seismic performance of rc structural squat walls with ... · reasonable strut-and-tie models for rc...
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This document is downloaded from DR‑NTU (https://dr.ntu.edu.sg)Nanyang Technological University, Singapore.
Seismic performance of RC structural squat wallswith limited transverse reinforcement
Xiang, Weizheng
2009
Xiang, W. (2009). Seismic performance of RC structural squat walls with limited transversereinforcement. Doctoral thesis, Nanyang Technological University, Singapore.
https://hdl.handle.net/10356/42146
https://doi.org/10.32657/10356/42146
Downloaded on 23 Feb 2021 19:56:34 SGT
SEISMIC PERFORMANCE OFRC STRUCTURAL SQUAT WALLS WITH
LIMITED TRANSVERSE REINFORCEMENT
XIANG WEIZHENG
School of Civil and Environmental Engineering
A thesis submitted to the Nanyang Technological University
in fulfillment of the requirement for the degree ofDoctor of Philosophy
2009
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ACKNOWLEDGEMENTS
First of all, I would like to express my deepest gratitude to Associate Professor Li Bing,
my supervisor, for his continuous guidance, valuable assistance and kind
encouragement throughout this research.
My sincere thanks also go to Mr. Wu Hui, Wang Zhiwei, Zhao Yiwen, Tang Haiyang,
Hang Hongsheng, Wang Wenyuan and Ms. Rong Haicheng, Cheng Qin for sharing
their valuable experiences on the finite element analysis and reinforced concrete design.
Their kind help in the research and friendship are really appreciated here.
I would also like to thank Mr. Leong Chee Lai and technicians in Protective
Engineering laboratory (PE Lab), for their kind assistance in the experimental program.
The research work was conducted at Nanyang Technological University (NTU).
Sincere thanks to the university for providing the research scholarship during his
candidature in these years of study.
Lastly, I wish to thank my parents and friends for their continuous encouragement and
support.
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ABSTRACT
In the last three decades, extensive research works has been conducted to assess the
validity of the design provisions of EC8 and ACI 318 for cyclic shear in reinforced
concrete (RC) structural walls with low aspect ratios. Significant progress has been
achieved in the understanding of global and local responses of such RC structural walls.
These squat walls are usually detailed according to current provisions and reinforced
against the shear by either conventionally or by adding additional cross-inclined
bidiagonal bars to achieve the full ductile behavior of the RC member. Previous
research, however, does not provide adequate and conclusive information of structural
squat walls with limited transverse reinforcement in boundary columns. Such RC walls
may exhibit only limited ductility and the sliding shear mode may dominate. Current
research is, therefore, initiated by the need to provide useful and conclusive
information related with the local and global responses of squat RC walls with limited
transverse reinforcement. In this study, both experimental and numerical investigations
of local and global responses of such RC walls under cyclic loadings have been
presented in detail.
An experimental program has been carried out to explore the local and global responses
of a total of eight squat RC structural walls with limited transverse reinforcement. The
global and local behavior of these RC walls from the experiments carried out is
described in detail. The influence of several design parameters such as axial
compression loads, transverse reinforcements in the wall boundary columns and the
presence of construction joints at the wall base, on the behavior of such RC walls under
cyclic loadings is also reported herein. Reasonable strut-and-tie models for RC
structural walls with and without axial loads are then developed to aid in better
understanding the force transfer mechanism and contribution of reinforcement in RC
walls based on the experiments carried out.
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Next, an analytical approach, combining the inelastic flexure and shear components of
deformation, is proposed to properly evaluate the initial stiffness of tested RC walls.
The use of this analytical approach which includes a comprehensive parametric study
with 180 combinations is carried out with the emphasis on four main parameters that
influence wall stiffness: yield strength of outermost longitudinal bars, axial loads,
aspect ratios and longitudinal reinforcement content in wall boundaries. These four
parameters are studied in detail and a simple expression based on this study is proposed
to determine the initial stiffness of squat RC walls as a function of three factors: yield
strength of the outermost longitudinal reinforcement, applied axial compression and
wall aspect ratios. Comparing with other stiffness prediction expressions, it is shown
that the initial stiffness determined by the proposed expression compares better with the
experimental results.
Finally, to aid in better understanding of global responses of such RC structural walls, a
nonlinear finite element analytical procedure for squat RC structural walls under cyclic
loadings is developed and verified with the experimental results. Global responses such
as strength capacity, stiffness characteristics and energy dissipation capacity etc. under
reversed seismic loadings are described in detail. Moreover, the influence of several
paramount parameters such as axial loads, longitudinal reinforcements in the wall
boundary elements, aspect ratio, area of boundary columns and the presence of
construction joints at the wall base on the global behavior of squat RC walls is also
investigated in detail by means of the proposed analytical procedure.
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TABLE OF CONTENTS
ACKNOWLEDGEMENTS .
ABSTRACT .
TABLE OF CONTENTS .
LIST OF TABLES .
LIST OF FIGURES .
CHAPTER ONE. INTRODUCTION .
1.1 General .
1.2 Objective of Present Research .
1.3 Outline of the Report .
CHAPTER TWO. SEISMIC PERFORMANCE OF LOW-RISE
STRUCTURAL WALLS WITH LIMITED TRANSVERSE
REINFORCEMENT .
Abstract .
2.1 Introduction and Background .
2.2 Experimental Program .
2.2.1 Material Properties .
2.2.2 Code Provisions for Confining Reinforcement in Plastic HingeRegions .
2.2.2.1 ACI 318 Code Provisions .
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2.2.2.2 NZS 3101:1995 Code Provisions........................ 12
2.2.3 Application of Design Procedures................................ 12
2.2.4 Experimental Set-up and Loading History.................................. 16
2.2.5 Instrumentation of Wall Specimens..................................... 18
2.3 Experimental Results of the Tested Specimens.......................... 19
2.3.1 Experimental Results of Specimen LWl......... 19
2.3.1.1 Global Behavior............................................................ 19
2.3.1.2 Local Response......................................................... 20
2.3.2 Experimental Results of Specimen LW2......... 22
2.3.2.1 Global Behavior........................................... 22
2.3.2.2 Local Response............................................................. 23
2.3.3 Experimental Results of Specimen LW3.................................... 24
2.3.3.1 Global Behavior............................................................ 24
2.3.3.2 Local Response............................................................. 25
2.3.4 Experimental Results of Specimen LW4.................................... 25
2.3.4.1 Global Behavior. 26
2.3.4.2 Local Response............................................................. 26
2.3.5 Experimental Results of Specimen LW5................................... 27
2.3.5.1 Global Behavior............................................................ 27
2.3.5.2 Local Response............................................................. 28
2.4 Discussion of Experimental Results................... 29
2.4.1 Crack Patterns and Failure Modes........................................... 29
2.4.2 Backbone Envelopes of Load-displacement Curves. 30
2.4.3 Components of Top Deformation............................................ 31
2.4.4 Curvature Distribution along the Wall Height.......................... .... 33
2.4.5 Stiffness Characteristics....................................................... 33
2.4.6 Energy Dissipation............................................................. 35
2.5 Extrapolation of Experimental Results... 36
2.6 Conclusions....... 38
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CHAPTER THREE. SEISMIC PERFORMANCE OF MEDIUM-RISE
STRUCTURAL WALLS WITH LIMITED TRANSVERSE
REINFORCEMENT ..
Abstract .
3.1 Introduction and Background .
3.2 Experimental Program ..
3.2.1 Material Properties ..
3.2.2 Code Provisions for Confining Reinforcement in Plastic HingeRegions .
3.2.2.1 ACI 318 Code Provisions ..
3.2.2.2 NZS 3101:1995 Code Provisions .
3.2.3 Details of Test Specimens .
3.2.4 Experimental Set-up and Loading History .
3.2.5 Instrumentation of Wall Specimens ..
3.3 Experimental Results .
3.3.1 Experimental Results of Specimen MWl.. ..
3.3.1.1 Global Behavior. .
3.3.1.2 Local Response .
3.3.2 Experimental Results of Specimen MW2 .
3.3.2.1 Global Behavior. .
3.3.2.2 Local Response ..
3.3.3 Experimental Results of Specimen MW3 .
3.3.3.1 Global Behavior. .
3.3.3.2 Local Response ..
3.4 Discussion of Experimental Results .
3.4.1 Crack Patterns and Failure Modes .
3.4.2 Backbone Envelopes of Load-displacement Curves .
3.4.3 Components of Top Deformation .
3.4.4 Curvature Distribution along the Wall Height. .
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3.4.5 Stiffness Characteristics .
3.4.6 Energy Dissipation ..
3.5 Extrapolation of Experimental Results .
3.6 Conclusions .
CHAPTER FOUR. STIFFNESS CHARACTERISTICS OF
STRUCTURAL WALLS WITH LIMITED TRANSVERSE
REINFORCEMENT .
Abstract .
4.1 Introduction and Background .
4.2 Previous Expressions in Evaluating the Cracked Stiffness of the
Walls .
4.2.1 Research Conducted by Fenwick and Bull ..
4.2.2 Research Conducted by Paulay and Priestley .
4.2.3 ACI 318-02 .
4.2.4 NZS 3101: 1995 .
4.2.5 FEMA 356 (FEMA 2000) .
4.3 Stiffness Characteristics .
4.3.1 Elastic Uncracked Stiffness ..
4.3.2 Analytical Cracked Stiffness .
4.3.3 Initial Stiffness .
4.3.3.1 Flexural Deformation Determination ..
4.3.3.2 Shear Deformation Determination .
4.3.3.3 Combination of Shear and Flexure Response .
4.3.4 Validation of the Proposed Approach .
4.4 Parametric Study for Initial Stiffness of Squat Structural Walls .
4.4.1 Influence of Aspect Ratio .
4.4.2 Influence of Axial Load .
4.4.3 Influence of Longitudinal Reinforcement Content in Wall
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Boundaries .
4.4.4 Influence of Yield Tensile Strength of Longitudinal Bars inWall Boundaries .
4.5 Proposed Equation for Moment of Inertia of Structural Walls .
4.6 Comparisons of Analytical Stiffness Ratios with Tested Results .
4.7 Conclusions .
CHAPTER FIVE. FINITE ELEMENT PARAMETRIC STUDY OF
THE BEHAVIOR OF STRUCTURAL WALLS WITH
LIMITED TRANSVERSE REINFORCEMENT .
Abstract .
5.1 Introduction and Background .
5.2 Description of Finite Element Models .
5.2.1 Constitutive Models for Concrete .
5.2.2 Constitutive Model for the Reinforcing Bars in Concrete .
5.2.3 Constitutive Model for Structural Interface .
5.3 Applications of the Finite Element Models .
5.3.1 Verification of the Finite Element Models ..
5.3.2 Numerical Investigations of Specimens Tested .
5.3.2.1 Predicted Global Response of Specimens Tested .
5.3.2.2 Predicted Local Response of Specimens Tested ..
5.3.3 Parametric Study of Squat Structural Walls .
5.3.3.1 Effect of Axial Loading ..
5.3.3.1.1 Effect of axial loading on wall strength .
5.3.3.1.2 Effect of axial loading on secant stiffness .
5.3.3.1.3 Effect of axial loading on energy dissipation .
5.3.3.1.4 Effect of axial loading on equivalent damping .
5.3.3.2 Effect of Longitudinal Reinforcement Content in BoundaryElement .
5.3.3.2.1 Effect oflongitudina1 reinforcement content on wallstrength .
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5.3.3.2.2 Effect of longitudinal reinforcement content on secantstiffuess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 199
5.3.3.2.3 Effect of longitudinal reinforcement content on energydissipation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 201
5.3.3.2.4 Effect of longitudinal reinforcement content on equivalentdamping.. . . .. . . . . . . . .. . . .. . . .. . . . . . .. 202
5.3.3.3 Effect of Boundary Columns.............................................. 203
5.3.3.3.1 Effect of boundary columns on wall strength..................... 204
5.3.3.3.2 Effect of boundary columns on secant stiffuess................... 204
5.3.3.3.3 Effect of boundary columns on energy dissipation............... 205
5.3.3.3.4 Effect of boundary columns on equivalent damping............. 206
5.3.3.4 Effect of Aspect Ratios.................................................... 207
5.3.3.4.1 Effect of aspect ratio on wall strength.............................. 208
5.3.3.4.2 Effect of aspect ratios on secant stiffuess....................... 208
5.3.3.4.3 Effect of aspect ratio on energy dissipation. . . . . . . . . . . . . . . . . . . . 209
5.3.3.4.4 Effect of aspect ratio on equivalent damping...................... 209
5.3.3.5 Effect of Construction Joints.............................................. 210
5.3.3.5.1 Effect of construction joints on wall strength..................... 210
5.3.3.5.2 Effect of construction joints on secant stiffness................... 211
5.3.3.5.3 Effect of construction joints on energy dissipation............... 211
5.3.3.5.4 Effect of construction joints on equivalent damping............. 211
5.4 Conclusions........................................................................ 212
CHAPTER SIX. CONCLUSIONS AND RECOMMENDATIONS....... 252
6.1 Conclusions........................................................................ 252
6.2 Recommendations for Future Works...... 259
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Table
Table 2.1
Table 2.2
Table 3.1
Table 3.2
Table 4.1
Table 4.2
Table 4.3
Table 4.4
Table 4.5
Table 4.6
Table 4.7
Table 4.8
Table 4.9
LIST OF TABLES
Observed strengths and ductility of specimens tested .
Strengths at onset of diagonal cracks of specimens tested andpredicted by analytical models .
Observed strengths and ductility of all specimens .
Strengths at onset of diagonal cracks of specimens testedand predicted by analytical models .
Effective section properties by New Zealand Standard(NZS 1995) .
Initial stiffness coefficients for linear analysis of walls inFEMA 356 ..
Stiffness evaluation of all tested walls ..
Flexural deformation determination .
Experimental and analytical results for initial stiffnessof the eight specimens tested .
Parameters investigated .
Stiffness ratio, Ele / EIg (%) for walls with Pb =1.4% .
Stiffness ratio, Ele / Eig (%) for walls with Pb =2.8% .
Stiffness ratio, Ele / Eig (%) for walls with Pb =4.2% .
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Table 4.10 Comparison of tested versus predicted stiffness. . . . . . . . . . . .. . . . . . . . . 154
Table 5.1
Table 5.2
Table 5.3
Table 5.4
Table 5.5
Coefficients for determination of the fracture energy .
Material properties and reinforcement ratio forUnit 1.0 and Specimen S-F1 ..
Comparisons of finite element predictions to test results ..
Concrete material properties used for finite element analysis .
Parameters investigated .
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Figure
Fig. 2.1
Fig. 2.2
Fig. 2.3
Fig. 2.4
Fig. 2.5
Fig. 2.6
Fig. 2.7
Fig. 2.8
Fig.2.9(a)
Fig.2.9(b)
Fig.2.10(a)
Fig. 2.1 O(b)
Fig. 2.11
Fig. 2.12
Fig. 2.13
Fig. 2.14(a)
Fig. 2.14(b)
Fig.2.15(a)
LIST OF FIGURES
Stress-strain relationship for steel reinforcements ..
Details of Specimen LWI .
Experimental set-up .
Applied loading history .
LVDTs support arrangements in all specimens tested .
Crack patterns of Specimen LWI ..
Lateral load - top displacement relationship of Specimen LWI
Strain distribution in outermost longitudinal bars ofSpecimen LW1 .Strain profiles of the vertical bars along section 1-1 ofSpecimen LW1 .
Strain profiles of the vertical bars along section 1-1 ofSpecimen LW1 .Strain distribution in the horizontal web bar (R bar) ofSpecimen LWI .Strain distribution in the horizontal web bar (T bar) ofSpecimen LWI .
Crack patterns of Specimen LW2 ..
Lateral load - top displacement relationship of Specimen LW2
Strain distribution in outermost longitudinal bars ofSpecimen LW2 .
Strain profiles of the vertical bars along section 1-1 ofSpecimen LW2 .
Strain profiles of the vertical bars along section 2-2 ofSpecimen LW2 .
Strain distribution in the horizontal web bar (R bar) ofSpecimen LW2 .
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Fig.2.15(b)
Fig. 2.16
Fig. 2.17
Fig. 2.18
Fig. 2.19(a)
Fig. 2.19(b)
Fig. 2.20(a)
Fig.2.20(b)
Fig. 2.21
Fig. 2.22
Fig. 2.23
Fig. 2.24(a)
Fig. 2.24(b)
Fig. 2.25(a)
Fig.2.25(b)
Fig. 2.26
Fig. 2.27
Fig. 2.28
Fig. 2.29(a)
Fig.2.29(b)
Fig. 2.30(a)
Fig.2.30(b)
Strain distribution in the horizontal web bar (T bar) ofSpecimen LW2 .
Crack patterns of Specimen LW3 .
Lateral load - top displacement relationship of Specimen LW3
Strain distribution in outermost longitudinal bars ofSpecimen LW3 .
Strain profiles of the vertical bars along section 1-1 ofSpecimen LW3 .
Strain profiles of the vertical bars along section 2-2 ofSpecimen LW3 .
Strain distribution in the horizontal web bar (R bar) ofSpecimen LW3 .
Strain distribution in the horizontal web bar (T bar) ofSpecimen LW3 .
Crack patterns of Specimen LW4 ..
Lateral load - top displacement relationship of Specimen LW4
Strain distribution in outermost longitudinal bars ofSpecimen LW4 .Strain profiles of the vertical bars along section 1-1 ofSpecimen LW4 .Strain profiles of the vertical bars along section 2-2 ofSpecimen LW4 .Strain distribution in the horizontal web bar (R bar) ofSpecimen LW4 .Strain distribution in the horizontal web bar (T bar) ofSpecimen LW4 .
Crack patterns of Specimen LW5 .
Lateral load - top displacement relationship of Specimen LW5
Strain distribution in outermost longitudinal bars ofSpecimen LW5 .Strain profiles of the vertical bars along section 1-1 ofSpecimen LW5 .Strain profiles of the vertical bars along section 2-2 ofSpecimen LW5 .Strain distribution in the horizontal web bar (R bar) ofSpecimen LW5 .Strain distribution in the horizontal web bar (T bar) ofSpecimen LW5 .
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Fig. 2.31
Fig. 2.32
Fig. 2.33
Fig. 2.34
Fig. 2.35
Fig. 2.36
Fig. 2.37
Fig. 2.38(a)
Fig.2.38(b)
Fig. 2.39
Fig. 2.40
Fig. 2.41
Fig. 2.42
Fig. 2.43
Fig. 2.44
Fig. 2.45
Fig. 2.46
Fig. 2.47
Fig. 3.1
Fig. 3.2
Fig. 3.3
Fig. 3.4
Fig. 3.5
Backbone envelopes of load-displacement curves for testedspecimens .
Contribution of various deformation modes to totaldisplacement of walls .
Wall curvature distribution of all specimens tested .
Secant stiffness of tested walls with respect to drift ratios .
Energy dissipation capacity of each specimen with respect tothe drift ratios .Flexure, shear and sliding displacements of Specimens LW2andLW4 .
