the longitduinal 6 mm diameter reinforcing bar on the right side of the web. The crack profile in the bottom web from Load Stage 8 is shown in Figure 4.1(c), while the specimen photo is shown in Figure E.6, while the hysteretic response is shown in Figure 4.3, as well as Figure E.14. 147 4.2.2 Specimen 2 Specimen 2 was tested at a much higher rate of loading (up to about 4 mm/s). The two actuators were brought to contact the specimen with a force of less than 1 kN, and were then tightened in place, to ensure a firm connection. Before imposing any displacement using the actuators, the data acquisition system was set to zero displacement and then recording started. For the first few Load Stages, the displacement was applied over a course of 4 seconds, and immediately reversed directions, unloading over a course of 4 seconds. As the magnitude protocol displacement became larger, the length of the load time increased as high as 8 seconds, due to limitations of the actuator velocity. The displacement coordinate targets were first used during this test, however, since the specimen was not held in place at the peak, it was difficult to mark/measure cracks and photograph the displaced shape targets. Residual crack widths were taken once all testing was completed. Once the protocol had finished and the displacement returned to zero, the data recording was stopped. The displacement recording was then set to zero before the protocol was applied in the other direction. Unlke the first specimen, the second specimen was loaded to three load load stages while remaining uncracked and in the linear elastic range. The magnitudes of the first three load stages were between mid-height displacements of +/- 1.0 mm, and base shear force demands of approximately -12 to +18 kN. The first cracks were observed during Load Stage 4, which were diagonal cracks in the lower web, minor diagonal cracking in the top web, as well as a few very minor flexural cracks along the flanges on each side of the specimen. Photos of the crack profiles in the bottom web of the specimen are shown in Figure 4.4(a), while full specimen photos are available in Figure E.16. The mid-height displacements of Load Stage 4 148 were between approximately -1.6 mm and +1.7 mm, and base shear force demands of between approximately -16 and +23 kN, which is shown in the hysteretic results availabile in Figure E.24. (a) (b) (c) Figure 4.4: Crack pattern of Specimen 2 test region for various stages of damage: (a) early cracking; (b) severe cracking; (c) crack pattern at failure. 149 As has previously been mentioned, only residual cracks were measured at the end of the test, once the load had been removed. The approximate location and width of measured residual diagonal cracks are shown in Figure 4.5. A full summary of measured crack widths, including a photo of the measured cracks is available in Appendix E. Figure 4.5: Approximate location and width of measured residual diagonal cracks during final load stage for Specimen 2. In Load Stages 5 through 7, diagonal cracks continued to develop in both webs, with more severe cracking in the bottom web, and there was very little additional flexural cracking in the flange. Due to the dynamic nature of the test it was not possible to measure crack widths during this test. The crack profile in the bottom web after Load Stage 6 is shown in Figure 4.4(b), while the full crack profiles of Load Stages 5 through 7 are shown in Figures E.17 through E.19. The peak base shear force strength was observed during Load Stage 7, and was equal approximately +/- 68 kN. Corresponding mid-height displacements of +/- 4.9 mm, and the hystertic data of Load Stages 5 through 7 is shown in Figures E.25 to E.27. 020040060080010001200140016000.00 0.30 0.60 0.90 1.20 1.50Height of Crack Measurement (mm)Measured Diagonal Crack Width (mm)0 0.3 0.6 0.9 1.2 1.5Measured Diagonal Crack Width (mm)LS8ALS8B150 (a) (b) Figure 4.6: Specimen 2 force-displacement results for: (a) base shear (Vb) vs mid-height displacement (D1); (b) mid-height moment (Mm) vs mid-height displacement (D1). 151 During Load Stage 8 the magnitude of the mid-height displacement was between -7.2 and +7.1 mm, and the corresponding shear strength at these ductility demands was between approximately +/- 42 kN, indicating a drop in strength of approximately 40% in each direction of loading. Due to the drop in shear strength the specimen was considered failed, with the failure mode being ductile shear as a result of diagonal concrete crushing in the bottom web after yielding in the stirrups occurred. The concrete crushing was observed along the longitduinal 6 mm diameter reinforcing bar on the right side of the web. The crack profile in the bottom web from Load Stage 8 is shown in Figure 4.4(c), while the specimen photo is shown in Figure E.20, while the hysteretic response is shown in Figure 4.6, as well as Figure E.28. 4.2.3 Specimen 3 Specimen 3 was tested in two phases, where the goal of the first phase was to yield the base of the specimen by applying first-mode shape displacements using only the top actuator. During the first phase, the top actuator was brought to contact the specimen with a force of less than 1 kN, and was then tightened in place, to ensure a firm connection. Before imposing any displacement using the hydraulic actuator, the data acquisition system was set to zero displacement and then recording started. The displacement was applied at the top only, to maximize the bending moment, over a length of 15 to 30 seconds. The length of time over which the displacement was applied increased as the imposed displacements became larger, due to limitations of the actuator velocity. The specimen was held in place at the peak displacement for 5 to 10 minutes to study the cracks and photograph the displaced shape 152 targets. Once the protocol had finished and the displacement returned to zero, the data recording was stopped. The second phase of the test began once the base of the specimen was determined to be yielded at the base, as the increase in displacement between each load stage began to decrease. The second phase was similar to the previous two tests, where the mid-height actuator was brought to make contact and was firmly tightened in place, and the second-mode displacement protocol was applied. From this point, each load stage was applied over a course of 15 to 30 seconds, held in place for 5 to 10 minutes to take crack measurements and photograph the displaced shape targets, and unloaded over 15 to 30 seconds. Once the protocol had finished and the displacement returned to zero, the data recording was stopped. A full table of all the displacement and force demands is available in Appendix E. During Phase 1, flexural cracks were observed first in the lower web and along the flanges on each side of the specimen that first developed during Load Stage 3, the crack profile is shown in the lower web in Figure 4.7(a), while the full specimen crack profile is shown in Figure E.30. During the seventh and final load stage of Phase 1, several diagonal cracks had developed in both the upper and lower webs, however the largest crack widths were measured near the base of the specimen. The largest crack width measured after Phase 1 was equal to 0.30 mm and was located approximately 125 mm from the base of the specimen. The full specimen crack profile after Phase 1 is available in Figure E.31. There were seven load stages in total during Phase 1, with maximum top specimen displacements between -28.9 and +28.6 mm and base shear demands between approximately -31 and +29 kN. There was very minor flexural yielding observed during the Phase 1, as can be seen in the hysteretic response shown in Figure 4.9 as well as E.38. 153 As the specimen had already been damaged during Phase 1, there was no observable increase in the size or number of cracks until Load Stage 5 of Phase 2. During Load Stage 5, the specimen had been cycled between mid-height displacements of +/- 2.3 mm, and base shear force demands of between -21 and +25 kN. The same crack that had a measured width of 0.30 mm during Phase 1 was measured to be 0.50 mm, and was the largest measured crack during Load Stage 5. The full specimen crack profile after Load Stage 5 of Phase 2 is shown in Figure E.32. (a) (b) (c) Figure 4.7: Crack pattern of Specimen 3 test region for various stages of damage: (a) early cracking; (b) severe cracking; (c) crack pattern at failure. 154 The approximate location and width of measured diagonal cracks are shown in Figure 4.8. Cracks measured during the first two load stages of Phase 2 were formed during Phase 1, and present at the beginning of Phase 2. A full summary of measured crack widths, including a photo of the measured cracks can be found in Appendix E. Figure 4.8: Approximate location and width of measured diagonal cracks during final load stage for Specimen 3. The largest base shear strength was observed during the first cycle of Load Stage 8, which was evaluated to be -64 kN in one direction, and +59 kN in the other. The corresponding mid-height displacements were equal to -6.5 and +5.7 mm. During the latter cycles of Load Stage 8, very minor degradation of the shear strength was observed, and is shown in the hysteretic response in Figure E.44. The large cracks that formed near the base during Phase 1 did not increase significantly in width, with the largest measured diagonal crack widths during Load Stage 8 being 1.25 mm at a height of approximately 300 mm from the base of the specimen in the middle of the web. The full specimen crack profile after Load Stage 8 of Phase 2 is shown in Figure E.35. 020040060080010001200140016000.00 0.30 0.60 0.90 1.20 1.50Height of Crack Measurement (mm)Measured Diagonal Crack Width (mm)0 0.3 0.6 0.9 1.2 1.5Measured Diagonal Crack Width (mm)LS8ALS8B155 (a) (b) Figure 4.9: Specimen 3 Phase 1 force-displacement results for: (a) base shear (Vb) vs mid-height displacement (D1); (b) base moment (Mb) vs top displacement (D2). 156 (a) (b) Figure 4.10: Specimen 3 Phase 2 force-displacement results for: (a) base shear (Vb) vs mid-height displacement (D1); (b) mid-height moment (Mm) vs mid-height displacement (D1). -80-60-40-20020406080-10 -8 -6 -4 -2 0 2 4 6 8 10Vb (kN)D1 (mm)L4AL9B2L8B2L6BL4BL9A2L8A2L6A157 During Load Stage 9 the magnitude of the mid-height displacement was between -8.1 and +7.4 mm, and the corresponding shear strength at these ductility demands was between approximately -38 and +47 kN, indicating a drop in strength of approximately 40% in the negative direction, and 20% in the positive direction. Due to the drop in shear strength in the negative direction, the specimen was considered failed, with the failure mode being ductile shear as a result of diagonal concrete crushing in the bottom web after yielding in the stirrups occurred. The concrete crushing was observed in the middle of the web, between the two longitduinal 6mm reinforcing bars. The crack profile in the bottom web from Load Stage 9 is shown in Figure 4.7(c), while the specimen photo is shown in Figure E.36, and the hysteretic response is shown in Figure 4.10 and Figure E.45. Figures 4.11 through 4.15 show the results of analyzing the displaced shape using the coordinates obtained from the photographed displacement targets posted on the specimen. From these coordinates, it was possible to calculate the rotation, curvature, flexural displacement, shear displacement and strain, and horizontal displacement and strain. The displacement profiles analyzed were chosen for selected load stages labelled in Figures 4.9 and 4.10, in this chapter only the results for “A” cycles are presented, with remaining results available in Appendix E. The process of converting the coordinates to these values was summarized in Chapter 3. Displacements labeled with “F” were Load Stages applied during Phase 1, while those labeled “L” were Load Stages in Phase 2. Figure 4.11 shows the total displacement profile of the specimen; the total displacement is considered the combination of both the shear and flexural components. The total displacement was determined in three different ways. The upper displacements were measured using the LVDT data and are denoted with squares. The total displacement was also calculated by taking 158 the sum of the flexural and shear displacements estimated by the displacement coordinates and is denoted by circles and connected plot points. Finally, the displaced shape could be determined simply by measuring the change in horizontal displacement of the displacement coordinates on the left-hand side (denoted by diamonds), and the right-hand side (denoted by triangles). In general, the results agree with each other, justifying the various methods used to obtain them. 159 (a) (b) Figure 4.11: Displaced shape determined from displacement coordinates in selected Load Stages for Specimen 3: (a) Phase 1; (b) Phase 2. 160 Figure 4.12 shows the flexural displacement along the height of the specimen, where the contrast between the displacements during Phase 1 and 2 can be seen. The flexural displacement in Phase 1 is relatively larger, while the flexural displacement in Phase 2 is responding in double curvature. The peak flexural displacement during Phase 2 was observed to be equal to approximately 3.5 mm at a height of 500 mm from the base of the specimen during Load Stage L9A2. (a) (b) Figure 4.12: Flexural displacement calculated from displacement coordinates in selected Load Stages for Specimen 3: (a) Phase 1; (b) Phase 2. 0100200300400500600700800-8 -6 -4 -2 0Height (mm)Flexural Displacement (mm)3-F4A3-F7A01002003004005006007008000 1 2 3 4Height (mm)Flexural Displacement (mm)3-L4A13-L6A13-L8A23-L9A2161 Figure 4.13 shows the estimated curvature profile along the height of the specimen, where the contrast between the displacements during Phase 1 and 2 can be seen. The curvature profiles are only estimates available as a constant value for each segment along the height, while in actuality the curvature would be varying along the entire height. The curvature profile in Phase 1 was entirely in one direction, while the profile in Phase 2 indicates the specimen is in double curvature. The peak curvature during Phase 2 was equal to -100 rad/km and occurred over the top 250 mm of the lower web. The large incremental curvature that occurs at each Load Stage can be attributed to flexural yielding due to the mid-height bending moment. (a) (b) Figure 4.13: Curvature profile calculated from displacement coordinates in selected Load Stages for Specimen 3: (a) Phase 1; (b) Phase 2. 0100200300400500600700800-60 -40 -20 0Height (mm)Curvature (rad/km)3-F4A3-F7A0100200300400500600700800-150 -100 -50 0 50 100Height (mm)Curvature (rad/km)3-L4A13-L6A13-L8A23-L9A2162 Figure 4.14 shows the calculated shear displacement and strain profile along the height of the specimen. The comparison in shear ductility demands between the two phases is displayed, where the strain observed during Phase 1 is similar to what was seen during the early load stages of Phase 2, despite that fact that the total displacement is much larger. A peak shear strain of just under 3% was observed at a height of roughly 250 mm. (a) (b) Figure 4.14: Shear parameters calculated from displacement coordinates during selected Load Stages for Specimen 3: (a) Shear displacement; (b) Shear strain. 0100200300400500600700800-15 -10 -5 0 5Height (mm)Shear Displacement (mm)3-F4A3-F7A3-L4A13-L6A13-L8A23-L9A20100200300400500600700800-4 -3 -2 -1 0 1Height (mm)Shear Strain (%)3-F4A3-F7A3-L4A13-L6A13-L8A23-L9A2163 Figure 4.15 shows the calculated horizontal displacement and strain profile along the height of the specimen. This displacement is defined as the change in length between the targets of equivalent heights on the left and right and side of the specimen and serve as a helpful estimate of the strain in the stirrups. The horizontal strain was obtained by dividing the displacement by 200 mm, which is the width of the web. The maximum positive strain was observed prior to reaching the peak base shear force demand in the specimen during Load Stage L6A1, at a height of approximately 250 mm. The maximum negative strain was observed during Load Stage L9A2, at a height of 750 mm, and was equal to just over -1.25%. (a) (b) Figure 4.15: Horizontal displacement calculated from displacement coordinates in selected Load Stages for Specimen 3: (a) Displacement; (b) Strain. 0100200300400500600700800-3 -2 -1 0 1 2Height (mm)Horizontal Displacement (mm)3-F4A3-F7A3-L4A13-L6A13-L8A23-L9A20100200300400500600700800-1.5 -1 -0.5 0 0.5 1Height (mm)Horizontal Strain (%)3-F4A3-F7A3-L4A13-L6A13-L8A23-L9A2164 4.2.4 Specimen 4 Specimen 4 was tested in two phases, where the goal of the first phase was to yield the base of the specimen by applying first-mode displacements using only the top actuator, and to apply a constant bending moment at the base of the specimen through a constant imposed displacement at the top of the specimen. The top actuator was brought to contact the specimen with a force of less than 1 kN, and was then tightened in place, to ensure a firm connection. Before imposing any displacement using the actuators, the data acquisition system was set to zero displacement and then recording started. The displacement was applied at the top only, in order to maximize the bending moment, over a length of 15 to 30 seconds. The length of time over which the displacement was applied increased as the imposed displacements became larger, due to limitations of the actuator velocity. The specimen was held in place at the peak displacement for 5 to 10 minutes to study the cracks and photograph the displaced shape targets. During the final half cycle, the top actuator was left at 30 mm, and not returned to zero, causing a constant displacement at the top of the specimen, and a bending moment at the base, meaning there was a load in the top actuator, and the mid-height actuator was still not connected. The second phase of the test began by bringing the mid-height actuator to contact the displaced specimen and was firmly tightened in place. While the specimen was in this displaced position, the position of the actuators was noted, and set as the relative zero-displacement point of the following second-mode displacement tests. From this point, each load stage was applied over a course of 15 to 30 seconds, held in place for 5 to 10 minutes to take crack measurements and photograph the displaced shape targets, and unloaded over 15 to 30 seconds. Once the protocol 165 had finished and the displacement returned to zero, the data recording was stopped. A full table of all the displacement and force demands is available in Appendix E. During Phase 1, flexural cracks were observed first in the lower web and along the flanges on each side of the specimen that first developed during Load Stage 2, the crack profile is shown in the lower web in Figure 4.16(a), while the full specimen crack profile is shown in Figure E.59. The corresponding top specimen displacements and shear forces after Load Stage 2 of Phase 1 were between -12.3 and +11.7 mm, and approximately -21 and +19 kN. During the fifth and final load stage of Phase 1, several diagonal cracks had formed in both the upper and lower webs, however the largest crack widths were measured near the base of the specimen. The largest crack measured after Phase 1 was equal to 0.50 mm and was located approximately 100 mm from the base of the specimen. The full specimen crack profile after Phase 1 is available in Figure E.60. There were seven load stages in total during Phase 1, with maximum top specimen displacements between -30.4 and +29.4 mm, and base shear demands between approximately -31 and +29 kN. There was very minor flexural yielding observed during Phase 1, as can be seen in the hystertic response in Figures 4.18 and E.38. During the final cycle, the imposed displacement held the specimen in place, resulting in Phase 2 having an initial base shear demand of approximately +29 kN. This position was assumed to be the new point of zero displacement for Phase 2. As the specimen had already been damaged during Phase 1, there was no observable increase in the size and number of cracks until Load Stage 4 of Phase 2. During Load Stage 4, the specimen had been cycled between mid-height displacements of - 2.6 and +1.8 mm from the point of imposed displacement at the end of Phase 1, and base shear force demands of between 166 approximately -10 and +58 kN. Due to the initial shear force and imposed displacement from the end of Phase 1, the specimen was only shear critical in the positive force direction, while flexural yielding controlled the response in the negative direction. This can be determined by evaluating the hysteretic responses given in Figure E.69. This was also determined based on large flexural cracks that developed when the mid-height actuator was pulling (negative shear force direction), and development of large diagonal cracks in the lower web when the mid-height actuator was pushing (positive shear force direction). The full crack profile after Load Stage 4 is shown in Figure E.61. (a) (b) (c) Figure 4.16: Crack pattern of Specimen 4 test region for various stages of damage: (a) early cracking; (b) severe cracking; (c) crack pattern at failure. 167 The approximate location and width of measured diagonal cracks are shown in Figure 4.17. Cracks measured during the first two load stages of Phase 2 were formed during Phase 1, and present at the beginning of Phase 2. A full summary of measured crack widths, including a photo of the measured cracks can be found in Appendix E. Figure 4.17: Approximate location and width of measured diagonal cracks during final load stage for Specimen 4. The largest base shear strength was observed during the first cycle of Load Stage 7, and was only reached in the positive shear force direction. The maximum shear strength was evaluated to be +61 kN, and the corresponding mid-height displacement was equal to +4.1 mm. During the second cycle of Load Stage 8, some degradation of the shear strength was observed and is shown in Figure E.72. The shear force observed in the second cycle was equal to 56 kN resulting in a decrease of approximately 8%. The large cracks that formed near the base during Phase 1 did not increase in size significantly, with the largest measured diagonal crack width during Load Stage 7 being 1.5 mm. The full specimen crack profile after Load Stage 7 is shown in Figure E.63. 02004006008001000120014000.00 0.30 0.60 0.90 1.20 1.50Height of Crack Measurement (mm)Measured Diagonal Crack Width (mm)0 0.3 0.6 0.9 1.2 1.5Measured Diagonal Crack Width (mm)LS7ALS7B168 (a) (b) Figure 4.18: Specimen 4 Phase 1 force-displacement results for: (a) (a) base shear (Vb) vs mid-height displacement (D1); (b) base moment (Mb) vs top displacement (D2). -40-30-20-10010203040-20 -15 -10 -5 0 5 10 15 20Vb (kN)D1 (mm)F5AF5B-60-40-200204060-40 -30 -20 -10 0 10 20 30 40Mb (kN-m)D2 (mm)169 (a) (b) Figure 4.19: Specimen 4 Phase 2 force-displacement results for: (a) base shear (Vb) vs mid-height displacement (D1); (b) mid-height moment (Mm) vs mid-height displacement (D1). -40-20020406080-10 -8 -6 -4 -2 0 2 4 6 8Vb (kN)D1 (mm)L3AL6BL5BL3BL8A1L7A2L7A1L6AL5AL8B1L7BL8A2L8B2170 During Load Stage 8 the magnitude of the mid-height displacement in the positive direction was +6.4 mm, and the corresponding shear strength at this displacement was approximately +47 kN, indicating a drop in strength of approximately 23% from the peak shear demand observed in the first cycle of Load Stage 7. Due to the drop in shear strength in the negative direction, the specimen was considered failed, with the failure mode being ductile shear as a result of diagonal concrete crushing in the bottom web after yielding in the stirrups occurred. The concrete crushing was observed along the longitduinal 6 mm diameter reinforcing bar on the right side of the web. The crack profile in the lower web after Load Stage 9 is shown in Figure 4.16(c), while the full specimen crack profile is shown in Figure E.64, and the hysteretic response is shown in Figures 4.19 and E.73. Figures 4.20 through 4.24 show the results of analyzing the displaced shape using the coordinates obtained from the photographed displacement targets posted on the specimen. The displacement profiles analyzed were chosen for selected load stages labelled in Figures 4.18 and 4.19, in this Chapter only the results for “A” cycles are presented, with remaining results available in Appendix E. Figure 4.20 shows the total displacement profile of the specimen; the total displacement is considered the combination of both the shear and flexural components. In general, the results agree with each other, justifying the various methods used to obtain them. 171 (a) (b) Figure 4.20: Displaced shape determined from displacement coordinates in selected Load Stages for Specimen 4: (a) Phase 1; (b) Phase 2. 020040060080010001200140016001800-40 -30 -20 -10 0Height (mm)Total Displacement (mm)4-F5A - LVDT Measurement - Calculated - Left Coordinates - Right Coordinates020040060080010001200140016001800-10 -5 0 5Height (mm)Total Displacement (mm)4-L3A4-L5A4-L6A4-L7A14-L7A24-L8A14-L8A2172 Figure 4.21 shows the flexural displacement along the height of the specimen, where the contrast between the displacements during Phase 1 and 2 can be seen. The flexural displacement in Phase 1 is relatively larger, while the flexural displacement in Phase 2 is responding in double curvature. The largest flexural displacement during Phase 2 was observed during Load Stage L7A1, and was equal to approximately 1.4 mm at a height of 500 mm. During the peak Load Stages (L8A1/2), the height of the peak displacement was still at a height of 500 mm, but the magnitude of the displacement had reduced. (a) (b) Figure 4.21: Flexural displacement calculated from displacement coordinates in selected Load Stages for Specimen 4: (a) Phase 1; (b) Phase 2. 0100200300400500600700800-10 -8 -6 -4 -2 0Height (mm)Flexural Displacement (mm)4-F5A0100200300400500600700800-1 -0.5 0 0.5 1 1.5Height (mm)Flexural Displacement (mm)4-L3A4-L5A4-L6A4-L7A14-L7A24-L8A14-L8A2173 Figure 4.22 shows the estimated curvature profile along the height of the specimen, where the contrast between the displacements during Phase 1 and 2 can be seen. The curvature profiles are only estimates available as a constant value for each segment along the height, while in actuality the curvature would be varying along the entire height. The curvature profile in Phase 1 was entirely in one direction, while the profile in Phase 2 indicates the specimen is in double curvature. A large increase in curvature was observed at the base, indicating base yielding of the flexural steel. (a) (b) Figure 4.22: Curvature profile calculated from displacement coordinates in selected Load Stages for Specimen 4: (a) Phase 1; (b) Phase 2. 0100200300400500600700800-80 -60 -40 -20 0Height (mm)Curvature (rad/km)4-F5A0100200300400500600700800-60 -40 -20 0 20 40Height (mm)Curvature (rad/km)4-L3A4-L5A4-L6A4-L7A14-L7A24-L8A14-L8A2174 Figure 4.23 shows the calculated shear displacement and strain profile along the height of the specimen. The comparison in shear ductility demands between the two phases is displayed, where the strain observed during Phase 1 is similar to what was seen during the early load stages of Phase 2, despite that fact that the total displacement is much larger. A peak shear strain of just over 1.5% was observed at a height of roughly 350 mm. (a) (b) Figure 4.23: Shear parameters calculated from displacement coordinates during selected Load Stages for Specimen 4: (a) Shear displacement; (b) Shear strain. 0100200300400500600700800-10 -8 -6 -4 -2 0Height (mm)Shear Displacement (mm)4-F5A4-L3A4-L5A4-L6A4-L7A14-L7A24-L8A14-L8A20100200300400500600700800-2 -1.5 -1 -0.5 0 0.5Height (mm)Shear Strain (%)4-F5A4-L3A4-L5A4-L6A4-L7A14-L7A24-L8A14-L8A2175 Figure 4.24 shows the calculated horizontal displacement and strain profile along the height of the specimen. This displacement is defined as the change in length between the targets of equivalent heights on the left and right and side of the specimen and serves as a helpful estimate of the strain in the stirrups. The horizontal strain was obtained by dividing the displacement by 200 mm, which is the width of the web. The maximum strain was observed during Load Stage L8A1, at a height of roughly 350 mm, with a decrease in strain the during the second cycle observed in L8A2. (a) (b) Figure 4.24: Horizontal displacement calculated from displacement coordinates in selected Load Stages for Specimen 4: (a) Displacement; (b) Strain. 0100200300400500600700800-1 0 1 2 3Height (mm)Horizontal Displacement (mm)4-F5A4-L3A4-L5A4-L6A4-L7A14-L7A24-L8A14-L8A20100200300400500600700800-0.5 0 0.5 1 1.5Height (mm)Horizontal Strain (%)4-F5A4-L3A4-L5A4-L6A4-L7A14-L7A24-L8A14-L8A2176 4.2.5 Specimen 5 Specimen 5 was tested in two phases, where the goal of the first phase was to form a plastic hinge at the base of the specimen by applying first-mode displacements using only the top actuator. The top actuator was brought to contact the specimen with a force of less than 1 kN, and was then tightened in place, to ensure a firm connection. The position of the actuator was recorded at this point, to ensure the actual zero displacement point of the specimen before any damage. Before imposing any displacement using the actuators, the data acquisition system was set to zero displacement and then recording started. The displacement was applied at the top only, to maximize the bending moment, over a length of 15 to 45 seconds. The length of time over which the displacement was applied increased as the imposed displacements became larger, due to limitations of the actuator velocity. The specimen was held in place at the peak displacement for 5 to 10 minutes to study the cracks and photograph the displaced shape targets. Once the first phase of the test was completed, the hydraulic system was turned off, and the pressure in the actuator was set to zero, leaving the specimen with a residual displacement due to the damage at the base. Before beginning the second phase, the top actuator was used to apply a displacement on the specimen that would overcome the residual displacement, returning it to the true zero position and leaving an initial load in the top actuator. The mid-height actuator was then brought to contact the displaced specimen and was firmly tightened in place. From this point, each load stage was applied over a course of 15 to 30 seconds, held in place for 5 to 10 minutes to take crack measurements and photograph the displaced shape targets, and unloaded over 15 to 30 seconds. Once the protocol had finished and the displacement returned to zero, the data 177 recording was stopped. A full table of all the displacement and force demands is available in Appendix E. During Phase 1, flexural cracks were observed first in the lower web and along the flanges on each side of the specimen. As the magnitude of the imposed displacement increased, the amount of flexural cracking increased significantly, with flexural cracks forming along the bottom three quarters of the specimen. There were also some diagonal cracks that formed in both the lower and top webs. The crack profile after Load Stage 4 of Phase 1 is shown in in Figure 4.25(a), and the full specimen crack profile is shown in Figure E.86. The corresponding top specimen displacements and shear forces after Load Stage 4 were between -24.8 and +24.2 mm, and approximately -29.5 and +29.3 kN. The final load stage of Phase 1 was cycled three times, in order to maximize flexural damage at the base of the specimen. Outside of approximately the bottom 350 mm of the specimen, the cracks no longer increased in size as the plasticity in the specimen was concentrated in the plastic hinge. Crack width measurements between 1.0 and 1.25 mm were observed in the plastic hinge region. The crack profile of the lower web after Phase 1 is shown in Figure 4.25(b), while the full specimen crack profile is shown in Figure E.87. There were eight load stages in total during Phase 1, with maximum top specimen displacements between -63.5 and +61.9 mm, and base shear demands between approximately -35 and +33 kN. There was major flexural yielding observed during the Phase 1, including the formation of a plastic hinge at the base of the specimen, as evidenced by the hysteretic data shown in Figures 4.27 and E.92. Due to the residual displacement at the end of Phase 1, a force of approximately 10 kN was applied using the top actuator to return the specimen to it original position of zero displacement. 178 (a) (b) (c) Figure 4.25: Crack pattern of Specimen 5 test region for various stages of damage: (a) early cracking; (b) severe cracking; (c) crack pattern at failure. The approximate location and width of measured diagonal cracks are shown in Figure 4.26. Cracks measured during the first two load stages of Phase 2 were formed during Phase 1, and present at the beginning of Phase 2. Large crack were measured near the base, however cracks in the upper portion of the specimen remained relatively small. A full summary of measured crack widths, including a photo of the measured cracks are presented in Appendix E. 179 Figure 4.26: Approximate location and width of measured diagonal cracks during final load stage for Specimen 5. As the specimen had already been severely damaged during Phase 1, there was little observable increase in the size and number of cracks until Load Stage 4 of Phase 2. The largest base shear strength was observed during the first cycle of Load Stage 4, and was only reached in the positive shear force direction. The maximum shear strength was evaluated to be +41 kN, and the corresponding mid-height displacement was equal to +5.0 mm. 020040060080010001200140016000.00 0.30 0.60 0.90 1.20 1.50Height of Crack Measurement (mm)Measured Diagonal Crack Width (mm)0 0.3 0.6 0.9 1.2 1.5Measured Diagonal Crack Width (mm)LS4ALS4B180 (a) (b) Figure 4.27: Specimen 5 Phase 1 force-displacement results for: (a) (a) base shear (Vb) vs mid-height displacement (D1); (b) base moment (Mb) vs top displacement (D2). 181 (a) (b) Figure 4.28: Specimen 5 Phase 2 force-displacement results for: (a) base shear (Vb) vs mid-height displacement (D1); (b) mid-height moment (Mm) vs mid-height displacement (D1). 182 Load Stage 4 was cycled three times in the negative direction, and four times in the positive direction. During the fourth cycle in the positive direction, the magnitude of the mid-height displacement in the positive direction was equal to +5.4 mm, and the corresponding shear strength was approximately +30 kN, indicating a drop in strength of 27% from the peak observed shear strength. Due to the drop in shear strength in the positive direction, the specimen was considered failed, with the failure mode being ductile shear as a result of diagonal concrete crushing in the bottom web after yielding in the stirrups occurred. The concrete crushing was observed along the longitduinal 6 mm diameter reinforcing bar on the right side of the web. The crack profile in the bottom web after Load Stage 4 of Phase 2 is shown in Figure 4.25(c), and the full specimen crack profile is shown in Figure E.90. The specimen was only shear critical in the positive force direction, while flexural yielding controlled the response in the negative direction. This was determined by evaluating the hysteretic responses given in Figures E.93 through E.96. Figures 4.29 through 4.33 show the results of analyzing the displaced shape using the coordinates obtained from the photographed displacement targets posted on the specimen. The displacement profiles analyzed were chosen for selected load stages labelled in Figures 4.27 and 4.28, in this Chapter only the results for “A” cycles are presented, with remaining results available in Appendix E. Figure 4.29 shows the total displacement profile of the specimen; the total displacement is considered the combination of both the shear and flexural components. In general, the results agree with each other, justifying the various methods used to obtain them. 183 (a) (b) Figure 4.29: Displaced shape determined from displacement coordinates in selected Load Stages for Specimen 5: (a) Phase 1; (b) Phase 2. 020040060080010001200140016001800-80 -60 -40 -20 0Height (mm)Total Displacement (mm)5-F3A5-F8A - LVDT Measurement - Calculated - Left Coordinates - Right Coordinates020040060080010001200140016001800-10 -5 0 5Height (mm)Total Displacement (mm)5-L1A15-L2A15-L3A15-L3A25-L4A3184 Figure 4.30 shows the flexural displacement along the height of the specimen, where the contrast between the displacements during Phase 1 and 2 can be seen. The flexural displacement in Phase 1 is relatively larger, while the flexural displacement in Phase 2 is responding in double curvature. The flexural displacements observed during Load Stage L4A3 were much larger than all previous Load Stages. The largest displacement was determined to be equal to approximately 1.7 mm at a height of 625 mm from the base of the specimen. (a) (b) Figure 4.30: Flexural displacement calculated from displacement coordinates in selected Load Stages for Specimen 5: (a) Phase 1; (b) Phase 2. 0100200300400500600700800-25 -20 -15 -10 -5 0Height (mm)Flexural Displacement (mm)5-F3A5-F8A0100200300400500600700800-0.5 0 0.5 1 1.5 2Height (mm)Flexural Displacement (mm)5-L1A15-L2A15-L3A15-L3A25-L4A3185 Figure 4.31 shows the estimated curvature profile along the height of the specimen, where the contrast between the displacements during Phase 1 and 2 can be seen. The curvature profiles are only estimates available as a constant value for each segment along the height, while in actuality the curvature would be varying along the entire height. The curvature profile in Phase 1 was entirely in one direction, while the profile in Phase 2 indicates the specimen is in double curvature. The curvature observed during Phase 1 was very large and concentrated at the base, indicating the formation of a plastic hinge. (a) (b) Figure 4.31: Curvature profile calculated from displacement coordinates in selected Load Stages for Specimen 5: (a) Phase 1; (b) Phase 2. 0100200300400500600700800-150 -100 -50 0Height (mm)Curvature (rad/km)5-F3A5-F8A0100200300400500600700800-20 -10 0 10 20 30 40Height (mm)Curvature (rad/km)5-L1A15-L2A15-L3A15-L3A25-L4A3186 Figure 4.32 shows the calculated shear displacement and strain profile along the height of the specimen. The comparison in shear ductility demands between the two phases is displayed, where the strain observed during Phase 1 is similar to what was seen during the early load stages of Phase 2, despite that fact that the total displacement is much larger. A uniform peak shear strain of roughly 2% was observed over the height of the first 350 mm of the specimen. (a) (b) Figure 4.32: Shear parameters calculated from displacement coordinates during selected Load Stages for Specimen 5: (a) Shear displacement; (b) Shear strain. 0100200300400500600700800-10 -8 -6 -4 -2 0Height (mm)Shear Displacement (mm)5-F3A5-F8A5-L1A15-L2A15-L3A15-L3A25-L4A30100200300400500600700800-2.5 -2 -1.5 -1 -0.5 0 0.5Height (mm)Shear Strain (%)5-F3A5-F8A5-L1A15-L2A15-L3A15-L3A25-L4A3187 Figure 4.33 shows the calculated horizontal displacement and strain profile along the height of the specimen. This displacement is defined as the change in length between the targets of equivalent heights on the left and right and side of the specimen and serve as a helpful estimate of the strain in the stirrups. The horizontal strain was obtained by dividing the displacement by 200 mm, which is the width of the web. The maximum strain was observed at a height of roughly 100 mm. (a) (b) Figure 4.33: Horizontal displacement calculated from displacement coordinates in selected Load Stages for Specimen 5: (a) Displacement; (b) Strain. 0100200300400500600700800-0.5 0 0.5 1 1.5 2Height (mm)Horizontal Displacement (mm)5-F3A5-F8A5-L1A15-L2A15-L3A15-L3A25-L4A30100200300400500600700800-0.5 0 0.5 1Height (mm)Horizontal Strain (%)5-F3A5-F8A5-L1A15-L2A15-L3A15-L3A25-L4A3188 4.3 Comparison of Test Results The testing procedures presented for each specimen are summarized in Table 4.6. This table contains information related to the number of phases (the first being application of flexural damage using only the top actuator, and the second phase being the second-mode displacement protocol), the rate of loading used to apply the displacement, the length of time the imposed displacement was paused at the peak, and the data acquired during the testing. Table 4.6: Summary of testing procedures for each specimen. Specimen # of Phases Time to Peak/Unload (s) Length of Pause at Peak (s) Crack Measurements Displaced Shape Target Photographs 1 1 600 300 - 600 Each Load Stage No targets 2 1 4 - 8 0 Residual See Note 1 3 2 15 - 30 300 - 600 Each Load Stage At peak displacement 4 2 15 - 30 300 - 600 Each Load Stage At peak displacement 5 2 15 - 45 300 - 600 Each Load Stage At peak displacement Note 1: Targets were placed on specimen, but difficulty taking photographs at peak displacement due to no pause led to unreliable data. 189 A comparison of the results of the first three specimens is shown in Figure 4.34, which demonstrates that the backbone curves of the base shear versus mid-height displacement of each specimen compare well, indicating that the rate of loading did not significantly affect the results of the test. The results are all quite similar, except for the initial stiffness of Specimen 3, which was reduced due to the initial damage that was applied during Phase 1, when only the top actuator was applying a displacement. Figure 4.34: Comparison of base shear (Vb) vs mid-height displacement (D1) backbone curves Specimens 1 through 3. -80-60-40-20020406080-8 -6 -4 -2 0 2 4 6 8Vb (kN)D1 (mm)Specimen 1Specimen 2Specimen 3190 Figure 4.35 compares the results of Specimens 4 and 5 to the previous results. From this comparison there are many contrasts between the results, such as the asymmetrical load-deformation relationship, and the reduction in strength observed with Specimen 5. Further discussion related to these results can be found in Chapter 5. Figure 4.35: Comparison of base shear (Vb) vs mid-height displacement (D1) backbone curves Specimens 1 through 5. The final cracked profile observed after diagonal concrete crushing for each specimen is shown in Figure 4.36, which shows that the approximate height where concrete crushing occurred was similar for each specimen, except for Test 5 where it occurred lower, in the plastic hinge region. Full photos of the crack profile of the specimen, including cracks in the upper portion and flanges, is available in Appendix E. -80-60-40-20020406080-8 -6 -4 -2 0 2 4 6 8Vb (kN)D1 (mm)Specimen 1Specimen 2Specimen 3Specimen 4Specimen 5191 (a) (b) (c) (d) (e) Figure 4.36: Comparison of crack patterns in Specimen test region, diagonal crushing for: (a) Specimen 1; (b) Specimen 2; (c) Specimen 3; (d) Specimen 4; (e) Specimen 5. 192 Figures 4.37 through 4.40 show the results of analyzing the displaced shape using the coordinates obtained from the photographed displacement targets posted on the specimen. The total displacement, flexural displacement, curvature, shear displacement and strain, and horizontal displacement and strain, were compared for each specimen. The displaced shape that corresponded with the “A” Load Cycle in which the peak base shear force demand was observed, was chosen for comparison for Specimens 3 through 5. Figure 4.37 shows the comparison of the total displacement profile of the specimen; the total displacement is considered the combination of both the shear and flexural components. The total displacement was determined in three different ways. The upper displacements were measured using the LVDT data and are denoted with squares. The total displacement was also calculated by taking the sum of the flexural and shear displacements estimated by the displacement coordinates and is denoted by circles and connected plot points. Finally, the displaced shape could be determined simply by measuring the change in horizontal displacement of the displacement coordinates on the left-hand side (denoted by diamonds), and the right-hand side (denoted by triangles). While the mid-height displacement is similar for each Specimen, the shape of the displacement profile varies. 193 Figure 4.37: Comparison of total displacement profile for Specimens 3 through 5 for Load Stage that corresponds with maximum observed base shear force. 020040060080010001200140016001800-15 -10 -5 0 5Height (mm)Total Displacement (mm)3-L8A24-L7A15-L4A3 - LVDT Measurement - Calculated - Left Coordinates - Right Coordinates194 Figure 4.38 shows the flexural displacement, and curvature, along the height of the specimen. The flexural displacement profile of each specimen indicates double curvature. The curvature profiles are only estimates available as a constant value for each segment along the height, while in actuality the curvature would be varying along the entire height. The curvature profile agrees with the flexural displacements, which indicates double curvature. (a) (b) Figure 4.38: Comparison of flexural ductility demands for Specimens 3 through 5 for Load Stage that corresponds with maximum observed base shear force: (a) Flexural displacement; (b) Curvature. 01002003004005006007008000 0.5 1 1.5 2Height (mm)Flexural Displacement (mm)3-L8A24-L7A15-L4A30100200300400500600700800-60 -40 -20 0 20 40Height (mm)Curvature (rad/km)3-L8A24-L7A15-L4A3195 Figure 4.39 shows the calculated shear displacement and strain profile along the height of the specimen. While the observed maximum values are similar, the profile of the demands varies between each specimen, in particular the height along the specimen where the peak shear strain is observed. (a) (b) Figure 4.39: Comparison of shear ductility demands for Specimens 3 through 5 for Load Stage that corresponds with maximum observed base shear force: (a) Shear displacement; (b) Shear strain. 0100200300400500600700800-10 -8 -6 -4 -2 0Height (mm)Shear Displacement (mm)3-L8A24-L7A15-L4A30100200300400500600700800-2.5 -2 -1.5 -1 -0.5 0 0.5Height (mm)Shear Strain (%)3-L8A24-L7A15-L4A3196 Figure 4.40 compares the calculated horizontal displacement and strain profile along the height of the specimen. While the maximum observed strains are similar, the height along the specimen where it was observed varies between each specimen. (a) (b) Figure 4.40: Comparison of horizontal ductility demands for Specimens 3 through 5 for Load Stage that corresponds with maximum observed base shear force: (a) Displacement; (b) Strain. All remaining data now shown, such as photos, load-deformation, and displacement coordinate results related to the experimental tests is available in Appendix E. The next chapter provides discussion related to the theoretical analysis of the experimental results. 0100200300400500600700800-2 -1 0 1 2 3Height (mm)Horizontal Displacement (mm)3-L8A24-L7A15-L4A30100200300400500600700800-1 -0.5 0 0.5 1 1.5Height (mm)Horizontal Strain (%)3-L8A24-L7A15-L4A3197 5. Discussion of Experimental Results ___________________________________________________________________________ 5.1 Overview This chapter presents the results of the experiment, with more insight regarding theories and analysis surrounding the outcomes. The results of each specimen were compared to predictions made with the procedures commonly used in industry. Section 5.2 will introduce the existing analytical tools used, including the methodology used during the analysis of the experimental results. Section 5.3 presents the predicted results relating to the strength of the specimens, as well as a comparison of these predictions to the actual results. Section 5.4 relates to discussion regarding the stiffness of the specimens, and what outcome is observed using standard industry analytical models, and how they compare to the observed results. 5.2 Analytical Methods To make an analytical prediction of the test results, it was necessary to develop a specific program that would allow for a displacement-based solution, that is solving for an imposed displacement at two nodes for each time-step. Many programs do not allow for a displacement-based solution to be calculated at multiple degrees of freedom, as it would be difficult to converge using typical algorithms paired with complex non-linear models. The program was developed using Excel, and due to the difficulty of convergence, was left to be simple in that it can solve for a linear and multi-linear backbone solution. While it would be ideal to develop a more complex analysis of the results in the future, the goal of this analysis was to use simple methods that would be comparable to what would be predicted using current typical code procedures. The program has an input for both multi-linear flexural and shear 198 stiffness models which are used to develop a predicted displacement. The technical aspects of the program will be presented in Appendix C. In addition to the program developed in Excel, other programs used include Response2000, which was used to develop a non-linear flexural model to be used by the developed program, as well as a prediction of the shear strength. 5.2.1 Non-Linear Flexural Model Modeling of flexural demands is well understood by designers and is typically accomplished through creation of a fibre-section model, in which the cross-section of the element is discretized into fibres that are distinguished by their material, i.e. concrete and steel. The characteristics of these fibres are defined by non-linear material models that simulate their behaviour under both tensile and compressive forces. Fibre-section models assume that plane sections remain plane and allow for the element to be analyzed along its height by calculating the response of the cross-section at various locations along the height. The program developed in excel is not capable of handling fibre-section models to generate a moment-curvature relationship used to predict the response along the cross-section, however, it does have the capability to use a tri-linear approximation of a moment-curvature relationship developed using another program, such as Response2000 or OpenSees. Thus, a moment-curvature relationship of the predicted specimen demands was developed using Response2000, and a tri-linear approximation that was used as the model for the specimen was completed as shown in Figure 5.1. Details of the model completed in Response2000, including how the cross-section and materials were modeled, can be found in 199 Appendix F. It should be noted that a tri-linear approximation can be used to predict a backbone curve of the response but would not be effective in predicting a cyclic response as there is no hysteretic behaviour in the model. (a) (b) 0102030405060700 20 40 60 80 100 120 140 160 180 200Moment (kN-m)Curvature (rad/km)Response2000Tri-Linear Approximation01020304050600 5 10 15 20 25 30 35 40 45 50Moment (kN-m)Curvature (rad/km)Response2000Tri-Linear Approximation200 (c) Figure 5.1: Comparison of Response2000 moment-curvature relationship and approximate tri-linear model up to maximum curvature of : (a) 200 rad/km; (b) 50 rad/km; (c) 12 rad/km. Table 5.1: Values used for tri-linear approximation of moment-curvature diagram for predictive model. Curvature (rad/km) Bending Moment (kN-m) Crack 0.79 8.6 Yield 10.0 47.5 Ultimate 190.0 65.0 It can be seen in Figure 5.1 that the tri-linear approximation fits to the moment-curvature relationship developed in Response2000 very well in the linear and cracked ranges and fits moderately well after yielding. As most flexural behaviour observed during the tests occurred in the cracked range, it is not expected that the fit of the approximation in the yielded range will have significant influence on the results. 