Strut-and-tie model of Specimen LW1 .
Forces acting at wall base section ..
Forces acting at base section of strut-and-tie modeL .
Strain history of gauge #T14 in horizontal bars ofSpecimen LW1 .
Strut-and-tie model of Specimen LW4 ..
Strain history of gauge #T14 in horizontal bars ofSpecimen LW4 .
Strut-and-tie model of Specimen LW2 .
Strain history of gauge #T 14 in horizontal bars ofSpecimen LW2 .
Strut-and-tie model of Specimen LW3 .
Strain history of gauge #T14 in horizontal bars ofSpecimen LW3 .
Strut-and-tie model of Specimen LW5 .
Strain history of gauge #T14 in horizontal bars ofSpecimen LW5 .
Stress-strain relationship for steel reinforcements .
Details of Specimen MW1 .
Experimental set-up .
Applied loading history .
LVDTs support arrangements in specimen walls .
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Fig. 3.6
Fig. 3.7
Fig. 3.8
Fig.3.9(a)
Fig.3.9(b)
Fig.3.9(c)
Fig. 3.10(a)
Fig.3.10(b)
Fig.3.10(c)
Fig. 3.11
Fig. 3.12
Fig. 3.13
Fig. 3.14(a)
Fig.3.14(b)
Fig.3.14(c)
Fig.3.15(a)
Fig. 3.15(b)
Fig. 3.16
Fig. 3.17
Fig. 3.18
Fig. 3.19(a)
Fig. 3.19(b)
Crack patterns of Specimen MW1 .
Lateral load - top displacement relationship ofSpecimen MW 1 .
Strain distribution in outermost longitudinal bars ofSpecimen MW1 .Strain profiles of the vertical bars along section 1-1 ofSpecimen MW1 .Strain profiles of the vertical bars along section 2-2 ofSpecimen MW1 .Strain profiles of the vertical bars along section 3-3 ofSpecimen MW1 .Strain distribution in the horizontal web bar (R bar) ofSpecimen MW1 .Strain distribution in the horizontal web bar (T bar) ofSpecimen MW1 .Strain distribution in the horizontal web bar (W bar) ofSpecimen MW1 .
Crack patterns of Specimen MW2 .
Lateral load - top displacement relationship ofSpecimen MW2 .
Strain distribution in outermost longitudinal bars ofSpecimen MW2 ..Strain profiles of the vertical bars along section 1-1 ofSpecimen MW2 .Strain profiles of the vertical bars along section 2-2 ofSpecimen MW2 .Strain profiles of the vertical bars along section 3-3 ofSpecimen MW2 .Strain distribution in the horizontal web bar (R bar) ofSpecimen MW2 .Strain distribution in the horizontal web bar (T bar) ofSpecimen MW2 .
Crack patterns of Specimen MW3 .
Lateral load - top displacement relationship ofSpecimen MW3 .
Strain distribution in outermost longitudinal bars ofSpecimen MW3 .Strain profiles of the vertical bars along section 1-1 ofSpecimen MW3 .Strain profiles of the vertical bars along section 2-2 ofSpecimen MW3 .
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Fig.3.19(c)
Fig. 3.20(a)
Fig.3.20(b)
Fig. 3.21
Fig. 3.22
Fig. 3.23
Fig. 3.24
Fig. 3.25
Fig. 3.26
Fig. 3.27
Fig. 3.28(a)
Fig.3.28(b)
Fig. 3.29
Fig. 3.30
Fig. 3.31
Fig. 3.32
Fig. 3.33
Fig. 4.1
Fig. 4.2
Fig. 4.3
Fig. 4.4
Fig. 4.5
Fig. 4.6
Strain profiles of the vertical bars along section 3-3 ofSpecimen MW3 .Strain distribution in the horizontal web bar (R bar) ofSpecimen MW3 .
Strain distribution in the horizontal web bar (T bar) ofSpecimen MW3 .
Backbone envelopes of load-displacement curves for testedspeCImens .Contribution of various deformation modes to totaldisplacement of walls .
Wall curvature distribution of tested walls .
Secant stiffness of tested walls with respect to drift ratios .
Energy dissipation capacity of each specimen with respect tothe drift ratios .Flexure, shear and sliding displacements of Specimens MWItoMW3 .
Strut-and-tie model of Specimen MW1 .
Forces acting at wall base section .
Forces acting at base section of strut-and-tie model .
Strain history of gauges in horizontal bars of Specimen MW1 ..
Strut-and-tie model of Specimen MW2 .
Strut-and-tie model of Specimen MW3 .
Strain history of the gauge T14 in selected horizontal bar ofSpecimen MW2 .
Strain history of the gauge T14 in selected horizontal bar ofSpecimen MW3 .
Initial stiffness determination [F1] .
Shear distortion of wall panel using analogous truss [P1] .
Compression strut of wall panel. .
Influence of wall aspect ratios on stiffness ratios .
Influence of axial load on wall stiffness ratios .
Influence of longitudinal reinforcement ratios in wallboundaries on wall stiffness ratios .
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Fig. 4.7
Fig. 4.8
Fig. 5.1
Fig. 5.2
Fig. 5.3
Fig. 5.4
Fig. 5.5
Fig. 5.6
Fig. 5.7
Fig. 5.8
Fig. 5.9
Fig. 5.10
Fig. 5.11
Fig. 5.12
Fig. 5.13
Fig. 5.14
Fig. 5.15
Fig. 5.16
Fig. 5.17
Fig. 5.18
Fig. 5.19
Fig. 5.20
Fig. 5.21
Comparisons of stiffness ratios proposed bythe parametric study and equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 160Comparison of initial stiffness between theanalytical results and tested data..................................... 161
Concrete in compression and tension............... 218
Shear friction hypothesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 218
Coulomb friction criterion.......................................... ... 219
Aggregate interlock relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
The overall dimensions and reinforcement layout of Unit 1.0... 219
The overall dimensions and reinforcement layout ofSpecimen S-Fl......................................................... 220
Finite element idealization and deformed shapes of Unit 1.0at a drift ratio of 1.0%......................... 219Finite element idealization and deformed shapes of SpecimenS-F1 at a drift ratio of 1.0%........................................................ 220
Experimental and analytical hysteretic responses of Unit 1.0... 220
Experimental and analytical hysteretic responses of SpecimenS-F1..................................................................... 221
Finite element idealization for all specimens tested............... 221
Experimental and analytical hysteretic responses of allspecimens tested. . . . . . . . . . . . . . . . . . . . . . . . . .. .. . . .... . .. . . .. .. . . . . .. . .. ... 223
Verification of longitudinal strain distribution along walllength.................. 224Verification of horizontal strain distribution in selectedhorizontal bars.......................................................... 226Representation ofthe secant stiffness, energy dissipationcapacity.................................................................. 226Effect of axial load on backbone curves of load-displacementloops.............................. 227
Contribution of axial load ratio to the wall strength............... 228
Effect of axial load on secant stiffness of walls studied.......... 229
Contribution of axial load ratio to the wall secant stiffness...... 230
Effect of axial load on energy dissipation of walls studied....... 231
Contribution of axial load ratio to the energy dissipation... ...... 232
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Fig. 5.22
Fig. 5.23
Fig. 5.24
Fig. 5.25
Fig. 5.26
Fig. 5.27
Fig. 5.28
Fig. 5.29
Fig. 5.30
Fig. 5.31
Fig. 5.32
Fig. 5.33
Fig. 5.34
Fig. 5.35
Fig. 5.36
Fig. 5.37
Fig. 5.38
Fig. 5.39
Fig. 5.40
Fig. 5.41
Fig. 5.42
Fig. 5.43
Fig. 5.44
Effect of axial load on equivalent damping of walls studied. . ... 233
Contribution of axial load ratio to the energy dissipation. . . . . ... 233
Effect of longitudinal reinforcement content on backbonecurves of load-displacement loops. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 234Contribution of longitudinal reinforcement content to the wallstrength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
Effect of longitudinal reinforcement content on secant stiffness 235
Contribution of longitudinal reinforcement content to secantstiffuess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 235Effect of longitudinal reinforcement content on energydissipation... .. . . . .. .. . . . .. . . . . . .. .. .. . . . . . . . . .. . . . . .. 236Contribution of longitudinal reinforcement content to theenergy dissipation...................................................... 236
Effect of longitudinal reinforcement content on equivalentdamping............................................................... ... 237
Contribution of longitudinal reinforcement content toequivalent damping.................................................... 237Effect of boundary columns on backbone curves of load-displacement loops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 238
Contribution of boundary columns to the wall strength........ ... 238
Effect of boundary columns on secant stiffuess................. ... 239
Contribution of boundary columns to secant stiffuess............ 239
Effect of boundary columns on energy dissipation. . . . . . . . . . . . . . .. 240
Contribution of boundary columns to the energy dissipation. . .. 240
Effect of boundary columns on equivalent damping........... ... 241
Contribution of boundary columns to equivalent damping. . . . . . 241
Effect of aspect ratio on backbone curves of load-displacementloops........... 242
Effect of aspect ratio on secant stiffuess. . . . . . . . . . . . . . . . . . . . . . . . . . .. 242
Effect of aspect ratio on energy dissipation. . . . . . . . . . . . . . . . . . . . . . . . 243
Effect of aspect ratio on equivalent damping. . . . . . ..... . . . . ... . . ... 243
Effect of construction joints on backbone curves of load-displacement loops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 244
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Fig. 5.45
Fig. 5.46
Fig. 5.47
Effect of construction joints on secant stiffness .
Effect of construction joints on energy dissipation .
Effect of construction joints on equivalent damping .
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244
245
245
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Chapter One
CHAPTER ONE
INTRODUCTION
1.1 General
Singapore and Peninsular Malaysia are in a seismic risk area with an active earthquake
belt comprising the Sumatra Fault and the subduction zone located at about 350 km
away from its closest point. Although there has never been any earthquake damage to
this region, ground tremors have been felt many times, and the incidents have increased
significantly in number over the last three decades. Strong tremors were felt in
buildings in Singapore from the recent north Sumatra Earthquake. In these low to
moderate seismic regions like Singapore and Peninsular Malaysia, reinforced concrete
(RC) structural walls are normally designed in accordance with the British Standard:
BS 8110 which excludes the influence of seismic loading. Hence, structural walls are
usually detailed non-seismically or detailed only to provide limited seismic resistance.
As such, the wall boundary elements contain a lack of confinement reinforcement and
are liable to extensive damages under earthquake excitation due to excessive shear
deformation and severe strength degradation. Therefore, it is of great concern that the
structural performance of these RC walls may not be adequate to sustain
earthquake-induced loads in regions of low to moderate seismicity like Singapore and
Peninsular Malaysia. Experimental and analytical studies of RC walls with limited
transverse reinforcement are therefore needed to explore their structural performance in
terms of strength and deformation characteristics, secant stiffness and energy
dissipation capacity.
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Cha12.ter One
In the field of experimental studies, extensive research have been conducted in recent
years to assess the validity of current design provisions such as ACI 318 and NZS 3101
for cyclic shear in squat RC structural walls. Usually, these walls are detailed according
to the current provisions and aimed at achieving its full ductile behavior. However, in
low to moderate seismicity regions such as Singapore and Peninsular Malaysia, the
ductility demands may not be the same as those required in higher seismicity zones. In
such situations, less ductility demand can be expected and thus the required quantities
of reinforcement, especially the transverse reinforcement in web and boundary element,
can be reduced. Therefore, it is considered to be necessary to understand the seismic
performance of squat RC structural walls with limited transverse reinforcements
located in these low to moderate seismicity regions. However, up to date only a few
experimental investigations [Ml, P2, Yl] of the behavior of such RC structural walls
under cyclic loadings have been conducted and from these investigations it is found
that there is still insufficient information regarding the structural performance such as
available ductility and energy dissipation capacity of such RC walls with axial
compression. Moreover, current available experimental data related to the behavior of
squat RC walls with weak interface like construction joints are rather inconclusive. To
provide adequate and conclusive information, experimental studies for such RC squat
walls under axial compression and the presence of construction joints should be carried
out to investigate their seismic performances.
In the field of analytical investigations of such RC walls, firstly it is found that previous
research in initial wall stiffness evaluation are either over-simplified which may lead to
inaccurate assessment of the wall stiffness or initial wall stiffness which can only be
justified for slender structural walls whose response is dominated by flexure. Thus,
proper stiffness evaluations applied to RC structural walls with low aspect ratios as
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Chapter One
their response may be controlled by shear deformations should be carried out to
improve wall stiffness predictions. Secondly, numerous analytical models in the past
decades were developed in the modeling of RC structural walls to explore their global
responses. Most of these were macroscopic models for RC structural walls due to their
easy application and in these studies great success has been achieved at the element
level. However, these analytical results are usually valid only for the specific
conditions upon which the derivation of the model is based upon. Moreover, most of
the previous work in the finite element analysis of the behavior of RC walls
concentrated on the development of the material models that could reproduce
experimental results and few research studies used the finite element method to
investigate behavior of RC walls other than that of the specimens tested in the
laboratory. Therefore, a nonlinear finite element analytical procedure incorporating
general-purpose microscopic models is needed to be developed to examine the
structural performances of squat RC structural walls under seismic loadings. An
extensive parametric study including several critical parameters: axial loads,
longitudinal reinforcements in the wall boundary elements, aspect ratio, area of
boundary columns and the presence of construction joints at the wall base are also
considered to be necessary to investigate their influence on the seismic performance of
such RC walls.
1.2 Objectives of Present Research
The main objectives of the present study for squat RC structural walls with limited
transverse reinforcement are to compile information to:
• examine the structural performance in terms of the displacement and strength
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Chal2.ter One
capacity, curvature distribution, the secant stiffness degradation and the energy
dissipation characteristics of RC structural walls with limited transverse
reinforcement.
• report the influence of several design parameters such as axial loads, transverse
reinforcements in the wall boundary elements, and the presence of construction
joints at the wall base on the seismic behavior ofRC walls tested.
• develop reasonable strut-and-tie models to aid in better understanding the force
transfer mechanism and contribution of reinforcement in RC walls tested.
• evaluate stiffness characteristics of squat RC structural walls with limited
transverse reinforcement.
• use a nonlinear finite element analytical procedure to aid in better
understanding the global responses of squat RC structural walls under several
critical parameters such as axial loads, longitudinal reinforcements in the wall
boundary elements, aspect ratio, area of boundary columns and the presence of
construction joints at the wall base.
1.3 Outline of the Report
The introduction to and objectives of this study is presented in this Chapter.
In Chapter 2, structural performances of five low-rise RC walls with limited transverse
reinforcement and an aspect ratio of 1.125, are examined by subjecting them to low
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Chapter One
levels of axial compression and cyclic lateral loading. The influence of axial loads,
transverse reinforcements in the wall boundary elements, and the effect of the presence
of construction joints at the wall base on the seismic behavior of walls are also reported
in this chapter. Towards the end of the chapter, reasonable strut-and-tie models are
developed to aid in understanding the force transfer mechanism and contribution of
reinforcement in walls tested.
In Chapter 3, three medium-rise RC structural walls with an aspect ratio of 1.625 are
tested to examine the structural performances of RC walls with limited transverse
reinforcement. The effect of axial loads, transverse reinforcements in the wall boundary
elements, and the effect of the presence of construction joints at the wall based on the
seismic behavior of walls are also reported from this chapter. Towards the end of the
chapter, reasonable strut-and-tie models are developed to aid in understanding the force
transfer mechanism and contribution of reinforcement in walls tested.
In Chapter 4, an analytical approach, combining inelastic flexure and shear components
of deformation, is proposed to properly evaluate the initial stiffness of the RC walls
tested. According to this verified analytical approach, an extensive parametric study
including a total of 180 combinations is carried out to investigate the influence of
several critical parameters on the initial stiffness of RC walls. Towards the end of the
chapter, a simple expression based on this parametric study is proposed to determine
the initial stiffness of walls as a function of three factors: yield strength of the
outermost longitudinal reinforcement, applied axial compression and wall aspect ratios.
In Chapter 5, a nonlinear finite element analytical procedure is provided to aid in better
understanding the global responses of squat RC structural walls with limited transverse
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Chal!.ter One
reinforcement. The global responses such as strength capacity, deformation
characteristics, energy dissipation capacity and equivalent damping of such structural
walls under reversed seismic loadings are described and investigated in detail.
Moreover, the influence of several paramount parameters such as axial loads,
longitudinal reinforcements in the wall boundary elements, aspect ratio, area of
boundary columns and the effect of the presence of construction joints at the wall base
on the seismic behavior of walls are also investigated by means of the developed finite
element models.
In Chapter 6, conclusions are drawn for this study and recommendations are presented
for future work.
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Chapter Two
CHAPTER TWO
SEISMIC PERFORMANCE OF
LOW-RISE STRUCTURAL WALLS WITH
LIMITED TRANSVERSE REINFORCEMENT
Abstract
The main objective of this study was to study the available ductility of low-rise
reinforced concrete (RC) walls containing less confining reinforcement than that
recommended by the New Zealand Concrete Design Code [NI] and American
Concrete Institute Code [AI]. Five RC walls, with an aspect ratio of 1.125, were
tested by subjecting them to low levels of axial compression loading and cyclic
lateral loading which simulated a moderate earthquake to examine the structural
performance of the walls with limited transverse reinforcement. Conclusions are
reached concerning the displacement capacity, strength capacity, curvature
distribution, the secant stiffness degradation and the energy dissipation
characteristics shown by the walls on the seismic behavior with limited transverse
reinforcement. The influence of axial loads, transverse reinforcements in the wall
boundary elements, and the presence of construction joints at the wall base on the
seismic behavior of walls are also reported in this study. Towards the end of the
chapter, reasonable strut-and-tie models are developed to aid in understanding the
force transfer mechanism and contribution of reinforcement in walls tested.
Keywords: Low-rise structural walls; Deformation capacity; Seismic performance;
Low to moderate seismic; Strut-and-tie model
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Chapter Two
2.1 Introduction and Background
RC structural walls are frequently used in buildings primarily to resist lateral loads
imposed by wind and earthquakes. The superior performance of buildings containing
structural walls in resisting earthquakes was well documented by Fintel [F I]. In
recent years, extensive research [LI, M2, PI-5, TI-2, and WI] have been conducted
to assess the validity of current design provisions [AI-2, CI, and NI] for cyclic
shear in squat reinforced concrete (RC) walls. Usually, these walls are detailed
according to the current provisions and aimed at achieving full ductile behavior of
structural walls. Presently, various countries have recommendations for confining
reinforcement to ensure that the required ductility demand can be achieved. This is
in light of design equations for the amount of confining reinforcement, the limitation
of stirrup spacing, and the length of confined regions. However, in low to moderate
seismicity regions such as Singapore and Malaysia, the ductility demands may not
be the same as that in higher seismicity zones. In such situations, less ductility
demand can be expected and therefore the required quantities of reinforcement,
especially the transverse reinforcement in web and boundary element, can be
reduced. In this research, the seismic performance of five low-rise structural RC
walls, which need only limited transverse reinforcements, is of interest.