051015202530354045500 2 4 6 8 10 12Moment (kN-m)Curvature (rad/km)Response2000Tri-Linear Approximation201 5.2.2 Shear Model While many structural engineers use complex and well understood models to simulate the flexural behaviour, they typically use simple methods regarding modelling of the shear demands. The shear response is typically modeled using a linear spring with a slope representative of a cracked shear modulus. This type of modelling would be effective in modelling structures in which the flexural stiffness dominates the behaviour and failure mode, while still capturing the influence of the shear stiffness in the response. However, it would not adequately simulate the response of a shear dominated structure. This is because it would not allow for any degradation in the shear strength in the material model, allowing for the structural model to potentially be pushed to artificially high shear demands. This becomes a problem when modelling high-rise reinforced concrete cantilever shear walls, as they are assumed to be flexure dominated structures. While this holds true under first mode demands, the experimental results of this thesis have shown that the behaviour is shear dominated when subjected to second-mode demands. One of the main reasons for this is a lack of extensive experimental data on which non-linear shear models can be developed. However, there are some models, including a tri-linear model developed by Gerin & Adebar (2004) that will be used to create a response that will be compared to the experimental results of this thesis. The tri-linear model is defined by three points, the cracking shear stress and strain, yield shear stress and strain, and the ultimate strain. The cracking shear stress can be assumed to be the concrete strength contribution to the total shear strength, which can be calculated many ways, but for this model was calculated as shown in Eq. 5.1, while the cracking shear strain is determined by dividing the stress by the initial shear stiffness, shown in Eq. 5.2. 202 𝑣𝑐𝑟 = 0.25√𝑓𝑐′ (5.1) 𝛾𝑐𝑟 = 𝑣𝑐𝑟𝐺𝑖 (5.2) Where 𝑣𝑐𝑟 is the cracking shear stress, 𝛾𝑐𝑟 is the cracking shear strain, 𝑓𝑐, is the concrete strength (in MPa), and 𝐺𝑖 is the initial shear modulus, which for an uncracked element is taken as roughly 40% of the elastic modulus. For the trilinear shear model created in this thesis, the concrete strength was taken as 40 MPa (based on cylinder test results), and the initial shear stiffness was taken as roughly 11,120 MPa. Similarly to the cracking shear stress, the yielding shear stress can be defined using several different tools, however for the model created for this thesis, a yielding shear stress of 5.44 MPa was assumed, as it corresponds with the approximate maximum base shear of roughly 68 kN that was observed in the experimental results. The yield and ultimate strains, which are more complex, were calculated in accordance to the model by Gerin and Adebar using Eqs. 5.3 and 5.4. 𝛾𝑦 = 𝑓𝑦𝐸𝑠+𝑣𝑦 − 𝑛𝜌𝑣𝐸𝑠+4𝑣𝑦𝐸𝑐 (5.3) 𝛾𝑢 = 𝛾𝑦 (4 − 12𝑣𝑦𝑓𝑐,) (5.4) Where 𝛾𝑦 is the yielding shear strain, 𝛾𝑢 is the ultimate shear strain, 𝑓𝑦 is the yield stress of the reinforcing steel, 𝐸𝑠 is the elastic modulus of the reinforcing steel, 𝑣𝑦 is the yielding shear stress, 𝑛 is the axial stress where compression is positive, 𝜌𝑣 is the vertical reinforcing ratio, 𝑓𝑐, is the concrete strength, and 𝐸𝑐 is the elastic modulus of the concrete. The yield stress of the 203 reinforcing steel was assumed 400 MPa based on experimental material testing, even though the test results shown in Chapter 3 show that there is not a clearly defined yield plateau in the reinforcing steel. There was no axial load in the specimen, and the vertical reinforcing ratio was equal to 3.11%. It should also be noted that the ultimate shear stress is assumed to be equal to the yielding shear stress. The yielding and ultimate shear strain were determined to be equal to approximately 0.37% and 0.86% respectively. Finally, the shear models that are used in the excel program are shown in Figure 5.2. The four models that are used include linear shear stiffnesses that are uncracked, 50% of uncracked, 10% of uncracked, and the tri-linear model that has been discussed in this subsection. The uncracked and 50% of uncracked linear shear stiffness models are recommended by both the Los Angeles Tall Buildings Structural Design Council (LATBSDC, 2017) and by the Pacific Earthquake Engineering Center Tall Buildings Initiative (PEER TBI, 2017). In both design guides, the uncracked shear stiffness model is recommended for service level-linear models, while the 50% of the uncracked linear shear stiffness model is recommended for Maximum Considered Earthquake (MCER) level models. It should be noted that PEER and LATBSDC recommend the effective stiffness in terms of the elastic modulus, E (40% and 20% respectively), which corresponds to the effective stiffnesses of 100% and 50% when expressed in terms of the concrete shear modulus, G, based on the relation ship between the elastic and shear moduli. 204 Figure 5.2: Comparison of various shear models used to analyze specimen results. 5.3 Discussion of Shear Strength As was mentioned in Section 5.2, there are many tools available that can be used to calculate the shear strength of an element. These tools typically have two unique components to represent the contribution of concrete and the steel to the total strength. The experimental results demonstrated that the shear strength and flexural strength of the five specimens was comparable and was not strongly influenced by the rate of loading. The only exception to this was the measured shear strength of Specimen 5, where it was observed that the plastic hinge at the base of the specimen had reduced the shear strength. In order to further understand these results, it is important to compare them to the values that would be obtained using standard analytical tools. 01234560 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01Shear Stress (MPa)Shear Strain (mm/mm)Uncracked50%10%Tri-Linear205 5.3.1 Shear Strength The most common tools used for calculating the shear capacity are the equations given in Clauses 11.3.4 and 11.3.4.1 of CSA A23.3-14 (CSA, 2014) which give the equations for the contributions from the concrete and steel to the total shear resistance. 𝑉𝑐 = 𝛽√𝑓𝑐′𝑏𝑤𝑑𝑣 (5.5) 𝑉𝑠 = 𝑓𝑦𝐴𝑣𝑠𝑑𝑣 cot 𝜃 (5.6) Where 𝑉𝑐 is the concrete shear strength, 𝑉𝑠 is the steel shear strength, 𝛽 is a factor to account for the shear resistance of cracked concrete, 𝑏𝑤 is the web thickness, 𝑑𝑣 is the effective shear depth, 𝐴𝑣 is the area of each stirrup, 𝑠 is the stirrup spacing, and 𝜃 is the angle of diagonal compressive stress, where all units are in metric. However, ACI 318-14 (ACI, 2014) gives slightly different equations which are given in Clauses 11.5.4.5 and 11.5.4.8 which give the following two equations for the concrete and steel contributions to the shear strength, where the units for stress must be in psi for Eq. 5.7. 𝑉𝑐 = 2√𝑓𝑐′ℎ𝑑 (5.7) 𝑉𝑠 =𝐴𝑣𝑓𝑦𝑡𝑑𝑠 (5.8) Where ℎ is the web thickness, 𝑑 is the effective shear depth, 𝑓𝑐′ is the concrete strength where the units are in psi, and 𝑓𝑦𝑡 is the tensile yield strength of the stirrups. In comparison of the 206 two methods, ACI 318-14 provides a much simpler set of equations in the concrete strength is not influenced by any variation in the 𝛽 factor, and the stirrup contribution is not influenced by variation of the stress angle 𝜃. According to Clause 21 of CSA A23.3-14, the 𝛽 factor must be assumed to be 0.18 (when the units for stress are in MPa) for the prediction of the strength observed in the first four specimens, which is allowable permitting the inelastic rotational demand is less than 0.005 radians. However, specimens in which the inelastic rotational demand is greater than 0.015 radians, the 𝛽 factor is equal to 0, effectively removing any concrete contribution from the specimen stiffness, which partially explains the decrease in strength observed during Specimen 5 where a plastic hinge was formed at the base of the specimen. It should be noted that using a value of 0.18 gives very similar results to what is obtained from Eq. 5.7. Clause 21 also requires that the stress angle must be assumed to be 45 degrees, if the axial compression in the wall is less than 10% of its axial resistance, which would remove any increase in strength due to the stress angle as the cotangent of 45 degrees is equal to 1, which would then result in Eqs. 5.6 and 5.8 giving the same result. However, using a more sophisticated analytical tool such as Response2000, it is possible to generate a more accurate prediction of the shear strength of the specimen. Assumption of values such as the principle tensile stress, stirrup stress, and the stress angle are given as a variable over the length of the cross-section, which can then be used to determine both the concrete and steel contributions to the total shear resistance. The stress angle was determined using the same Response2000 model that was used to generate the moment-curvature diagram discussed in Section 5.2, and details relating to creation of the model are available in Appendix F. 207 The variation of the principle tensile stress and stirrup stress over the depth of the cross-section, as calculated in Response2000, are given in Figure 5.3 and 5.4 respectively, where the distinction between the stress in the web and flanges is shown. Figure 5.3: Change in principle tensile stress along depth of cross-section for Response2000 model subjected to pure shear. Figure 5.4: Change in stirrup stress along depth of cross-section for Response2000 model subjected to pure shear. -150-100-500501001500 0.5 1 1.5 2 2.5 3 3.5 4Depth (mm)Principle Tensile Stress (MPa)FlangeWebFlange0.42-150-100-50050100150350 370 390 410 430 450Depth (mm)Stirrup Stress (MPa)StressStress at CrackFlangeWebFlange431.8414.2208 Figure 5.5 shows the variation of the stress angle over the depth of the cross-section, where it can be observed that the minimum angle is 23.3 degrees, which is significantly less than the assumed angle of 45 degrees by the code-based approaches. Figure 5.5: Change in stress angle along depth of cross-section for Response2000 model subjected to pure shear. The concrete and steel stress contributions to the total shear resistance are then determined by calculating Eqs. 5.9 and 5.10 along the depth of the cross-section. 𝑣𝑐 = 𝑓𝑐1𝑐𝑜𝑡𝜃 (5.9) 𝑣𝑠 =𝑓𝑠𝐴𝑣𝑐𝑜𝑡𝜃𝑏𝑤𝑠 (5.10) The variation of the concrete and steel contributions to the steel stress is then shown in Figure 5.6 and 5.7 respectively. The area of the stress over the depth of the cross-section can be -150-100-500501001500 10 20 30 40 50 60 70 80 90Depth (mm)Angle (Degrees)FlangeWebFlange23.3 209 integrated to determine the concrete and steel contributions to the strength. It should be noted that while the depth is shown, the width of the cross-section is not. The width in the flange is equal to 175 mm, while the width in the web is equal to 50 mm. Integrating the stress over the cross-section gave a concrete contribution to the strength of 29.8 kN, which averaged over the length of effective shear depth of 250 mm, and assuming a width of 50 mm, gives a stress of 2.38 MPa. Figure 5.6: Change in concrete shear stress along depth of cross-section for Response2000 model subjected to pure shear; flange width is 175 mm; web with is 50 mm. The same process was completed for the stirrrup contribution, integrating the stress over the cross-section gave a total steel contribution to the strength of 38.9 kN, which averaged over the length of effective shear depth of 250 mm, and assuming a width of 50 mm, gives a stress of 3.11 MPa. -150-100-500501001500 0.5 1 1.5 2 2.5 3 3.5 4Depth (mm)Concrete Shear Stress (MPa)FlangeWebFlangeArea = Vc = 29.8 kNVc/bwdv = 2.38 MPa210 Figure 5.7: Change in steel shear stress along depth of cross-section for Response2000 model subjected to pure shear; flange width is 175 mm; web with is 50 mm. The total shear resistance is then determined based on the sum of the two contributions. Which gave a total strength of 68.7 kN, and 5.48 MPa average over the effective shear depth. Figure 5.8: Change in total shear stress along depth of cross-section for Response2000 model subjected to pure shear; flange width is 175 mm; web with is 50 mm. -150-100-500501001500 1 2 3 4 5 6Depth (mm)Shear Stress Resisted by Stirrups (MPa)FlangeWebFlangeArea = Vs = 38.9 kNVs/bwdv = 3.11 MPa-150-100-500501001500 1 2 3 4 5 6 7 8Depth (mm)Shear Stress (MPa)FlangeWebFlangeArea = Vr = Vc + Vs = 68.7 kNVr/bwdv = 5.48 MPa211 The strength components determined using Respone2000 can then be compared to what was determined using the two code-based approaches. The value of all the variables used in both the Response2000 model as well as the code-based approaches are summarized in Table 5.2. Table 5.2: Summary of values used to determine shear strength of specimen using standard code procedures. Variable CSA A23.3 ACI 318 Response2000 𝑓𝑐′ (MPa) 40 (5800 psi) 40 (5800 psi) 40 (5800 psi) 𝛽 0.18 - - 𝑏𝑤 , ℎ (mm) 50.0 50.0 50.0 𝑑𝑣 , 𝑑 (mm) 250.0 250.0 250.0 𝑠 (mm) 125.0 125.0 125.0 𝐴𝑣 (mm2) 30.0 30.0 30.0 𝜃 (degrees) 45.0 45.0 Figure 5.5 𝑓𝑦 (MPa) 414.2 414.2 Figure 5.4 The shear strength calculated using each method is shown in Table 5.3. It should be noted that for the concrete shear strength calculated using the ACI 318-14 (2014) method, the stress was calculated in psi and converted to MPa. Table 5.3: Comparison of maximum base shear strengths calculated using standard code procedures to observed peak base shear. CSA A23.3 ACI 318 Response2000 Vc (kN) 14.2 13.1 29.8 Vs (kN) 24.9 24.9 38.9 Vr (kN) 38.5 37.4 68.7 Observed Vr (kN) 68.0 Error (%) 43.8 45.4 1.2 212 Based on the results shown in Table 5.3, there is very little difference in the concrete contribution between the two code-based approaches, and the steel contribution is the same. Both methods underestimate the observed shear strength of the specimen by roughly 40-45%. The prediction made using Response2000 is much more accurate and predicts a total shear strength within 1.2% of the observed value. Both the concrete and steel shear strength contributions are greater than those calculated using the code-based methods. It should be noted that this conservatism is required by code-based predictions in order to help prevent shear failures. 5.3.2 Bending Moment – Shear Force Interaction Figure 5.9 shows how the specimen test results fit within the moment to shear interaction diagram. The bending moment – shear force interaction curve was generated using the model created in Response2000, that can be found in Appendix F. Instead of plotting the base shear versus either the base or mid-height moments, it was plotted versus the estimated moment force that occurred at the approximate height where diagonal concrete crushing was observed, which was 375 mm from the base of the specimen, compared to the effective shear depth of 250 mm. This was a simple approximation and was determined by taking the average of the base and mid-height bending moments. It can be observed that the experimental results all fit within the interaction diagram, which confirms that Response2000 not only gives a good prediction of the maximum shear strength, but also of its relationship with the bending moment demands. 213 Figure 5.9: Comparison of moment-shear interaction diagram to base shear vs estimated moment at height of diagonal concrete crushing. -80-60-40-20020406080-60 -40 -20 0 20 40 60Base Shear (kN)Moment (kN-m)Specimen 1Specimen 2Specimen 3Specimen 4Specimen 5214 5.4 Effect of Shear Stiffness Model on Prediction The Excel program discussed in Appendix C was used along with the material models discussed in Section 5.2 to generate a force-displacement prediction of the specimen, along with the corresponding experimental results. A prediction was completed for each of the four shear models that were discussed, however the flexural model was left constant as the tri-linear approximation of the moment-curvature relationship developed using Response2000. Figures 5.10 through 5.14 show the comparison of the predicted results using each shear model to the observed experimental results of each specimen. The predictions using linear shear models continue to increase past the expected failure point, this is because the only non-linearity in these predictions is from the flexural model. The specimen is shear critical, a much greater top and mid-height force combination is required to fail the specimen in flexure than it is in shear, which is why the amount of non-linearity observed in the response is so small. Figure 5.10: Prediction of load-deformation backbone results of base shear (Vb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 1. 010203040506070800 1 2 3 4 5 6 7 8Vb (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear215 In Figure 5.10, the prediction obtained by the uncracked and linear shear models underestimate the shear ductility. The predictions obtained using the 10% linear shear model give a result that would be sufficient in predicting the peak load observed from the test results, however there is no degradation in the response. As the actual specimen experimental results begin to degrade, the demands of the linear model continue to increase. Finally, the tri-linear model provides the best estimate of the result. Relative to the other three models, it limits the maximum stress and strain, however it still does not model any degradation. Figure 5.11: Prediction of load-deformation backbone results of base shear (Vb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 2. The load-displacement results from Specimen 2, shown in Figure 5.11, were very similar to those observed in Specimen 1, thus the comparison of the predictions to the results has not changed. The only exception would be the fit of the tri-linear model post yielding to the experimental results, where the ultimate shear strain matches very well with the displacement observed before significant degradation. 010203040506070800 1 2 3 4 5 6 7 8Vb (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear216 Figure 5.12: Prediction of load-deformation backbone results of base shear (Vb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 3. Again, the results in Figure 5.12 are similar to what has previously been shown. The major exception in these results is the application of damage during Phase 1 testing. The initial stiffness was reduced because of the Phase 1 damage, and thus the tri-linear model over predicts the strength, while the 10% linear shear model provides a very good prediction of the initial stiffness, but still underestimates the ductility (albeit much less severely than the Uncracked, and 50% models). 010203040506070800 1 2 3 4 5 6 7 8Vb (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear217 Figure 5.13: Prediction of load-deformation backbone results of base shear (Vb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 4. It is difficult to compare the predictions to the results obtained for Specimen 4 before the peak shear force was observed, due to the initial shear force in the specimen. It was not possible to adjust the predictions for the scenario of a initial force based on limitations of the created program, instead the predicted responses were offset by -1.0 mm. Similar conclusions as was discussed for the previous three specimens can be observed in regards to the comparison of the tri-linear model, and the 50% and uncracked linear models to the specimen results in the post yield region. 01020304050607080-2 -1 0 1 2 3 4 5 6 7 8Vb (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear218 Figure 5.14: Prediction of load-deformation backbone results of base shear (Vb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 5. Finally, it was difficult to draw conclusions to the comparison of the predictions to the results of Specimen 5, as the models were not adjusted to reflect the reduction in strength and ductility of this specimen that was a result of the damage applied in Phase 1. During Phase 1, the top actuator was used to apply large bending moments at the base of the specimen to form a plastic hinge. While the only figures shown in this subsection were of the prediction of base shear versus mid-height moment, the prediction was also completed for mid-height shear, mid-height moment, and base moment, all versus mid-height displacement. These figures are available in Appendix F. 01020304050607080-2 -1 0 1 2 3 4 5 6 7 8Vb (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear219 5.5 Conclusion It was demonstrated that both American and Canadian code-based approaches used to predict the nominal shear resistance of a wall element subjected to second-mode shear demands underestimate the shear resistance that was observed during the experimental phase. However, the prediction completed by Response2000 gave a much more accurate result and displayed that the contribution of both the concrete and the steel was greater than what was calculated using the code-based approaches. The shear demands were within the moment to shear interaction diagram developed using Response2000 when plotting the base shear against the bending moment at 375 mm from the base of the specimen. Models common in industry practice were used, such as a fibre model for axial load and bending, and a simple linear shear model. The uncracked and 50% of uncracked linear shear stiffness models recommended by LATBSDC (2017) and PEER TBI (2017) were used, giving much higher resistance at displacements equal to those observed from the test specimen results. The 10% of uncracked linear shear stiffness model gave a better prediction of the cracked stiffness, and the tri-linear model gave relatively good results overall. A prediction that was representative of the initial loading conditions for Specimens 4 and 5 could not be completed, and it is recommended that one be completed in the future that accounts for the initial shear force and displacement in the specimen when the second-mode shear demands were applied. 220 6. Conclusion and Recommendations ___________________________________________________________________________ 6.1 Overview This chapter summarizes all the conclusions drawn from the research completed as part of this thesis, as well as gives recommendations for further research. Section 6.2 gives an overall summary of the work done, Section 6.3 summarizes the conclusions from the analyses presented in Chapter 2, Section 6.4 presents the results from the experimental program (Chapters 3 and 4), Section 6.5 summarizes the conclusions from the comparison of analytical predictions with test results in Chapter 5, and finally Section 6.6 gives recommendations for future research. 6.2 Overall Summary The objective of the current thesis was to investigate the shear resistance of reinforced concrete cantilevered shear walls subjected to the typical shear demands from an earthquake. It was determined that the demands from the first and second-modes of a tall cantilever shear wall represents a very large portion of the total demands. Thus, the current study involved the analytical and experimental investigation of a cantilever wall subjected to first and second-mode demands. Before the experiment was carried out, a series of dynamic analyses of a simplified high-rise cantilever shear wall were conducted, to determine how the modes of vibration affect the behaviour of the structure as it is pushed increasingly to greater non-linear demands. The 221 findings from this series of analyses influenced how the experimental study was carried out, particularly the specimen design and the loading protocol that was used. Five identical specimens were constructed, and these were subjected to different demands (loading protocols) during the test. An innovative approach was used to precast the wall segments of the specimens separately from the foundation so that one foundation could be used for all specimens. Two hydraulic actuators were used to impose displacements on the specimen forcing it into a second-mode displaced shape. Through the different protocols, the effects of rate of loading were explored, as well as the effects of forcing the specimen into its second-mode after pre-existing damage such as the onset of flexural yielding at the base and the formation of a significant plastic hinge. The results of each specimen were compared to predictions made with the procedures commonly used in industry, such as a fibre model for axial load and bending, and a simple linear shear model. The results were also compared with a more refined trilinear shear model. 6.3 Summary of Dynamic Analyses A simplified model was created of a typical high-rise building shear wall core. In order to focus the study on the first two modes, the model included only two concentrated masses, with one located at the mid-height and the other at the top of the wall. The masses were adjusted such that the period of the first mode was equal to that of the prototype building, and the second-mode period was equal to approximately 1/6th of the first mode period, which is typical for cantilever walls. 