Besides, many wall structures before the capacity design procedures were
introduced may possess considerable amount of inherent lateral strength which is
excess of that predicted for fully ductile systems. This suggests that the ductility
demand in such "strong" structures will be less and a trade-off between the ductility
and strength should be considered. In practice, such type of walls should not be
considered to be dangerous during earthquakes as long as the shear strength of the
walls is kept greater than that needed for a fully ductile structure and energy
dissipation is considered to be acceptable. As such, the potential strength of
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Chapter Two
structural walls is in excess of that required when considering fully ductile response
to the design earthquake. Thus, it is important to identify the strength capacity,
ductility capacity and energy dissipation capacity of such type of walls when
displaced its elastic limit.
Currently, although several experimental investigations [G1, M1, P2, Y1] of the
behavior of low-rise RC walls with limited transverse reinforcements under
simulated seismic loading have been conducted, there is still insufficient information
regarding the available ductility of walls with axial compression. Moreover, current
available experimental data related to the behavior of squat reinforced concrete
walls with weak interface like construction joints are rather inconclusive. The study
was motivated by the need to better understand the basic behavior of such walls.
Conclusions are reached concerning the displacement capacity, strength capacity,
curvature distribution, the secant stiffness degradation and the energy dissipation
characteristics shown by the seismic behavior of walls with limited transverse
reinforcement.
The present study for low-rise structural walls with limited transverse
reinforcements compiles information for the economical design of structures which
fall between full ductility and elasticity: structures with strengths greater than that
required by seismic loading for fully ductile behavior, or less important structures
which do not warrant detailing for full ductility. Besides, this comprehensive
experimental program can present better understanding on the basic behavior of
limited ductile structural walls with the presence of axial loadings, various quantities
of transverse reinforcements at boundaries, and the presence of construction joints.
Finally, the proposed strut-and-tie models can offer insights into the concept of shear
transfer and the contribution of reinforcement in reinforced concrete squat walls.
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Chapter Two
2.2 Experimental Program
The five low-rise structural walls, referred to as Specimens LWI-LW5, were
constructed and tested as isolated cantilever walls with an aspect ratio of 1.125. The
experimental program presented herein is aimed at investigating the performance of
squat reinforced concrete walls with limited transverse reinforcement. The effects of
reinforcement detailing, axial load and construction joints at wall base on strength
capacity, stiffness characteristics, and energy dissipation capacity of walls with
limited transverse reinforcement were investigated. Based on this, the experimental
procedure includes the following objectives:
(1) The basic performance such as the strength capacity, available ductility, and
energy dissipation capacity of the prototype walls should be the main
concern of this research;
(2) The effects of main test parameters on the performance of the structural
walls with limited transverse reinforcement;
(3) The load path and crack patterns should be presented at each ductility/drift
level to study the shear transfer mechanisms and failure modes;
(4) Deformations of the test specimens due to shear, flexure, and sliding should
be measured to investigate mechanisms of shear and flexure contributions.
2.2.1 Material Properties
Ready mixed concrete with 13 mm maximum aggregate specified by a characteristic
strength of 35 MPa was used to cast the specimens. A total of thirty-three
150x150x150 mm cubes and 150x300 mm cylinders were cast and tested
according to the standard procedures. The average concrete cylinder compressive
strength fe' for different specimens observed after 28 days was varied between
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Chapter Two
(2.1)
(2.2)A h = O.09sh Ie's e I
yh
Two types of steel bars, high yield steel bar (T bar) with the nominal yield strength
of 460 Mpa and mild steel bar (R bar) with the nominal yield strength of 250 Mpa,
were used in all specimens. Fig. 2.1 displays the typical stress-strain relationships of
32.43 Mpa and 38.81 Mpa.
2.2.2 Code Provisions for Confining Reinforcement in Plastic Hinge Regions
the bars. Among all bars, TID, RIO, and R6 were used in the walls while T13 and
T20 were used at the top and base beams.
Due to the existence of different requirements for the amount of confining
reinforcement in walls in the ACI 318 [AI] and NZS 3101 [N1] codes, design
equations in both design codes to ensure adequate ductility are described as follows.
2.2.2.1 ACI 318 Code Provisions
The required area of hoop reinforcement is given by the larger of
and
transverse reinforcement measured along the longitudinal aXIS of the structural
where Ash is the total cross-sectional area of transverse confining reinforcement
within spacing s and perpendicular to dimension he; s is the spacing of
member; he is the cross-sectional dimension of column core measured
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T_____ n____ .....
Chapter Two
center-to-center of confining reinforcement; Ag is the gross area of section; Ach
is the cross-sectional area of a structural member measured out-to-out of transverse
reinforcement;
2.2.2.2 NZS 3101:1995 Code Provisions
Where the neutral axis depth in the potential yield regions of a wall, computed for
the approximate design forces for the ultimate limit state, exceeds:
Cc = (0.3¢o / f.J)L w (2.3)
where cc is the distance of the critical neutral axis from the compression edge of
the wall section at the ultimate limit state; ¢Jo is the ratio of moment of resistance at
overstrength to moment resulting from specified earthquake forces, where both
moments refer to the base section of wall. f.1 is the displacement ductility capacity
relied on in the design; L w is the horizontal length of wall.
The following requirements of the transverse reinforcing steel shall be satisfied in
that part of the wall section which is subjected to compression strains due to the
design forces.
A = (l::.+O.I)s h A g Ie.' (~-0.07Jsh 40 h c A f L
c yh w
2.2.3 Application of Design Procedures
(2.4)
In the following, the flexure and shear design strategy for structural walls of limited
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Chapter Two
transverse reinforcements is presented in succession. As a general rule, the test walls
are proportioned to ensure an eventual flexure rather than shear failure, that is to
provide flexure strength less than the shear strength. The flexure strength of the
specimen is calculated by means of the conventional sectional analysis, taking into
account the effect of reinforcement hardening. The target for shear design of walls is
to prevent wall shear failure by providing shear resistance greater than the shear
demand. At the design stage, the shear demand is unknown, thus the designer tries to
provide shear supply greater than the factored design shears which is an estimate of
the shear demand and obtained from the results of earthquake response analysis. In
this case, prior to the application of demand analysis, the required lateral force is
given by Eq. (2.5) in terms of the pre-assumed displacement ductility capacity, 11;1. ,
of 3 which is the lower-bound value recommended by NZS 3101 code [N1] for the
fully ductile walls.
(2.5)
where VE is the shear demand derived from code-specified lateral static forces for
a given displacement ductility; OJv is the dynamic shear magnification factor and
t/Jo,w is the over-strength factor for a wall. This equation indicates that the ideal
shear strength of the wall need not be taken larger than that corresponding to elastic
response because limited inelastic deformations of the walls can be expected for
walls with limited transverse reinforcement. It is a conservative limit which is based
on the "equal-displacement concept". However, due to the presence of short period
for the assumed walls, the "equal-energy concept" should be applied as a more
practical limit. In this study, the reference wall is designed to resist a shear force of
VE = 320 kN according to a displacement ductility capacity of Il!),. = 3 .
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Chapter Two
For the determination of the required amount of shear reinforcement, the
contribution of concrete in the total shear resistance should be estimated. As seen in
the literature review, there existed a large number of empirical equations for the
contribution of concrete to shear strength, each with its own range of applicability.
However, none of the equations apply precisely to the present case, low-rise walls
with limited transverse reinforcements under seismic loading. Thus, the calculation
of the ideal shear strength for the present test units was somewhat undefined. In this
case, the shear strength of low-rise walls is evaluated by the method proposed by
ACI 318 [AI] or NZS 3101 [N1]. As stipulated in these codes, the shear strength of
low-rise walls is calculated by the sum of the shear contributions from the concrete
and web shear reinforcements. Represented by the concrete shear stress, the
contribution of concrete [N1] is taken as lesser of following:
..,.
~ N uv =0.27"Jfc +4A
uc g
or
I (0 1 ~f' Nvue =O.os!i: + w • VJe +O.2f)
(~u _I; J g
(2.6)
(2.7)
where N u is the axial load (negative for tension), and M u ' Vu are the moment and
shear force respectively at the section at the ultimate limit state.
Based on the modified truss analogy and by assuming the strut angle to be 45°, the
contribution ofweb transverse reinforcement at a spacing of s is given by
v _ Atfytdwus -
s
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(2.8)
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Chapter Two
where At is the area of transverse reinforcement within a distance s, d w is the
effective length of wall and normally set to be 80% of the whole wall length.
The controlled wall LWI, which is designed based on aforementioned
considerations, is shown in Table 2.1. All five specimens have the same web
reinforcement, consisting of two curtains (orthogonal grids) of 10 mm diameter high
yield steel bars spaced at 250 mm. This gives a reinforcement ratio of 0.50 percent.
The 150 mmx 300 mm boundary elements are reinforced with eight mild steel
bars of 10 mm diameter; giving a reinforcing ratio of 1.4 percent (minimum code
requirement is 1.0 percent). Reduced confinement in the boundary elements is
provided by 6.0 mm diameter closed stirrups (hoops) spaced at 75 mm
corresponding to 70% of the transverse confining reinforcement required by NZS
3101 at a limited displacement ductility of 3.0 and 25% of that required by ACI 318
as shown in Eq. (2.4) and Eq. (2.1) respectively. While for Specimen LW3, the
vertical spacing of the hoops is meant to be 200 mm which corresponds to 30% and
10% of the transverse confining reinforcement required by NZS 3101 and ACI 318
respectively for seismic detailing of fully ductile walls. The overall dimensions and
reinforcement details of the controlled specimen LWI are shown in Fig. 2.2. Each
low-rise wall specimen in this test was 2000 mm wide, 2000 mm high and 120 mm
thick. This provided the value of aspect ratios for all five specimens with 1.125
which was calculated according to the respective wall height of 2250 mm measured
by the vertical distance between the lateral loading point and the wall base.
The specimens were cast monolithically in the vertical direction except that for
Specimens LW4 and LW5 with construction joints at the base, they were kept to
stand for three days after the concrete for the base beam had been poured, vibrated,
and leveled off. Just before the upper part of concrete was poured, the hardened
concrete and the reinforcing bars in the construction joint area were brushed to
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..
Chapter Two
remove any loose particles. Then, the base beam concrete surface was moistened
and the fresh concrete was poured into the upper part of the moulds. After seven
days, the moulds were removed and the specimens were allowed to expose to
laboratory environment until just before testing. In the end, a day prior to testing the
outer surface of every specimen was made extra smooth for drawing crack patterns
during testing.
2.2.4 Experimental Set-up and Loading History
The test rig used in this study is shown in Fig. 2.3. It consisted of two main
independent systems: an in-plane loading system and an in-plane base beam reaction
system. The in-plane loading system comprises one horizontal hydraulic actuator
which was fixed to the reaction wall and two vertical actuators connected to the
strong floor. The test units were subjected to in-plane, reversed cyclic loading from
the horizontal double-acting actuator applied at the level of the top steel transfer
beam (transfer beam 2) as shown in Fig. 2.3. The hydraulic actuator with 1000 mm
stroke possessed a capacity of 1000 kN in compression and tension. The axial
loading was applied through two vertical actuators, each with a compression
capacity of 1000 kN and 500 mm stroke, attached to the top beam system, as shown
in Fig. 2.3. Fig. 2.3 also shows two steel transfer beams, transfer beam 1 and transfer
beam 2 which were conservatively designed and built to transfer the lateral loading
to the test specimen. It should be noted that high yield steel bolts were provided in
the connection and tightened to efficiently transfer the lateral loading to the test
specimen. Two levels of constant axial load were adopted in the testing program.
They corresponded to 0.0 and 0.05 of the cylinder compressive strength of the wall
cross section that is equal to fc'Ag
which was considered representative of the
amount of axial load at the base of the wall of a single story and a low-rise building,
respectively. After the total constant axial force was applied, the horizontal loading
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Chapter Two
would be introduced through the top steel beam (transfer beam 2) of the specimens.
The base beam of the specimens was anchored to the laboratory floor with twelve
high strength rods that prevent uplifting of the specimens and horizontal sliding of
the units along the floor during the application of the horizontal loading. Every high
strength rod was pre-stressed to efficiently restrain the rotation and sliding of the
specimen during the test. During the experimental set-up, laser point machines were
employed to ensure that the center lines of the three actuators and the specimen
tested are in the same vertical plane. This is to effectively avoid out-of-plane
buckling of walls during testing.
In the previous tests of ductile members [M1], the displacement loading, in which
the level of displacement increases according to the ductility factor (f.1tJ. =~/~y ),
controls the subsequent cycles. However, since the present test units were expected
to exhibit only limited ductility, it was thought more valuable to displace the units to
deflections corresponding to selected values of drift due to the fact that the ductility
factor (f.1tJ. =~/~ y ) depends heavily upon the definition of ~ y which is not readily
identified. As mentioned in the previous research [M1], a value of ~/ hw
=0.01 is
considered a practical limit on the drift as this value realistically is expected in
low-rise structural wall buildings. Herein the end of the test was reached at a drift
ratio of 1.0% or the strength dropped to less than 80% of the recorded maximum
loading. The test specimen was subjected to two cycles at each displacement level
except that only one cycle was applied to the specimen at a drift ratio of 1/2000. Fig.
2.4 demonstrates the detailed loading sequences during the testing. The axial loading,
whenever present, would be kept constant during the entire test by applying load
control to the vertical actuator. Moreover, hinged connections at the tips of both the
vertical and the horizontal actuator prevent any substantial restraint to the rotation of
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the top of the wall, thus insuring cantilever behavior.
2.2.5 Instrumentation of Wall Specimens
For measurement of top deflection, flexure deformations and shear deformations,
two types of Linear Variable Differential Transducers (LVDT), with 100 mm travel
and 50 mm travel, were introduced as shown in Fig. 2.5. One LVDT numbered as
L14 with 100 mm travel was installed at the top of the specimen to monitor the top
displacement. A total of ten LVDTs were arranged along the two vertical edges of
the specimens to measure the flexure deformation. The panel shear deformations
were detected by two LVDTs (L4 and L5) distributed along diagonal directions of
the panels. Two inclined LVDTs (L2 and L3) with one end of the steel rods at the
base beam were used to the measure the sliding deformations of the wall panels. The
sliding of base beam was detected by one LVDT named as Ll with 50 mm travel
positioned at the strong floor.
FLA type 5 mm-gauge length strain gauges with 10m vinyl-insulated lead wires
were used to measure the local strains of the selected reinforcing bars. The strain
gauges were attached to merely one layer of the reinforcing nets such as only the
outer layer for the bars of boundary elements was considered to attach the strain
gauges due to the symmetric configurations of the wall units. During testing, the
strains of the bars were recorded by an automatic data-logger, and a strain gauge was
deemed no longer reliable when the strain exceeded 0.02.
2.3 Experimental Results of the Tested Specimens
The global behavior, represented by crack patterns and hysteretic loops, and local
response such as longitudinal and transverse bar strain distribution of the tested
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specimens are presented in the following figures. For those concerning the crack
pattern at different drift ratios, it should be noted that the dashed lines in the grids,
indicating the spacing of the reinforcement, represent the negative cracks opened
during negative loading while the continuous lines refer to the positive cracks in the
positive loading. The blackish areas represent the splitting of the concrete.
2.3.1 Experimental Results of Specimen LWI
Unlike the applied loading history of other specimens as shown in Fig. 2.4, the
reference wall LWI as shown in Fig. 2.2 was firstly displaced with one cycle at drift
ratios of 0.1 %, 0.25% and 0.33% and then it was applied with two cycles at each
continued drift ratio of 0.5% or 1.0%. However, such type of loading history was
later found to be providing inadequate data for analyzing the seismic performance
and that in the other specimens, minor modifications in the loading sequences were
made. The modified loading history is illustrated in Fig. 2.4.
2.3.1.1 Global Behavior
Fig. 2.6 demonstrates the crack patterns and failure modes of the reference wall
LWI at drift ratios of 0.1 %, 0.5% and 1.0% corresponding to the initial cracking
stage, crack development stage and final failure stage, respectively. As shown in Fig.
2.6(a) the initial flexure cracks at the lower part of the reference wall were observed
at a lateral displacement of approximately 2.2 mm corresponding to a drift ratio of
0.1 %. With increasing lateral displacements, shear cracks propagated from the wall
boundaries toward the opposite side and from the bottom upward. Displacements up
to 5.0 mm first-yield of flexure reinforcements in wall boundary elements occurred
and thereafter a number of web cracks and their apparent inclination to the
horizontal increased due to the joining of web cracks with cracks originating higher
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up in the boundary elements were observed. Fig. 2.6(b) presented the post-yield
crack patterns at a lateral displacement of 11.2 mm corresponding to a drift ratio of
0.5%. By a displacement of 18 mm, the web cracks propagated more extensively
toward the opposite side in the lower part of the wall and several vertical cracks also
appeared at the boundaries near the wall base which indicated spalling of concrete
cover at these locations. At a displacement level of 22.5 mm the concrete in the
lower boundary elements of the wall was spalling considerably and the web cracks
in the lower part of the wall opened more widely.
The lateral load versus top displacement relationship for wall LW 1 is shown in Fig.
2.7. The ideal strength of ~ = 321 kN was obtained from rational section analysis
by use of computer program RESPONSE-2000 and exceeded by 12% and 2°~ for
the positive and negative loadings, respectively. It was observed almost linear elastic
behavior up to a magnitude of base shear of 304 kN (5 mm) which is close to the
predicted yielding of the outermost rebars. For the positive loadings, this presented
an initial stiffness of 60.6 kN/mm and hereafter the slope of the response curve
changed significantly which indicated considerably degradation of the wall secant
stiffness. The maximum base shear of 361 kN was achieved at 18 mm top lateral
displacement. At this time, a number of diagonal struts were formed and spalling of
the concrete cover at the wall base was observed. Increasing the top displacement to
22.5 mm resulted in a reduction of its shear capacity by 17% to a magnitude of 300
kN. At this last stage, it could be seen from Fig. 2.6© that the wall crack pattern was
dominated by flexure and hence a flexure - shear failure mode as expected was
developed.
2.3.1.1 Local Response
Fig. 2.8 shows the longitudinal strain distribution of the outermost rebar (#A bar)
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with the variation of wall drift ratios in three different positions along the wall
height: 50 mm, 550 mm, and 1550 mm above the wall base. Longitudinal strains
under both positive and negative loading directions are demonstrated. Under the
positive loading direction, it is observed that the first yield of outermost bar occurs
at a drift ratio of 0.25% and thereafter the strain increases more rapidly. However, in
the negative loading direction, the strain of the longitudinal bar under compression
is observed to be not yielding. This reduction of the compression strain can be due to
the effect of existing residual tensile strain of the longitudinal bars during testing.
Under the negative loading direction, the strain distribution of other longitudinal
bars along two different sections is also illustrated in Fig. 2.9(a) and Fig. 2.9(b),
respectively. The two wall sections, section 1-1 and section 2-2 as shown in Figs.