222 The wall was modeled using fibre-sections and non-linear force-based elements, which resulted in a realistic curvature distribution and flexural behaviour along the height of the model. In addition, a simplified sectional hysteretic tri-linear shear stiffness model was used. The model was excited using sinusoidal ground acceleration, where the period of this ground acceleration varied, with the most significant results being when the period of the ground acceleration was equal to either the first or the second period of vibration of the structure. The response of the model was decomposed into its two modal contributions to determine how each mode influenced the response of the model during free vibration. Both linear elastic and non-linear versions of the model were used to understand how the change in material behaviour influenced the effect of the first and second-mode demands. The conclusions from the analyses presented in Chapter 2 can be summarized as follows: a) When the linear-elastic model was excited at various periods it was observed that the response in free vibration was mainly composed of the first mode, unless the period of excitation is approximately equal to the period of the second-mode, in which case the response is composed of both the first and second-modes. b) When the second-mode is present in the linear-elastic free vibration response of the structure, it strongly influences the mid-height displacement, mid-height bending moment, and the shear forces in both the top and base elements. c) When the non-linear model was excited, flexural yielding was observed at the base regardless of the period of excitation, and ultimately led to formation of a plastic hinge. It was found that this could be accounted for in the modal decomposition by modifying the first mode shape to represent a pinned base, however modifying the 223 second-mode had very little influence on the decomposition of the displacement demands. d) No rational modification for modal decomposition of the force demands of either mode could be determined. e) When the non-linear displacement response was decomposed using the modified first mode displaced shape and the linear elastic second-mode displaced shape, it was observed that the second-mode displacements behaved predictably, while the force demands behaved unpredictably. f) The outcome of (e) led to the observation that the second-mode shear demands could be applied as a pair of imposed lateral displacements (in opposite directions) at the mid-height and the top of the wall, with a constant ratio between the two regardless of the level of nonlinearity in the wall. The results of the dynamic analyses informed the specimen design, as it was determined that if the first mode is present in the response, the maximum applied forces was limited by flexural yielding at the base, however the desired form of failure during the tests was diagonal shear failure. Thus, only the second-mode shape was imposed on the specimen, and not a combination of the first and second-mode, which maximized the shear force demands while minimizing the bending moment demands. 6.4 Summary of Experimental Results The wall specimens were precast without foundations and then were connected to a reusable foundation by grouting the specimens into a recess in the foundation. This novel approach 224 reduced the amount of resources required to build the specimens, made construction of the specimens more efficient and reduced the storage space required in the lab. The method for connecting and separating the wall specimens from the foundation was refined during the first few specimens. The optimum connection design, which provides a stiff and strong connection; but is easy to separate, is documented in Chapter 3. The walls were tested using two hydraulic actuators that forced the specimens into a second-mode shape by imposing a displacement at the top and mid-height locations. The first two specimens were tested using the same displacement protocol – increasing second-mode displacement without any prior damage due to first mode demands; however, different rates of loading were used. Specimen 1 was loaded at a very slow rate of loading (between 0.01 mm/s and 0.03 mm/s), while Specimen 2 was loaded at up to a 400 times higher rate of loading (up to about 4 mm/s). Surprisingly, no significant difference was observed between the results of the two specimens. In all five specimens, ductile shear failures were observed. While it is often thought that shear failures are always brittle, the specimens exhibited significant yielding of the horizontal steel before diagonal compression failure of the concrete. No yielding of the vertical reinforcement was observed while the specimen was subjected to second-mode displacements. Flexural yielding did occur during the first phase of Specimens 3, 4 and 5 while the specimens were subjected to first-mode displacements. Specimens 1 and 2 exhibited almost identical behaviour up until the maximum shear capacity was reached. Significant shear yielding started at a mid-height specimen displacement of about 4 mm. Specimen 2 (higher rate of loading) exhibited ductile shear behaviour up to a mid-height 225 displacement of 6.6 mm, while Specimen 1 started to degrade at a mid-height displacement of 5.2 mm. Specimens 3, 4 and 5 were tested in two phases. During the first phase, the walls were subjected to a first-mode displacement protocol using only the hydraulic actuator at the top of the wall in order to create various levels of flexural damage in the cantilever walls before the second-mode displacement protocol was applied in Phase 2. During these tests, the rate of loading varied between 0.2 and 0.6 mm/s. Specimen 3 was subjected to a maximum top displacement of 30 mm during Phase 1 (first-mode displacements). The specimen had significant flexural and diagonal cracking, and the vertical reinforcement was just at the onset of yielding by the end of Phase 1. When the second-mode displacements were applied (Phase 2), the specimen had a lower stiffness than Specimens 1 and 2 due to the pre-existing diagonal cracks. The peak shear load occurred at a mid-height displacement of about 4 mm, and the mid-height displacement at the end of the shear yielding range was about 6 mm. From the displacement target data, shear strains up to 1.0% existed when the mid-height displacement was 4 mm, and shear strains up to 1.7% existed when the mid-height displacement was about 6 mm. The maximum shear strains measured with the target data during the test was 2.85% at a mid-height displacement of 7.4 mm, and the largest measured average horizontal strain in the “web” of the wall was 0.83% when the mid-height displacement was 4 mm. Phase 1 of Specimen 4 was very similar to Phase 1 of Specimen 3; however, the beginning of Phase 2 was very different. While Phase 2 of Specimen 3 began with a zero top-wall displacement, Phase 2 of Specimen 4 started with the top-wall displacement equal to the maximum value of +30 mm. A force of approximately +30 kN had to be applied at the top of 226 the wall to maintain that initial displacement. When the mid-height displacement added additional shear to the base of the wall, the maximum shear force that could be resisted was very similar to what was observed in the previous three tests, and the level of shear ductility was similar. In the reverse direction, where the mid-height actuator force is in the opposite direction to the +30 kN top actuator force, the specimen stiffness was very much reduced, and the maximum applied shear force was much smaller. The shear force – shear displacement relationship in the positive shear direction was very similar to the previous three tests, except that the curve was offset along the displacement axis. Specimen 5 was subjected to a maximum top-wall displacement of 60 mm in Phase 1. Approximately half this displacement was the result of a plastic hinge forming at the base of the wall. At the end of Phase 1, the specimen had a residual top-wall displacement that required a top actuator force of +10 kN to return the top of the wall to zero displacement. The shear force – shear displacement relationship measured during Phase 2 was highly unsymmetrical. In the positive shear direction, the maximum applied shear force was 41 kN, which is considerably smaller than the shear strengths of the Specimens 1 to 4, which varied from 65 kN to 70 kN. In the negative shear direction (opposite direction to the +10 kN shear force applied at the top to eliminate the residual displacement), very little shear force developed in the specimen. In all five specimens, the observed failure mode was diagonal concrete crushing shear failure after significant yielding of the horizontal reinforcement. In Specimens 1 to 4, the diagonal crushing occurred at approximately the same height – between 250 and 350 mm from the base of the wall, while the diagonal crushing occurred in Specimen 5 between 125 and 250 mm from the base. Diagonal crushing occurred in Specimens 1 to 4 further from the base because 227 of the reduced restrained to horizontal expansion, while it occurred in the location of the plastic hinge in Specimen 5. An interesting observation is that in most specimens, the concrete crushing occurred along the 6 mm diameter vertical reinforcing bar in the web where the concrete cover was slightly lower. The maximum observed shear strength in Specimens 1 through 4 ranged from 65 to 70 kN, which gives a shear stress (𝑉/ 𝑏𝑤𝑑𝑣) of 5.2 to 5.6 MPa. From the concrete cylinder compression tests available in Chapter 3, a strength of 40 MPa was assumed. Using this value, the shear stress ratio (𝑣/𝑓𝑐′) ranged from 0.13 and 0.14. This corresponds well to the shear stress ratio in the Clause 21 of the 2004 and 2014 editions of CSA A23.3-14 (CSA, 2014), which limits the shear stress ratio to 0.15 for walls with small inelastic rotational demands. Specimen 5, which was subjected to large inelastic rotations, had a shear strength of about 41 kN, which results in a shear stress of 3.3 MPa, and a shear stress ratio of 0.08. This also corresponds reasonably well with the limit of 0.10 in CSA A23.3-14 for walls subjected to large inelastic rotation demands. 6.5 Summary of Analytical Predictions Predictions of the test results were made using commonly used analytical methods. The CSA A23.3-14 (CSA, 2014) and ACI 318-14 (ACI, 2014) approaches to calculating the nominal shear strength were compared to the results from computer program Response2000. CSA A23.3-14 and ACI 318-14 both assume a 45-degree angle for the principal compression stress, while Response2000 predicts a minimum stress angle of about 23 degrees at the centre of the wall cross-section. This resulted in a stirrup shear strength contribution that was much higher than what was predicted using the code-based approaches. Response2000 gives a very accurate 228 prediction of the shear strength, while both CSA A23.3-14 and ACI 318-14 both give very conservative predictions of shear strength. A prediction of the load-displacement response was completed using a tri-linear approximation of a non-linear moment-curvature response for the flexural stiffness model, and various types of shear models. The linear shear models include assuming an uncracked section, and using 50% of the uncracked shear stiffness, which are recommended by LATBSDC (2017) and PEER TBI (2017) for service level and MCE level analysis, respectively. These models over-predicted the shear force that is induced in the shear wall specimen when subjected to an imposed second-mode displacement. A cracked shear stiffness equal to 10% of the uncracked shear stiffness and a tri-linear shear model that was used gave better predictions of the induced shear force. The multi-linear shear model provided a much-improved prediction of the results and demonstrates that using more complex models for the shear stiffness would limit the over-prediction of the shear demands that were observed when common linear models are used. While additional work is needed to develop a comprehensive shear model that will improve the estimate of higher mode shear demands, it should be acknowledged that at least the current methods are conservative in over-predicting demands and under-predicting resistance. 6.6 Recommendations for Future Research While the completed tests provided very useful results regarding the strength and ductility of reinforced concrete cantilever shear walls subjected to second-mode demands, additional tests could be done to investigate a number of issues. While shear walls in high-rise buildings are subjected to significant axial compression, the test specimens had no applied axial 229 compression. Additional tests could be done to investigate the influence of the axial compression, which will increase both the flexural capacity and the shear capacity of the shear walls. The specimens had a height-to-length ratio of five. Additional tests could be done on more slender specimens. Finally, all the test specimens had the same reinforcement, concrete dimensions and concrete strengths. Additional tests could be done to investigate specimens with different characteristics. 230 Bibliography Adebar, P., Dezhdar, E. (2015). Effective stiffness for linear dynamic analysis of concrete shear walls buildings: CSA A23.3 – 2014. 11th Can. Conf. on Earthquake Engineering, Victoria, July 2015, 8 pp. ACI. (2014). ACI 318-14: Building Code Requirements for Structure Concrete. American Concrete Institute, Farmington Hills, MI. Bentz, E. (2000). Response2000, Retrieved December 2018, from http://www.ecf.utoronto.ca/~bentz/r2k.htm Beyer, K., Dazio, A., & Priestly, M.J.N. (2011). Shear deformations of slender reinforced concrete walls under seismic loading. ACI Structural Journal, March-April 2011, 167-177. Birely, A.C. (2012). Seismic Performance of Slender Reinforced Concrete Structural Walls. PhD Thesis, University of Washington. Blakely, R.W.G., Cooney, R.C., & Megget, L.M. (1975). Seismic shear loading at flexural capacity in cantilever wall structures. Bulletin of the New Zealand Society for Earthquake Engineering, 8, 278-290. CSA. (1994). A23.3-94: Design of Concrete Structures. Canadian Standards Association, Mississauga, ON. CSA. (2014). A23.3-14: Design of Concrete Structures. Canadian Standards Association, Mississauga, ON. Eberhard, M.O. & Sozen, M.A. (1993). Behavior-Based Method to Determine Design Shear in Earthquake-Resistant Walls. Journal of Structural Engineering, 119, 619-940. El-Azizy, O.A., Shedid, M.T, El-Dakhakhni, W. & Drysdale, R.G. (2015). Experimental Evaluation of the Seismic Performance of Reinforced Concrete Structural Walls with Different End Configurations. Engineering Structures, 101:246-263. GB 1499.2. (2007). Steel for the Reinforcement of Concrete – Part 2: Hot Rolled Ribbed Bars, National Standard of the People’s Republic of China. 231 Gerin, M. & Adebar, P. (2004). Accounting for Shear in Seismic Analysis of Concrete Structures. 13th World Conference on Earthquake Engineering, Paper No. 1747. Ghosh, S.K., & Markevicius, V.P. (1990). Design of Earthquake Resistant Shear Walls to Prevent Shear Failure. Proceedings of Fourth U.S. National Conference on Earthquake Engineering (Volume 2), USA, 905-913. Hiraishi, H. (1984). Evaluation of Shear and Flexural Deformations of Flexural Type Shear Walls. Bulletin of the New Zealand Society for Earthquake Engineering, V. 17, No. 2, 135-144. Keintzel, E. (1992). Advances in the Design for Shear of RC Structural Walls Ender Seismic Loading. In H. Krawinkler & P. Fajfar (Ed.), Nonlinear Seismic Analysis and Design of Reinforced Concrete Buildings (pp. 171-180). New York: Elsevier. LATBSDC. (2017). An Alternative Procedure for Seismic Analysis and Design of Tall Buildings Located in the Los Angeles Region, Los Angeles Tall Buildings Structural Design Council, Los Angeles, CA. Mitchell, D., Paultre, P., Adebar, P. (2016). Chapter 11 – Seismic Design, Concrete Design Handbook. Fourth Edition, Cement Association of Canada, Ottawa, 2016, pp. 11-1 to 11-64. NRC. (2015). National Building Code of Canada 2015. National Research Council Canada, Ottawa, ON. NZS. (1982). Part 1: Code of Practice for the Design of Concrete Structures. Wellington: Standards Association of New Zealand Orakcal, K. & Wallace, J.W. (2006). Flexural Modeling of Reinforcing Concrete Walls – Experimental Verification. ACI Structural Journal, March-April 2006 pp. 196-206. Oyen, P.E. (2006). Evaluation of Analytical Tools for Determining the Seismic Response of Reinforced Concrete Shear Walls. MSc Thesis, University of Washington. Priestly, N. & Amaris, A. (2003). Dynamic Amplification of Seismic Moments and Shear Forces in Cantilever Walls. Proceedings, FIB Symposium, Concrete Structures in Seismic Regions, Athens. 232 Pugh, J. (2012). Numerical Simulation of Walls and Seismic Design Recommendations for Walled Buildings. Diss. University of Washington. Rutenberg, A. (2013). Seismic Shear Forces on RC Walls: Review and Bibliography. Bulletin of Earthquake Engineering, Vol. 11, No. 5, pp. 1727-1752 Tall Buildings Initiative. (2017). Guidelines for Performance Based Seismic Design of Tall Buildings, v 2.03. Pacific Earthquake Engineering Center, Report No. 2017/06. UC Berkeley. (2017). OpenSees – Open System for Earthquake Engineering (version 2.5.0). Retrieved April 2017, from http://opensees.berkeley.edu/index.php. 233 A. – Linear Elastic Response Spectrum Analysis Procedure The prototype building modeled was the “30-Story Residential Building in Vancouver” example taken from Section 11.5 of the CAC Handbook (CSA, 2014). A simple model was created of the structure, by creating a lumped mass stick model using the software SAP2000. Each story of the structure was modelled using a frame element to represent the shear wall core in the cantilevered direction, as this thesis is only concerned with cantilevered shear walls, and not with shear walls connected by coupling beams. The example states that the average mass per story is 505,000 kg, which were placed at each story in the form of a lumped mass. The strength of the concrete was 30 MPa for levels greater than 20, 35 MPa for story’s 11-20, and 45 MPa for story’s below 11. The modal analysis was completed and gave the results for the first 8 modes, which is available in Table A.1. Table A.1: Modal periods, participation factors, and mass ratios of 30 story building. Mode Period (s) Participation Factor Mass Ratio 1 3.506 12418.66 0.630 2 0.595 6947.90 0.200 3 0.221 -4127.89 0.069 4 0.118 3026.24 0.037 5 0.075 -2312.92 0.022 6 0.053 1872.97 0.014 7 0.040 1551.50 0.010 8 0.032 -1284.60 0.007 SUM: 0.989 From these results the fundamental period is equal to 3.51 seconds, which is about 5% off of the period given in the example which is equal to 3.71 seconds. The fact that the period of the simple model is that close to the example tells us that the results of this simple model will be 234 sufficient for demonstrating the effect that higher modes have on the response of the structure. The first 8 modes account for 98.9% of the total mass participation of the structure, however modes 6, 7, and 8 together account for only 3.1%, and thus they were not included in the response spectrum analysis. The remaining 5 modes of vibration account for 95% of the mass participation and thus provides reliable results. The spectral acceleration response spectrum used was taken from NBCC 2015, and is for Site Class C, assuming 5% damping, in the City of Vancouver, and can be seen in Figure A.1, where the values of the first 5 periods are also shown. The corresponding spectral acceleration values for each mode can be seen in Table A.2. Figure A.1: Vancouver spectral acceleration response spectrum assuming 5% damping used for analysis, with periods of first five modes shown. 00.10.20.30.40.50.60.70.80.90 0.5 1 1.5 2 2.5 3 3.5 4Spectral Acceleration (g)Period (s)T1T2T3, T4, T5235 Table A.2: Spectral acceleration and displacement values of first five modes of building in cantilevered direction. Mode Period Spectral Acceleration (g) Spectral Displacement (m) 1 3.506 0.168 3.227 2 0.595 0.689 0.381 3 0.221 0.841 0.064 4 0.118 0.848 0.018 5 0.075 0.848 0.007 From the spectral accelerations found in Figure A.1, the corresponding spectral displacements associated with each mode of vibration can be found based on Eq. A.1. Where 𝑆𝑎𝑖 is the spectral acceleration, 𝑔 is the gravitational constant (taken as 9.81 m/s2), and 𝑇𝑖 is the respective mode shape. 𝑆𝑑𝑖 = 𝑆𝑎,𝑖𝑔𝑇𝑖22𝜋 (A.1) The displacement response along the height of the structure from each mode can then be calculated by multiplying the respective spectral displacement, modal participation factor, and mode shape together. This leaves us with the displacement response from each mode shape, which can all be combined to give the total modal envelope response of the structure. To combine the results of each mode into an envelope of the maximum displacement, the square root of sum of squares, or SRSS, method can be used. The SRSS is a simple method which is known to give a reasonable degree of accuracy when predicting the maximum forces and displacements of a time-history analysis. The SRSS of all five modes, as well as the SRSS of only the first two modes, was calculated and compared with each other, and can be seen in Figure A.2. 236 Next, the forces at each story can be calculated based on Eq. A.2, where Γ𝑖 is the modal participation factor if the ith mode, 𝑆𝑎,𝑖 is the corresponding spectral acceleration value of the ith mode, 𝑔 is the gravitational constant, 𝑀 is the global mass matrix of the model, and Φ𝑖 is the mode shape vector of the ith mode. It should be noted that the modal participation factor is dependent on how the mode shapes are normalized. 𝐹𝑖 = Γ𝑖𝑆𝑎,𝑖𝑔[𝑀][Φ𝑖] (A.2) From these forces the bending moments and shear forces at each story can be calculated for each mode shape individually, and the total envelope was again calculated using the SRSS method. The bending moment response can be seen in Figure A.3, while the shear force response is shown in Figure A.4. Figure A.2: Absolute displacement results of linear elastic response spectrum analysis of 30 story reinforce concrete shear wall core structure. 01020304050607080901000 1 2 3 4 5 6Height Above Grade (m)Displacement (m)Mode 1 Mode 2Mode 3 Mode 4Mode 5 SRSSSRSS (1,2)237 Figure A.3: Absolute bending moment results of linear elastic response spectrum analysis of 30 story reinforce concrete shear wall core structure. Figure A.4: Absolute shear force results of linear elastic response spectrum analysis of 30 story reinforce concrete shear wall core structure. 01020304050607080901000 200000 400000 600000 800000 1000000 1200000 1400000Height Above Grade (m)Moment (kN-m)Mode 1 Mode 2Mode 3 Mode 4Mode 5 SRSSSRSS (1,2)01020304050607080901000 5000 10000 15000 20000 25000 30000Height Above Grade (m)Shear (kN)Mode 1 Mode 2Mode 3 Mode 4Mode 5 SRSSSRSS(1,2)238 B. – Data from Non-Linear Dynamic Analysis B.1 Linear Elastic Model Excited by Sinusoidal Ground Acceleration (a) (b) (c) Figure B.1: Displacement response when excited at a period of 0.25s: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -15-10-5051015200 1 2 3 4 5 6 7 8 9 10Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-15-10-5051015200.5 1.5 2.5 3.5 4.5D2(mm)Time (s)TotalMode 1Mode 2-6-4-2024680.5 1.5 2.5 3.5 4.5D1(mm)Time (s)TotalMode 1Mode 2239 (a) (b) (c) Figure B.2: Static force response when excited at a period of 0.25s: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -1500-1000-5000500100015000 2 4 6 8 10Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-600-400-20002004006008000.5 1.5 2.5 3.5 4.5F2(kN)Time (s)TotalMode 1Mode 2-1500-1000-5000500100015000.5 1.5 2.5 3.5 4.5F1(kN)Time (s)TotalMode 1Mode 2240 (a) (b) (c) Figure B.3: Shear force response when excited at a period of 0.25s: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -1200-1000-800-600-400-20002004006008000 2 4 6 8 10Shear Force (kN)Time (s)Base (Vb)Top (Vt)-1200-1000-800-600-400-20002004006008000.5 1.5 2.5 3.5 4.5Vb(kN)Time (s)TotalMode 1Mode 2-800-600-400-20002004006000.5 1.5 2.5 3.5 4.5Vt(kN)Time (s)TotalMode 1Mode 2241 (a) (b) (c) Figure B.4: Bending moment response when excited at a period of 0.25s: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -40000-30000-20000-10000010000200003000040000500000 2 4 6 8 10Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-40000-30000-20000-10000010000200003000040000500000.5 1.5 2.5 3.5 4.5Mb(kN-m)Time (s)TotalMode 1Mode 2-20000-15000-10000-50000500010000150002000025000300000.5 1.5 2.5 3.5 4.5Mm(kN-m)Time (s)TotalMode 1Mode 2242 (a) (b) (c) (d) Figure B.5: Modal trend of response excited at period of 0.25s: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.3254xy = 0.3257x-5-4-3-2-10123456-20 -10 0 10 20D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.4008xy = 0.4252x-20000-15000-10000-5000050001000015000200002500030000-40000 -20000 0 20000 40000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.5293xy = 0.7397x-600-400-2000200400600-1000 -500 0 500 1000Vt(kN)Vb (kN)TotalMode 1Mode 2y = 0.0725xy = 0.352x-1000-800-600-400-2000200400600800-500 0 500 1000F1(kN)F2 (kN)TotalMode 1Mode 2243 (a) (b) (c) Figure B.6: Displacement response when excited at a period of 0.5s: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -30-20-100102030400 1 2 3 4 5 6 7 8 9 10Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-30-20-100102030401.5 2.5 3.5 4.5 5.5D2(mm)Time (s)TotalMode 1Mode 2-15-10-5051015201.5 2.5 3.5 4.5 5.5D1(mm)Time (s)TotalMode 1Mode 2244 (a) (b) (c) Figure B.7: Static force response when excited at a period of 0.5s: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -6000-4000-2000020004000600080000 2 4 6 8 10Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-2500-2000-1500-1000-500050010001500200025001.5 2.5 3.5 4.5 5.5F2(kN)Time (s)TotalMode 1Mode 2-6000-4000-2000020004000600080001.5 2.5 3.5 4.5 5.5F1(kN)Time (s)TotalMode 1Mode 2245 (a) (b) (c) Figure B.8: Shear force response when excited at a period of 0.5s: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -5000-4000-3000-2000-1000010002000300040000 2 4 6 8 10Shear Force (kN)Time (s)Base (Vb)Top (Vt)-5000-4000-3000-2000-1000010002000300040001.5 2.5 3.5 4.5 5.5Vb(kN)Time (s)TotalMode 1Mode 2-2500-2000-1500-1000-500050010001500200025001.5 2.5 3.5 4.5 5.5Vt(kN)Time (s)TotalMode 1Mode 2246 (a) (b) (c) Figure B.9: Bending moment response when excited at a period of 0.5s: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -100000-500000500001000001500000 2 4 6 8 10Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-100000-500000500001000001500001.5 