2.9(a) and 2.9(b), are located at wall heights of 50 mm and 550 mm, respectively
above the wall base. As indicated in Figs. 2.9(a) and 2.9(b), with the increase of wall
drift ratios, the neutral axis depth for both two wall sections is observed to shift from
the section middle to around 200 mm calculated from the left flange. The flexure
plane section hypothesis can be applied to wall section 1-1 till a drift ratio of 0.33%,
while it is not suitable for wall section 2-2.
Figs. 2.1 O(a) and 2.1O(b) depict the strain distribution with the variation of the wall
drift ratios for two different horizontal web bars: R bar and T bar, respectively. The
two horizontal bars locate at the wall heights of 250 mm and 750 mm from the base
and in each horizontal bar, the strain values obtained from four strain gauges are
plotted under both positive and negative loading directions. In general, the strains in
the specified horizontal bars seldom reach yielding. In the positive loadings, larger
strain values are observed to occur along the diagonal strut (R7, R8 in R bar and T14,
T15 in T bar) and under the negative loadings, it is likewise obvious (R5, R6 in R
bar and T13, T14 in T bar).
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2.3.2 Experimental Results of Specimen LW2
The geometry layout and reinforcement details of specimen LW2 are presented to be
the same as that of specimen LW 1. The test of specimen LW2 is intended to
investigate the effect of low to moderate axial load on the global behavior and local
response of low-rise structural walls. Different from that corresponding to specimen
LW1, the applied loading history with the increase of wall drift ratios is illustrated in
Fig. 2.4.
2.3.2.1 Global Behavior
The crack patterns and failure modes of specimen LW2 subjected to axial loadings
are shown in Fig. 2.11. Fig. 2.12 presents the typical hysteretic loops of specimen
LW2. As shown in Fig. 2.11, two initial flexure cracks located at the lower part of
each wall boundary element were observed at a drift ratio of 0.1 % for a base shear
of approximately 276 kN. With increasing cycle numbers, the outermost flexure
reinforcement in left boundary elements experienced yielding at a drift ratio of 0.2%
(4.5 mm) corresponding to a base shear of 400 kN and based on this, a magnitude of
yield displacement of 5.9 mm was obtained as shown in Fig. 2.12. For a
displacement up to 11.25 mm, the flexure cracks in the boundaries became denser
and extended up to the top of the wall. At this stage, almost all shear cracks in the
web penetrated into the opposite side of the wall were developed. The maximum
base shear of 579.9 kN was achieved at a displacement of 22.68 mm corresponding
to a displacement ductility of 3.8 for the positive loadings. With increasing
displacements, it was observed that the inclined web cracks opened more widely and
formed several diagonal struts apparently.
Final failure occurred with the observed behaviors that the small quantity of
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concrete in the right flange near the bottom was spalling and four shear cracks in the
lower part of web opened significantly wide. By comparison with the crack patterns
of the reference wall LWI, less shear cracks in the web of the wall LW2 was
observed due to the presence of the axial loadings which played a beneficial role in
controlling wall cracking.
2.3.2.2 Local Response
A typical strain distribution of an outermost longitudinal bar (#A bar) in left wall
flange along the wall drift ratios is shown in Fig. 2.13. As indicated in Fig. 2.13, first
yielding of the longitudinal bar occurs at a drift ratio of approximately 0.2%.
Maximum strain, which is less than that of specimen LW 1, is obtained for strain
gauge A31 located at 550 mm from the base at ultimate state. This may be induced
by the presence of low to moderate axial loads in the wall during testing which
inhibits the development of the flexure yielding of longitudinal bars at the base. Figs.
2.14(a) and 2.14(b) present the strain profiles of the vertical bars along section 1-1
and section 2-2, respectively. As can be seen from these two figures, plane section
hypothesis can not be applied to wall section 1-1 after the attainment of a wall drift
ratio of 0.5% corresponding to the crack development stage.
The strain distribution of two horizontal web bars with the variation of the drift
ratios is illustrated in Figs. 2.15(a) and 2.15(b). It can be seen that the strains are
affected greatly by the presence of nearby cracks. A rapid increase in the plot
generally results from a crack having crossed the horizontal bar at the position of the
strain gauge. The stress in the surrounding region of concrete is effectively
transferred to the reinforcing bar at the crack.
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2.3.3 Experimental Results of Specimen LW3
The main purpose of this testing was to investigate the confinement effect of
transverse reinforcements in wall boundaries on the seismic performance of the
specimen. For specimen LW3, the vertical spacing of transverse reinforcements in
wall boundaries was modified from 75 mm, which was applied in wall LW2, to 200
mm. This led to the confinement reinforcement ratio represented by volumetric ratio
of transverse reinforcement in wall boundaries changing from 0.85% to 0.35%.
2.3.3.1 Global Behavior
The crack patterns and hysteretic loops of specimen LW3 are shown in Fig. 2.16 and
Fig. 2.17, respectively. Hairline flexure cracks in the boundaries were initiated at a
drift ratio of approximately 0.12% corresponding to a base shear of 270.4 kN which
is close to that recorded by wall LW2. The first post-cracking patterns of wall LW3
at a drift ratio of 0.17% is presented in Fig. 2.16(a). Cycling the wall to a drift ratio
of 0.22%, yielding of the outermost flexure reinforcement in wall left boundary
occurred and the recorded base shear at this stage reached at 402.3 kN. Fig. 2.16(b)
shows the post-yielding crack patterns of wall LW3 at a drift ratio of 0.5% which
were rather similar to those of wall LW2 with the observed behavior that almost
horizontally flexure cracks extended up to the top of wall and inclined shear cracks
propagated from the wall boundaries towards the opposite side. At that stage in
testing, the maximum base shear of 523.8 kN was achieved corresponding to a
lateral displacement of 11.25 mm for the positive loadings. When the displacements
were further increased to 15 mm, severe deterioration of the concrete within the left
wall flange at a wall height of approximately 500-1000 mm was observed and the
buckling of flexure reinforcements in this area followed. The final crack patterns and
failure modes of wall LW3 are shown in Fig. 2.16(c).
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2.3.3.2 Local Response
The plot of strain of the longitudinal bar (#K bar) in the vertical boundary elements
of the specimen subjected to both positive and negative loading directions is shown
in Fig. 2.18. Strains reach approximately yield strain at a drift ratio of 0.22% just
before the attainment of flexure strength. In the positive loading direction, the bar is
subjected to compression and the measured strain is less than that under the negative
loading direction. In the tensile boundary element, strains are not distributed
uniformly or changed rapidly with height above the base level. This mixed pattern of
strain distribution is clearly indicative of both the flexure and shear interaction. The
strain distribution of the vertical bars in wall flanges and web along the wall sections
under the negative loadings is also illustrated in Figs. 2.19(a) and 2.19(b). It is seen
that in moving from horizontal section 1-1 (Fig. 2. 19(a)) to horizontal section 2-2
(Fig. 2.19(b)), the neutral axis shifts toward the tension face of the wall. At section
1-1, compressive strains at ultimate state are observed within approximately 200 mm
( O.ILw ) of the compression face. At section 2-2, compressive strains are attained
within 500 mm ( 0.25Lw ) of the compression face.
The strains along two different selected horizontal web bars at the various drift
peaks are shown in Figs. 2.20(a) and 2.20(b). In general, the strains are observed to
be small and are concentrated along the main diagonals (R8, T15 for positive
loading direction and R6, T14 for negative loading direction).
2.3.4 Experimental Results of Specimen LW4
The specimen LW4 was chosen to study the effect of existing dry construction joints
on the seismic behavior of the walls. The overall dimension and reinforcement
detailing of the specimen were the same as that of the reference wall LW1 except
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that there were dry construction joints at the base of wall LW4.
2.3.4.1 Global Behavior
The crack patterns and failure modes of wall LW4 are shown in Fig. 2.21. It can be
seen that the specimen experienced first cracking at a drift ratio of 0.1 % for a base
shear of approximately 203.5 kN. In light of recorded strain of flexure
reinforcements in the boundary elements, the first yielding of those bars was
observed at a drift ratio of 0.2% for a base shear of 300.6 kN. With increasing cycle
numbers, similar developing patterns of the flexure and shear cracks with those of
the reference wall LW 1 were observed until the wall LW4 was displaced to a drift
ratio of 0.67% corresponding to a maximum base shear of 365.1 kN. After that stage
in testing, the flexure cracks at the bottom of the wall interconnected roughly and
the sliding shear failure occurred when the lateral displacements of the wall reached
at a drift ratio of 1.0%.
By contrast with the wall LWl, the hysteretic loops of the wall LW4 as shown in Fig.
2.22 depict more degree of pinching due to the presence of dry construction joints at
the wall base which resulted in a reduction of the sliding shear strength and hence
the sliding shear becomes more important.
2.3.4.2 Local Response
Fig. 2.23 plots the strain distribution in the outermost longitudinal bar (#A bar) in
vertical flange of specimen LW4. First yield strain is reached at a drift ratio of
0.25°A> under the positive loading direction. In the compressive flange, strains are
small and less than that in the tensile flange for a given drift ratio. After the
attainment of drift ratio of 0.5%, it is observed from Fig. 2.23 that almost the whole
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bar along the wall height experiences yielding since the ideal flexure strength is
attained. Moreover, rapid increase in longitudinal strain at wall height 50 mm
(position A29) occurs at a drift ratio of 0.67% which can be induced by the strain
gauge crossed by a crack. The strains in other vertical bars of the wall flanges and
web along with the wall section 1-1 and section 2-2 are also presented in Figs.
2.24(a) and 2.24(b), respectively. It can be seen that plane section 1-1 does not
remain plane after a drift ratio of 0.5%. This indicates an increasing influence of
shear on the wall behavior. When there is a change of wall sections from section 1-1
to section 2-2, the neutral axis moves away from the compression face of the wall.
Strains along two different horizontal web bars, R bar and T bar, at the various drift
ratios are shown in Figs. 2.25(a) and 2.25(b), respectively. In general, the strains are
small and increase with added wall drift ratios. The strains are mainly concentrated
along the main diagonals.
2.3.5 Experimental Results of Specimen LW5
For the purpose of investigating the behavior of walls with existing construction
joints at the base and at the same time exposed to a medium level of axial loadings,
the wall LW5 was tested to make a comparison with the experimental results of the
wall LW2 which was subjected to a same level of axial loading but with no
construction joints at the wall base.
2.3.5.1 Global Behavior
The crack pattern and final failure mode of the tested wall are shown in Fig. 2.26.
During testing, the initial flexure cracks in the lower part of the wall developed at a
displacement of 3.25 mm for a base shear of approximately 310.4 kN. Increasing
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displacement to 4.8 mm resulted in the yielding of the flexure reinforcements in the
boundary elements. Specimen LW5 developed its maximum base shear of 520 kN at
15.39 mm top lateral displacement. At this time, a large portion of the cracks was
fonned and several diagonal struts crossed in the middle of the wall web. Further
cycling the wall LW5 to 22 mm lateral displacements resulted in several diagonal
cracks extended further to form diagonal comer to corner cracks and cracks at the
lower part of the wall were found to open more widely. Fig. 2.27 illustrates the
lateral load versus top displacement relationship of the specimen LW5. By contrast,
it can be seen that the crack pattern prior to failure was very similar to that of
specimen LW2 but the load- carrying capacity of specimen LW2 was found to be
more significant than that of specimen LW5 after maximum base shear was attained.
2.3.5.2 Local Response
Fig. 2.28 plots the strain distribution in the outennost longitudinal bars of specimen
LW5. Strains of other longitudinal bars in the wall flange and web along two wall
sections, section 1-1 and 2-2, are also presented in Fig. 2.29(a) and 2.29(b),
respectively. As shown in Fig. 2.28, the strains are approximately unifonn with
height above the base level. This can be due to the increasing influence of
strut-and-tie action on the wall behavior. The plot of Fig. 2.29(a) shows that the
plane section 1-1 does not remain plane after wall drift ratio of 0.67% is attained. As
the test progressed, the neutral axis shifted from the wall middle towards the
compression face of the wall.
Figs. 2.30(a) and 2.30(b) illustrate the strain distribution in the two horizontal web
bars, R bar and T bar, respectively. It can be seen that the strains are concentrated
along the main diagonal strut. The plot of Fig. 2.30(b) shows that the strains in the
center of the T bar (T14 and T15) are large under both loading directions since this
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portion of wall is situated along the main diagonals for both loading directions. The
bars are stressed in tension for both loading directions.
2.4 Discussion of Experimental Results
The main experimental results for each specimen tested are listed in Table 2.1. In
this section, a simple discussion of experimental results is presented in terms of the
observed behavior of each specimen tested. Note that different methods of defining
the yield displacements of walls existed. In this testing, the measured displacement
at yield state is determined through electrical strain gauges, i.e. when the strain of
flexure reinforcements reached the value of 0.2 percent, the displacement and load
reading are recorded.
2.4.1 Crack Patterns and Failure Modes
It was observed that the initial flexure cracks located at the lower part of specimens
occurred at a drift ratio ranging from 0.1 % to 0.17% within the length of the
boundary elements. This value of drift ratio may be considered as a serviceability
limit of the structural element. With regards to specimens without axial loadings
(LW1 and LW4), these initial flexure cracks were developed at a base shear of
approximately 204 kN while for specimens subjected to axial loadings the observed
base shear corresponding to first cracking of the walls was approximately 270 kN.
Further increasing the wall top displacements resulted in all specimens experiencing
the yielding of flexure reinforcements in the boundaries at a drift ratio of
approximately 0.2%. Moreover, a base force of approximately 300 kN was achieved
at the time of flexure yielding for specimens LW1 and LW4 without axial loadings
while the observed shear force for corresponding specimens subjected to axial
compression was approximately 400 kN. With increasing lateral displacements up to
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reaching at the maximum base shear, the flexure-shear cracks of all specimens were
extended up to the wall top and a number of diagonal struts were formed to
efficiently transfer the lateral loading from the wall top to the bottom. Further
cycling the walls to maximum lateral displacements observed no more emergence of
new cracks but existing cracks especially those located at the lower part of the walls
were found to be widely opened and meanwhile spalling of the concrete cover at the
wall base occurred.
In general, all specimens behaved in a flexure manner, characterized by concrete
crushing and reinforcement buckling at the wall boundaries. Meanwhile, moderate
diagonal cracking of the web and shear sliding at the base for almost all specimens
were also observed. It should be noted that the damage of the specimens was mainly
located at the lower part of the walls except for wall LW3, whose the damage zone
extended upwards and occurred within the left wall flange with a height range from
approximately 500 mm to 1000 mm.
2.4.2 Backbone Envelopes of Load-displacement Curves
The backbone envelopes of load-displacement curves have long been recognized to
be a critical feature in modeling the inelastic behavior of RC walls. It was generally
generated with the curve determined from a monotonic test and herein was
constructed by connecting the peaks of recorded lateral load versus top displacement
hysteretic loops for the first cycle at each deformation level of the tested specimens.
Fig. 2.31 shows the backbone envelopes of load-displacement curves of all
specimens tested along with the estimated average flexure capacities (refer to Table
2.1). It can be seen clearly from the figure that the presence of axial loadings
significantly increases the strength and the stiffness of the tested walls. Interestingly,
for tested walls with or without axial loadings, similar top drift was achieved for
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tested walls with only one exception (Specimen LW3). For this specimen, the
smallest drift capacity was observed due to the fact that inadequate transverse
reinforcements were provided to prevent the longitudinal reinforcements in the
boundaries from buckling.
As observed from Table 2.1, all five specimens were capable of developing their
flexure strength prior to failure, which is a prerequisite of adequate seismic
performance. Moreover, in the case of specimens with construction joints, similar
maximum flexure strength was developed by comparing with that of corresponding
specimens without construction joints. This observation does clearly indicate that for
the specimens tested, sliding shear will not inhibit the development of flexure
capacity. Meanwhile, a good agreement between the calculated maximum strength
and measured maximum flexure strength was observed which indicates that
maximum strengths for all specimens were governed by the maximum flexure
strength obtained from inelastic section analysis.
2.4.3 Components of Top Deformation
The top deformation of walls in this testing is mainly caused by flexure
displacement, panel shear displacement, and sliding displacement. Note that the
flexure component of the total deformation also included the contribution from bond
slip in longitudinal bars at the base of the wall. From figures of strain profiles such
as Figs. 2.9(a) and 2.9(b) etc. for all five tested specimens, it is found that flexure
deflection measured by LVDTs at the left and right side of tested specimens could be
overestimated and as such the effect of internal strains should be considered to
accurately evaluate the flexure deflections. For this purpose, the plane section of
tested specimens is assumed to deflect with respect to the best fit line of internal
strain distributions and in this study nonlinear strain distributions along the wall
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sections 1-1 and 2-2 for each tested specimen are represented by linear trend lines
which consider the internal strains mostly. It is found by comparing the results
considering the effects of internal strains that the flexure deformations of tested
specimens by use of this kind of LVDT measure arrangement are overestimated by a
percentage value ranging from 4.87 to 7.97. Fig. 2.32 illustrates the ratios of three
displacement components to the total deformation with respect to wall drift ratios by
considering the effect of internal strains on flexure deflection of tested specimens.
From the figure, the relative contribution of three displacement components varied
in terms of the wall drift ratios with the observed behavior that with increasing
lateral displacements up to wall failure the contribution of flexure deformations
decreased while the ratios of the other two components tended to rise slightly. In
general, the flexure deformations dominated the response because it accounted for
more than 50% of the total displacement up to the final loading stage for all
speCImens.
For the reference wall LW1, the sliding shear components accounted for less than
5% initially in both loading directions, but for 15% close to failure. This varied trend
is more pronounced for the specimen LW4 with construction joints at the base due to
the fact that the sliding components made up approximately 23% at the final stage
and meanwhile, exceeded the contributions from shear displacements which was
generally observed to be higher than the sliding contributions in other specimens.
However, in the case of walls subjected to axial loading including the Specimen
LW5 with construction joints at the base, the contribution of sliding shear did not
exceed 13%. This indicated that the axial loading played a favorable role on
controlling the wall sliding deformations as expected and under axial compression,
the presence of the construction joints at the wall base had a minor effect with
respect to sliding, as only up to approximately 13% of the total displacement of
Specimen LW5 was due to this mode compared with 10% of that contribution in the
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comparable Specimens LW2 and LW3.
In contrast with the contributions of various deformation modes for Specimen LW3
as shown in Fig. 2.32©, the flexure displacement contribution of Specimen LW2 as
seen from Fig. 2.32(b) tended to be greater (73% compared with 62% in the positive
loading direction) at a drift ratio of 0.67% while the shear components of total
displacements becomes lesser at that stage (12% compared with 21 % of the total
displacements). This suggested that for Specimen LW2 with more content of the
transverse reinforcements in wall boundaries, the flexure contribution of the total
deformation became greater while the sliding component of both walls was observed
to be almost the same. Hence, it can be considered that the content of transverse
reinforcement for flanged specimens could have an important effect in achieving a
more ductile hysteretic response.
2.4.4 Curvature Distribution along the Wall Height
The wall curvature distribution along the wall height, but only the first cycle at
certain drift level is shown in Fig. 2.33 for all specimens tested. It is observed that
the curvature was highly concentrated at the bottom region and, for wall region
higher than 500 mm from the wall base, the curvature remains constant at a lower
level. In negative and positive loading directions, the bottom curvatures of the
specimen were observed to be close with respect to certain drift ratios. After
attaining the wall drift ratio of approximately 0.33%, the bottom curvatures
increased significantly. Moreover, at a drift ratio of 1.0% the average curvature of
Specimen LWI as shown in Fig. 2.33(a) was observed to be approximately 20%
higher than that of Specimen LW2 as shown in Fig. 2.33(b). Similar conclusion can
be made by comparing the bottom curvatures of Specimen LW4 as shown in Fig.
2.33(d) with that of Specimen LW5 as shown in Fig. 2.33(e). This indicates that
higher level of wall average curvatures can be attained for the specimen without
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axial loadings (Specimen LW4) than that corresponding to specimens subjected to
axial compression (Specimen LW5). Moreover, the rate of increasing wall
curvatures tends to rise with respect to the increasing drift ratios of the walls.
2.4.5 Stiffness Characteristics
Previous research [AI, P4, T2] indicated that the true stiffness of the wall elements
is significantly lower than that corresponding to gross section properties, even at the
serviceability limit state. It is, therefore, essential to evaluate realistic stiffness
properties of wall elements which can lead to more accurate modeling and analysis
of RC buildings with structural walls. The values of initial stiffness for each
specimen in both loading directions, which defined at the first yielding of
longitudinal reinforcements at wall boundaries, are listed in Table 2.1. It can be seen
from the table that the average values of initial stiffness for the selected specimens
were attained to be approximately 78.4 kN/mm for walls subjected to axial
compressions, and 60.0 kN/mm for the specimens without axial loadings. Fig. 2.34
demonstrates the detailed stiffness properties of the walls which were evaluated
using secant stiffness at the peak of first cycle at each deformation amplitude. As
expected, all specimens experienced considerable reduction in stiffness with
increasing wall deformations. The secant stiffness of each specimen rapidly dropped
to about 55% of its uncracked stiffness by a drift ratio of approximately 0.2%
corresponding to the yielding state of the wall. With the increase of the wall top drift,
the stiffness of each specimen further decreased and at the final stage in testing
which only accounted for 15% of its uncracked stiffness.
By comparing with two other specimens subjected to axial loadings, more severe
degradation of stiffness for Specimen LW3 with inadequate transverse
reinforcements provided at wall boundaries was observed at a drift ratio of 0.67%.
- 34-
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Chapter Two
This indicated that such type of wall suffered more apparent strength degradation
when the same displacement amplitude was reached. The degradation ratio of secant
stiffness, which evaluated by dividing the values of secant stiffness at the initial
loading stages by those corresponding to the final loading stages, was achieved to be
about 85% for all specimens with or without axial loadings. This suggested that such
low level of axial loadings had a minor effect on the degradation rate of secant
stiffness despite the fact that the presence of axial compressions in specimens can
lead to higher secant stiffness in contrast with those without axial loadings subjected
to at the same drift ratios.
2.4.6 Energy Dissipation
The energy dissipation capacity for each specimen which is calculated from the
inner area of load-displacement curves has long been recognized to be of paramount
importance in the evaluation of the seismic performance of RC walls. Fig. 2.35 plots
the energy dissipation capacity of each specimen with respect to its drift ratios. The
observations indicate that prior to yield, rather small amount of energy was
dissipated, and thereafter the increase rate of energy dissipation for all specimens
tended to rise with increasing top drift ratios. With respect to specimens subjected to
axial loadings, Specimens LW2 and LW3 as well as LW5, the amount of energy
dissipated was larger than that corresponding to specimens without axial
compression. This can be due to the favorable effect of the axial compression with
regard to controlling the pinching of hysteretic loops. Moreover, for Specimens LW4
and LW5 with construction joints at the wall base, as expected, lower amount of
energy was dissipated in contrast to that dissipated by Specimens LWI and LW2,
respectively which was especially obvious when the wall drift ratio was higher than
0.5%. This can be explained by the presence of construction joints at the wall base
which presented excessive sliding shear displacements and therefore led to
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J
Chapter Two
significant pinching of hysteretic loops of specimens at the last stages in testing.
Meanwhile the amount of energy dissipated by Specimen LW2, which had more
dense transverse reinforcement at wall boundaries than corresponding Specimen
LW3, was observed slightly larger than that by Specimen LW3 up to a wall drift
ratio of 0.5%. But thereafter the increasing rate of energy dissipation for Specimen
LW3 was more rapid than that corresponding to Specimen LW2 up to the drift ratio
of 0.67%.
Meanwhile, for the purpose of comparing the amount of energy dissipated by
individual components for each specimen, the flexure deformations, the shear
deformations, and the sliding deformations of all specimens are separated from their
top displacements and are plotted against the lateral loadings; Fig. 2.36 only shows
the loops of Specimens LW2 and LW4. From the figure, it is evident that the energy
dissipated by the flexure deformation is much higher than that from shear or sliding
deformations. However for Specimen LW4, the sliding shear displacements
contribute more to the total displacement and may significantly influence the flexure
behavior of the structure and thus decrease the energy dissipation capacity.
2.5 Extrapolation of Experimental Results
The shear force transfer mechanism of low-rise structural walls has been
investigated by many researchers [PI-3, WI, TI] and was well outlined by Park et al.
[P1] in which the shear force is transferred to the wall base by a middle strut and a
truss in the triangular region beside the strut. This mechanism of shear force transfer
in low-rise walls tested can be verified by the observations from previous figures
(For example, Figs. 2.IO(a) and 2.IO(b)) which show strain histories of gauges in
web horizontal reinforcement of tested specimens. It can be observed apparently in
these figures that the tensile strain of the horizontal bars in gauges along the
- 36-
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Chapter Two
diagonal strut is normally larger than that of gauges away from the strut.
As such, a strut-and-tie analytical model shown in Fig. 2.37 is proposed to simulate
the behavior of the reference wall LW1. The concrete contribution is provided by a
direct strut (dashed line) from the loading point to the base of the wall and is kept
constant after the onset of diagonal cracking. The inclination of the struts varies
from 28.4 to 50.6 degree which is close to the expected inclination of shear-flexure
cracks. The cross section of the concrete struts converging to the base of the wall is
approximately equal to Ac =1.4· c . b, where c is the depth of the compression
zone as shown in Fig. 2.38(a) calculated by the bending theory, and b is the wall
width. The area of the two outer struts, outer strut 1 and 2 as shown in Fig. 2.38(b),
is assumed to be 1/3 and 'l4 the area of the inner struts, respectively (Fig. 2.38(a))
since the shear force is mainly transferred by the inner diagonal strut. The eight
longitudinal bars in the web plus eight longitudinal bars in the flange are clustered in
the vertical member AE in the center of the flange. This consideration can be
validated from Figs. 2.9(a) and 2.9(b) since the strains of those bars included were
observed to be beyond yield strains at the maximum load. The transverse
reinforcement within a distance of 1000 mm, including six web horizontal bars in
the lower part of the wall, is concentrated in horizontal member CD. The failure load
of the truss is assumed when the longitudinal reinforcement is observed to be
yielded during tests.
The concrete contribution for shear in tested specimens can be estimated by the
strength at the onset of the diagonal cracks Vcr which can be detected by the strain
gauge attached to the horizontal reinforcement because significant tensile strain is
developed at this stage. When the horizontal reinforcement is sufficient to resist the
applied shear, the tensile strain of the reinforcement will be stable at a certain level
which is generally less than its yield strain, as shown in Fig. 2.1 O(a). The strengths
detected by the gauges attached to the web horizontal bars of tested specimens are
listed in Table 2.2. It can be seen that the detected strengths are close to that
- 37-
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Chapter Two
calculated by Eqs. (2.6) and (2.7) and it is within the range of approximately 10%.
Provided that the direct strut takes the shear force equal to the diagonal crack
strength 252.4 kN listed in Table 2.2, the other member forces can be determined.
The member forces at the maximum negative strength are presented in Fig. 2.37. By
employing this model to Specimen LW1, it can be observed that the average tensile
strain predicted by the assumed strut-and-tie model agrees well with the tensile
strain history of web horizontal bars in walls tested, as shown in Fig. 2.39. This can
be further verified by employing this model to Specimen LW4 as shown in Figs.
2.40 and 2.41.
For Specimen LW2 subjected to axial loads, the proposed strut-and-tie model is
appropriately modified as shown in Fig. 2.42. The six longitudinal bars in the web
plus eight longitudinal bars in the flange are clustered in the vertical member AF in
the center of the flange. This consideration can be validated from Figs. 2.14(a) and
2.14(b) since the strains of those bars included were observed to be beyond yield
strains at the maximum load. The contribution of web horizontal reinforcements is
considered to be within a distance of 1125 mm which includes 8 horizontal bars in
wall web. The angle of struts AC and DE is 31.30 that is close to the average angles
of diagonal cracks in the lower part of the wall. The area of the two outer struts,
outer strut 1 and 2 as shown in Fig. 2.38(b), is assumed to be ~ and ~ the area of the
inner struts (Fig. 2.38(a)), respectively. By use of this model, the average strain of
tie CD is evaluated and depicted in Fig. 2.43 which also shows tested strain history
of horizontal reinforcement in the lower part of the wall against the applied shear
force. It can be observed from Fig. 2.43 that the assumed strut-and-tie model is a
reasonable model for the flow of forces. This can be further verified by employing
this model to Specimens LW3 and LW5 with axial compression, as shown in Figs.
2.44 - 2.47. The maximum loads for all specimens calculated by the proposed model
are listed in Table 2.2. It can be seen that the analytical maximum loads show good
correlation with the tested maximum loads. Thus, the proposed model may provide
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Chapter Two
insights into the force transfer mechanism of low-rise walls with or without axial
loadings.
2.6 Conclusions
The experimental tests conducted on five RC walls with an aspect ratio of 1.125,
with low axial compression load and limited transverse reinforcement, subjected to
cyclic lateral loading simulating a moderate earthquake, showed that:
1. In general, all tested specimens with limited transverse reinforcement behave
in a flexure - shear manner and are capable of developing their flexure
capacity prior to failure. Values of drift at initial cracking range from 0.1 % to
0.17%. Specimens LWl, LW2, LW4 and LW5 generally exhibit more ductile
behavior than expected, even if they have insufficient confinement
reinforcement corresponding to 70% and 25% of the NZS 3101 and ACI 318
specified quantity of confining reinforcement, respectively. As shown in
Table 1, the displacement ductility factors of the four specimens are more
than 3.0 and can generally experience average story drift of at least 1%
without significant strength degradation. By contrast, Specimen LW3,
containing 30% and 10% of the NZS 3101 and ACI 318 specified quantity of
confining reinforcement respectively, shows quite critical seismic
performance with respect to the strength and deformation capacities achieved.
The displacement ductility of Specimen LW3 is observed to be less than 3.0
which proves to exhibit only limited ductile behavior of the wall as stipulated
in NZS 3101 code [N1]. As such, it is evident that the NZS 3101 and
ACI-318 requirements for wall boundary element confinement can be
relaxed for structural walls to some extent.
- 39-
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Chapter Two
2. Comparing the results considering internal strains, the flexure deformations
of tested specimens by use of this kind of LVDT measure arrangement are
overestimated by a percentage value ranging from 4.87 to 7.97.
3. It is also found in the testing that the level of axial compression loadings has
a minor effect on the degradation rate of secant stiffness despite the fact that
the presence of axial compressions in specimens can lead to higher secant
stiffness in contrast with those without axial loadings. For the wall
specimens subjected to axial compression loadings, the amount of energy
dissipated is larger than that corresponding to specimens without axial
compression due to the favorable effect of the axial compression with regard
to controlling the pinching of hysteresis loops. The amount of energy
dissipation due to shear components does not change much under the
condition of axial loadings on the specimen. With regards to the energy
dissipation contributed by sliding components, it is found to have increased
slightly due to the presence of construction joints at the wall base, but still
remains at a low level up to the final stage in testing which is especially
apparent in specimens subjected to axial loadings.
4. Two strut-and-tie analytical models for low-rise structural walls with and
without axial load respectively, accounting for different contribution of
horizontal and longitudinal web reinforcement, are developed to accurately
reflect the force transfer mechanisms of low-rise structural walls under cyclic
loadings. The tensile strains in horizontal web bars of low-rise structural
walls can be predicted by use of the assumed strut-and-tie model which
agrees well with the tested data. This provides evidence that the assumed
strut-and-tie models are reasonable models for the flow of forces and
contribution of web reinforcements in walls tested.
- 40-
...
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Tab
le1.
1-
Obs
erve
dst
reng
ths
and
duai
lity
of
spec
imen
ste
sted
No.
I(a
)I
(b)
I(c
)I
(d)
I(e
)I
(f)
I(g
)(1
)(2
)(3
)(4
)I
(5)
I(6
)I
(7)
I(8)
pJ'-;
rf,"~(ttn)
~/:mx(C(lj'.)
r.~~
..~y
PhI
P"I
PbI
PsI
P.\
ZS
P~.j
CIif
~4)
r~:......
.~K
iIJ1~
5~
~/o
~/o
~/o
~'o
%~
~,~
kNkN
kNn
un
kN'n
un
~"o
210.
036
1.0
321.
01.
12"\
...60
.64.
31.
0L
'V1
I0.
50-'.~
0.50
1.40
0.88
1.20
3.64
0.0
-204
.3-3
41.0
-321
.01.
06-5
.558
.4-4
.1-1
.0
1.20
I3.
64I
0.05
!27
5.9
579.
950
0.0
1.15
5.9
84.7
4.9
1.3
L'V
2I
0.50
I0.
50I
1.40
I0.
88I
-264
.0-5
62.7
-500
.01.
12-6
.675
.:-3
.3-1
.0
~L
'V3
I0.
50I
0.50
I1.
40I
0.33
I1.
20I
3.64
I0.0
5!
270.
452
3.8
500.
01.
046.
073
.52.
40.
67
-258
.5-5
61.0
-500
.01.
12-6
.283
.3~~
-0.5
-.........
0.0
~20
3.5
365.
132
1.0
1.14
5.5
58.4
4.0
1.0
L'V
4I
0.50
I0.
50I
1.40
I0.
88I
1.20
I3.
64I
-198
.3-3
63.4
-321
.01.
13-6
.053
.5-3
.7-1
.0
1.20
I3.
64I
0.05
:31
0.4
520.
050
0.0
1.04
6.5
76.9
3.3
1.0
L'V
5I
0.50
I0.
50I
1.40
I0.
88I
-323
.2-5
41.2
-500
.01.
08-6
.083
.3-3
.4-1
.0
Not
e:(a
)H
oriz
onta
lw
ebre
info
rcem
entr
atio
;(b
)V
erti
cal
web
rein
forc
emen
trat
io;
(c)
Fle
xm
alre
info
rcem
entr
atio
inbO
'Wld
aly
elem
ent;
(d)
Vol
umet
ric
rati
oo
ftr
ansv
erse
rein
forc
emen
tin
boun
dary
elem
em;
(e)
\·bl
umet
ric
rati
oo
ftr
ansv
erse
rein
forc
emem
requ
ired
by
NZ
Sco
de;
(f)
\·bh
unet
ric
rati
oo
ftr
ansv
erse
rein
forc
emen
treq
uire
db
yA
CI
code
;(g
)..o
\.."I
{ial
load
rati
o;
(1)
Obs
erve
dsh
ear
forc
eat
firs
tcr
acki
ng;
(2)~'Iaximun
obse
rved
stre
ngth
duri
ngth
ete
st;
(3)~faximum
idea
lfl
exur
alst
reng
th;
(4)
The
rati
oo
fm
a."I
{im
un
obse
rved
shea
rst
reng
thto
idea
lfl
exm
alst
reng
th;
(5)
Yie
lddi
spla
cem
emw
'hen
ou
ter
botm
dary
long
itud
inal
rein
forc
emem
syi
eld;
(6)
The
init
ial
stif
fnes
sfo
rth
e
ith
spec
imen
;(7
)r.,·
Iaxi
mun
disp
lace
men
tdu
ctil
ity
leve
l;(8
)r.
,'!ax
imum
top
drif
trat
ioac
hiev
ed.
9 ~ ~ ~ c
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Chapter Two
Table 2.2 - Strengths at onset of diagonal cracks of specimens testedand predicted by analytical models
..
~ LWI LW2 LW3 LW4 LW5
268.8 481.5 387.6 268.8 376.3Vcr (kN)
-252.4 -441.0 -414.5 -252.4 -380.9
Vc (kN)295.3 437.3 437.3 295.3 437.3-295.3 -437.3 -437.3 -295.3 -437.3
V IV0.91 1.10 0.89 0.91 0.86
cr c 0.86 1.01 0.95 0.86 0.87
~russ (kN)388.8 574.0 573.4 389.7 573.3
-388.8 -574.0 -573.4 -389.7 -573.3
Vmax I ~russ'0.93 0.99 0.91 0.94 0.91
0.88 0.98 0.98 0.93 0.94
- 42-
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0.1
R6
0.09
.Jl Q _
R10
0.07 0.08
T13
T20
0.05 0.060.040.03
02R10
R6@75 ~
]I 4R10
7TlO@250 R6@75
20 2RlO
0.020.01
0
~
""'i'-i'-
" TlOH&V ~N
~0
~[--
0
~
800 5f 1700 ,5C 800
~-=~--------=----------------------
~------------------------
eneCi)------~
O-t----r-------,----.-------,-------,-----,-----,------r------,------j
o
Fig. 2.1 - Stress-strain relationship for steel reinforcements
Strain
Chapter Two
Fig. 2.2 - Details of Specimen LWI
- 43-
100 ------------------------------------------------------
300 -------------------------------
700 -,-----,ti:--------------------------------,
0..
~
500
600
400
200 -----------------------------------------------------
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-wChapter Two
Fig. 2.3 - Experimental set-up
I_ ...l,_
IIII1
-"1
IIII
__ ...l
IIIII
I
11
I___ 1 1 1 I I J J -.L 1.