2.5 3.5 4.5 5.5Mb(kN-m)Time (s)TotalMode 1Mode 2-100000-80000-60000-40000-200000200004000060000800001000001200001.5 2.5 3.5 4.5 5.5Mm(kN-m)Time (s)TotalMode 1Mode 2247 (a) (b) (c) (d) Figure B.10: Modal trend of response excited at period of 0.5s: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.3223xy = 0.3257x-15-10-505101520-40 -20 0 20 40D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.1411xy = 0.4252x-100000-80000-60000-40000-20000020000400006000080000100000120000-100000 -50000 0 50000 100000 150000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = -0.1784xy = 0.7397x-2500-2000-1500-1000-50005001000150020002500-6000 -4000 -2000 0 2000 4000Vt(kN)Vb (kN)TotalMode 1Mode 2y = -1.5692xy = 0.352x-6000-4000-200002000400060008000-3000 -2000 -1000 0 1000 2000 3000F1(kN)F2 (kN)TotalMode 1Mode 2248 (a) (b) (c) Figure B.11: Displacement response when excited at a period of 0.62s: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -40-30-20-10010203040500 1 2 3 4 5 6 7 8 9 10Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-40-30-20-10010203040501.8 2.8 3.8 4.8 5.8D2(mm)Time (s)TotalMode 1Mode 2-15-10-505101520251.8 2.8 3.8 4.8 5.8D1(mm)Time (s)TotalMode 1Mode 2249 (a) (b) (c) Figure B.12: Static force response when excited at a period of 0.62s: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -10000-8000-6000-4000-20000200040006000800010000120000 2 4 6 8 10Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-4000-3000-2000-1000010002000300040001.8 2.8 3.8 4.8 5.8F2(kN)Time (s)TotalMode 1Mode 2-10000-8000-6000-4000-20000200040006000800010000120001.8 2.8 3.8 4.8 5.8F1(kN)Time (s)TotalMode 1Mode 2250 (a) (b) (c) Figure B.13: Shear force response when excited at a period of 0.62s: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -8000-6000-4000-200002000400060000 2 4 6 8 10Shear Force (kN)Time (s)Base (Vb)Top (Vt)-8000-6000-4000-200002000400060001.8 2.8 3.8 4.8 5.8Vb(kN)Time (s)TotalMode 1Mode 2-4000-3000-2000-1000010002000300040001.8 2.8 3.8 4.8 5.8Vt(kN)Time (s)TotalMode 1Mode 2251 (a) (b) (c) Figure B.14: Bending moment response when excited at a period of 0.62s: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -150000-100000-500000500001000001500002000000 2 4 6 8 10Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-150000-100000-500000500001000001500002000001.8 2.8 3.8 4.8 5.8Mb(kN-m)Time (s)TotalMode 1Mode 2-150000-100000-500000500001000001500001.8 2.8 3.8 4.8 5.8Mm(kN-m)Time (s)TotalMode 1Mode 2252 (a) (b) (c) (d) Figure B.15: Modal trend of response excited at period of 0.62s: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.3197xy = 0.3257x-15-10-50510152025-40 -20 0 20 40 60D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.0452xy = 0.4252x-150000-100000-50000050000100000150000-200000 -100000 0 100000 200000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = -0.2618xy = 0.7397x-4000-3000-2000-100001000200030004000-10000 -5000 0 5000 10000Vt(kN)Vb (kN)TotalMode 1Mode 2y = -1.8822xy = 0.352x-10000-8000-6000-4000-2000020004000600080001000012000-4000 -2000 0 2000 4000F1(kN)F2 (kN)TotalMode 1Mode 2253 (a) (b) (c) Figure B.16: Displacement response when excited at a period of 0.75s: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -50-40-30-20-1001020304050600 1 2 3 4 5 6 7 8 9 10Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-50-40-30-20-1001020304050602 3 4 5 6D2(mm)Time (s)TotalMode 1Mode 2-20-15-10-5051015202 3 4 5 6D1(mm)Time (s)TotalMode 1Mode 2254 (a) (b) (c) Figure B.17: Static force response when excited at a period of 0.75s: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -8000-6000-4000-200002000400060008000100000 2 4 6 8 10Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-3000-2000-100001000200030002 3 4 5 6F2(kN)Time (s)TotalMode 1Mode 2-8000-6000-4000-200002000400060008000100002 3 4 5 6F1(kN)Time (s)TotalMode 1Mode 2255 (a) (b) (c) Figure B.18: Shear force response when excited at a period of 0.75s: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -6000-5000-4000-3000-2000-10000100020003000400050000 2 4 6 8 10Shear Force (kN)Time (s)Base (Vb)Top (Vt)-6000-4000-200002000400060002 3 4 5 6Vb(kN)Time (s)TotalMode 1Mode 2-3000-2000-100001000200030002 3 4 5 6Vt(kN)Time (s)TotalMode 1Mode 2256 (a) (b) (c) Figure B.19: Bending moment response when excited at a period of 0.75s: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -150000-100000-500000500001000001500002000000 2 4 6 8 10Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-150000-100000-500000500001000001500002000002 3 4 5 6Mb(kN-m)Time (s)TotalMode 1Mode 2-150000-100000-500000500001000001500002 3 4 5 6Mm(kN-m)Time (s)TotalMode 1Mode 2257 (a) (b) (c) (d) Figure B.20: Modal trend of response excited at period of 0.75s: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.3233xy = 0.3257x-20-15-10-505101520-60 -40 -20 0 20 40 60D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.2703xy = 0.4252x-100000-50000050000100000150000-200000 -100000 0 100000 200000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.0057xy = 0.7397x-3000-2500-2000-1500-1000-5000500100015002000-6000 -4000 -2000 0 2000 4000 6000Vt(kN)Vb (kN)TotalMode 1Mode 2y = -0.9846xy = 0.352x-8000-6000-4000-200002000400060008000-2000 -1000 0 1000 2000 3000F1(kN)F2 (kN)TotalMode 1Mode 2258 (a) (b) (c) Figure B.21: Displacement response when excited at a period of 1.0s: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -60-40-200204060800 1 2 3 4 5 6 7 8 9 10Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-60-40-200204060803 4 5 6 7D2(mm)Time (s)TotalMode 1Mode 2-25-20-15-10-505101520253 4 5 6 7D1(mm)Time (s)TotalMode 1Mode 2259 (a) (b) (c) Figure B.22: Static force response when excited at a period of 1.0s: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -4000-3000-2000-10000100020003000400050000 2 4 6 8 10Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-2000-1500-1000-500050010001500200025003 4 5 6 7F2(kN)Time (s)TotalMode 1Mode 2-4000-3000-2000-10000100020003000400050003 4 5 6 7F1(kN)Time (s)TotalMode 1Mode 2260 (a) (b) (c) Figure B.23: Shear force response when excited at a period of 1.0s: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -3000-2000-100001000200030000 2 4 6 8 10Shear Force (kN)Time (s)Base (Vb)Top (Vt)-3000-2000-100001000200030003 4 5 6 7Vb(kN)Time (s)TotalMode 1Mode 2-2500-2000-1500-1000-50005001000150020003 4 5 6 7Vt(kN)Time (s)TotalMode 1Mode 2261 (a) (b) (c) Figure B.24: Bending moment response when excited at a period of 1.0s: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -150000-100000-500000500001000001500002000000 2 4 6 8 10Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-150000-100000-500000500001000001500002000003 4 5 6 7Mb(kN-m)Time (s)TotalMode 1Mode 2-100000-80000-60000-40000-200000200004000060000800001000003 4 5 6 7Mm(kN-m)Time (s)TotalMode 1Mode 2262 (a) (b) (c) (d) Figure B.25: Modal trend of response excited at period of 1.0s: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.3254xy = 0.3257x-25-20-15-10-50510152025-100 -50 0 50 100D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.411xy = 0.4252x-80000-60000-40000-20000020000400006000080000100000-200000 -100000 0 100000 200000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.6032xy = 0.7397x-2500-2000-1500-1000-500050010001500-4000 -2000 0 2000 4000Vt(kN)Vb (kN)TotalMode 1Mode 2y = 0.1706xy = 0.352x-3000-2000-10000100020003000-2000 -1000 0 1000 2000 3000F1(kN)F2 (kN)TotalMode 1Mode 2263 (a) (b) (c) Figure B.26: Displacement response when excited at a period of 1.5s: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -150-100-500501001500 2 4 6 8 10 12Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-150-100-500501001504.2 5.2 6.2 7.2 8.2D2(mm)Time (s)TotalMode 1Mode 2-40-30-20-100102030404.2 5.2 6.2 7.2 8.2D1(mm)Time (s)TotalMode 1Mode 2264 (a) (b) (c) Figure B.27: Static force response when excited at a period of 1.5s: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -3000-2000-100001000200030000 2 4 6 8 10 12Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-3000-2000-100001000200030004.2 5.2 6.2 7.2 8.2F2(kN)Time (s)TotalMode 1Mode 2-3000-2000-1000010002000300040004.2 5.2 6.2 7.2 8.2F1(kN)Time (s)TotalMode 1Mode 2265 (a) (b) (c) Figure B.28: Shear force response when excited at a period of 1.5s: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -4000-3000-2000-1000010002000300040000 2 4 6 8 10 12Shear Force (kN)Time (s)Base (Vb)Top (Vt)-4000-3000-2000-1000010002000300040004.2 5.2 6.2 7.2 8.2Vb(kN)Time (s)TotalMode 1Mode 2-3000-2000-100001000200030004.2 5.2 6.2 7.2 8.2Vt(kN)Time (s)TotalMode 1Mode 2266 (a) (b) (c) Figure B.29: Bending moment response when excited at a period of 1.5s: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -300000-200000-10000001000002000003000000 2 4 6 8 10 12Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-300000-200000-10000001000002000003000004.2 5.2 6.2 7.2 8.2Mb(kN-m)Time (s)TotalMode 1Mode 2-150000-100000-500000500001000001500004.2 5.2 6.2 7.2 8.2Mm(kN-m)Time (s)TotalMode 1Mode 2267 (a) (b) (c) (d) Figure B.30: Modal trend of response excited at period of 1.5s: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.3256xy = 0.3257x-40-30-20-10010203040-150 -100 -50 0 50 100 150D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.4243xy = 0.4252x-150000-100000-50000050000100000150000-400000 -200000 0 200000 400000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.7285xy = 0.7397x-3000-2000-10000100020003000-4000 -2000 0 2000 4000Vt(kN)Vb (kN)TotalMode 1Mode 2y = 0.3355xy = 0.352x-1500-1000-5000500100015002000-4000 -2000 0 2000 4000F1(kN)F2 (kN)TotalMode 1Mode 2268 (a) (b) (c) Figure B.31: Displacement response when excited at a period of 2.0s: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -200-150-100-500501001502000 2 4 6 8 10 12 14 16 18 20Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-200-150-100-500501001502005.7 6.7 7.7 8.7 9.7D2(mm)Time (s)TotalMode 1Mode 2-60-40-2002040605.7 6.7 7.7 8.7 9.7D1(mm)Time (s)TotalMode 1Mode 2269 (a) (b) (c) Figure B.32: Static force response when excited at a period of 2.0s: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -4000-3000-2000-1000010002000300040000 5 10 15 20Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-4000-3000-2000-1000010002000300040005.7 6.7 7.7 8.7 9.7F2(kN)Time (s)TotalMode 1Mode 2-3000-2000-100001000200030005.7 6.7 7.7 8.7 9.7F1(kN)Time (s)TotalMode 1Mode 2270 (a) (b) (c) Figure B.33: Shear force response when excited at a period of 2.0s: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -5000-4000-3000-2000-10000100020003000400050000 5 10 15 20Shear Force (kN)Time (s)Base (Vb)Top (Vt)-5000-4000-3000-2000-10000100020003000400050005.7 6.7 7.7 8.7 9.7Vb(kN)Time (s)TotalMode 1Mode 2-4000-3000-2000-1000010002000300040005.7 6.7 7.7 8.7 9.7Vt(kN)Time (s)TotalMode 1Mode 2271 (a) (b) (c) Figure B.34: Bending moment response when excited at a period of 2.0s: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -400000-300000-200000-10000001000002000003000004000000 5 10 15 20Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-400000-300000-200000-10000001000002000003000004000005.7 6.7 7.7 8.7 9.7Mb(kN-m)Time (s)TotalMode 1Mode 2-150000-100000-500000500001000001500002000005.7 6.7 7.7 8.7 9.7Mm(kN-m)Time (s)TotalMode 1Mode 2272 (a) (b) (c) (d) Figure B.35: Modal trend of response excited at period of 2.0s: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.3256xy = 0.3257x-60-40-200204060-200 -100 0 100 200D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.4252xy = 0.4252x-200000-150000-100000-50000050000100000150000200000-400000 -200000 0 200000 400000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.7378xy = 0.7397x-4000-3000-2000-100001000200030004000-6000 -4000 -2000 0 2000 4000 6000Vt(kN)Vb (kN)TotalMode 1Mode 2y = 0.3469xy = 0.352x-1500-1000-5000500100015002000-4000 -2000 0 2000 4000F1(kN)F2 (kN)TotalMode 1Mode 2273 (a) (b) (c) Figure B.36: Displacement response when excited at a period of 2.5s: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -300-200-10001002003000 2 4 6 8 10 12 14 16 18 20Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-300-200-10001002003007.3 8.3 9.3 10.3 11.3D2(mm)Time (s)TotalMode 1Mode 2-100-80-60-40-200204060801007.3 8.3 9.3 10.3 11.3D1(mm)Time (s)TotalMode 1Mode 2274 (a) (b) (c) Figure B.37: Static force response when excited at a period of 2.5s: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -6000-4000-2000020004000600080000 2 4 6 8 10 12 14 16 18 20Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-6000-4000-2000020004000600080007.3 9.3 11.3F2(kN)Time (s)TotalMode 1Mode 2-3000-2000-100001000200030007.3 9.3 11.3F1(kN)Time (s)TotalMode 1Mode 2275 (a) (b) (c) Figure B.38: Shear force response when excited at a period of 2.5s: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -10000-8000-6000-4000-2000020004000600080000 2 4 6 8 10 12 14 16 18 20Shear Force (kN)Time (s)Base (Vb)Top (Vt)-10000-8000-6000-4000-2000020004000600080007.3 9.3 11.3Vb(kN)Time (s)TotalMode 1Mode 2-8000-6000-4000-200002000400060007.3 9.3 11.3Vt(kN)Time (s)TotalMode 1Mode 2276 (a) (b) (c) Figure B.39: Bending moment response when excited at a period of 2.5s: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -600000-400000-20000002000004000006000008000000 2 4 6 8 10 12 14 16 18 20Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-600000-400000-20000002000004000006000008000007.3 9.3 11.3Mb(kN-m)Time (s)TotalMode 1Mode 2-300000-200000-10000001000002000003000007.3 9.3 11.3Mm(kN-m)Time (s)TotalMode 1Mode 2277 (a) (b) (c) (d) Figure B.40: Modal trend of response excited at period of 2.5s: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.1008xy = 0.3257x-100-80-60-40-20020406080100-400 -200 0 200 400D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.4252xy = 0.4252x-300000-200000-1000000100000200000300000-1000000 -500000 0 500000 1000000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.7394xy = 0.7397x-8000-6000-4000-20000200040006000-10000 -5000 0 5000 10000Vt(kN)Vb (kN)TotalMode 1Mode 2y = 0.3506xy = 0.352x-2500-2000-1500-1000-50005001000150020002500-10000 -5000 0 5000 10000F1(kN)F2 (kN)TotalMode 1Mode 2278 (a) (b) (c) Figure B.41: Displacement response when excited at a period of 3.0s: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -600-400-20002004006000 2 4 6 8 10 12 14 16 18 20Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-600-400-20002004006008.5 9.5 10.5 11.5 12.5D2(mm)Time (s)TotalMode 1Mode 2-200-150-100-500501001502008.5 9.5 10.5 11.5 12.5D1(mm)Time (s)TotalMode 1Mode 2279 (a) (b) (c) Figure B.42: Static force response when excited at a period of 3.0s: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -15000-10000-50000500010000150000 2 4 6 8 10 12 14 16 18 20Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-15000-10000-50000500010000150008.5 9.5 10.5 11.5 12.5F2(kN)Time (s)TotalMode 1Mode 2-5000-4000-3000-2000-10000100020003000400050008.5 9.5 10.5 11.5 12.5F1(kN)Time (s)TotalMode 1Mode 2280 (a) (b) (c) Figure B.43: Shear force response when excited at a period of 3.0s: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -20000-15000-10000-5000050001000015000200000 2 4 6 8 10 12 14 16 18 20Shear Force (kN)Time (s)Base (Vb)Top (Vt)-20000-15000-10000-5000050001000015000200008.5 9.5 10.5 11.5 12.5Vb(kN)Time (s)TotalMode 1Mode 2-15000-10000-50000500010000150008.5 9.5 10.5 11.5 12.5Vt(kN)Time (s)TotalMode 1Mode 2281 (a) (b) (c) Figure B.44: Bending moment response when excited at a period of 0.62s: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -1500000-1000000-5000000500000100000015000000 2 4 6 8 10 12 14 16 18 20Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-1500000-1000000-5000000500000100000015000008.5 9.5 10.5 11.5 12.5Mb(kN-m)Time (s)TotalMode 1Mode 2-600000-400000-20000002000004000006000008.5 9.5 10.5 11.5 12.5Mm(kN-m)Time (s)TotalMode 1Mode 2282 (a) (b) (c) (d) Figure B.45: Modal trend of response excited at period of 3.0s: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.3256xy = 0.3257x-200-150-100-50050100150200-1000 -500 0 500 1000D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.4253xy = 0.4252x-600000-400000-2000000200000400000600000-2000000 -1000000 0 1000000 2000000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.7401xy = 0.7397x-15000-10000-5000050001000015000-20000 -10000 0 10000 20000Vt(kN)Vb (kN)TotalMode 1Mode 2y = 0.3509xy = 0.352x-5000-4000-3000-2000-1000010002000300040005000-15000 -10000 -5000 0 5000 10000 15000F1(kN)F2 (kN)TotalMode 1Mode 2283 (a) (b) (c) Figure B.46: Displacement response when excited at a period of 3.5s: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -800-600-400-200020040060080010000 2 4 6 8 10 12 14 16 18 20Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-800-600-400-200020040060080010009.6 10.6 11.6 12.6 13.6D2(mm)Time (s)TotalMode 1Mode 2-300-200-10001002003009.6 10.6 11.6 12.6 13.6D1(mm)Time (s)TotalMode 1Mode 2284 (a) (b) (c) Figure B.47: Static force response when excited at a period of 3.5s: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -20000-15000-10000-5000050001000015000200000 2 4 6 8 10 12 14 16 18 20Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-20000-15000-10000-5000050001000015000200009.6 10.6 11.6 12.6 13.6F2(kN)Time (s)TotalMode 1Mode 2-8000-6000-4000-2000020004000600080009.6 10.6 11.6 12.6 13.6F1(kN)Time (s)TotalMode 1Mode 2285 (a) (b) (c) Figure B.48: Shear force response when excited at a period of 3.5s: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -25000-20000-15000-10000-500005000100001500020000250000 2 4 6 8 10 12 14 16 18 20Shear Force (kN)Time (s)Base (Vb)Top (Vt)-30000-25000-20000-15000-10000-500005000100001500020000250009.6 10.6 11.6 12.6 13.6Vb(kN)Time (s)TotalMode 1Mode 2-20000-15000-10000-5000050001000015000200009.6 10.6 11.6 12.6 13.6Vt(kN)Time (s)TotalMode 1Mode 2286 (a) (b) (c) Figure B.49: Bending moment response when excited at a period of 3.5s: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -2000000-1500000-1000000-50000005000001000000150000020000000 2 4 6 8 10 12 14 16 18 20Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-2000000-1500000-1000000-50000005000001000000150000020000009.6 10.6 11.6 12.6 13.6Mb(kN-m)Time (s)TotalMode 1Mode 2-800000-600000-400000-200000020000040000060000080000010000009.6 10.6 11.6 12.6 13.6Mm(kN-m)Time (s)TotalMode 1Mode 2287 (a) (b) (c) (d) Figure B.50: Modal trend of response excited at period of 3.5s: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.3256xy = 0.3257x-300-200-1000100200300-1000 -500 0 500 1000D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.4254xy = 0.4252x-800000-600000-400000-20000002000004000006000008000001000000-2000000 -1000000 0 1000000 2000000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.7402xy = 0.7397x-20000-15000-10000-500005000100001500020000-30000 -20000 -10000 0 10000 20000 30000Vt(kN)Vb (kN)TotalMode 1Mode 2y = 0.3509xy = 0.352x-8000-6000-4000-200002000400060008000-20000 -10000 0 10000 20000F1(kN)F2 (kN)TotalMode 1Mode 2288 (a) (b) (c) Figure B.51: Displacement response when excited at a period of 3.71s: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -1000-800-600-400-200020040060080010000 2 4 6 8 10 12 14 16 18 20Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-1000-800-600-400-2000200400600800100010 11 12 13 14D2(mm)Time (s)TotalMode 1Mode 2-300-200-100010020030040010 11 12 13 14D1(mm)Time (s)TotalMode 1Mode 2289 (a) (b) (c) Figure B.52: Static force response when excited at a period of 3.71s: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -20000-15000-10000-5000050001000015000200000 2 4 6 8 10 12 14 16 18 20Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-20000-15000-10000-50000500010000150002000010 11 12 13 14F2(kN)Time (s)TotalMode 1Mode 2-8000-6000-4000-20000200040006000800010 11 12 13 14F1(kN)Time (s)TotalMode 1Mode 2290 (a) (b) (c) Figure B.53: Shear force response when excited at a period of 3.71s: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -30000-20000-1000001000020000300000 2 4 6 8 10 12 14 16 18 20Shear Force (kN)Time (s)Base (Vb)Top (Vt)-30000-20000-10000010000200003000010 11 12 13 14Vb(kN)Time (s)TotalMode 1Mode 2-20000-15000-10000-50000500010000150002000010 11 12 13 14Vt(kN)Time (s)TotalMode 1Mode 2291 (a) (b) (c) Figure B.54: Bending moment response when excited at a period of 3.71s: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -2000000-1500000-1000000-500000050000010000001500000200000025000000 2 4 6 8 10 12 14 16 18 20Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-2000000-1500000-1000000-5000000500000100000015000002000000250000010 11 12 13 14Mb(kN-m)Time (s)TotalMode 1Mode 2-1000000-800000-600000-400000-2000000200000400000600000800000100000010 11 12 13 14Mm(kN-m)Time (s)TotalMode 1Mode 2292 (a) (b) (c) (d) Figure B.55: Modal trend of response excited at period of 3.71s: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.3256xy = 0.3257x-300-200-1000100200300400-1000 -500 0 500 1000D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.4254xy = 0.4252x-1000000-800000-600000-400000-20000002000004000006000008000001000000-2000000 0 2000000 4000000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.7402xy = 0.7397x-20000-15000-10000-500005000100001500020000-40000 -20000 0 20000 40000Vt(kN)Vb (kN)TotalMode 1Mode 2 y = 0.3509xy = 0.352x-8000-6000-4000-200002000400060008000-20000 -10000 0 10000 20000F1(kN)F2 (kN)TotalMode 1Mode 2293 (a) (b) (c) Figure B.56: Displacement response when excited at a period of 4.0s: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -1000-800-600-400-200020040060080010000 2 4 6 8 10 12 14 16 18 20Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-1000-800-600-400-2000200400600800100010.6 11.6 12.6 13.6 14.6D2(mm)Time (s)TotalMode 1Mode 2-300-200-100010020030040010.6 11.6 12.6 13.6 14.6D1(mm)Time (s)TotalMode 1Mode 2294 (a) (b) (c) Figure B.57: Static force response when excited at a period of 4.0s: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -20000-15000-10000-5000050001000015000200000 2 4 6 8 10 12 14 16 18 20Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-20000-15000-10000-50000500010000150002000010.6 11.6 12.6 13.6 14.6F2(kN)Time (s)TotalMode 1Mode 2-8000-6000-4000-2000020004000600080001000010.6 11.6 12.6 13.6 14.6F1(kN)Time (s)TotalMode 1Mode 2295 (a) (b) (c) Figure B.58: Shear force response when excited at a period of 4.0s: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -30000-20000-1000001000020000300000 2 4 6 8 10 12 14 16 18 20Shear Force (kN)Time (s)Base (Vb)Top (Vt)-30000-20000-10000010000200003000010.6 11.6 12.6 13.6 14.6Vb(kN)Time (s)TotalMode 1Mode 2-20000-15000-10000-50000500010000150002000010.6 11.6 12.6 13.6 14.6Vt(kN)Time (s)TotalMode 1Mode 2296 (a) (b) (c) Figure B.59: Bending moment response when excited at a period of 4.0s: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -2000000-1500000-1000000-500000050000010000001500000200000025000000 2 4 6 8 10 12 14 16 18 20Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-2000000-1500000-1000000-5000000500000100000015000002000000250000010.6 11.6 12.6 13.6 14.6Mb(kN-m)Time (s)TotalMode 1Mode 2-1000000-800000-600000-400000-2000000200000400000600000800000100000010.6 11.6 12.6 13.6 14.6Mm(kN-m)Time (s)TotalMode 1Mode 2297 (a) (b) (c) (d) Figure B.60: Modal trend of response excited at period of 4.0s: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.3256xy = 0.3257x-300-200-1000100200300-1000 -500 0 500 1000D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.4254xy = 0.4252x-1000000-800000-600000-400000-20000002000004000006000008000001000000-2000000 -1000000 0 1000000 2000000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.7402xy = 0.7397x-20000-15000-10000-500005000100001500020000-30000 -20000 -10000 0 10000 20000 30000Vt(kN)Vb (kN)TotalMode 1Mode 2y = 0.3509xy = 0.352x-8000-6000-4000-200002000400060008000-20000 -10000 0 10000 20000F1(kN)F2 (kN)TotalMode 1Mode 2298 B.2 Non-Linear Model Excited by Sinusoidal Ground Acceleration (a) (b) (c) Figure B.61: Displacement response when excited at the period of the first mode – Trial 1: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -80-60-40-200204060801000 2 4 6 8 10 12 14 16 18 20Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-80-60-40-2002040608010010 11 12 13 14D2(mm)Time (s)TotalMode 1Mode 2-30-20-10010203010 11 12 13 14D1(mm)Time (s)TotalMode 1Mode 2299 (a) (b) (c) Figure B.62: Static force response when excited at the period of the first mode – Trial 1: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -2000-1500-1000-50005001000150020000 2 4 6 8 10 12 14 16 18 20Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-2000-1500-1000-500050010001500200010 11 12 13 14F2(kN)Time (s)TotalMode 1Mode 2-600-400-200020040060080010 11 12 13 14F1(kN)Time (s)TotalMode 1Mode 2300 (a) (b) (c) Figure B.63: Shear force response when excited at the period of the first mode – Trial 1: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -3000-2500-2000-1500-1000-500050010001500200025000 2 4 6 8 10 12 14 16 18 20Shear Force (kN)Time (s)Base (Vb)Top (Vt)-3000-2500-2000-1500-1000-5000500100015002000250010 11 12 13 14Vb(kN)Time (s)TotalMode 1Mode 2-2000-1500-1000-500050010001500200010 11 12 13 14Vt(kN)Time (s)TotalMode 1Mode 2301 (a) (b) (c) Figure B.64: Bending moment response when excited at the period of first mode – Trial 1: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -200000-150000-100000-500000500001000001500002000000 2 4 6 8 10 12 14 16 18 20Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-200000-150000-100000-5000005000010000015000020000010 11 12 13 14Mb(kN-m)Time (s)TotalMode 1Mode 2-80000-60000-40000-2000002000040000600008000010000010 11 12 13 14Mm(kN-m)Time (s)TotalMode 1Mode 2302 (a) (b) (c) (d) Figure B.65: Modal trend of response when excited at the period of first mode – Trial 1: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.326xy = 0.3257x-30-20-100102030-100 -50 0 50 100D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.425xy = 0.4252x-80000-60000-40000-20000020000400006000080000100000-200000 -100000 0 100000 200000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.7391xy = 0.7397x-2000-1500-1000-5000500100015002000-3000 -2000 -1000 0 1000 2000 3000Vt(kN)Vb (kN)TotalMode 1Mode 2y = 0.3528xy = 0.352x-600-400-2000200400600800-2000 -1000 0 1000 2000F1(kN)F2 (kN)TotalMode 1Mode 2303 (a) (b) (c) Figure B.66: Displacement response when excited at the period of the first mode – Trial 2: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -300-200-10001002003004000 2 4 6 8 10 12 14 16 18 20Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-300-200-100010020030040010 11 12 13 14 15D2(mm)Time (s)TotalMode 1Mode 2-100-5005010015010 11 12 13 14 15D1(mm)Time (s)TotalMode 1Mode 2304 (a) (b) (c) Figure B.67: Static force response when excited at the period of the first mode – Trial 2: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -5000-4000-3000-2000-10000100020003000400050000 2 4 6 8 10 12 14 16 18 20Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-5000-4000-3000-2000-100001000200030004000500010 12 14F2(kN)Time (s)TotalMode 1Mode 2-4000-3000-2000-10000100020003000400010 12 14F1(kN)Time (s)TotalMode 1Mode 2305 (a) (b) (c) Figure B.68: Shear force response when excited at the period of the first mode – Trial 2: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -6000-4000-200002000400060000 2 4 6 8 10 12 14 16 18 20Shear Force (kN)Time (s)Base (Vb)Top (Vt)-6000-4000-2000020004000600010 12 14Vb(kN)Time (s)TotalMode 1Mode 2-5000-4000-3000-2000-100001000200030004000500010 12 14Vt(kN)Time (s)TotalMode 1Mode 2306 (a) (b) (c) Figure B.69: Bending moment response when excited at the period of first mode – Trial 2: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -2000000-1500000-1000000-500000050000010000001500000200000025000000 2 4 6 8 10 12 14 16 18 20Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-2000000-1500000-1000000-5000000500000100000015000002000000250000010.6 11.6 12.6 13.6 14.6Mb(kN-m)Time (s)TotalMode 1Mode 2-1000000-800000-600000-400000-2000000200000400000600000800000100000010.6 11.6 12.6 13.6 14.6Mm(kN-m)Time (s)TotalMode 1Mode 2307 (a) (b) (c) (d) Figure B.70: Modal trend of response when excited at the period of first mode – Trial 2: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.3709xy = 0.375x-100-50050100150-400 -200 0 200 400D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.4155xy = 0.4252x-200000-150000-100000-50000050000100000150000200000-400000 -200000 0 200000 400000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.684xy = 0.7397x-5000-4000-3000-2000-1000010002000300040005000-10000 -5000 0 5000 10000Vt(kN)Vb (kN)TotalMode 1Mode 2y = 0.333xy = 0.352x-4000-3000-2000-100001000200030004000-6000 -4000 -2000 0 2000 4000 6000F1(kN)F2 (kN)TotalMode 1Mode 2308 (a) (b) (c) Figure B.71: Displacement response when excited at the period of the first mode – Trial 3: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -800-600-400-20002004006008000 2 4 6 8 10 12 14 16 18 20Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-800-600-400-200020040060080011 13 15 17D2(mm)Time (s)TotalMode 1Mode 2-300-200-100010020030011 13 15 17D1(mm)Time (s)TotalMode 1Mode 2309 (a) (b) (c) Figure B.72: Static force response when excited at the period of the first mode – Trial 3: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -8000-6000-4000-2000020004000600080000 2 4 6 8 10 12 14 16 18 20Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-8000-6000-4000-2000020004000600011 13 15 17F2(kN)Time (s)TotalMode 1Mode 2-8000-6000-4000-20000200040006000800011 13 15 17F1(kN)Time (s)TotalMode 1Mode 2310 (a) (b) (c) Figure B.73: Shear force response when excited at the period of the first mode – Trial 3: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -10000-8000-6000-4000-2000020004000600080000 2 4 6 8 10 12 14 16 18 20Shear Force (kN)Time (s)Base (Vb)Top (Vt)-8000-6000-4000-2000020004000600011 13 15 17F2(kN)Time (s)TotalMode 1Mode 2-8000-6000-4000-20000200040006000800011 13 15 17F1(kN)Time (s)TotalMode 1Mode 2311 (a) (b) (c) Figure B.74: Bending moment response when excited at the period of first mode – Trial 3: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -600000-400000-20000002000004000006000000 2 4 6 8 10 12 14 16 18 20Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-600000-400000-200000020000040000060000011 13 15 17Mb(kN-m)Time (s)TotalMode 1Mode 2-300000-200000-100000010000020000030000011 13 15 17Mm(kN-m)Time (s)TotalMode 1Mode 2312 (a) (b) (c) (d) Figure B.75: Modal trend of response when excited at the period of first mode – Trial 3: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.392xy = 0.3973x-300-200-1000100200300-1000 -500 0 500 1000D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.4069xy = 0.4252x-250000-200000-150000-100000-50000050000100000150000200000250000300000-500000 0 500000 1000000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.6008xy = 0.7397x-6000-4000-20000200040006000-10000 -5000 0 5000 10000Vt(kN)Vb (kN)TotalMode 1Mode 2y = 0.2081xy = 0.352x-8000-6000-4000-200002000400060008000-10000 -5000 0 5000 10000F1(kN)F2 (kN)TotalMode 1Mode 2313 (a) (b) (c) Figure B.76: Displacement response when excited at the period of the first mode – Trial 4: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -1500-1000-500050010001500200025000 2 4 6 8 10 12 14 16 18 20Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-1500-1000-5000500100015002000250010 12 14 16 18 20D2(mm)Time (s)TotalMode 1Mode 2-600-400-2000200400600800100010 12 14 16 18 20D1(mm)Time (s)TotalMode 1Mode 2314 (a) (b) (c) Figure B.77: Static force response when excited at the period of the first mode – Trial 4: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -10000-8000-6000-4000-20000200040006000800010000120000 2 4 6 8 10 12 14 16 18 20Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-8000-6000-4000-2000020004000600080001000010 15 20F2(kN)Time (s)TotalMode 1Mode 2-10000-500005000100001500010 15 20F1(kN)Time (s)TotalMode 1Mode 2315 (a) (b) (c) Figure B.78: Shear force response when excited at the period of the first mode – Trial 4: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -12000-10000-8000-6000-4000-2000020004000600080000 2 4 6 8 10 12 14 16 18 20Shear Force (kN)Time (s)Base (Vb)Top (Vt)-12000-10000-8000-6000-4000-20000200040006000800010 15 20Vb(kN)Time (s)TotalMode 1Mode 2-10000-8000-6000-4000-20000200040006000800010 15 20Vt(kN)Time (s)TotalMode 1Mode 2316 (a) (b) (c) Figure B.79: Bending moment response when excited at the period of first mode – Trial 4: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -600000-400000-20000002000004000006000000 2 4 6 8 10 12 14 16 18 20Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-600000-400000-200000020000040000060000080000010 15 20Mb(kN-m)Time (s)TotalMode 1Mode 2-400000-300000-200000-100000010000020000030000040000010 15 20Mm(kN-m)Time (s)TotalMode 1Mode 2317 (a) (b) (c) (d) Figure B.80: Modal trend of response when excited at the period of first mode – Trial 4: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.4447xy = 0.4426x-600-400-20002004006008001000-1000 0 1000 2000 3000D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.406xy = 0.4252x-300000-200000-1000000100000200000300000-1000000 -500000 0 500000 1000000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.653xy = 0.7397x-8000-6000-4000-20000200040006000-10000 -5000 0 5000 10000Vt(kN)Vb (kN)TotalMode 1Mode 2 y = 0.3696xy = 0.352x-8000-6000-4000-20000200040006000-10000 -5000 0 5000 10000F1(kN)F2 (kN)TotalMode 1Mode 2318 (a) (b) (c) Figure B.81: Displacement response when excited at the period of the second-mode – Trial 1: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -40-30-20-10010203040500 1 2 3 4 5 6 7 8 9 10Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-40-30-20-10010203040501.8 2.8 3.8 4.8 5.8D2(mm)Time (s)TotalMode 1Mode 2-15-10-5051015201.8 2.8 3.8 4.8 5.8D1(mm)Time (s)TotalMode 1Mode 2319 (a) (b) (c) Figure B.82: Static force response when excited at the period of the second-mode – Trial 1: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -10000-8000-6000-4000-200002000400060008000100000 2 4 6 8 10Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-4000-3000-2000-1000010002000300040001.8 2.8 3.8 4.8 5.8F2(kN)Time (s)TotalMode 1Mode 2-10000-8000-6000-4000-200002000400060008000100001.8 2.8 3.8 4.8 5.8F1(kN)Time (s)TotalMode 1Mode 2320 (a) (b) (c) Figure B.83: Shear force response when excited at the period of the second-mode – Trial 1: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -8000-6000-4000-200002000400060000 2 4 6 8 10Shear Force (kN)Time (s)Base (Vb)Top (Vt)-8000-6000-4000-200002000400060001.8 2.8 3.8 4.8 5.8Vb(kN)Time (s)TotalMode 1Mode 2-4000-3000-2000-1000010002000300040001.8 2.8 3.8 4.8 5.8Vt(kN)Time (s)TotalMode 1Mode 2321 (a) (b) (c) Figure B.84: Bending moment response when excited at the period of second-mode – Trial 1: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -150000-100000-500000500001000001500002000000 2 4 6 8 10Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-150000-100000-500000500001000001500002000001.8 2.8 3.8 4.8 5.8Mb(kN-m)Time (s)TotalMode 1Mode 2-150000-100000-500000500001000001500001.8 2.8 3.8 4.8 5.8Mm(kN-m)Time (s)TotalMode 1Mode 2322 (a) (b) (c) (d) Figure B.85: Modal trend of response when excited at the period of second-mode – Trial 1: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.3207xy = 0.3257x-15-10-505101520-40 -20 0 20 40 60D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.0769xy = 0.4252x-150000-100000-50000050000100000150000-200000 -100000 0 100000 200000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = -0.2404xy = 0.7397x-4000-3000-2000-100001000200030004000-10000 -5000 0 5000 10000Vt(kN)Vb (kN)TotalMode 1Mode 2y = -1.7916xy = 0.352x-10000-8000-6000-4000-20000200040006000800010000-4000 -2000 0 2000 4000F1(kN)F2 (kN)TotalMode 1Mode 2323 (a) (b) (c) Figure B.86: Displacement response when excited at the period of the second-mode – Trial 2: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -150-100-500501001502000 1 2 3 4 5 6 7 8 9 10Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-150-100-500501001502001.8 2.8 3.8 4.8 5.8D2(mm)Time (s)TotalMode 1Mode 2-60-40-200204060801.8 2.8 3.8 4.8 5.8D1(mm)Time (s)TotalMode 1Mode 2324 (a) (b) (c) Figure B.87: Static force response when excited at the period of the second-mode – Trial 2: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -20000-15000-10000-500005000100001500020000250000 2 4 6 8 10Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-8000-6000-4000-2000020004000600080001.8 2.8 3.8 4.8 5.8F2(kN)Time (s)TotalMode 1Mode 2-20000-15000-10000-500005000100001500020000250001.8 2.8 3.8 4.8 5.8F1(kN)Time (s)TotalMode 1Mode 2325 (a) (b) (c) Figure B.88: Shear force response when excited at the period of the second-mode – Trial 2: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -15000-10000-50000500010000150000 2 4 6 8 10Shear Force (kN)Time (s)Base (Vb)Top (Vt)-15000-10000-50000500010000150001.8 2.8 3.8 4.8 5.8Vb(kN)Time (s)TotalMode 1Mode 2-8000-6000-4000-2000020004000600080001.8 2.8 3.8 4.8 5.8Vt(kN)Time (s)TotalMode 1Mode 2326 (a) (b) (c) Figure B.89: Bending moment response when excited at the period of second-mode – Trial 2: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -400000-300000-200000-10000001000002000003000004000000 2 4 6 8 10Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-400000-300000-200000-10000001000002000003000004000001.8 2.8 3.8 4.8 5.8Mb(kN-m)Time (s)TotalMode 1Mode 2-400000-300000-200000-10000001000002000003000004000001.8 2.8 3.8 4.8 5.8Mm(kN-m)Time (s)TotalMode 1Mode 2327 (a) (b) (c) (d) Figure B.90: Modal trend of response when excited at the period of second-mode – Trial 2: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.3407xy = 0.3435x-60-40-20020406080-200 -100 0 100 200D1(mm)D2 (mm)TotalMode 1Mode 2 y = 0.2739xy = 0.4252x-400000-300000-200000-1000000100000200000300000400000-400000 -200000 0 200000 400000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = -0.0367xy = 0.7397x-8000-6000-4000-200002000400060008000-15000 -10000 -5000 0 5000 10000 15000Vt(kN)Vb (kN)TotalMode 1Mode 2y = -1.0917xy = 0.352x-20000-15000-10000-50000500010000150002000025000-10000 -5000 0 5000 10000F1(kN)F2 (kN)TotalMode 1Mode 2328 (a) (b) (c) Figure B.91: Displacement response when excited at the period of the second-mode – Trial 3: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -300-200-10001002003004005000 1 2 3 4 5 6 7 8 9 10Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-300-200-10001002003004005001.8 3.6 5.4 7.2D2(mm)Time (s)TotalMode 1Mode 2-150-100-500501001502002501.8 3.6 5.4 7.2D1(mm)Time (s)TotalMode 1Mode 2329 (a) (b) (c) Figure B.92: Static force response when excited at the period of the second-mode – Trial 3: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -30000-20000-1000001000020000300000 2 4 6 8 10Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-10000-8000-6000-4000-200002000400060008000100001.8 3.6 5.4 7.2F2(kN)Time (s)TotalMode 1Mode 2-30000-20000-1000001000020000300001.8 3.6 5.4 7.2F1(kN)Time (s)TotalMode 1Mode 2330 (a) (b) (c) Figure B.93: Shear force response when excited at the period of the second-mode – Trial 3: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -25000-20000-15000-10000-5000050001000015000200000 2 4 6 8 10Shear Force (kN)Time (s)Base (Vb)Top (Vt)-25000-20000-15000-10000-5000050001000015000200001.8 3.6 5.4 7.2Vb(kN)Time (s)TotalMode 1Mode 2-10000-8000-6000-4000-200002000400060008000100001.8 3.6 5.4 7.2Vt(kN)Time (s)TotalMode 1Mode 2331 (a) (b) (c) Figure B.94: Bending moment response when excited at the period of second-mode – Trial 3: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -500000-400000-300000-200000-10000001000002000003000004000005000006000000 2 4 6 8 10Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-500000-400000-300000-200000-10000001000002000003000004000005000006000001.8 3.6 5.4 7.2Mb(kN-m)Time (s)TotalMode 1Mode 2-500000-400000-300000-200000-10000001000002000003000004000005000001.8 3.6 5.4 7.2Mm(kN-m)Time (s)TotalMode 1Mode 2332 (a) (b) (c) (d) Figure B.95: Modal trend of response when excited at the period of second-mode – Trial 3: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.3826xy = 0.3964x-150-100-50050100150200250-400 -200 0 200 400 600D1(mm)D2 (mm)TotalMode 1Mode 2 y = 0.2969xy = 0.4252x-500000-400000-300000-200000-1000000100000200000300000400000500000-600000-400000-200000 0 200000 400000 600000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = -0.0195xy = 0.7397x-10000-8000-6000-4000-20000200040006000800010000-20000 -10000 0 10000 20000Vt(kN)Vb (kN)TotalMode 1Mode 2y = -1.0449xy = 0.352x-20000-15000-10000-5000050001000015000200002500030000-10000 -5000 0 5000 10000F1(kN)F2 (kN)TotalMode 1Mode 2333 (a) (b) (c) Figure B.96: Displacement response when excited at the period of the second-mode – Trial 4: (a) Top and mid-height displacement; (b) Modal decomposition of top displacement (D2); (c) Modal decomposition of mid-height Displacement (D1). -1000-50005001000150020000 2 4 6 8 10 12Displacement (mm)Time (s)Top (D₂)Mid-Height (D₁)-1000-50005001000150020001.8 4.8 7.8 10.8D2(mm)Time (s)TotalMode 1Mode 2-400-200020040060080010001.8 4.8 7.8 10.8D1(mm)Time (s)TotalMode 1Mode 2334 (a) (b) (c) Figure B.97: Static force response when excited at the period of the second-mode – Trial 4: (a) Top and mid-height force; (b) Modal decomposition of top force (F2); (c) Modal decomposition of mid-height force (F1). -40000-30000-20000-100000100002000030000400000 2 4 6 8 10 12Static Force (kN)Time (s)Top (F₂)Mid-Height (F₁)-15000-10000-50000500010000150001.8 4.8 7.8 10.8F2(kN)Time (s)TotalMode 1Mode 2-40000-30000-20000-100000100002000030000400001.8 4.8 7.8 10.8F1(kN)Time (s)TotalMode 1Mode 2335 (a) (b) (c) Figure B.98: Shear force response when excited at the period of the second-mode – Trial 4: (a) Base and top Shear; (b) Modal decomposition of base shear (Vb); (c) Modal decomposition of top shear (Vt). -25000-20000-15000-10000-500005000100001500020000250000 2 4 6 8 10 12Shear Force (kN)Time (s)Base (Vb)Top (Vt)-25000-20000-15000-10000-500005000100001500020000250001.8 4.8 7.8 10.8Vb(kN)Time (s)TotalMode 1Mode 2-15000-10000-50000500010000150001.8 4.8 7.8 10.8Vt(kN)Time (s)TotalMode 1Mode 2336 (a) (b) (c) Figure B.99: Bending moment response when excited at the period of second-mode – Trial 4: (a) Base and mid-height moment (b) Modal decomposition of base moment (Mb); (c) Modal decomposition of mid-height moment (Mm). -600000-400000-20000002000004000006000000 2 4 6 8 10 12Moment (kN-m)Time (s)Base (Mb)Mid-Height (Mm)-600000-400000-20000002000004000006000008000001.8 4.8 7.8 10.8Mb(kN-m)Time (s)TotalMode 1Mode 2-500000-400000-300000-200000-10000001000002000003000004000005000001.8 4.8 7.8 10.8Mm(kN-m)Time (s)TotalMode 1Mode 2337 (a) (b) (c) (d) Figure B.100: Modal trend of response when excited at the period of second-mode – Trial 4: (a) Displacement; (b) Bending moment; (c) Shear force; (d) Equivalent static forces. y = 0.4484xy = 0.452x-400-20002004006008001000-1000 -500 0 500 1000 1500 2000D1(mm)D2 (mm)TotalMode 1Mode 2y = 0.3331xy = 0.4252x-500000-400000-300000-200000-1000000100000200000300000400000500000-500000 0 500000 1000000Mm(kN-m)Mb (kN-m)TotalMode 1Mode 2y = 0.1203xy = 0.7397x-15000-10000-5000050001000015000-30000 -20000 -10000 0 10000 20000Vt(kN)Vb (kN)TotalMode 1Mode 2y = -0.7261xy = 0.352x-30000-20000-10000010000200003000040000-15000 -10000 -5000 0 5000 10000 15000F1(kN)F2 (kN)TotalMode 1Mode 2338 C. – Prediction Program To predict the expected forces that the specimens will be under during the imposed second-mode displacement protocols, a program was created in Microsoft Excel using Visual Basic. This program allows the user to determine the curvature profile and shear strains to result in the given displacement, and from this the forces can be determined based on an input stress-strain model of the shear and flexural stiffness. Based on the assumption that the only applied demands on the specimen are from point loads at the location of the actuators, the two required point loads that would generate the given bending moment and shear forces at each time-step can be determined. This program was created because the response of the specimen displacement is being controlled from two separate nodes (at the top, and at the mid-height), and software such as OpenSees only allows displacement-based analysis to be controlled from one node. Visual Basic was used as this prediction program will be used for a very specific purpose, however for more in-depth analysis in the future it is recommended that a more rigorous program be created and implemented into something such as the OpenSees environment. The program works two ways, either a force history can be implemented in order to determine the resulting displacements, or vice versa. The former is a very simple process as it is simple to determine the bending moments and shear forces that would be caused by the two given forces. However, to go from displacements to forces is a more computationally demanding process, as the program must iterate until it is able to reduce the error between the solved and target displacements below a selected tolerance. 339 The displacements are calculated from the forces based on the following methodology. A point load is applied at the top and mid-height of the wall. These two point forces result in a bending moment profile along the wall, and from these moments the resulting curvature profile can be generated, based on a given moment-curvature relationship for the cross-section. The curvature profile was discretized along the length as can be seen in Figure C.1, and it was found that roughly 15 discrete sections along the element gave accurate results within roughly 1%. The curvature profile along the length of the beam can be integrated to solve for the displacements. The curvature profile shown in Figure C.1 is an example of what would be expected when a second mode displaced shape is imposed on the specimen in the linear elastic range. 340 Figure C.1: Visualization of how curvature distribution is discretized in program. As it can be seen the bending moment in each section is calculated based on the two forces, however the bending moments in the bottom half of the wall are influenced by both forces, while the bending moments in the top half are influenced only by the top force. This corresponds with Eq. C.1 and C.2, where 𝑀𝑖 is the moment at length 𝑥, F1 and F2 are the forces at the mid-height and top of the wall respectively, 𝑙𝑤 is the length of the wall, and 𝑥 is the length along the wall. Curvature LengthCurvature 341 𝑀𝑖 = 𝐹2 ∗ (𝑙𝑤 − 𝑥), 𝑖𝑓 𝑥 > 𝑤2 (C.1) 𝑀𝑖 = 𝐹2 ∗ (𝑙𝑤 − 𝑥) + 𝐹1 ∗ ( 𝑤2− 𝑥), 𝑖𝑓 𝑥 < 𝑤2 (C.2) After the program calculates the bending moment at each discrete section of the wall, it is able to assign a given curvature value based on the input moment-curvature relationship of the section. Based on these values a curvature profile is generated which can be integrated to calculate the corresponding displacements at the mid-height and top of the wall. The displacements at the mid-height, denoted by D1, are calculated by taking the area of the section of the curvature profile, multiplied by its respective distance from the point of interest, and curvature profiles in the top half do not influence the deflection at the middle, as can be seen in Eq. C.3. The top displacement, denoted by D2, are calculated by the same process, except the curvatures along the length of the entire wall are integrated, as can be seen in Eq. C.4, where 𝜑𝑖 is the curvature of the section at length 𝑥, and 𝑙𝑖 is the length of the discrete section at length 𝑥. 𝐷1 = ∑𝜑𝑖𝑙𝑖 ( 𝑤2− 𝑥) , 𝑖𝑓 𝑥 < 𝑤2 (C.3) 𝐷2 = ∑𝜑𝑖𝑙𝑖 (𝑙𝑤 − 𝑥) (C.4) Generally, calculating the displacements along the wall given a set of two forces is simple process, however, to generate forces based on a set of given displacements is a more complex iterative process. The first step is that a pair of given forces is assumed, for the first time-step it is assumed zero, however to increase the speed of the prediction, all subsequent time-steps use an initial assumed force pair equal to the final result of the previous time-step. From these 342 assumed forces, the process discussed earlier in this section is completed, and a resulting mid-height and top displacement are calculated. These displacements are known as the predicted displacements and are used by the program to check if the given target displacements have been reached or not. If the difference between the two is not within the desired tolerance, the program generates a new set of forces and iterate the process again. The new assumed force is determined by adding the difference between the target and predicted displacement, multiplied by a variable, 𝑖 ,to the assumed force of the previous time-step, as can be seen in Eq. C.5 and C.6, where 𝑛 is the current time step. 𝐹1𝑛+1 = 𝐹1𝑛 + 𝑖 ∗ (𝐷1𝑇𝑎𝑟𝑔𝑒𝑡 − 𝐷1𝑃𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑) (C.5) 𝐹2𝑛+1 = 𝐹2𝑛 + 𝑖 ∗ (𝐷2𝑇𝑎𝑟𝑔𝑒𝑡 − 𝐷2𝑃𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑) (C.6) This process is effective due to the fact that if the predicted force is too large, the difference will be negative, which reduces the assumed force, and if the predicted displacement is too small the next assumed force will be larger. The variable is used to control the size of the increase to the assumed force, in order to allow the program to converge upon the correct answer more quickly. When the difference between the target and assumed force is large, 𝑖 is equal to 2, which allows the program to close the gap between the predicted and target displacement efficiently. However, as the predicted displacement gets close to the target displacement the program can get stuck in a loop, especially after yielding in the steel when the displacement is much more sensitive to an increase in the force and will not converge. It was found that reducing the size of the incremental force helped the program converge upon the target solution. Hence, the 𝑖 variable was used to achieve this, and after set numbers of iterations ,𝑖 , can be reduced smaller and smaller as desired. 343 However, what makes the process complicated, is that having both forces change in the same iteration did not work well, therefore in any given iteration either only the top, F2, or mid-height force, F1, is changed. If neither the top nor the mid-height displacement are within the tolerance limit, then the forces take turns going back and forth each iteration. However, sometimes the top displacement will be within the tolerance and the mid-height displacement will not be. In this case only the mid-height force will update, and the top force will be left alone, until its respective displacement is outside of the tolerance limit. Once the flexural displacement is calculated, the shear displacement is then determined based on the forces used to calculate the flexural displacement. While the flexural displacement is more complicated to calculate based on the curvature gradient along the specimen, the shear stiffness is much easier to determine based on the fact that each element is assumed to be subjected to a uniform shear stress and strain. The displacement at the mid-height, and the displacement at the top, are calculated using Eqs. C.7 and C.8 respectively. 