IIIII I 1 I I I I 1 I I I I I
- - - r - - -1- - - -1- - - -1- - - -1- - - "1 - - - "1 - - - T - - - T - - - r - - - r - - -1- - - -I-
I I I I I I I I I I I I II I I I I I I I I I I I II I I I I I I I I I I I II I I I I I I I I I I I I I I
___ 1 1 1 I I ...l .J J. .1.. L L 1 1 1 I
I I I I I I I I I I I I I I II I I I I I 1 I I I I I I I II 1 I I I I I I I I I I I I II I I I I 1 1 I I I I I I I II 1 I I I I I I I I I I I I I
:: : I I I I I 1I I I I I I I I I I I I I I
___ ~ : : ' Dri~ ratio = :::::::: 8 =,/75: : : , : :- __ ~ ~ ~ ~ ~ : :~_=l(l~O_: __ ~I I I I I I I I I I I(J = 1~ 150 I I I ~ I - -I I I I I I I I I I I I
___ ~---:----:----:---~---J---J~~-l~~O-O-~~~~[~O-O-: : :~:~: :I I I (J _ 1Y600 : (J =1V400 I I I I ~ - - - 1- - - - 1- - - 1- - -I - -I I I - I I I I I I I I
I f) l' 1 I I I I I I I I I I I I I
___ L _ == _/1_ QQQ 1 I I I I I I I I I I I I
8=11kooo : : :- - - -: - - - : - - - : - -1\-: - -1\-7-~~ -A~-~:--~:-- :-- :--I I I /\1 I I 1 I I
I I /\1 /\1 /\1 I I I I I I II AI I I : : I I I I :: I
40
30
20
E 10SC(1)
E 0(1)C,,)coC.eno -10
-20
-30
-40Cycle Number
Fig. 2.4 - Applied loading history
- 44-
.. ~
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Fig. 2.5 - LVDTs support arrangements in all specimens tested
Chapter Two
80050
3600
1700
- 45 -
50800
sot!1600
~so
L14
or> or>
C L9 L13
~ L8
~
or> L7 LII
'"' L6
LI5
~
~LJ
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--.Chapter Two
~~I L,',-
~I/I J/I.·-'
o DIIIllIIJJDM~/L'
~I,t·.~
u·...
n1!'- 0'J J
f',-
~;l .. • " • .J ...~
; I~"
.,.-..r't,·--~
h11\1 ....
J-I.·"'
." - ~
(a)At a drift ratio of 0.1 % (b)At a drift ratio of 0.5% ©At a drift ratio of 1.0%
Fig. 2.6 - Crack patterns of Specimen LWI
25
1 1.0%
201510
- - i- - - - - 1- - - - -' - - - -
1 1
I 1
5
Ductility
Drift ratio
0,1 % 0.~5% 0.33% 0.5% I
o-5
1
1 1 I~~L L 1 1 _
1 1 1 1
I 1 I 1
1 1 1 I-- - - - - I - - - - r - - - -1- - - - -1- - - - -
1 I 1 1
1 I 1
- . V j=321 kN - ;- - - - - ;- - - - -: - - - -
Specimen LW1
Drift ratio1
-10
, 1 , 1
0.5% q.33% 0.~5% 0.1%'
-15-20
1.0%
---lIPC
.) _.
_ Ductility - -1- ~ -_'L -'- - '___ ,,11 L 1-,, __ 'D _.Vi=~21kN. ,_: ,-13--:-n--
I-----4-
1 --
I
J1
-500
-25
-400
500
400
300
200
~ 100'"0ro.9 0"';~fl) -100~.-:l
-200
-300
Displacement (mm)
Fig. 2.7 - Lateral load - top displacement relationship of Specimen LW 1
E.E-mIIIell
CD
~
1<l:1:Cl
'OJI
1500
1000
500
0.006
#A bar
0.004
I I
;;!JL-OA31A29 -
0.002
j j I 0
0.008
A3~'"!I'.'A31A29 -
l:::::l#A bar
,9 I
~Specimen LW1
C,)
I.9 ---1%fl)~ A31:::lro
0
A29
J l I j
-0.008 -0.006 -0.004 -0.002
StrainFig. 2.8 - Strain distribution in uUlcl11lu~llongitudinalbars of Specimen LWI
- 46-
II
'IL ~
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Chapter Two
2000
R8
.. - ........
18001600
Rfl R7
14001200
Drift ratio = A,'"---~.~.~.'\
1000
RR
Gauge location
800
Wall width marked from the left (mm)
600
R6 R7
.. -"
R5
f----'Ii I I II'Specimen LW1
f----
Drift Ratio =-
+I"'.
RBar ~10~+
'" ~ 1%
~~~ ~~ 0.5% ~~
------.::::/~V-- / ,,~~30 // ,,~
~~250 / ""'~~.25% '\.""----.... 0,1% / "0.1%
Specimen LW1
1.0%~-'-'·-······-···'·····
0.5% . 0.33% ~--------+--------:::,.....:;.....-:..:..::....;-:----.-..~.=.-.-=•.:;;".-.-::-.r-. 0.25% •• -
0.1% --
-0.004
-0.002
0.005
Drift ratio =
n' --0.10%0.004
0.25%
_0.33%0.003
- -•• ·0.50%
;:::: Specimen LW1 ----1.00%
.; 0.002.t:i .•...-r/)
0.001
400 600 800 1000 1200 1400 1600 1800 2000
-0.001 Wall width marked from the left (mm)
0.01
0.008
0.006
;:::: 0.004
.;;...;
V) 0.002
0,002
0.0018
0.0016
0.0014
00012;::::.;
0,001;...;
V)0.0008
0.0006
0.0004
0.0002
-0.0002
Fig. 2.9(a) - Strain profiles of the vertical bars along section I-I of Specimen LWI
Fig. 2.9(b) - Strain profiles of the vertical bars along section 2-2 of Specimen LW I
Fig. 2.IO(a) - Strain distribution in the horizontal web bar (R bar) of Specimen LWI(Note: R6 in Figs. 2.IO(a), 2.15(a), 2.20(a), 2.25(a) and 2.30(a) represents gauge
location rather than bar type)
- 47 -
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]aI I 11'=f-~ Specimen LW1
"--- TL:l T'. + 1% ~%11 .... TL':ll
.......... r........ Drift Ratio = T14 T~
f- .......~ ""/
/1~ ~ 0.5% :\TBar """'-II 03~~ ,,~ \
II /V / ~25~ \1/ o250/'lV '" \\
1/ / 1'\\ \ill V 0.1% 0.1% ~....::::::---
0.002
0.0018
0.0016
0.0014
0.0012
l=l 0.001';.b 0.0008r/)
0.0006
0.0004
0.0002
0
-0.0002T13 T14 T15 T16 T13 T14 T15 T16
Chapter Two
~
Gauge location
Fig. 2.10(b) - Strain distribution in the horizontal web bar (T bar) of Specimen LW1
I
I'-'-.
•.,
" ,~: :. ~-".' .,('"-
, -- -: ': ~:"::I
I
""'-
(a)At a drift ratio of 0.1 % (b)At a drift ratio of 0.5% ©At a drift ratio of 1.00%
Fig. 2.11 - Crack patterns of Specimen LW2
- 48-
~
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
Chapter Two
30
1.3%
252015105
---,---I---~-------+---
_ V,=500kN _ ~,:,~,:irll~~ _L~2 _
_ _~~~i~ _ ~ ~ J ~ _I I I I
DriflJ ratio I I I
o
I I I
~~~~-~---~---~---+---I I I I
---~---~---~---~---~---I I I I II I I I- ---1-------1---1---1---I I I I
-5-10-15-20-25
I I- - -f- - - -I- - - -I--
I I I- - _1- 1 _
I II I
- - -1- - - -I-
I I---1----
I I
- 0.-..- fl- ~.c:::::o""~'7"!$""__4~: 1 _
I
I 1.0% 10.67%1 0.5%, 0.33%:01..0.05%
I I I ft: ratio I---l!'-'-Ductility - - -
~~~ .0. ~~i~5~0~N~~I I I I
r--' I I---, , ,·--'1----1---
I I I I I
---1----1----1----1----1---I I I I I
- - - 1- 1 1 _I I
I I
-700
-30
700
600
500
400
Z 300
C 200."
'" 100.Q~ 0....2 -100
'"-J -200
-300
-400
-500
-600
Displacement (mm)
Fig. 2.12 - Lateral load - top displacement relationship of Specimen LW2
m00m
<II
~m>a.0<l:
t·iiiI
1000
1500
500
0.006
#A bN
0.0040.002-lJ.002-lJ.004-0006
#A ''''I
Specimen LW2
-0.008
I i I
~ '~I Ct-_Dn_.ft_ratriO_=--.----..<-t"to:----..:.:::;]I Il~~ - --0.10% ~~ -
.. -..- -0.17%___ 0.25%
- -~ -0.33%
-0.50%
'''0.67%
-1%
A31f---t-----+------j-----N'1f't\-----,?-m-+-+-+---+--t----j
A21=:::;::::=l:=;::::::::J=:::;::::=t~;::::J=~:::t:~;::::::::J=::;:::=l:==t00008
Strain
Fig. 2.13 - Strain distribution in outermost longitudinal bars of Specimen LW2
Wall width marked from the left (mm)
O.OOS
0.004 n0.003
>=0,002
.C;;Specimen LW2-l:l 0,001
r/l
-0.001
-0.002
-0.003
1000
1.0%
1200 1400 1600 1800 21)00
Fig. 2.14 (a) - Strain profiles of the vertical bars along section 1-1 of Specimen LW2
- 49-
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
-wChapter Two
-t:=.t1800 2000160014001200
Driflratio'"
• -to- -0.17% 0.25%
_ 0.33% . - - 0.50%
-0.67% -1.00%
-0.05% -0.10%
Specimen LW2
n'0.007
0.006
0.005
0.004
;:: 0.0030;~ 0.002I/)
0.001
-0.001 Wall width marked from the left (mm)
Fig. 2.14(b) - Strain profiles of the vertical bars along section 2-2 of Specimen LW2
I--Dl I I DCSpecimen LW2
I--
Drift Ratio =I--
+j~0.67% R Bar
111\\~% 1413 %~~VI7 "- '\
oF1%
~"rTf o~'\. ~~670 ~017~17 ~ ~~---...:..:
0.1%~~
0.1%-Q.25%-=:P'""
0.002
0.0018
00016
0.0014
0.0012;::0; 0001l-<
V) 0.0008
0.0006
0.0004
0.0002
0
-00002R5 R6 R7 R8 R5 R6 R7 R8
Gauge location
Fig. 2.15(a) - Strain distribution in the horizontal web bar (R bar) of Specimen LW2
It iI'I---
Specimen LW2I--
I-- ""co + Drift Ratio = TWT1~ _
~1%+
JJ!~]olo TBar ~
7 05~ 1\ ~.II / ~~ r\.\ 0.5% ~
}IT/ ~ \.~33% ~~ 1// ./ 025"-.\ ",
r/ ~ 01%-0.17~ -7r-----=::=:::.: ---0.1%-Q.17%-----10-""
0.002
0.0018
0.0016
0.0014
0.0012
0.001;::0; 0.0008
~0.0006I/)
0.0004
0.0002
0
-0.0002T13 T14 T15 T16 T13 T14 T15 T16
Gauge location
Fig. 2.15(b) - Strain distribution in the horizontal web bar (T bar) of Specimen LW2
- 50-
.
................., .--
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
Chapter Two
20
©At a drift ratio of 0.67%
15
0.5% 0.67%
0.006
10
0.004
5
0.002
0.05%-0.25% 0.33%
o
1 1-,-----1-----1-----1 1 1- -----1-----,-----,-----I 1 I
-----~-----;-----;-----
I 1 1
-----~-----4-----4-----
Displacement (mm)
-5
(b)At a drift ratio of 0.5%
Drift ratio__1_ - - - - ...J-=-±-:::t-::-I:=-::r-=t=-±~:::r-±--=_-=_-=_:-::_~_~_-=_-=_-=_~J
Ductility
_ V j=500kN Specimen LW3
Ductility: 1 1
10.33% 0.25%-0.05% --Drift;ati~-----;-----;-----
-10
-0.004 -0.002
-15
13
0.67% 1 0.5%
Strain
- 51 -
Fig. 2.16 - Crack patterns of Specimen LW3
1
I- -1- - - --
I 1 1-----,-----1------1----- -
1 1 1- - - - - I- - - - - -1- - - - - -1- - - --
1 I 1
I--~-+----r--+_____r-+----r--;----,,....----+--.---__+_-_r___+-_r___+_O
0.008-0.008 -0.006
A29t----+---+---<_------lM-----...........-l---;--l---l---+-<~-~--~r---__I
Fig. 2.17 - Lateral load - top displacement relationship of Specimen LW3
-400
-500
-600
-700
-20
P(+) P(-) 1500I I
A33 Xl K34 Driflratio = n::K32--0.10%
K30 _ K30- ••- '0.17%
EI
#K ____ 0.25% #K bar 1000 ..§..~
Specimen LW3 • -)llo -0.33% ~.g lD
ro -0.50%~U
·0.67%.9 Ql
l1) A31 j0I:l::::s 500ro
~V
700
600
500
400
300
Z 200
C 100"'0ro 0.9"'@ -100~l1) -200~....:l -300
Fig. 2.18 - Strain distribution in outermost longitudinal bars of Specimen LW3
(a)At a drift ratio of 0.17%
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
-wChapter Two
Wall width marked from the left (mm)
Drift ratio =
1200 1400 1600 1600 2000
0.25% 0.16% '<~"~:~.!/ ~ //0.1% ~ /-........r--
~O.OS%l::i-
0.33%
800 1000600
JlP<_,
0 _1 ~"""'- ~/--""0.67% /_____ '~Specimen LW3 ,,0
0.5
% ,
"
-{l.001
0.005
0.004
0.003
l::0.002.c;
"""~0.001
-{l.002
Fig. 2.19(a) - Strain profiles of the vertical bars along section 1-1 of Specimen LW3
Wall width marked from the left (mm)
/ /////-4. 0.5%
/~ 0.67% ' 'y--t/
Drift ratio =
0.16% ~.0.1% "---
. ~o.OS% _____
:.-://SpecimenLW3
g'It=::;:::Zi __L? --- 600 .....~O· -' 1000 1200 1400 1600 1800 2000
O.OOS
0.004
0.003
0.002
l::.c;0.001tl
f/)
-{l.001
-{l.002
Fig. 2.19(b) - Strain profiles of the vertical bars along section 2-2 of Specimen LW3
ISpecimen LW3
0002
0.0018
0.0016
0.0014
00012
0.001
l:: 0.0008.c;0.0006tl
f/)0.0004
0.0002
0
-0.0002
-0.0004R5 R6 R7 R8 R5 R6 R7 R8
Gauge location
Fig. 2.20(a) - Strain distribution in the horizontal web bar (R bar) of Specimen LW3
- 52-
~-------~------- ~
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
.{).0004..1....------'------'--------'------'------'--------'-----'--------1--
,,\ ',~
~ \. \... // J
~i'-" '; / " .< '. ""X...,.r-.. ......... ,,_., .' .,--::f.... '-.' x' - .~
r....')'-' ".. t'-.:~.' :-..~~~:.-
T16
©At a drift ratio of 1.0%
T14 T15T13T16
0-""'" _JJ f ; ..
r"'-...,.""'I-,. ... ,:~, .'-----"-. ......... )(.' .~
"\.. 1"';.·'''\
Gauge location
(b)At a drift ratio of 0.5%
T14 T15T13
.. ,....
- 53 -
Fig. 2.21 - Crack patterns of Specimen LW4
Chapter Two
0.002 -r------r---.----.----,------.------.---,--------,--n,t jg(.Il0.0018 f-- - -
0.0016 f-- '" '" -+--+/J+;~:~.----o--specimenLW3----+----+ '" no -_
0.0014 f-- ". m v/o", nrift ~"tin=
0.0012 +---+--~'I-/-_+_----"O'\..____+_--___+---__l_;_-t----__+_-0.001 +---+_~/'------+___ _j;;:_0.-.50-Yo_'\-+_T_Ba_r--+__~/'__t\_+-t-----+-
0.0008 +---+-~/'---__+__+_l/~~I"f.;_--~;:--t---_+_-0.6~7yl<---__+____\_\_+_--___+_-0.0006 / / 0 __~ r\. / \
0.0004 f-- _ / 1~·25% lJ\.. \0.0002 +--t-....~;:::---~/'_,1-+/---_1_--+_"'~~~0~.3~3%~""'~~:s;;<--;::::l--
~Ij / ~ 0.25''" //n----_=_.{).0002+---+~__~f_17--+--0.1-%-+_--_+_0.-10I.-o.{)-.17'l-V.~~V-_+_--___j--
(a)At a drift ratio of 0.1 %
Fig. 2.20(b) - Strain distribution in the horizontal web bar (T bar) of Specimen LW3
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
...Chapter Two
2520
!I
- -I-
II
1-----1---_I II II I
15105o-5
Duc~i1ity
V j=321kN
-10
II
I I I I-----------------~----
II I I I
------~---~----~---~----
I I I II I I I
- - Vj=321kN ~. Specimen LW4I I
__ Ducti lity ; ~ : _I II I II I I I I
Drift: ratio I I II I I
-15-20
~~~~n-}
o
100
300
500
200
400
-300
-100
-400
-200
-500
-25
zC"0
C';j
..8~~
2C';j
.....:l
Displacement (mm)
Fig. 2.22 - Lateral load - top displacement relationship of Specimen LW4
E~
.s(J)(J)ltl
CD
~g;o
.D
iOl
0Qi
I
500
1000
1500
~
#A bar
p(~n)D'A33 _
A31 _A29 _
Drift ratio =
---0.50%'0.67%
-1%
#A bar
ISpecimen LW4
A311
t Mol 1 .. i .... '. Jr I
I
A331-!'311-D¢::;.P(-l
A31A29 -
A29j I '." f. I I ..... I It .....
.§~u
..8(J)~::::lC';j
o
0.0060.0040.002-0.002-0.004-0.006
j j i j i J I I j I J 0
0.008-0.008
Strain
Fig. 2.23 - Strain distribution in outermost longitudinal bars of Specimen LW4
- 54-
I!il'
~-------....
M······,::>·<~·::I~;:
~
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
Fig. 2.24(a) - Strain profiles of the vertical bars along section 1-1 of Specimen LW4
Fig. 2.24(b) - Strain profiles of the vertical bars along section 2-2 of Specimen LW4
Chapter Two
2000
R8
1800
.....
1600
R6 R7
0.5%
1400
0.1% --:=::-:-:O.O:::::5'\::-Y.""*=~::::;
R5
__ 0.05% ---0.10%
• -... - 0.17% 0.25%
_ 0.33% - -e- ·0.50%
--0.67% _ 1.00%
Drift ratio =
Drift ratio =
1200
0.16%
1000
1000 1200 1400 1600 1800 2000
R8
- 55 -
Gauge location
800
Wall width marked from the left (mm)
600
Wall width marked from the left (mm)
Specimen LW4
R6 R7
JCl-'
400
R5
f----11I I n'Specimen LW4
f----
Drift Ratio =f----
RBar +
/\ 1% J+ \\ 1%
/ \ 0.670/.j;~ ",\\ h~
/~~\~p ~~.25% .67Y~.