𝐷1𝑣 =𝑉𝐵𝐿2𝐺𝐴𝑣 (C.7) 𝐷2𝑣 =𝑉𝑇𝐿2𝐺𝐴𝑣+𝑉𝐵𝐿2𝐺𝐴𝑣 (C.8) Where 𝐷1𝑣 and 𝐷2𝑣 are the shear displacements at the mid-height and top respectively, 𝑉𝐵 and 𝑉𝑇 are the shear forces in the bottom and top element respectively, 𝐿 is the length of the entire specimen, 𝐴𝑣 is the shear area of the specimen, and 𝐺 is the shear modulus which is determined based on a defined shear stress-strain relationship. Each time-step is concluded with one of two possible outcomes, the first is that the mid-height and top displacement are both within the tolerance limit, and the other is that the total iteration 344 limit is reached. The iteration limit serves to limit the amount of time the program takes, and prevent infinite loops. A flowchart explaining the general algorithm of how the program determines the forces and displacements can be seen in Figure C.2. Figure C.2: Prediction program flowchart of process. Assume initial forces at top and mid-height Determine resulting curvature profile from bending moments Integrate curvature profile and add shear displacement Top displacement within tolerance? No Yes Mid-height displacement within tolerance? Yes No Advance to next displacement target Adjust forces at top and mid-height 345 D. – Test Setup Design Drawings D.1 Specimen and Foundation Formwork Figure D.1: Precast speciment formwork – top view of bottom form (dimensions in mm). 346 Figure E.2: Precast speciment formwork – top view of upper form (dimensions in mm). Figure D.3: Precast speciment formwork – side view (dimensions in mm). 347 Figure D.4: Segmental foundation formwork – top view (dimensions in mm). D.2 Specimen and Foundation Reinforcement Detail Figure D.5: Reinforcing detail of segmental founation – top view (dimensions in mm). 348 (a) (b) Figure D.6: Reinforcing detail of segmental founation – end view: (a) no box; (b) side of box (dimensions in mm). (a) (b) Figure D.7: Reinforcing detail of segmental founation – side view: (a) no box; (b) side of box (dimensions in mm). 349 Figure D.8: Reinforcing detail along height of specimen. 350 Figure D.9: Reinforcing detail of specimen cross-section. 351 D.3 Setup Components Figure D.10: Test setup design – side view (dimensions in mm). Figure D.11: Test setup design – top view (dimensions in mm). 352 E. – Summary of Experimental Tests This Appendix presents the results and observations of each experimental specimen that was tested. There were five specimens in total, which were tested under several load stages that progressively increase until the specimen strength degraded. While each specimen was constructed identically, the way they were tested varies. Each summary includes the peak recorded displacement and force at the mid-height and top of the Specimen for of each Load Stage, data relating to the crack patterns at the end of each Load Stage, measured crack widths, the load displacement relationship at the end of each Load Stage. E.1 Summary of Specimen 1 The tests were completed in several load stages using half cycles, i.e. the specimen was loaded to a specified displacement, and then unloaded. Cycles denoted “A” are when the mid-height actuator is pushing on the specimen, while Cycles denoted “B”, are when the mid-height actuator is pulling on the specimen. While the displacement protocol is controlled based on the displacement in the actuator, there is a discrepency between this displacement and the actual movement of the specimen as shown in Table E.1. In load stages 1, 2, and 3-1A, the specimen displacement was recorded using string-pots, which were replaced with LVDT’s, in order to provide more reliable data collection. 353 Table E.1: Specimen 1 peak displacement test data. Load Stage Cycle Mid-Height Actuator Displacement (mm) Mid-Height Specimen Displacement (mm) Top Specimen Displacement (mm) Mid-Height Force (kN) Top Force (kN) 1 A1 + 1.150 + 0.249 - 0.105 + 31.42 - 13.25 A2 + 0.341 - 0.036 + 29.58 - 12.05 B1 - 1.150 - 0.374 + 0.043 - 34.13 + 11.71 B2 - 0.279 + 0.008 - 34.17 + 12.06 B3 - 0.354 - 0.040 - 35.55 + 12.66 B4 - 0.384 + 0.048 - 36.89 + 12.84 2 A1 + 1.500 + 0.344 - 0.220 + 37.95 - 15.71 B1 - 1.500 - 0.421 + 0.034 - 42.21 + 15.02 3 A1 + 2.000 + 0.697 - 0.374 + 49.01 - 20.42 A2 + 0.840 - 0.377 + 48.12 - 19.45 B1 - 2.000 - 0.980 + 0.153 - 51.69 + 19.12 B2 - 0.990 + 0.208 - 51.71 + 19.48 4 A1 + 3.000 + 1.490 - 0.540 + 67.95 - 27.52 A2 + 1.510 - 0.673 + 63.06 - 25.24 B1 - 3.000 - 1.710 + 0.387 - 63.18 + 24.49 B2 - 1.720 + 0.378 - 66.18 + 25.47 5 A1 + 4.000 + 2.228 - 0.894 + 82.77 - 34.39 B1 - 4.000 - 2.370 + 0.620 - 78.54 + 31.09 6 A1 + 6.000 + 3.700 - 1.640 + 109.08 - 46.72 A2* + 3.480 - 1.520 + 97.59 - 42.39 A3* + 3.600 - 1.630 + 107.67 - 48.03 A4 + 3.680 - 1.640 + 107.85 - 49.01 B1 - 6.000 - 3.800 + 1.260 - 96.82 + 39.22 B2 - 3.620 + 1.190 - 92.89 + 35.96 7 A1 +8.000 + 5.200 - 2.350 + 123.32 - 57.29 A2 + 5.450 - 2.570 + 106.66 - 54.84 B1 - 8.000 - 5.360 + 1.650 - 111.01 + 44.82 B2 - 5.320 + 1.670 - 107.20 + 43.57 8 A1 + 10.000 + 7.130 - 3.140 + 113.45 - 61.68 A2 + 7.370 - 3.230 + 89.39 - 56.14 A3 + 7.470 - 3.310 + 82.40 - 54.46 B1 - 10.000 - 7.150 + 2.570 - 102.40 + 47.52 B2 - 7.320 + 2.590 - 90.96 + 47.55 * Test ended abruptly prior to completion due to hydraulic system failure. 354 E.1.1 Specimen 1 Crack Profile and Measurements Figure E.1: S1 – LS3 crack profile. 355 Figure E.2: S1 – LS4 crack profile. 356 Figure E.3: S1 – LS5 crack profile. 357 Figure E.4: S1 – LS6 crack profile. 358 Figure E.5: S1 – LS7 crack profile. 359 Figure E.6: S1 – LS8 crack profile. 360 Figure E.7: Specimen 1 crack labels. 1A 2A 3A 4A 1B 2B 3B 4B 5A 6A 7A 8A 5B 6B 7B 8B 9B 9A 361 Table E.2: Specimen 1 crack width measurements. Crack Width (mm) LS1/2 LS3 LS4 LS5 LS6 LS7 LS8 1A - - - - 0.15 0.30 0.30 2A - - 0.05 0.2 0.30 0.60 0.80 3A - < 0.05 0.05 0.2 0.50 0.80 1.00 4A - < 0.05 0.05 0.05 0.05 0.15 0.10 5A - < 0.05 0.2 0.35 0.80 1.00 1.00 6A - < 0.05 0.2 0.35 0.95 1.25 1.30 7A - - - 0.2 0.70 1.00 1.50 8A - - - 0.2 0.25 0.30 0.40 9A - - - 0.2 0.15 0.30 0.25 1B - - - - 0.30 0.15 0.60 2B - < 0.05 0.05 0.15 0.10 0.50 0.15 3B - - - 0.15 0.20 0.15 0.30 4B - < 0.05 0.05 0.15 0.25 0.33 0.30 5B - - 0.2 0.3 0.20 0.30 0.15 6B - - 0.25 0.3 0.80 1.00 1.50 7B - - - 0.25 0.55 1.25 1.25 8B - - - - 0.35 1.25 1.50 9B - - < 0.05 0.05 0.15 1.00 0.60 Table E.2 shows the measured crack widths for labelled cracks according to Figure E.7. Only major diagonal cracks were labelled and measured consistently. There is no photo or measurements for load stages 1 and 2, as no cracking had occurred, and the specimen was still in the linear elastic range. Crack widths were estimated using a crack comparator and measured as close to the middle of the web as possible. 362 E.1.2 Specimen 1 Load-Displacement Relationship LS2 (a) -20-15-10-505101520-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5Vb (kN)D1 (mm)(Corrected) 363 (b) Figure E.8: S1 – LS2 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. As load stages 1 and 2 were both located in the linear elastic range, they were included together in the figures above. The original specimen displacement data for this load stage was recorded using string-pots attached to the specimen, which did not provide reliable data. To correct this the data above was manipulated slightly in order to represent the results of the test more accurately. The data above are the manipulated results. -10-8-6-4-20246810-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5Mm (kN-m)D1 (mm)(Corrected) 364 LS3 (a) (b) Figure E.9: S1 – LS3 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -40-30-20-10010203040-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Vb (kN)D1 (mm)-20-15-10-505101520-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Mm (kN-m)D1 (mm)365 LS4 (a) (b) Figure E.10: S1 – LS4 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -50-40-30-20-1001020304050-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Vb (kN)D1 (mm)-25-20-15-10-50510152025-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Mm (kN-m)D1 (mm)366 LS5 (a) (b) Figure E.11: S1 – LS5 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -60-40-200204060-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5Vb (kN)D1 (mm)-40-30-20-100102030-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5Mm (kN-m)D1 (mm)367 LS6 (a) (b) Figure E.12: S1 – LS6 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -80-60-40-20020406080-4 -3 -2 -1 0 1 2 3 4Vb (kN)D1 (mm)-50-40-30-20-10010203040-4 -3 -2 -1 0 1 2 3 4Mm (kN-m)D1 (mm)368 LS7 (a) (b) Figure E.13: S1 – LS7 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -80-60-40-20020406080-6 -4 -2 0 2 4 6Vb (kN)D1 (mm)-60-50-40-30-20-1001020304050-6 -4 -2 0 2 4 6Mm (kN-m)D1 (mm)369 LS8 (a) (b) Figure E.14: S1 – LS8 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -80-60-40-20020406080-8 -6 -4 -2 0 2 4 6 8Vb (kN)D1 (mm)-60-50-40-30-20-1001020304050-8 -6 -4 -2 0 2 4 6 8Mm (kN-m)D1 (mm)370 E.1.3 Specimen 1 Additional Observations (a) (b) Figure E.15: Specimen 1 comparison of observed specimen displacement vs target actuator displacement for: (a) Mid-height displacement; (b) Top displacement. y = 0.6848xR² = 0.9902-10-7.5-5-2.502.557.510-10 -7.5 -5 -2.5 0 2.5 5 7.5 10Mid-Height Specimen Displacement (mm)Mid-Height Actuator Displacement (mm)y = 0.7662xR² = 0.9561-4-3-2-101234-4 -3 -2 -1 0 1 2 3 4Top Specimen Displacement (mm)Top Actuator Displacement (mm)371 E.2 Summary of Specimen 2 The test was completed in several load stages using half cycles, i.e. the specimen was loaded to a specified displacement, and then unloaded. Cycles denoted “A” are when the mid-height actuator is pushing on the specimen, while Cycles denoted “B”, are when the mid-height actuator is pulling on the specimen. While the displacement protocol is controlled based on the displacement in the actuator, there is a discrepency between this displacement and the actual movement of the specimen as shown in Table E.3. All specimen displacements were measured using LVDT’s. The protocol was imposed dynamically, with the length of each cycle of Load Stage 1 through 6 being 4 seconds, long, while cycles in Load Stage 7 are 6 seconds long, and cycles in Load Stage 8 are 8 seconds long. 372 Table E.3: Specimen 2 peak displacement test data. Load Stage Cycle Mid-Height Actuator Displacement (mm) Mid-Height Specimen Displacement (mm) Top Specimen Displacement (mm) Mid-Height Force (kN) Top Force (kN) 1 A1* + 1.00 - - - A2 + 0.42 - 0.06 + 30.72 - 14.24 B1* - 1.00 - - - - B2 - 0.52 - 0.39 - 25.55 + 6.65 2 A1 + 1.50 + 0.72 - 0.01 + 43.13 - 18.61 A2 + 0.73 - 0.06 + 41.00 - 17.76 B1 - 1.50 - 0.73 - 0.32 - 34.99 + 10.58 B2 - 0.70 - 0.22 - 37.14 + 11.55 3 A1 + 2.00 + 0.97 - 0.26 + 49.94 - 21.42 A2 + 1.04 - 0.27 + 48.89 - 22.15 B1 - 2.00 - 1.02 - 0.25 - 42.71 + 14.29 B2 - 1.00 - 0.12 - 45.44 + 14.48 4 A1 + 3.00 + 1.67 - 0.60 + 67.1 - 28.82 A2 + 1.59 - 0.66 + 63.68 - 27.57 B1 - 3.00 - 1.59 + 0.20 - 59.23 + 20.90 B2 - 1.62 - 0.21 - 57.72 + 19.70 5 A1 + 4.00 + 2.06 - 0.96 + 80.79 - 35.02 A2 + 2.20 - 0.96 + 76.69 - 33.49 B1 - 4.00 - 2.18 + 0.33 - 73.1 + 25.39 B2 - 2.24 - 0.12 - 71.54 + 24.46 6 A1 + 6.00 + 3.41 - 1.73 + 109.02 - 46.91 A2 + 3.42 - 1.76 + 103.93 - 45.18 B1 - 6.00 - 3.47 + 0.93 - 95.54 + 35.41 B2 - 3.49 + 1.09 - 93.15 + 34.82 7 A1 + 8.00 + 4.86 - 2.54 + 124.78 - 55.62 A2 + 4.93 - 2.57 + 117.33 - 53.47 B1 - 8.00 - 4.96 + 1.85 - 111.79 + 43.89 B2 - 4.93 - 1.84 - 111.21 + 44.18 8 A1 + 10.00 + 6.57 - 3.47 + 127.65 - 60.81 A2 + 6.92 - 3.42 + 108.76 - 57.15 A3 + 7.09 - 3.41 + 97.40 - 54.87 B1 - 10.00 - 6.81 + 2.80 - 106.59 + 49.17 B2 - 7.22 + 2.76 - 89.62 + 46.40 * Data not recorded 373 E.2.1 Specimen 2 Crack Profile and Measurements Figure E.16: S2 – LS4 crack profile. 374 Figure E.17: S2 – LS5 crack profile. 375 Figure E.18: S2 – LS6 crack profile. 376 Figure E.19: S2 – LS7 crack profile. 377 Figure E.20: S2 – LS8 crack profile. 378 Figure E.21: Specimen 2 crack labels. 2A 1A 3A 2B 3B 4A 1B 4B 379 Table E.4: Specimen 2 crack width measurements. Crack Residual Width (mm) LS8 1A 0.60 2A 1.00 3A 0.15 4A 0.15 1B 0.15 2B 0.30 3B 0.60 4B 0.30 Table E.4 shows the measured crack widths for labelled cracks according to Figure E.21. Only major diagonal cracks were labelled and measured consistently. Due to the dynamic protocol, it was not possible to measure crack widths at the time of the maximum displacement. Residual crack widths were measured at the end of the test. Crack widths were estimated using a crack comparator and measured as close to the middle of the web as possible. 380 E.2.2 Specimen 2 Load-Displacement Relationship LS2 (a) (b) Figure E.22: S2 – LS2 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -30-20-100102030-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Vb (kN)D1 (mm)-20-15-10-5051015-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Mm (kN-m)D1 (mm)381 LS3 (a) (b) Figure E.23: S2 – LS3 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -40-30-20-10010203040-1.5 -1 -0.5 0 0.5 1 1.5Vb (kN)D1 (mm)-20-15-10-5051015-1.5 -1 -0.5 0 0.5 1 1.5Mm (kN-m)D1 (mm)382 LS4 (a) (b) Figure E.24: S2 – LS4 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -50-40-30-20-1001020304050-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Vb (kN)D1 (mm)-30-25-20-15-10-505101520-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Mm (kN-m)D1 (mm)383 LS5 (a) (b) Figure E.25: S2 – LS5 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -60-40-200204060-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5Vb (kN)D1 (mm)-40-30-20-100102030-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5Mm (kN-m)D1 (mm)384 LS6 (a) (b) Figure E.26: S2 – LS6 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -80-60-40-20020406080-4 -3 -2 -1 0 1 2 3 4Vb (kN)D1 (mm)-50-40-30-20-10010203040-4 -3 -2 -1 0 1 2 3 4Mm (kN-m)D1 (mm)385 LS7 (a) (b) Figure E.27: S2 – LS7 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -80-60-40-20020406080-6 -4 -2 0 2 4 6Vb (kN)D1 (mm)-50-40-30-20-10010203040-6 -4 -2 0 2 4 6Mm (kN-m)D1 (mm)386 LS8 (a) (b) Figure E.28: S2 – LS8 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -80-60-40-20020406080-8 -6 -4 -2 0 2 4 6 8Vb (kN)D1 (mm)-60-50-40-30-20-1001020304050-8 -6 -4 -2 0 2 4 6 8Mm (kN-m)D1 (mm)387 E.2.3 Specimen 2 Additional Observations (a) (b) Figure E.29: Specimen 2 comparison of observed specimen displacement vs target actuator displacement for: (a) Mid-height displacement; (b) Top displacement. y = 0.6639xR² = 0.9819-10-7.5-5-2.502.557.510-10 -7.5 -5 -2.5 0 2.5 5 7.5 10Mid-Height Specimen Displacement (mm)Mid-Height Actuator Displacement (mm)y = 0.8507xR² = 0.9258-4-3-2-101234-4 -3 -2 -1 0 1 2 3 4Top Specimen Displacement (mm)Top Actuator Displacement (mm)388 E.3 Summary of Specimen 3 The test was completed in several load stages using half cycles, i.e. the specimen was loaded to a specified displacement, and then unloaded. Cycles denoted “A” are when the mid-height actuator is pushing on the specimen, while Cycles denoted “B”, are when the mid-height actuator is pulling on the specimen. This test was completed in two phases, in the first phase only the top actuator was used, in order to apply large bending moments at the base of the specimen. This moment was applied through several load stages, with the purpose of damaging the specimen until the point where the flexural steel begins to yield. Load Stages during Phase 1 are denoted “F”. During the second phase of the test, the mid-height actuator was connected to the specimen to apply the second mode displacement protocol that was applied in the first two tests. Table E.5: Phase 1 - Specimen 3 peak displacement test data. Load Stage Cycle Top Actuator Displacement (mm) Mid-Height Specimen Displacement (mm) Top Specimen Displacement (mm) Top Force (kN) 1 A1 + 4.00 + 0.78 + 3.31 + 8.80 B1 - 4.00 - 0.76 - 4.57 - 7.77 2 A1 + 8.00 + 0.88 + 7.71 + 15.26 B1 - 8.00 - 1.66 - 8.85 - 14.29 3 A1 + 12.00 + 4.71 + 11.32 + 19.01 B1 - 12.00 - 5.06 - 12.31 - 21.11 4 A1 + 16.00 + 6.47 + 15.22 + 23.19 B1 - 16.00 - 6.85 - 16.06 - 25.58 5 A1 + 20.00 + 8.18 + 19.09 + 26.16 B1 - 20.00 - 8.67 - 20.40 - 29.24 6 A1 + 24.00 + 9.95 + 22.43 + 28.00 B1* - 24.00 - - - 7 A1 + 30.00 + 12.92 + 28.56 + 29.39 B1 - 30.00 - 13.49 - 28.89 - 30.89 * Data not recorded for this half-cycle. 389 Table E.6: Phase 2 - Specimen 3 peak displacement test data. Load Stage Cycle Mid-Height Actuator Displacement (mm) Mid-Height Specimen Displacement (mm) Top Specimen Displacement (mm) Mid-Height Force (kN) Top Force (kN) 1 A1 + 1.00 + 0.65 - 0.12 + 13.89 - 7.11 B1 - 1.00 - 0.54 + 0.19 - 9.12 + 3.78 2 A1 + 1.50 + 0.87 - 0.26 + 23.56 - 11.44 B1 - 1.50 - 0.78 + 0.47 - 19.80 + 8.55 3 A1 + 2.00 + 1.08 - 0.57 + 31.55 - 15.07 B1 - 2.00 - 1.11 + 0.25 - 26.39 + 11.23 4 A1 + 3.00 + 1.75 - 0.65 + 47.66 - 22.42 B1 - 3.00 - 1.75 + 0.38 - 43.57 + 18.54 5 A1 + 4.00 + 2.28 - 1.03 + 61.57 - 29.34 A2 + 2.25 - 1.00 + 64.96 - 30.68 B1 - 4.00 - 2.20 + 0.69 - 56.35 + 23.31 B2 - 2.28 + 0.54 - 60.45 + 24.94 6 A1 + 6.00 + 3.41 - 1.60 + 91.08 - 42.28 A2 + 3.45 - 1.62 + 86.39 - 40.43 B1 - 6.00 - 3.60 + 1.06 - 84.72 + 35.47 B2 - 3.50 + 1.06 - 85.50 + 35.86 7 A1 + 8.00 + 4.64 - 2.28 + 106.06 - 49.78 A2 + 4.67 - 2.24 + 100.16 - 47.57 B1 - 8.00 - 5.01 + 1.62 - 102.57 + 43.59 B2 - 5.00 + 1.61 - 99.52 + 42.79 8 A1 + 10.00 + 5.71 - 2.95 + 112.78 - 54.02 A2 + 5.79 - 2.93 + 104.61 - 51.26 A3 + 5.83 - 2.87 + 101.60 - 50.24 B1 - 10.00 - 6.48 + 2.15 - 113.06 + 49.28 B2 - 6.43 + 2.23 - 110.19 + 48.46 B3 - 6.39 + 2.19 - 108.44 + 47.92 9 A1 + 12.00 + 6.13 - 3.73 + 109.51 - 54.75 A2 + 7.43 - 4.12 + 101.80 - 54.07 B1 - 12.00 - 7.91 + 2.57 - 111.95 + 51.20 B2 - 8.10 + 2.66 - 96.40 + 48.04 390 E.3.1 Specimen 3 Crack Profile and Measurements Figure E.30: S3 – F3 crack profile. 391 Figure E.31: S3 – F7 crack profile. 392 Figure E.32: S3 – LS5 crack profile. 393 Figure E.33: S3 – LS6 crack profile. 394 Figure E.34: S3 – LS7 crack profile. 395 Figure E.35: S3 – LS8 crack profile. 396 Figure E.36: S3 – LS9 crack profile. 397 Figure E.37: Specimen 3 crack labels. 1A 2A 3A 4A 1B 2B 3B 4B 5A 6A 7A 8A 6B 7B 8B 9B 10B 9A 10A 11B 5B 398 Table E.7: Specimen 3 crack width measurements. Crack Width (mm) LS1/2 LS3 LS4 LS5 LS6 LS7 LS8 1A - - - - - 0.20 0.60 2A 0.10 0.20 0.25 0.30 0.35 0.40 0.60 3A - - - - - 0.20 0.35 4A 0.10 0.10 0.15 0.15 0.25 0.25 0.30 5A - - - - 0.10 0.15 0.30 6A - 0.10 0.25 0.30 0.30 0.60 0.80 7A 0.10 0.15 0.25 0.30 0.45 0.80 1.15 8A - - - 0.15 0.30 1.00 1.25 9A 0.20 0.25 0.30 0.40 0.45 1.25 1.25 10A 0.30 0.35 0.35 0.50 0.40 0.60 0.60 1B 0.05 0.05 0.10 0.10 0.15 0.25 0.35 2B 0.05 0.10 0.15 0.20 0.20 0.10 0.35 3B <0.05 0.05 0.15 0.25 0.30 0.40 0.50 4B - - - - 0.15 0.15 0.20 5B 0.15 0.15 0.15 0.15 0.15 0.15 0.15 6B - - - - 0.10 0.30 0.35 7B 0.10 0.30 0.30 0.40 0.50 1.00 1.00 8B - - - - - - 0.10 9B - - - 0.10 0.25 0.80 1.25 10B - - - 0.10 0.35 0.60 0.80 11B 0.30 0.30 0.30 0.35 0.35 0.35 0.30 Table E.7 shows the measured crack widths for labelled cracks according to Figure E.37. Crack widths were estimated using a crack comparator and measured as close to the middle of the web as possible. Cracks measured during Load Stage 1 and 2 were formed during Phase 1, and present at the beginning of Phase 2. 399 E.3.2 Specimen 3 Load-Displacement Relationship Phase 1 (a) (b) Figure E.38: S3 – F7 Phase 1 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Base moment (Mb) vs top displacement (D2). -40-30-20-10010203040-15 -10 -5 0 5 10 15Vb (kN)D1 (mm)-60-40-200204060-40 -30 -20 -10 0 10 20 30 40Mb (kN-m)D2 (mm)400 LS3 (a) (b) Figure E.39: S3 – LS3 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -20-15-10-505101520-1.5 -1 -0.5 0 0.5 1 1.5Vb (kN)D1 (mm)-15-10-5051015-1.5 -1 -0.5 0 0.5 1 1.5Mm (kN-m)D1 (mm)401 LS4 (a) (b) Figure E.40: S3 – LS4 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -30-20-100102030-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Vb (kN)D1 (mm)-25-20-15-10-505101520-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Mm (kN-m)D1 (mm)402 LS5 (a) (b) Figure E.41: S3 – LS5 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -50-40-30-20-10010203040-3 -2 -1 0 1 2 3Vb (kN)D1 (mm)-30-20-100102030-3 -2 -1 0 1 2 3Mm (kN-m)D1 (mm)403 LS6 (a) (b) Figure E.42: S3 – LS6 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -60-40-200204060-4 -3 -2 -1 0 1 2 3 4Vb (kN)D1 (mm)-40-30-20-10010203040-4 -3 -2 -1 0 1 2 3 4Mm (kN-m)D1 (mm)404 LS7 (a) (b) Figure E.43: S3 – LS7 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -80-60-40-20020406080-6 -4 -2 0 2 4 6Vb (kN)D1 (mm)-50-40-30-20-1001020304050-6 -4 -2 0 2 4 6Mm (kN-m)D1 (mm)405 LS8 (a) (b) Figure E.44: S3 – LS8 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -80-60-40-20020406080-8 -6 -4 -2 0 2 4 6 8Vb (kN)D1 (mm)-50-40-30-20-1001020304050-8 -6 -4 -2 0 2 4 6 8Mm (kN-m)D1 (mm)406 LS9 (a) (b) Figure E.45: S3 – LS9 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -80-60-40-20020406080-10 -8 -6 -4 -2 0 2 4 6 8 10Vb (kN)D1 (mm)-60-50-40-30-20-1001020304050-10 -8 -6 -4 -2 0 2 4 6 8 10Mm (kN-m)D1 (mm)407 E.3.3 Specimen 3 Displacement Coordinate Results (a) (b) Figure E.46: S3 – Selected Load Stages for displaced shape analysis: (a) Phase 1; (b) Phase 2. -40-30-20-10010203040-15 -10 -5 0 5 10 15Vb (kN)D1 (mm)F7BF4BF7AF4A-80-60-40-20020406080-10 -8 -6 -4 -2 0 2 4 6 8 10Vb (kN)D1 (mm)L4AL9B2L8B2L6BL4BL9A2L8A2L6A408 (a) (b) Figure E.47: S3 – Total displacement - “A” Load Stages: (a) Phase 1; (b) Phase 2. 020040060080010001200140016001800-30 -20 -10 0Height (mm)Total Displacement (mm)3-F4A3-F7A - LVDT Measurement - Calculated - Left Coordinates - Right Coordinates020040060080010001200140016001800-20 -15 -10 -5 0 5 10Height (mm)Total Displacement (mm)3-L4A13-L6A13-L8A23-L9A2409 (a) (b) Figure E.48: S3 – Flexural displacement – “A” Load Stages: (a) Phase 1; (b) Phase 2. 0100200300400500600700800-8 -6 -4 -2 0Height (mm)Flexural Displacement (mm)3-F4A3-F7A01002003004005006007008000 1 2 3 4Height (mm)Flexural Displacement (mm)3-L4A13-L6A13-L8A23-L9A2410 (a) (b) Figure E.49: S3 – Curvature distribution – “A” Load Stages: (a) Phase 1; (b) Phase 2. 0100200300400500600700800-60 -40 -20 0Height (mm)Curvature (rad/km)3-F4A3-F7A0100200300400500600700800-150 -100 -50 0 50 100Height (mm)Curvature (rad/km)3-L4A13-L6A13-L8A23-L9A2411 (a) (b) Figure E.50: S3 – Shear ductility demands for Phase 1 and 2 – “A” Load Stages: (a) Shear displacement; (b) Shear strain. 0100200300400500600700800-15 -10 -5 0 5Height (mm)Shear Displacement (mm)3-F4A3-F7A3-L4A13-L6A13-L8A23-L9A20100200300400500600700800-4 -3 -2 -1 0 1Height (mm)Shear Strain (%)3-F4A3-F7A3-L4A13-L6A13-L8A23-L9A2412 (a) (b) Figure E.51: S3 – Horizontal ductility demands for Phase 1 and 2 – “A” Load Stages: (a) Horizontal displacement; (b) Horizontal strain. 0100200300400500600700800-3 -2 -1 0 1 2Height (mm)Horizontal Displacement (mm)3-F4A3-F7A3-L4A13-L6A13-L8A23-L9A20100200300400500600700800-1.5 -1 -0.5 0 0.5 1Height (mm)Horizontal Strain (%)3-F4A3-F7A3-L4A13-L6A13-L8A23-L9A2413 (a) (b) Figure E.52: S3 – Total displacement - “B” Load Stages: (a) Phase 1; (b) Phase 2. 0200400600800100012001400160018000 10 20 30 40Height (mm)Total Displacement (mm)3-F4B3-F7B - LVDT Measurement - Calculated - Left Coordinates - Right Coordinates020040060080010001200140016001800-5 0 5 10 15Height (mm)Total Displacement (mm)3-L4B13-L6B13-L8B23-L9B2414 (a) (b) Figure E.53: S3 – Flexural displacement – “B” Load Stages: (a) Phase 1; (b) Phase 2. 01002003004005006007008000 2 4 6 8 10Height (mm)Flexural Displacement (mm)3-F4B3-F7B0100200300400500600700800-1.5 -1 -0.5 0Height (mm)Flexural Displacement (mm)3-L4B13-L6B13-L8B23-L9B2415 (a) (b) Figure E.54: S3 – Curvature distribution – “B” Load Stages: (a) Phase 1; (b) Phase 2. 01002003004005006007008000 20 40 60Height (mm)Curvature (rad/km)3-F4B3-F7B0100200300400500600700800-40 -20 0 20 40Height (mm)Curvature (rad/km)3-L4B13-L6B13-L8B23-L9B2416 (a) (b) Figure E.55: S3 – Shear ductility demands for Phase 1 and 2 – “B” Load Stages: (a) Shear displacement; (b) Shear strain. 01002003004005006007008000 5 10 15Height (mm)Shear Displacement (mm)3-F4B3-F7B3-L4B13-L6B13-L8B23-L9B201002003004005006007008000 0.5 1 1.5 2 2.5Height (mm)Shear Strain (%)3-F4B3-F7B3-L4B13-L6B13-L8B23-L9B2417 (a) (b) Figure E.56: S3 – Horizontal ductility demands for Phase 1 and 2 – “B” Load Stages: (a) Horizontal displacement; (b) Horizontal strain. 0100200300400500600700800-1 -0.5 0 0.5 1Height (mm)Horizontal Displacement (mm)3-F4B3-F7B3-L4B13-L6B13-L8B23-L9B20100200300400500600700800-0.4 -0.2 0 0.2 0.4 0.6Height (mm)Horizontal Strain (%)3-F4B3-F7B3-L4B13-L6B13-L8B23-L9B2418 E.3.4 Specimen 3 Additional Observations (a) (b) Figure E.57: Comparison of Specimen 5, Phase 1 and 2 Load Displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -80-60-40-20020406080-15 -10 -5 0 5 10 15Vb (kN)D1 (mm)Phase 1Phase 2-60-50-40-30-20-1001020304050-15 -10 -5 0 5 10 15Mm (kN-m)D1 (mm)Phase 1Phase 2419 (a) (b) Figure E.58: Specimen 3 comparison of observed specimen displacement vs target actuator displacement for: (a) Mid-height displacement; (b) Top displacement. y = 0.5965xR² = 0.9943-14-11.2-8.4-5.6-2.802.85.68.411.214-14 -11.2 -8.4 -5.6 -2.8 0 2.8 5.6 8.4 11.2 14Mid-Height Specimen Displacement (mm)Mid-Height Actuator Displacement (mm)y = 0.7202xR² = 0.9709-5-4-3-2-1012345-5 -4 -3 -2 -1 0 1 2 3 4 5Top Specimen Displacement (mm)Top Actuator Displacement (mm)420 E.4 Summary of Specimen 4 The test was completed in several load stages using half cycles, i.e. the specimen was loaded to a specified displacement, and then unloaded. Cycles denoted “A” are when the mid-height actuator is pushing on the specimen, while Cycles denoted “B”, are when the mid-height actuator is pulling on the specimen. This test was completed in two phases, in the first phase only the top actuator was used, in order to apply displacement at the top of the specimen. This was applied through several load stages, with the purpose of damaging the specimen until the point where the flexural steel begins to yield. Load Stages during Phase 1 are denoted “F”. The amount of damage applied is equal to that applied in Test 3. During the second phase of the test, the specimen was held at a position of + 30 mm top displacement (which corresponds with the yield moment), the mid-height actuator was then connected to the specimen to apply the second mode displacement protocol. Table E.8: Phase 1 - Specimen 4 peak displacement test data. Load Stage Cycle Top Actuator Displacement (mm) Mid-Height Specimen Displacement (mm) Top Specimen Displacement (mm) Top Force (kN) 1 A1 + 6.00 + 2.41 + 5.74 + 10.84 B1 - 6.00 - 2.51 - 6.06 - 11.99 2 A1 + 12.00 + 4.94 + 11.71 + 19.19 B1 - 12.00 - 5.12 - 12.28 - 20.63 3 A1 + 18.00 + 7.76 + 17.82 + 25.20 B1 - 18.00 - 8.12 - 18.26 - 27.67 4 A1 + 24.00 + 10.84 + 23.51 + 28.50 B1 - 24.00 - 11.44 - 24.30 - 29.66 5 A1 + 30.00 + 14.14 + 29.37 + 29.39 B1 - 30.00 - 14.69 - 30.40 - 30.62 421 Table E.9: Phase 2 - Specimen 4 peak displacement test data. Load Stage Cycle Mid-Height Actuator Displacement (mm) Mid-Height Specimen Displacement (mm) Top Specimen Displacement (mm) Mid-Height Force (kN) Top Force (kN) 1 A1 + 1.00 + 0.38 - 0.23 + 17.67 + 19.73 A2 + 0.34 - 0.25 + 16.17 + 19.66 B1 - 1.00 - 0.45 + 0.20 - 20.71 + 35.33 B2 - 0.45 + 0.21 - 21.54 + 35.33 2 A1 + 2.00 + 0.76 - 0.59 + 33.63 + 11.68 A2 + 0.79 - 0.49 + 34.20 + 10.83 B1 - 2.00 - 1.10 + 0.39 - 40.02 + 41.90 B2 - 1.05 + 0.39 - 40.30 + 41.69 3 A1 + 3.00 + 1.27 - 0.78 + 46.37 + 5.02 A2 + 1.25 - 0.74 + 45.96 + 4.42 B1 - 3.00 - 1.81 + 0.65 - 49.88 + 45.98 B2 - 1.81 + 0.66 - 49.27 + 45.32 4 A1 + 4.00 + 1.80 - 0.91 + 54.25 + 0.27 A2 + 1.77 - 0.86 + 52.97 - 0.19 B1 - 4.00 - 2.63 + 0.87 - 54.03 + 47.77 B2 - 2.59 + 0.86 - 55.52 + 47.64 5 A1 + 5.00 + 2.31 - 1.16 + 62.90 - 5.11 A2 + 2.31 - 1.14 + 61.14 - 5.15 B1 - 5.00 - 3.27 + 1.07 - 65.41 + 51.34 B2 - 3.28 + 1.09 - 63.65 + 50.20 6 A1 + 6.00 + 2.89 - 1.38 + 69.82 - 9.79 A2 + 2.90 - 1.31 + 67.95 - 9.86 B1 - 6.00 - 4.04 + 1.24 - 72.49 + 53.26 B2 - 4.04 + 1.21 - 70.71 + 52.13 7 A1 + 8.00 + 4.10 - 1.99 + 79.08 - 18.17 A2 + 4.33 - 1.65 + 74.80 - 18.36 B1 - 8.00 - 5.72 + 1.15 - 83.75 + 55.78 B2 - 5.85 + 0.97 - 78.38 + 53.33 8 A1 + 10.00 + 5.68 - 2.15 + 80.06 - 24.73 A2 + 6.18 - 1.53 + 74.56 - 23.84 A3 + 6.43 - 1.25 + 69.06 - 22.63 B1 - 10.00 - 7.69 + 0.19 - 84.91 + 54.85 B2 - 7.81 + 0.07 - 78.87 + 52.75 Summary of data values at peak load for Phase 2 of Test 4. 