~0.25%- .50'\ ~ .~~V7 '"~ 10.17%'~ ""\V
0.17% ,
f--0.1% 0.1%
-0.001
-0.002
-0.004
0.002
0.0018
0.0016
0.0014
0.0012
~0.001
'a 0.0008~
(/)0.0006
0.0004
0.0002
-0.0002
0.005
0.004
0.003
~.; 0.002
.t:lrJ)
0.001
0.01
0.008 £1-0.006
Specimen LW4
~0.004
.;.1:J
0.002rJ)
Fig. 2.25(a) - Strain distribution in the horizontal web bar (R bar) of Specimen LW4
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11Chapter Two
11 +
1~[,-+ Specimen LW4
"" '. I \1%.... ,-:. Drift Ratio = 1% TUTn _
- J;I \\II ~ TSar ~67%
II \\~ ~"'"0.5%
I / - /'/ /~""0.33%
1/ //().25% / 0.25%-{).~~1/ / /0.17%
0.17% ~\L / / 0.1% 0.1% "\' ==------
0.002
0.0018
0.0016
0.0014
0.0012
~ . 0.001"; jI-; : 0.0008
~0.0006
0.0004
0.0002
0
-0.0002T13 T14 T15 T16 T13 T14 T15 T16
Gauge location
Fig. 2.25(b) - Strain distribution in the horizontal web bar (T bar) of Specimen LW4
. -" ~""'l_,"
~"'·"'.'''i•• . .11·.. -
II/ " .i
: ,,', '~::~,,::., . ~"" .,,~' ,,~
• ~#' - -~ • ,-,.. .~
~
,\J 17<DJ··..[/~- "·'·r'-.
,," ~ ~ -"
,.'''",.J'',.,,,I ..
n~(a)At a drift ratio of 0.17% (b)At a drift ratio of 0.5% ©At a drift ratio of 1.0%
Fig. 2.26 - Crack patterns of Specimen LW5
252015105o-5-10-15
I' I 'I
0.67% I 0.5% 0133% 0.215-0.05%
-20
1.0%
____: • '.. I Drift ratioIJrPH ----1-----1 1-1 I-____:D Ductility 41- 1-100-1 I ozI Ii __ I
____ I Vj=500kNI ---1---
- - - _I II, ;
--:- ~ ~ ~ J~ ~ ~ ~ t~ ~ ~ ~ i~ -----I 'I I
I I I_ - __1_ - _ - ~ - _ - _ .l-
- - - -:- - - - ~ -~1
- - - -1- --II
I , I
-~----~---_I-_--~----, I I
--~~~~~~~~~t~~~~t~~~j~~~~I I I
____ l L ~ J __
r4~B/:if"; - ir~ -- -r-V,=500kN - Specimen LW5
I I 'I I' I I! I I __ ~u~t~lity I I I~ L L __ ~ _
Drift fatio
-600
-700
-25
700
600
500
400
300Z 200C"'0 100~0
--< 0c;I-;
-100f1)
~....:l -200
-300
-400
-500
Displacement (mm)
Fig. 2.27 - Lateral load - top displacement relationship of Specimen LW5
- 56-
1-..-.. -----.-..------------- ~
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
Chapter Two
1500
1000
500
..... IIl
1800 20001600
0.006
#A bar
... -.... -.,,'1400
0.004
Drift ratio =
1200
. -...
0.002
1000
SpecimenLW5
~-=------ 0.1% ~'"
c.-::..-------- 0.05% <::=ij , , 'i ,
800 1000 1200 1400 1600 1800 2000
800
Wall width marked from the left (mm)
-0.002
600
Wall width marked from the left (mm)
Drift ratio =
--0.10%• -... -0.17%
-0.25%- -:t:- -0.33%
-0.50%·0.67%
-----1%
-0.004-0.006
#A bar
ISpecimen LW5
t----,--+-~-_+_-r__-+-_r_-+--,-__r--___r_-+----r--+-__.____-+O
0.008-0.008
-0.001
A3 A~3_DIt=P(-)
A31A29 -
0.005
Drift ratio =
0.004_0.05% -0.10%
••.., ·0.17% 0.25%
_0.33% - ... -0.50%0.003
_0.67% -1.00%
.SC'd 0.002l-<
~
0.001
Strain
- 57-
A2-1---_+_---+----+.....-r--I-----A-+--4.-.,..---t-----+----
~o
";gC,)
..9
~ A3 f-------+----+-----+--~.....-\_;._M!-t___-*_--_+__--___1;:jC'do
0.006
0.005
0.004
0.003
.S0.002C'd
,tjrJ) 0.001
-0.001
-0.002
-0.003
Fig. 2.28 - Strain distribution in outermost longitudinal bars of Specimen LW5
Fig. 2.29(a) - Strain profiles of the vertical bars along section 1-1 of Specimen LW5
Fig. 2.29(b) - Strain profiles of the vertical bars along section 2-2 of Specimen LW5
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
-wChapter Two
R8R6 R7R5R8
n"'4 J I -
ItS,...? _
0.002
0.0018
0.0016
0.0014
0.0012
0.001
I=: t I 0.67%
'a 0.0008
~0.0006r./)
0.0004
0.0002
0
,0.0002R5 R6 R7
Gauge location
Fig. 2.30(a) - Strain distribution in the horizontal web bar (R bar) of Specimen LW5
Fig. 2.30(b) - Strain distribution in the horizontal web bar (T bar) of Specimen LW5
T16T14 T15
0.1%-0.17%
T13
- - - - - - - -1- - - - - - - - -1- - - - - - - --I
0.25% 10.5% 1 1.0% 1
101
- - - 1- - - - - - _:_ - - - - 1
V, = 321 kN : ----- LW1 - - - -i________ : - -• -- LW2 1
V, =500kN: --.-LW3------ i
1 1
________ : ••• E) •• ' LW4 :
= Drift ratio: -.- LW5 :1 I
I
T16
Gauge location
0.1%-0.17%
fA' I I I rnr'=~
T14 T15
0.002
0.0018
0.0016
0.0014
0.0012
0.001I=:.c; 0.0008;..,.
~ 0.0006
0.0004
0.0002
0
-0.0002T13
: 1.0% : 0.5%: 0.25%----------------------~8Se
r--------r--------r--~-8ee1 1 Drift ratio = Z1 1 ~
1 1 1-
L--------L--------L--i sseI 1 1 a1 I 1-
: : : e~--------~--------~- S 4seI 1 1 Ctl1 I I.....J1 I I
~--------~--------~----2eeI 1 1
1 I I1 1 I
Fig. 2.31 - Backbone envelopes of load-displacement curves for tested specimens
- 58-
~
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
Fig. 2.32 - Contribution of various deformation modes to total displacement of walls
1 1 I 1- I - -I - -I - - - -1- - - - - -
I 1 I I-,- -I - -1- - - -1- - - - --
_.J _I 1 I 1 I I
_ :.J _ -.J __I __I __I __ 1__ 1__
1 I 1 1 1 1 1- -l - -1- -1- -1- -1- -1- -I--
I 1 I I I I I- --i -..., - -1- -1- -1- -1- -I--
I 1 Specinen LW5 I I- -j - -j - ,- -1- -1- -
1 I I I 1I -I--
I
I I 1 I I- I - "I - "I - -I - -I-
I I I 1 I I I- , - -I - I - -I - -, - -1- -1- -
=~=:==:==I I I 1 I I 1
- -< - -I - -I - -I - -I - -I - -I - -I I I 1 1 I I
- -+ - -I - -I - -I - -I - -I - -I - -
1 1 Specinen LW3 I 1-4----j-, ,--1--1--
-~-:--:---~-:--:--
, I I I 1 I I
(c)
.Q.88 -0.75 .Q,63 .Q,5 .Q,38 -D.25 -D13 0 013 0.25 038 0.5 0,63 0.75 0,88 1
(e)
Drift ratio (%)
.Q.88 -0.75 .Q,63 .Q,5 .Q,38 -0.25 .Q,13 0 0,13 0.25 0,38 0,5 0,63 0.75 0,88 1
Drift ratio (%)
I I I I I I I-'-'-I-I-T-'-I-
1 1 I 1 I I I-'-'-1-1-1-'-'-
=t=t~=I I I I I I I
- ~ - ~ - _ FleXlJ"e ~ - ~ - ~ -
- t- - +- _ .....-Shear + - + - -+-I I --Siding I I I
-r-r-T-T-i-i-~-
-~-~~-
- ~ - ~ -~~ - ~ -~-II~;'I;-
I I 1 1 1 I 1-,-,-1'-,-,-1-1-
1 1 I I 1 1 1-1-,-1-----1-1'-_ L_I 1_1 -.1_
I I 1 I I 1_ L_L_.L_l._l._J._J._
I I I 1 I I 1
- ~ - ~ - __ FleXlJ"e ~ - ~ - ~ -
- t- - t- ----Shear +- - -+ - -+ -I I __ Siding I 1 I
-t--r-T-T-i-"t-4-
I I-,-,-T-T-T I
..........---lo_..............._1-'-'-I-T- I 1
I I 1 I I 1
100
-- 90
~ 80rJJ;:: 70.8"E 60
.0 50'C1:: 400() 30
4-i0 20;::.g 10
() 0C'::ll-< -1u..
(a)
.Q.88 -0.75 .Q.63 .Q.5 .Q38 .Q25 -D.13 0 0.13 025 038 0.5 063 0.75 0.88 1
1 I I I 1 I 1 1 I I I 1- , - , - , - T - T - 1 - 1 - - I - -I - - - -1- -1- -1- -
I 1 I I I 1 I I I I 1 I- , - , - "I - 1 - 1 - 1 - 1 - -, - -I - - - -1- - - -1- -1- -
_L_L I 1_ _ _1 __1 1__ 1__
I 1 I I I I I I I I_ L _ L _ L _ 1. _ 1. _ ..1. _ ..1. _ _.J _ -.J __I __I __I _
1 I I I I I I I I I I I I 1
- ~ - ~ - _FklxlJ"e ~ - ~ - ~ - - ~ - ~ - -: - -: - -:- -:- -:- -
- t- - t- - --- Shear +- - -+ - -+ - - --i - ..., - -I - -I - -I - -1- -1- -
I I ......... Sliding I I I 1 1 Specimen LW1 I I-r-r-r-T-i-~--t- --j----j- ,--1--1--
I I I 1 1 I I I I I I I-r-T- -,- -,;:L=.....:1.-~-I--I--
I I I I I I I-r-r-r -,- -,-
I I I
1 I I I- -I - - - -, - - - - - - 1- - 1- -
1 I I I- -I - - - -1- - - - 1- - ,- -
I II I 1 1 I
__1 1__ 1__ 1__ L _ L _
1 I I I I I I- -I - -1- -1- -1- -1- - I- - I- -
1 I 1 1 I I I- -I - -I - -I - -1- - 1- - I- - I- -
I I Specimen LW2 I I- -I - -I -, ,- - 1- - r -
I I I I I 1 I-1--1--1--1--,-
100-- 90;!2..~
80rJJ;::
700
'"5 60.0'C 50
1:: 400()
304-i0
20;::.g 10
()0C'::l
l-< -1u..
Drift ratio (%)
I I- -I-I-
I I-,-,-
- 59-
(b)
-0.88 .Q.75 -0.63 .Q5 .Q.38 .Q25 .Q.13 0 013 0.25 0.38 0.5 0.63 0.75 0,88 1
Drift ratio (%)
Chapter Two
...-.-T'"_~I I_ l.._l.._~_~_-l_-l_
I I I I 1 I I
- ~ - ~ - -- FleXlJ"e ~ - ~ - ~ -
- t- - +- - -- Shear -+ - -+ - --i -I I __ Sliding 1 I I
-r-r- I I ;-4-4-
1 I I- -,-
III
00 100-- I 1 1 1 I I I I 1 I --~ 90 -'-'-'-1-'-'-1- - I - I - - - - - - - - - -1- -
~90
rJJ 80 I I I 1 I I I 1 I I 1 I 80;:: -1"-1"---------1- - 'I - - - -I - -I - - - -I - -1- -
rJJ
.2 70 _ .L_l.._l._l._ 1- _1 __ ;:: 70
"E 1 I I I I I I I I I I 1 I .860 -/--/---1---1--+-+--+- - -j - -1- -1- -1- -1- -1- -1-- "E 60
.0 1 I I I I I I I I 1 I I I I
'B 50 - t- - r - __ RexlJ"e T - T - -t - - --i - ---j - -1- -j - -1- -1- -1-- .050
;::- ~ - ~ ---Shear
I I 1 1 1 I I I 1 I 'B0 40 T-T-,- - , - -, - -1- ,- -1- -1- -1- - ;:: 40() _!..- _!..- _ ......... Slidng I 1 1 _ -.! _ -.! _Specimen Lwtl __ I__ 1__ 0
4-i 30 I I I I ,-i-I- I I 1 I 1() 30
0- L -~ - t _l_l~_ _ J J __I _ I
4-i;:: 20 0 200
10 _ L_L_ ..1. _ ;::'..0 1 1 1
,g 10C,)C'::l 0 () 0l-< C'::lu.. -1 .{Ja8 -0.75 .{J,63 .{J,5 .{J,38 .{J,25 .{J,13 0 0.13 0,25 0,38 0.5 0,63 0.75 0,88 1 l-< -1u..
Drift ratio (%)
(d)
00-- 90
~rJJ
80
;::70.2
"E 60
.050'C
1:: 400()
304-i0 20;::.g 10
() 0C'::ll-< -1u..
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
~..Chapter Two
III
I 1 I 1 I I
-~-_:_~s -~-1 I - . II I 4 4 I
I I 3 3 I1 I 2 "II 1 II I I
- -t - -1- - t- - -t - -1- - t- -I I I I I II I 1 I I II I I 1 1 I1 I 1 I I II I I I I 1I I 1 I I I
- I - -1- - I" - "1 - -1- - I" -1 I I I I
~ I I Drift ratioII~I
~ I I /' ..... Ij J \ '\ ""o -1% I I -o15~/.-C.25% '-o,f% 0'0), 0.25b.-0." ,.
-0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
500
1000
2000 i L
1500s§~CJj
C':l.D
'";~~
>o~~
g.sCJja
Curvature (rad/m)
(a)
2000 ,----,--,--,--r---r--=-,.--r---r--,--,.----,
0.01 0.02 0.03 0,04 0.05-0,04 -0.03 -0.02 -0.01
II
: Specimen LW3 I__ .1. __ -l I 1 _
1 I I II 1 I I1 I I II 1 I 1I I I I
- - T - - ., - - -I - - - 1- - -
1 I I II 1 1 I1 I I II I I II I I I
- - 'I D~,ft~ti~ - - -1- - - 1- -
I I1
I
II
__1 !!II ::-- : : -~--
: ~ ;:1 I
- - ~ - - -: - - - 1- - - ~ - -
: : : :I 1
- - ~ - _: - - - 1- __ ~ __
I Drift ratio I
o I .. .; , :I"'~ ~I I0.05%-.33'1. 1 :o~.."~° .5% 0.67% I
500
2000 l--,--r---,----r----;;~--,---,-----,-----,--
1000
1500
s§~CJj
C':l.D
-;~~
>o.D
C':l~C,)
BCJj
a0.01 0,02 0.03 0.04 0.05 0,06
I I1 1I I I 1 1 I
- ~ - - ~ - _: - - ~ - _:- - - - ~ - ~~'"' - ~ - -1 Specimen LW2 I I I 4 4 II I I I I I I, J I1 I I I I I I. II 1 I I I 'I 1I I I I I I 1 I
- -I - - I - -1- - T - - 1- - - r - -I - - r - -I - - T - -
I I I I 1 I I I I I1 1 1 I I I I I I II I I I 1 I I I 1 II I I I I I I I 1 I
- -< - - +- - -1- - + - -1- - I- - -< - - l- - -1- - + - -I I I I I I 1 I
Drift ratio 1 I I I Drift ratioI I
1 I I Io ..- HJ.67% I I 0.67°/d 1-1 %
-0.06 -0.05 -0.04 -0.03 -0.02 -0.01
500
1000
1500
s§~CJj
C':l.D
'";~~
>o.D
C':l~
g.sCJj
aCurvature (rad/m)
(b)Curvature (rad/m)
(g
S 2000 S2000
I I 1 I
§ 1 I § I I
I I I I 1 I I 1 I I 1 I 1~ I 1 I 1 I I -_:-_:-~ -:-- ~
, , I I I
--~-_:_~~--CJj __I __ L __ 1__ .1 __ 1__C':l 1500
- .... - .... - -I __ 1__ '- _ .... _ CJj 1500I I! 5 I.D 1 I Specmen LW4 1 I I I 4 4 I
C':l I Specimen LW5 I
I I I I I 1 1 1 3 3 1.D 1 1 1 I· • 1
'"; 1 1 I I 1 I I I I -; 1 I I I ~ :l I
~ I I 1 1 1 1 1 1 I 1 I I I 1
I I I I I I I I I ~ I I 1 I I I' I~ 1000 - T - '1 - -I - - 1- - r - T - . - -I - - 1- - r - I - I - -I - -
~1000 - -I - - I - -1- - "t - - 1- - - r - -I - - T - -1- - "t - -
> I I 1 I I I I 1 I 1 I I > I I I I I I I I I I0.D I I I 1 , I I I I I 1 I 0 , I I 1 I 1 I I 1 1
.D I 1 I I I I IC':l I I 1 I 1 1 1 I 1 1 I I C':lI I I
~, 1 I I 1 I I I I I 1 I I , I 1 I I I I , I
C,)500 - .... - .... - -< - - 1- - '- - ... - - -1- -1- - +- - ... - ~ - -1--
~500
__I __ L __ 1__ .1 __ ,__ _ L __I __ 1. __ 1__ .1 __C,)
B I I I I I I I 1 I I ;::::1 I I I I I I I I 1 I
Driftlratio I I 1 I IDriftrl!tio .s I Driftlao I I 1 I Driftrao ICJj
I I I I 1 CJj I I I I Ia I I I a I I I
-1% -015% 10.67%10 0
-0,07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0,04 0,05 0,06 0,07 -0.06 -0.05 -0,04 -0.03 -0,02 -0,01 0 0.01 0.02 0.03 0.04 0.05 0,06
Curvature (rad/m) Curvature (rad/m)
(d) (e)
Fig. 2.33 - Wall curvature distribution of all specimens tested
- 60-
Iii
1- - .
........'.'.""', .--
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
1.2
1.21
0.80.6
0.8
0.40.2
- - - -, - - - - r - - - -1- - - - -, - - - - r - - - -I
I 1 I 1 1 1
1 1 I I 1 1
I 1 1 1 1 1
- - -j - - - - t- - - - -1- - - - -t - - - - t- - - - -I
I 1 1 1 1 I
1 1 1 I 1 I
1 1 1 1 I 1__ -.J L 1 -.J L I
1 I 1 1 1 I
1 I 1 1 I I
1 1 1 1 1 1
1 1 I 1
-:----:--- ---LW1 ":----:I 1 1 1
~ - - - - ~ - - - ---+- LW2 "~ :
: - -.- - LW3: :\. 1 1 1
~ l," - - ~ - - - _ LW4 "~- - - -:1 1 1
~ -0- LW5 "~ :1 1
1 1
1 I____ J L I
1 1 1
1 I
I1
1
1
1
1
Chapter Two
1
I1
___ J _I1
1
1 I---1----1--------
1 I1 1
1 1
o
0.60.40.2
-0.8 -0.6 -0.4 -0.2
•- - - - - - - --..-. LW 1 -- - - - - - - - - - - - - - - - - - - - -",.! - - - - - - - - --
- -.- - LW2________~LW3 ~~ _
LW4~LW5
-1
o
o-1.2
- 61 -
Drift ratio (%)
Fig. 2.34 - Secant stiffness of tested walls with respect to drift ratios
o
2000
8000
4000
6000
10000
12000
180
160
--- 140ee
120-........