422 E.4.1 Specimen 4 Crack Profile and Measurements Figure E.59: S4 – F2 crack profile. 423 Figure E.60: S4 – F5 crack profile. 424 Figure E.61: S4 – LS4 crack profile. 425 Figure E.62: S4 – LS5 crack profile. 426 Figure E.63: S4 – LS7 crack profile. 427 Figure E.64: S4 – LS8 crack profile. 428 Figure E.65: Specimen 4 crack labels. 1A 2A 1B 2B 3B 4B 3A 4A 5A 6A 6B 7B 8B 7A 8A 5B 429 Table E.10: Specimen 4 crack width measurements. Crack Width (mm) LS1 LS2 LS3 LS4 LS5 LS6 LS7 1A 0.15 0.15 0.15 0.15 0.15 0.15 0.25 2A 0.05 0.05 0.05 0.05 0.05 0.05 0.50 3A 0.15 0.25 0.40 0.50 0.80 0.80 1.00 4A 0.10 0.25 0.35 0.50 0.60 0.80 1.25 5A 0.15 0.25 0.30 0.50 1.00 1.00 1.50 6A 0.25 0.35 0.60 0.90 1.15 1.25 1.30 7A 0.50 0.50 0.50 0.50 0.50 0.50 0.50 8A 0.15 0.10 0.20 0.20 0.20 0.20 0.20 1B - - - - 0.15 0.15 0.15 2B 0.24 0.25 0.60 0.80 0.80 1.00 1.00 3B 0.10 0.25 0.40 0.50 0.80 0.80 0.80 4B 0.05 0.05 0.10 0.20 0.10 0.10 0.10 5B 0.05 0.05 0.10 0.15 0.20 0.20 0.20 6B - - - - - - 0.50 7B 0.25 0.25 0.25 0.25 0.25 0.25 0.40 8B 0.25 0.25 0.25 0.25 0.25 0.25 0.80 Table E.10 shows the measured crack widths for labelled cracks according to Figure E.54. Crack widths were estimated using a crack comparator and measured as close to the middle of the web as possible. Cracks measured during Load Stage 1 and 2 were formed during Phase 1, and present at the beginning of Phase 2. 430 E.4.2 Specimen 4 Load-Displacement Relationship Phase 1 (a) (b) Figure E.66: S4 – F5 Phase 1 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Base moment (Mb) vs top displacement (D2). -40-30-20-10010203040-20 -15 -10 -5 0 5 10 15 20Vb (kN)D1 (mm)-60-40-200204060-40 -30 -20 -10 0 10 20 30 40Mb (kN-m)D2 (mm)431 LS2 (a) (b) Figure E.67: S4 – LS2 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. 05101520253035404550-1.5 -1 -0.5 0 0.5 1Vb (kN)D1 (mm)0510152025303540-1.5 -1 -0.5 0 0.5 1Mm (kN-m)D1 (mm)432 LS3 (a) (b) Figure E.68: S4 – LS3 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -100102030405060-2 -1.5 -1 -0.5 0 0.5 1 1.5Vb (kN)D1 (mm)0510152025303540-2 -1.5 -1 -0.5 0 0.5 1 1.5Mm (kN-m)D1 (mm)433 LS4 (a) (b) Figure E.69: S4 – LS4 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -20-10010203040506070-4 -3 -2 -1 0 1 2 3Vb (kN)D1 (mm)-5051015202530354045-4 -3 -2 -1 0 1 2 3Mm (kN-m)D1 (mm)434 LS5 (a) (b) Figure E.70: S4 – LS5 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -20-10010203040506070-4 -3 -2 -1 0 1 2 3Vb (kN)D1 (mm)-1001020304050-4 -3 -2 -1 0 1 2 3Mm (kN-m)D1 (mm)435 LS6 (a) (b) Figure E.71: S4 – LS6 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -30-20-10010203040506070-5 -4 -3 -2 -1 0 1 2 3 4Vb (kN)D1 (mm)-20-1001020304050-5 -4 -3 -2 -1 0 1 2 3 4Mm (kN-m)D1 (mm)436 LS7 (a) (b) Figure E.72: S4 – LS7 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -40-20020406080-8 -6 -4 -2 0 2 4 6Vb (kN)D1 (mm)-20-1001020304050-8 -6 -4 -2 0 2 4 6Mm (kN-m)D1 (mm)437 LS8 (a) (b) Figure E.73: S4 – LS8 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -40-20020406080-10 -8 -6 -4 -2 0 2 4 6 8Vb (kN)D1 (mm)-30-20-1001020304050-10 -8 -6 -4 -2 0 2 4 6 8Mm (kN-m)D1 (mm)438 E.4.3 Specimen 4 Displacement Coordinate Results (a) (b) Figure E.74: S4 – Selected Load Stages for displaced shape analysis: (a) Phase 1; (b) Phase 2. -40-30-20-10010203040-20 -15 -10 -5 0 5 10 15 20Vb (kN)D1 (mm)F5AF5B-40-20020406080-10 -8 -6 -4 -2 0 2 4 6 8Vb (kN)D1 (mm)L3AL6BL5BL3BL8A1L7A2L7A1L6AL5AL8B1L7BL8A2L8B2439 (a) (b) Figure E.75: S4 – Total displacement - “A” Load Stages: (a) Phase 1; (b) Phase 2. 020040060080010001200140016001800-40 -30 -20 -10 0Height (mm)Total Displacement (mm)4-F5A - LVDT Measurement - Calculated - Left Coordinates - Right Coordinates020040060080010001200140016001800-10 -5 0 5Height (mm)Total Displacement (mm)4-L3A4-L5A4-L6A4-L7A14-L7A24-L8A14-L8A2440 (a) (b) Figure E.76: S4 – Flexural displacement – “A” Load Stages: (a) Phase 1; (b) Phase 2. 0100200300400500600700800-10 -8 -6 -4 -2 0Height (mm)Flexural Displacement (mm)4-F5A0100200300400500600700800-1 -0.5 0 0.5 1 1.5Height (mm)Flexural Displacement (mm)4-L3A4-L5A4-L6A4-L7A14-L7A24-L8A14-L8A2441 (a) (b) Figure E.77: S4 – Curvature distribution – “A” Load Stages: (a) Phase 1; (b) Phase 2. 0100200300400500600700800-80 -60 -40 -20 0Height (mm)Curvature (rad/km)4-F5A0100200300400500600700800-60 -40 -20 0 20 40Height (mm)Curvature (rad/km)4-L3A4-L5A4-L6A4-L7A14-L7A24-L8A14-L8A2442 (a) (b) Figure E.78: S4 – Shear ductility demands for Phase 1 and 2 – “A” Load Stages: (a) Shear displacement; (b) Shear strain. 0100200300400500600700800-10 -8 -6 -4 -2 0Height (mm)Shear Displacement (mm)4-F5A4-L3A4-L5A4-L6A4-L7A14-L7A24-L8A14-L8A20100200300400500600700800-2 -1.5 -1 -0.5 0 0.5Height (mm)Shear Strain (%)4-F5A4-L3A4-L5A4-L6A4-L7A14-L7A24-L8A14-L8A2443 (a) (b) Figure E.79: S4 – Horizontal ductility demands for Phase 1 and 2 – “A” Load Stages: (a) Horizontal displacement; (b) Horizontal strain. 0100200300400500600700800-1 0 1 2 3Height (mm)Horizontal Displacement (mm)4-F5A4-L3A4-L5A4-L6A4-L7A14-L7A24-L8A14-L8A20100200300400500600700800-0.5 0 0.5 1 1.5Height (mm)Horizontal Strain (%)4-F5A4-L3A4-L5A4-L6A4-L7A14-L7A24-L8A14-L8A2444 (a) (b) Figure E.80: S4 – Total displacement - “B” Load Stages: (a) Phase 1; (b) Phase 2. 0200400600800100012001400160018000 10 20 30 40Height (mm)Total Displacement (mm)4-… - LVDT Measurement - Calculated - Left Coordinates - Right Coordinates020040060080010001200140016001800-5 0 5 10Height (mm)Total Displacement (mm)4-L3B4-L5B4-L6B4-L7B14-L8B14-L8B2445 (a) (b) Figure E.81: S4 – Flexural displacement – “B” Load Stages: (a) Phase 1; (b) Phase 2. 01002003004005006007008000 2 4 6 8Height (mm)Flexural Displacement (mm)4-F5B0100200300400500600700800-2 -1.5 -1 -0.5 0 0.5Height (mm)Flexural Displacement (mm)4-L3B4-L5B4-L6B4-L7B14-L8B14-L8B2446 (a) (b) Figure E.82: S4 – Curvature distribution – “B” Load Stages: (a) Phase 1; (b) Phase 2. 01002003004005006007008000 20 40 60Height (mm)Curvature (rad/km)4-F5B0100200300400500600700800-20 -15 -10 -5 0 5 10Height (mm)Curvature (rad/km)4-L3B4-L5B4-L6B4-L7B14-L8B14-L8B2447 (a) (b) Figure E.83: S4 – Shear ductility demands for Phase 1 and 2 – “B” Load Stages: (a) Shear displacement; (b) Shear strain. 01002003004005006007008000 2 4 6 8 10Height (mm)Shear Displacement (mm)4-F5B4-L3B4-L5B4-L6B4-L7B14-L8B14-L8B20100200300400500600700800-0.5 0 0.5 1 1.5 2Height (mm)Shear Strain (%)4-F5B4-L3B4-L5B4-L6B4-L7B14-L8B14-L8B2448 (a) (b) Figure E.84: S4 – Horizontal ductility demands for Phase 1 and 2 – “B” Load Stages: (a) Horizontal displacement; (b) Horizontal strain. 0100200300400500600700800-0.8 -0.6 -0.4 -0.2 0 0.2 0.4Height (mm)Horizontal Displacement (mm)4-F5B4-L3B4-L5B4-L6B4-L7B14-L8B14-L8B20100200300400500600700800-0.4 -0.2 0 0.2Height (mm)Horizontal Strain (%)4-F5B4-L3B4-L5B4-L6B4-L7B14-L8B14-L8B2449 E.4.4 Specimen 4 Additional Observations (a) (b) Figure E.85: Specimen 4 comparison of observed specimen displacement vs target actuator displacement for: (a) Mid-height displacement; (b) Top displacement. y = 0.646xR² = 0.9718-10-7.5-5-2.502.557.510-10 -7.5 -5 -2.5 0 2.5 5 7.5 10Mid-Height Specimen Displacement (mm)Mid-Height Actuator Displacement (mm)y = 0.4049xR² = 0.7299-4-3-2-101234-4 -3 -2 -1 0 1 2 3 4Top Specimen Displacement (mm)Top Actuator Displacement (mm)450 E.5 Summary of Specimen 5 The test was completed in several load stages using half cycles, i.e. the specimen was loaded to a specified displacement, and then unloaded. Cycles denoted “A” are when the mid-height actuator is pushing on the specimen, while Cycles denoted “B”, are when the mid-height actuator is pulling on the specimen. This test was completed in two phases, in the first phase only the top actuator was used, in order to impose displacements at the top of the specimen to maximize the bending moment at the base of the specimen. This moment was applied through several load stages, with the purpose of damaging the specimen until the point where the specimen forms a well defined plastic hinge at the base. Load Stages applied during Phase 1 are denoted “F”. During the second phase of the test, the specimen was held at its original position, the mid-height actuator was then connected to the specimen to apply the second mode displacement protocol. 451 Table E.11: Phase 1 - Specimen 5 peak displacement test data. Load Stage Cycle Top Actuator Displacement (mm) Mid-Height Specimen Displacement (mm) Top Specimen Displacement (mm) Top Force (kN) 1 A1 + 6.00 + 2.35 + 5.64 + 12.62 B1 - 6.00 - 2.47 - 6.09 - 10.25 2 A1 + 12.00 + 5.00 + 11.97 + 20.57 B1 - 12.00 - 5.03 - 12.12 - 19.16 3 A1 + 18.00 + 7.77 + 18.01 + 27.11 B1 - 18.00 - 7.95 - 18.52 - 26.13 4 A1 + 24.00 + 10.87 + 24.25 + 29.33 B1 - 24.00 - 11.33 - 24.84 - 29.48 5 A1 + 30.00 + 14.18 + 30.52 + 30.27 B1 - 30.00 - 14.71 - 31.13 - 30.59 6 A1 + 40.00 + 19.69 + 41.07 + 31.59 B1 - 40.00 - 20.00 - 41.15 - 31.71 7 A1 + 50.00 + 25.01 + 51.47 + 32.55 B1 - 50.00 - 25.54 - 52.79 - 33.87 8 A1 + 60.00 + 30.06 + 61.86 + 33.38 A2 + 30.05 + 61.87 + 33.39 A3 + 30.06 + 61.83 + 33.18 B1 - 60.00 - 30.88 - 63.46 - 35.43 B2 - 30.91 - 63.38 - 34.80 B3 - 31.01 - 63.32 - 34.68 452 Table E.12: Phase 2 - Specimen 5 peak displacement test data. Load Stage Cycle Mid-Height Actuator Displacement (mm) Mid-Height Specimen Displacement (mm) Top Specimen Displacement (mm) Mid-Height Force (kN) Top Force (kN) 1 A1 + 2.00 + 1.32 - 0.38 + 21.76 - 0.31 A2 + 1.20 - 0.35 + 20.72 - 0.33 B1 - 2.00 - 1.46 + 0.65 - 19.67 + 20.00 B2 - 1.45 + 0.68 - 19.67 + 19.86 2 A1 + 4.00 + 2.54 - 1.02 + 42.49 - 11.13 A2 + 2.44 - 1.04 + 41.19 - 11.77 B1 - 4.00 - 3.14 + 1.54 - 27.26 + 26.22 B2 - 3.13 + 1.52 - 27.18 + 26.07 3 A1 + 6.00 + 3.69 - 1.79 + 60.35 - 22.64 A2 + 3.75 - 1.80 + 57.47 - 23.12 B1 - 6.00 - 4.77 + 2.29 - 35.48 + 32.26 B2 - 4.77 + 2.25 - 35.18 + 31.91 4 A1 + 8.00 + 5.04 - 2.65 + 73.21 - 33.35 A2 + 5.16 - 2.71 + 65.91 - 31.82 A3 + 5.27 - 2.70 + 63.45 - 31.38 A4 + 5.36 - 2.70 + 60.77 - 30.70 B1 - 8.00 - 6.30 + 2.97 - 46.80 + 39.09 B2 - 6.29 + 2.92 - 46.33 + 38.56 B3 - 6.32 + 2.94 - 45.39 + 37.99 453 E.5.1 Specimen 5 Crack Profile and Measurements Figure E.86: S5 – F4 crack profile. 454 Figure E.87: S5 – F7 crack profile. 455 Figure E.88: S5 – LS2 crack profile. 456 Figure E.89: S5 – LS3 crack profile. 457 Figure E.90: S5 – LS4 crack profile. 458 Figure E.91: Specimen 5 crack labels. 1A 2A 1B 2B 3B 4B 3A 4A 5A 6A 6B 7B 8B 7A 8A 5B 459 Table E.13: Specimen 5 crack width measurements. Crack Width (mm) LS1 LS2 LS3 LS4 1A 0.05 0.10 0.20 0.25 2A 0.05 0.10 0.20 0.25 3A 0.05 0.10 0.20 0.20 4A 0.10 0.15 0.20 0.15 5A 1.00 1.25 1.50 1.50 6A 1.25 1.50 > 1.50 > 1.50 7A 1.00 1.25 1.40 1.50 8A 1.00 1.00 1.00 1.00 1B 0.05 0.15 0.20 0.25 2B 0.10 0.20 0.25 0.30 3B < 0.05 < 0.05 < 0.05 0.05 4B < 0.05 < 0.05 < 0.05 0.05 5B < 0.05 < 0.05 < 0.05 0.40 6B 0.15 0.15 0.15 1.50 7B 0.10 0.10 0.10 0.30 8B 0.15 0.15 0.15 0.30 Table E.13 shows the measured crack widths for labelled cracks according to Figure E.69. Crack widths were estimated using a crack comparator and measured as close to the middle of the web as possible. Cracks measured during Load Stage 1 were formed during Phase 1, and present at the beginning of Phase 2. 460 E.5.2 Specimen 5 Load-Displacement Relationship Phase 1 (a) (b) Figure E.92: S5 – F7 Phase 1 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Base moment (Mb) vs top displacement (D2). -50-40-30-20-10010203040-40 -30 -20 -10 0 10 20 30 40Vb (kN)D1 (mm)-80-60-40-20020406080-80 -60 -40 -20 0 20 40 60 80Mb (kN-m)D2 (mm)461 LS1 (a) (b) Figure E.93: S5 – LS1 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. 0510152025-2 -1.5 -1 -0.5 0 0.5 1 1.5Vb (kN)D1 (mm)-2024681012141618-2 -1.5 -1 -0.5 0 0.5 1 1.5Mm (kN-m)D1 (mm)462 LS2 (a) (b) Figure E.94: S5 – LS2 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -50510152025303540-4 -3 -2 -1 0 1 2 3Vb (kN)D1 (mm)-15-10-50510152025-4 -3 -2 -1 0 1 2 3Mm (kN-m)D1 (mm)463 LS3 (a) (b) Figure E.95: S5 – LS3 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -20-1001020304050-7 -5 -3 -1 1 3 5 7Vb (kN)D1 (mm)-40-30-20-10010203040-7 -5 -3 -1 1 3 5 7Mm (kN-m)D1 (mm)464 LS4 (a) (b) Figure E.96: S5 – LS4 force-displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -20-1001020304050-7 -5 -3 -1 1 3 5 7Vb (kN)D1 (mm)-40-30-20-10010203040-7 -5 -3 -1 1 3 5 7Mm (kN-m)D1 (mm)465 E.5.3 Specimen 5 Displacement Coordinate Results (a) (b) Figure E.97: S5 – Selected Load Stages for displaced shape analysis: (a) Phase 1; (b) Phase 2. -50-40-30-20-10010203040-40 -30 -20 -10 0 10 20 30 40Vb (kN)D1 (mm)F3BF8AF8BF3A-20-1001020304050-7 -5 -3 -1 1 3 5 7Vb (kN)D1 (mm)L1AL2AL3A2L4A3L3A1L4BL2B466 (a) (b) Figure E.98: S5 – Total displacement - “A” Load Stages: (a) Phase 1; (b) Phase 2. 020040060080010001200140016001800-80 -60 -40 -20 0Height (mm)Total Displacement (mm)5-F3A5-F8A - LVDT Measurement - Calculated - Left Coordinates - Right Coordinates020040060080010001200140016001800-10 -5 0 5Height (mm)Total Displacement (mm)5-L1A15-L2A15-L3A15-L3A25-L4A3467 (a) (b) Figure E.99: S5 – Flexural displacement – “A” Load Stages: (a) Phase 1; (b) Phase 2. 0100200300400500600700800-25 -20 -15 -10 -5 0Height (mm)Flexural Displacement (mm)5-F3A5-F8A0100200300400500600700800-0.5 0 0.5 1 1.5 2Height (mm)Flexural Displacement (mm)5-L1A15-L2A15-L3A15-L3A25-L4A3468 (a) (b) Figure E.100: S5 – Curvature distribution – “A” Load Stages: (a) Phase 1; (b) Phase 2. 0100200300400500600700800-150 -100 -50 0Height (mm)Curvature (rad/km)5-F3A5-F8A0100200300400500600700800-20 -10 0 10 20 30 40Height (mm)Curvature (rad/km)5-L1A15-L2A15-L3A15-L3A25-L4A3469 (a) (b) Figure E.101: S5 – Shear ductility demands for Phase 1 and 2 – “A” Load Stages: (a) Shear displacement; (b) Shear strain. 0100200300400500600700800-10 -8 -6 -4 -2 0Height (mm)Shear Displacement (mm)5-F3A5-F8A5-L1A15-L2A15-L3A15-L3A25-L4A30100200300400500600700800-2.5 -2 -1.5 -1 -0.5 0 0.5Height (mm)Shear Strain (%)5-F3A5-F8A5-L1A15-L2A15-L3A15-L3A25-L4A3470 (a) (b) Figure E.102: S5 – Horizontal ductility demands for Phase 1 and 2 – “A” Load Stages: (a) Horizontal displacement; (b) Horizontal strain. 0100200300400500600700800-0.5 0 0.5 1 1.5 2Height (mm)Horizontal Displacement (mm)5-F3A5-F8A5-L1A15-L2A15-L3A15-L3A25-L4A30100200300400500600700800-0.5 0 0.5 1Height (mm)Horizontal Strain (%)5-F3A5-F8A5-L1A15-L2A15-L3A15-L3A25-L4A3471 (a) (b) Figure E.103: S5 – Total displacement - “B” Load Stages: (a) Phase 1; (b) Phase 2. 0200400600800100012001400160018000 20 40 60 80Height (mm)Total Displacement (mm)5-F3B5-F8B - LVDT Measurement - Calculated - Left Coordinates - Right Coordinates020040060080010001200140016001800-5 0 5 10Height (mm)Total Displacement (mm)5-L2B15-L4B1472 (a) (b) Figure E.104: S5 – Flexural displacement – “B” Load Stages: (a) Phase 1; (b) Phase 2. 01002003004005006007008000 5 10 15 20 25Height (mm)Flexural Displacement (mm)5-F3B5-F8B0100200300400500600700800-4 -3 -2 -1 0Height (mm)Flexural Displacement (mm)5-L2B15-L4B1473 (a) (b) Figure E.105: S5 – Curvature distribution – “B” Load Stages: (a) Phase 1; (b) Phase 2. 01002003004005006007008000 50 100 150Height (mm)Curvature (rad/km)5-F3B5-F8B0100200300400500600700800-30 -20 -10 0 10 20Height (mm)Curvature (rad/km)5-L2B15-L4B1474 (a) (b) Figure E.106: S5 – Shear ductility demands for Phase 1 and 2 – “B” Load Stages: (a) Shear displacement; (b) Shear strain. 01002003004005006007008000 5 10 15Height (mm)Shear Displacement (mm)5-F3B5-F8B5-L2B15-L4B10100200300400500600700800-1 0 1 2 3Height (mm)Shear Strain (%)5-F3B5-F8B5-L2B15-L4B1475 (a) (b) Figure E.107: S5 – Horizontal ductility demands for Phase 1 and 2 – “B” Load Stages: (a) Horizontal displacement; (b) Horizontal strain. 0100200300400500600700800-1 -0.5 0 0.5 1 1.5Height (mm)Horizontal Displacement (mm)5-F3B5-F8B5-L2B15-L4B10100200300400500600700800-0.5 0 0.5 1Height (mm)Horizontal Strain (%)5-F3B5-F8B5-L2B15-L4B1476 E.5.4 Specimen 5 Additional Observations (a) (b) Figure E.108: Comparison of Specimen 5, Phase 1 and 2 Load Displacement: (a) Base shear (Vb) vs mid-height displacement (D1); (b) Mid-height moment (Mm) vs mid-height displacement. -50-40-30-20-1001020304050-40 -30 -20 -10 0 10 20 30 40Vb (kN)D1 (mm)Phase 1Phase 2-40-30-20-10010203040-40 -30 -20 -10 0 10 20 30 40Mm (kN-m)D1 (mm)Phase 1Phase 2477 (a) (b) Figure E.109: Specimen 5 comparison of observed specimen displacement vs target actuator displacement for: (a) Mid-height displacement; (b) Top displacement. y = 0.6972xR² = 0.9894-9-6-30369-9 -6 -3 0 3 6 9Mid-Height Specimen Displacement (mm)Mid-Height Actuator Displacement (mm)y = 0.9656xR² = 0.9886-4-3-2-101234-4 -3 -2 -1 0 1 2 3 4Top Specimen Displacement (mm)Top Actuator Displacement (mm)478 F. – Analytical Prediction of Experimental Results Figure F.1: Cross section of Response2000 model. (a) (b) Figure F.2: Material models used in Response2000 model: (a) Concrete model; (b) Reinforcing steel model for longitudinal and transverse (stirrup) steel. 479 Figure F.3: Input for full member properties of Response2000 model. Figure F.4: Shear to moment interaction diagram from Response2000 model. 010203040506070800 10 20 30 40 50 60Shear (kN)Moment (kN-m)480 Figure F.5: Prediction of load-deformation backbone results of base shear (Vb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 1. Figure F.6: Prediction of load-deformation backbone results of top shear (Vt) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 1. 0204060801001201401600 1 2 3 4 5 6 7 8Vb (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear-80-70-60-50-40-30-20-1000 1 2 3 4 5 6 7 8Vt (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear481 Figure F.7: Prediction of load-deformation backbone results of mid-height moment (Mm) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 1. Figure F.8: Prediction of load-deformation backbone results of base moment (Mb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 1. -70-60-50-40-30-20-1000 1 2 3 4 5 6 7 8Mm (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear-30-20-10010203040506070-8 -6 -4 -2 0 2 4 6 8Mb (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear482 Figure F.9: Prediction of load-deformation backbone results of base shear (Vb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 2. Figure F.10: Prediction of load-deformation backbone results of top shear (Vt) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 2. 0204060801001201401600 1 2 3 4 5 6 7 8Vb (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear-80-70-60-50-40-30-20-1000 1 2 3 4 5 6 7 8Vt (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear483 Figure F.11: Prediction of load-deformation backbone results of mid-height moment (Mm) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 2. Figure F.12: Prediction of load-deformation backbone results of base moment (Mb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 2. -70-60-50-40-30-20-1000 1 2 3 4 5 6 7 8Mm (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear-30-20-10010203040506070-8 -6 -4 -2 0 2 4 6 8Mb (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear484 Figure F.13: Prediction of load-deformation backbone results of base shear (Vb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 3. Figure F.14: Prediction of load-deformation backbone results of top shear (Vt) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 3. 0204060801001201401600 1 2 3 4 5 6 7 8Vb (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear-80-70-60-50-40-30-20-1000 1 2 3 4 5 6 7 8Vt (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear485 Figure F.15: Prediction of load-deformation backbone results of mid-height moment (Mm) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 3. Figure F.16: Prediction of load-deformation backbone results of base moment (Mb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 3. -70-60-50-40-30-20-1000 1 2 3 4 5 6 7 8Mm (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear-20-10010203040506070-8 -6 -4 -2 0 2 4 6 8Mb (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear486 Figure F.17: Prediction of load-deformation backbone results of base shear (Vb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 4. Figure F.18: Prediction of load-deformation backbone results of top shear (Vt) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 4. 0102030405060708090100110120130140150160-2 -1 0 1 2 3 4 5 6 7 8Vb (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear-75-60-45-30-15015304560-8 -6 -4 -2 0 2 4 6 8Vt (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear487 Figure F.19: Prediction of load-deformation backbone results of mid-height moment (Mm) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 4. Figure F.20: Prediction of load-deformation backbone results of base moment (Mb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 4. -60-50-40-30-20-1001020304050-8 -6 -4 -2 0 2 4 6 8Mm (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear010203040506070-8 -6 -4 -2 0 2 4 6 8Mb (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear488 Figure F.21: Prediction of load-deformation backbone results of base shear (Vb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 5. Figure F.22: Prediction of load-deformation backbone results of top shear (Vt) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 5. 020406080100120140160-2 -1 0 1 2 3 4 5 6 7 8Vb (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear-75-60-45-30-150153045-8 -6 -4 -2 0 2 4 6 8Vt (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear489 Figure F.23: Prediction of load-deformation backbone results of mid-height moment (Mm) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 5. Figure F.24: Prediction of load-deformation backbone results of base moment (Mb) vs mid-height displacement (D1) using a tri-linear flexural model and various shear models for Specimen 5. -60-50-40-30-20-10010203040-8 -6 -4 -2 0 2 4 6 8Mm (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear-10010203040506070-8 -6 -4 -2 0 2 4 6 8Mb (kN)D1 (mm)ResultsUncracked Shear50% Shear10% ShearTri-Linear Shear
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Resistance of reinforced concrete cantilever shear walls to seismic shear demands Young, Mitchell P. 2019
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Title | Resistance of reinforced concrete cantilever shear walls to seismic shear demands |
Creator |
Young, Mitchell P. |
Publisher | University of British Columbia |
Date Issued | 2019 |
Description | Reinforced concrete shear wall core structures are very common among high-rise buildings in Vancouver, and increasingly so elsewhere. While the flexural behaviour of these structures is well understood, the shear behaviour is not. Much of the research regarding the shear behaviour is related to the amplifications of demands due to higher mode seismic shear, however, there has been little research regarding the resistance of these structures to higher mode seismic shear demands. It is theorized that due to the lack of experimental data on which more complex shear models could be based on, structural engineers have resorted to using building models with complex non-linear fibre section flexural stiffness, but a linear elastic shear stiffness. Which may have lead to higher mode shear demands to be overestimated. Therefore, the goal of this thesis is to complete an experimental program in which scaled shear wall core specimens are tested under higher mode demands in the cantilevered direction. Through the experimental program, topics that are investigated include: the effect of the rate of loading on the shear resistance, the effect of existing flexural base yielding on the shear resistance, and the presence of a plastic hinge at the base on the shear resistance. In addition to the experimental program, a series of dynamic analyses were completed on a simplified model, in order to better understand the behaviour of a high-rise reinforced concrete shear wall structure to higher mode effects, and the results of the experimental program are also compared to predictions made using common analytical tools. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2019-02-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0376559 |
URI | http://hdl.handle.net/2429/68440 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2019-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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