~'-
V'J 100V'J(])
~~ 80
V'J
"Sc::l 60u(])
(/)
40
20
Drift ratio (%)
Fig. 2.35 - Energy dissipation capacity of each specimen with respect to the driftratios
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Chapter Two
1:0 1,5 ~a____ L ..:.. 1 I
Flexural displacement (mml:1 1 1
----r---T----I----I1 1 11 1 1
_ L ~ I I
1 1 11 1 11 1 1
-iO -1 5L ~ __
1 11 1
1 1r - -1
1
L _1
1 1 1 1
~ - - - - - - - -1- -40-9
r - - - I - - - - 1- - - .;;. - - 40e - - - -- - - - I - - - -I - -
r 1 r:Z 1 11 1 1 ~L .J 1 "C - -JOe
: : :.21 1 1 car---.,----I---.! --20B1 Ilea1 1 I...J
L---~----I----~-roe1 1 1 11 1 11 1 1
I I I I I I--...,---r--~---r--...,---I
1 FI"xural cilisplaqemenl (mm) 1--,---r--!---r--~---I
I I I I I I--...,---r--,---r--""---I
I I I I I I--,---r--...,---r--""---j
I I I I I I--...,---r--i---r--""'---I
I I I I I I
'--'---1--,--70-0-i I I Z Ir--....,---r~...,-6ao-
1,- -~---~~~_5eo_I I I..S! I
~ - - ~ - - - ~ 'E- ~ -4110-r--I---r~...,-3ao-
I, __ ~ ~ ~_ ~ -zel}-I I I I1--...,---1----,--11 1 1
- - - - - - - _1- __ 1 1 1 1 I
: Shear displacement (mm) 11 1 I 1 1
- -I- - -1- - -1- - -1- - -1- - -1- - -1- --I
1 1 1 1 1 I I 11 1 1 1 1 1 1 1I 1 1 I 1 1 I 1
- -1-- -1- - -1- - -1- - -1- - -1- - -1- --I
1 1 1 I 1 1 11 1 1 1 I 1
r - -....,- - - T" - - -1- - - T - - -1-7Oe-I I I I I Z I
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~: ~ ~ ~ ~ ~~~ :~::I I I I 1.2 II - - -I - - - T - - -1- - - T -n; -1-40&
~ -- -I - - - ~ - - -: - - - ~ - ~ - :-30&I I I I I...J Ir - - -I - - - r - - -1- - - T - - -1-2{)e
~ - - -:- - - ~ - - -:- - -+- - -:-WG, I I I I
1
- - - -1- - - - --
1 11---1---1 1I - - -1-
~-- 1---:-509
~ - - -1- - - f - - -:- - - +- - -:-6tll}~ - - _I ~ 1 .!. 1_
1OG
---r--""---,---I1 1
- - -I
1- - -I
1- --,
1
1
1
- --,1
r - --,1 1
1
I I I IShean displacement (mm)1- - - 1- - - -I - - - I" - - -I
I I I I- - - 1- - - -I - - - I" - - -I
I I 1 1- - - 1- - - -I - - - 1 - - -I
1 1 1 1- - - 1- - - -I - - - 1 - - -I
___ I__ -...!- __ ~ I
r--r--r--r41 , 121
1 1 1 CIL __ L.. __ L-cLa
: : 1.2 :1 Ilea 1
r--~--~ i ~
1 1 ...J 1
L--~--I---I_41 1 I 11 1 11 1
$ t
r - - - - r - - -~ , - -:rStt-1 1:Z 1r - - - - r - - ~ r - -6t1o-1 1 "'C 1r - - - - r - - -~ r - ~Stt-
1 1 - 1
~ ~ ~ ~ ~ ~ ~ ~1~ ~ ::tt-I I...J 11----1----1--281 1I----I----T--1 1
-~ - - - --~1
1- - --
1r--1
I· -
,- 1
rl... - - - - ~ - - - - ~ - ..(ieo-'- - - - - '- - - - - '- - -:rstt-
----T----T----i----i1 1
"""~;;"_~:::I-r---_-,,_i, - - - - ~
- - - 1
1- -- - 1
1- - - -I
1-,1 11----11 1
----~----~-·-·t1 1 1 1
- - - - : Siidlng ~i;pla-c;.;je-;,t·(;;,;;,)-:----1----1----1----1
1 I 1 1----1----,----1----1
1 I 1 1----1----1----1-----1____ l l l J
1 1 1 1____ 1 1__ J
r---r---r-~r4GO
liZ 11 1 I:' 1L L __ L_"'C
: : : ~1 1 1 ~r---r---r- .!1 Ilea1 1 I...J 1
L - - -I-- - - - I-- - - -1-_4001 1 1 1I I I I1 1 1
-Jl -31 11----1-1 11
11- --
1
1
~
1
1 1 1 1
l... - - - 1- - - _ 1- :-.-400
- - - -1- - - -,'- - - -1- - - -1- - --,
1 1 1 11
3 'I1 1 1 I 1
- - -1- - - -1- 1 1__ --I
1 Sliding displacement (mm);1 '
1- - -1- - -
11 1 1 1
- - -1- - - -1- - - -1- - - -1- - --I
1 1 1 1 11 1 1 1
- - _1- 1 1 1 I
(a) Specimen LW2 (b) Specimen LW4
Fig. 2.36 - Flexure, shear and sliding displacements of Specimens LW2 and LW4
- 62-
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kN
Strain
pP(-l
Outer strut 1
Outer strut 2
Tie
V
---.t'--N2 ---:h:.::..-- N_ltFig. 2.38(b) - Forces acting at base
section of strut-and-tie model
FlL
Chapter Two
-400 Strut-and-Tie predictions
- 63 -
Fig. 2.37 - Strut-and-tie model of Specimen LWI
400
The area ofconcrete struts
IA P(-)=341.0
~'/I--- ------c;~
Or)N,
I~tf~,
" ~,1' ,," ..c
,,' ~b,1/ cf.,
I-c ..c 0(") 0
.~~Po "",;'C //] ~ / / S, /:;r I
~"k ,~ 230 /",,-"") r
B 1-0 1> 1111 ,. ".-r-rl.I1~) C.
" ,- /1--kJ
" ,,/ ~,/
I I ,"
~~~lr"\0..c I /
,,,\ f6 ~ 0cf., If ,/
N 0Xi
I,
/r- .J/ o::t S/ ~/ -' /11
I ..,-' r/ "
r , , ';}, I " /
D / ,- ) E1->" .1> ,. ./ ,. //'
-
~ ~~ ~ •341
1850
Fig. 2.39 - Strain history of gauge #T14 in horizontal bars of Specimen LWI
Fig. 2.38(a) - Forces acting at wall basesection
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-wChapter Two
IA P(-)=363.4
~~l1li1---
0
~VI
~ N
" ~_tI~" ~
" J
~,,1" ~~1~N
0:xl-' ~ 0
.LLJ ~~,[f',Jt;/~J s,4" III L
~"U;! ~~'252.B ... ~ ~ ~,/~ l..'<- - c-i, - ~'--' III ~/ ~-
-j,;/ .--:// ,./I //~ /)1~A
.-r
I ,,~~ N 0
VI " o.:t 0r-
JJ'~ - o.:t SI ~,
~III /
I -' / // -
I,~ ~ '"tJ /
( ED ~~/ /" (t.¥
~ ICf~
363.4
1850
kN
Fig. 2.40 - Strut-and-tie model of Specimen LW4
.....
0.00145 0.00175
Strain
Gauge #T14
pP<-)
T13 T16- T14 T15 -
Specimen LW4
0.001150.00085
Strut-and-Tie predictions
400
300
200
Z 100C~C,)
"""~"""
-0.00C'::I~ -100..c:
C/)
-200
-300
-400
Fig. 2.41 - Strain history of gauge #T14 in horizontal bars of Specimen LW4
- 64-
111
1
1
i__--------- ..'
':i.''c'' "'; .
~
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Fig. 2.43 - Strain history of gauge #T14 in horizontal bars of Specimen LW2
Chapter Two
0.0011
Strain
0.0009
P(-)=562.7 kN
Specimen LW2
Tl3 T16 Gauge #T14Tl4 Tl5
pPH
A
0.0007
294kN
- 65 -
1850
Strut-and-Tie predictions
294kN
B
Fig. 2.42 - Strut-and-tie model of Specimen LW2
600
-600
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294kN
B
294kN
A P(-)=561.0 kN
Chapter Two
'TI ,/ /IW~ I " ""t' '<;), ~/ g:: '"NN I 1\" "1----: ~~ g - - N
\.,," 0" '<;)" /,," 00' / V"l/
~ 146.5 & ~14.5 ////
C I ~~-;t 1465/\"'", /,~AH 0l // ~
.. /;/ /<:>~>nr~ I /' /" ~ , -- oC;:: I, I /,' -<,,,,,,~"C / __ ;!i '"I" / ~"/ ~ / /- ~
? ,," /146.5 ~ /" - /"/ /7 \>d-
~\~/ /", ~/ ,/1" / / /",/ I I I F
561
1850
Fig. 2.44 - Strut-and-tie model of Specimen LW3
600 Strut-and-Tie predictions
Specimen LW3
z~'-'
IJ)C,)~
~ -0.0004~IJ)
..s::::r./)
-600
0.0012
. '," ..
Strain
0.0016 0.002
uge #T14T13 T16
Tl4 TIS
Fig. 2.45 - Strain history of gauge #T14 in horizontal bars of Specimen LW3
- 66-
III:
1:1
11:1
II'!
1:-_-------- ~
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0.00195
Gauge #T14
#T bnr
Strain
~l3 __ Tl~
Tl4 TIS
0.00155
Specimen LW5
P(-)=541.2 kN
F
Strut-and-Tie predictions
D
A
0.00115
Chapter Two
- 67-
~... '\Y 160.3' \':11' ;-'.-."
1850
- , ~::- .~
I "~I--- ~f- /
r M~~ 160.3 ~ 380. /
1294 kN
c
Fig. 2.46 - Strut-and-tie model of Specimen LW5
600
400 -. -
200ZCfl)C,)~
eB 5~fl)
..dr./)
-400
-600
Fig. 2.47 - Strain history of gauge #T14 in horizontal bars of Specimen LW5
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Chapter Two
REFERENCES
[AI] ACI Committee 318, "Building Code Requirements for Structural Concrete
(ACI 318-02) and Commentary (318R-02)," American Concrete Institute,
Farmington Hills, Mich., 2002,391 pp.
[A2] ACI Committee 318, "Building Code Requirements for Structural Concrete
(ACI 318-99) and Commentary (318R-99)," American Concrete Institute,
Farmington Hills, Mich., 1999,391 pp.
[C 1] CEN Technical Committee 250/SC8, "Eurocode 8: Earthquake Resistant
Design of Structures - Part 1: General Rules (ENV 1998 1-1, 1-2, and 1-3),"
CEN, Brussels, 1995.
[F1] Fintel M., "Shearwalls - An Answer for Seismic Resistance?", Concrete
International, VoU3, No.7, July 1991, pp.48-53.
[G 1] Greifenhagen, C. and Lestuzzi, P., "Static Cyclic Tests on Lightly Reinforced
Concrete Shear Walls," Engineering Structures, V 27, pp. 1703-1712,2005.
[Ll] Lefas, L.D., Kotsovos, M.D. and Ambraseys, N.N., "Behavior of Reinforced
Concrete Structural Walls: Strength, Deformation Characteristic, and Failure
Mechanism," ACI Structural Journal, V87, No.1, Jan-Feb 1990, pp.23-31.
[M1] Mestyanek 1. M. "The Earthquake of Resistance of Reinforced Concrete
Structural Walls of Limited Ductiltiy," Master thesis, University of
Canterbury, Christchurch, New Zealand, 1986.
[M2] Maier J. and Thurlimann B., "Shear Wall Tests," The Swiss Federal Institute
of Technology, Zurich, Switzerland, 1985, 130pp.
[N1] New Zealand Standard Code of Practice for the Design of Concrete
Structures, "NZS 3101: Part 1, 185 p.; Commentary NZS 3101: Part 2,247
p.;" Standard Association of New Zealand, Wellington, New Zealand.
[PI] Park, R., and Paulay, T., "Reinforced Concrete Structures," John Wiley &
Sons, New York, 1975, 769 pp.
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ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
Chapter Two
[P2] Paulay, T., and Priestley, M. J. N., "Seismic Design of Reinforced Concrete
and Masonry Buildings," John Wiley & Sons, New York, 1992,744 pp.
[P3] Paulay, T., Priestley, M. J. N., and Synge, A. J., "Ductility in Earthquake
Resisting Squat Shearwalls," American Concrete Institute, Detroit,
July-August, 1982, pp. 257-269.
[P4] Pilakoutas, K. and Elnashai, A. "Cyclic Behavior of Reinforced Concrete
Cantilever Walls, Part I: Experimental Results," ACI Material Journal, V.92,
No.3, May-June, 1995, pp. 271-281.
[P5] Pilakoutas, K. and Elnashai, A. "Cyclic Behavior of Reinforced Concrete
Cantilever Walls, Part II: Discussions and Theoretical Comparisons," ACI
Material Journal, V.92, No.4, May-June, 1995, pp. 425-434.
[Tl] Thomas N. Salonikios, Andreas J. Kappos, loannis A. Tegos, and Georgios G.
Penelis, "Cyclic Load Behavior of Low-Slenderness Reinforced Concrete
Walls: Design Basis and Test Results," ACI Structural Journal, Y.96, No.4,
July-August 1999, pp. 649-660.
[T2] Thomas N. Salonikios, Andreas J. Kappos, loannis A. Tegos, and Georgios G.
Penelis, "Cyclic Load Behavior of Low-Slenderness Reinforced Concrete
Walls: Failure Modes, Strength and Deformation Analysis, and Design
Implications," ACI Structural Journal, V.97, No.1, Jan.-Feb. 2000, pp.
132-142.
[WI] Wood, S.L., "Shear Strength of Low-Rise Reinforced Concrete Walls," ACI
Journal, V87, No.1, Jan-Feb 1990, pp.99-107.
[Yl] Young-Hun Oh, Sang, W. H., and Lee L. H., "Effect of Boundary Element
Details on the Seismic Deformation Capacity of Structural Walls,"
Earthquake Engineering and Structural Dynamics, 2002, 31: 1583-1602.
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Ag
Aeh
Ash
Ce
fe'
f y
he
K i
L w
Vcr
Vrnax(test)
Vrnax(test)
f.1
f.1~ max
~y
s
¢o
Chapter Two
NOTATIONS
Gross area of section
Cross-sectional area of a structural member measured out-to-out of
transverse reinforcement
Total cross-sectional area of transverse confining reinforcement
within spacing s
Distance of the critical neutral axis from the compression edge of the
wall section
Cylinder strength of concrete
Yielding stress of reinforcing steel bar
Cross-sectional dimension of column core measured center-to-center
of confining reinforcement
The initial stiffness for the ith specimen
Horizontal length of wall
Observed shear force at first cracking
Maximum observed strength during the test
Maximum ideal flexure strength
Displacement ductility factor
Maximum displacement ductility
Yield displacement of the walls
Spacing of transverse reinforcement
Ratio of moment of resistance at overstrength to moment
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Chapter Two
Horizontal web reinforcement ratio
Vertical web reinforcement ratio
Flexure reinforcement ratio in boundary element
Volumetric ratio of transverse reinforcement in boundary element
Maximum top drift ratio achieved
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Chapter Three
CHAPTER THREE
SEISMIC PERFORMANCE
OF MEDIUM-RISE STRUCTURAL WALLS
WITH LIMITED TRANSVERSE REINFORCEMENT
Abstract
This study is intended to examine available ductility of medium-rise reinforced
concrete (RC) structural walls containing less confining reinforcement than that
recommended by the New Zealand Concrete Design Code [N1] and American
Concrete Institute [A2]. Three RC structural walls with an aspect ratio of 1.625
were tested subjected to low levels of axial compression loading and cyclic lateral
loading which simulated a moderate earthquake to examine the structural
performance of medium-rise walls with limited transverse reinforcement.
Conclusions are reached concerning the displacement capacity, strength capacity,
curvature distribution, the secant stiffness degradation and the energy dissipation
characteristics shown by the walls on the seismic behavior with limited transverse
reinforcement. The influence of axial loads, transverse reinforcements in the wall
boundary elements, and the presence of construction joints at the wall based on the
seismic behavior of walls are also reported from this study. Towards the end of the
chapter, reasonable strut-and-tie models are developed to aid in understanding the
force transfer mechanism and contribution of reinforcement in walls tested.
Keywords: Medium-rise structural walls; Boundary elements; Deformation
capacity; Limited transverse reinforcement; Seismic performance; Strut-and-tie
model
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Chapter Three
3.1 Introduction and Background
For the past two decades, significant progress has been achieved in understanding
the performance of structural walls with full ductility. However, in some cases (such
as in low to moderate seismicity) the ductility demand may not be as high as that
required by fully ductile response and thus the design concept may not be the same
as that for fully ductile walls. Besides in existing structural walls, many of them
may possess inherent excessive strength compared with that corresponding to the
fully ductile response. To take full advantage of the residual strength, and design
with a simple and economical procedure, the structural wall with limited transverse
reinforcement, which is expected to exhibit limited ductile behavior, has attracted
increasing awareness in recent years. At the same time, some structural walls may
possess weak interface like cracks initiated by shrinkage, creep etc or construction
joints which may have a significant effect on the structural behavior of walls.
However, until now, rather limited research is conducted on the analytical and
experimental investigations related to medium-rise walls with or without weak
interface though it is a fact that many structural walls existed with such problems.
The supenor performance of RC structural walls in buildings has long been
recognized [FI] in seismic regions and much more attention has been given to the
behavior of short and slender walls, either isolated or coupled with structural frame.
A complete literature review of experimental results of 134 short walls with aspect
ratio less than 2.0 was given in the research conducted by Wood [WI]. Among the
total tested walls reviewed, more than 90% of them had aspect ratios less than 1.0
and only three specimens tested had aspect ratios larger than 1.5. Moreover, most of
the aforementioned studies focused on a single aspect ratio and thus the effect of
this crucial parameter on the failure mode could only be estimated by comparing
results for similar walls tested in different programs.
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Chapter Three
Based on literature review, Penelis [P6] concluded that the behavior of properly
designed walls with aspect ratios larger than 2.0 is dominated by flexure while that
of walls with aspect ratios less than 1.0 is dominated by shear. Aspect ratios around
1.5 typically result in complicated predictable behavior, either flexure or shear, or a
mixed mode of failure under seismic loading. Recently, Thomas et al. [T1-2] tested
eleven wall specimens with rectangular sections, six with shear span ratio of 1.5 and
five with 1.0, detailed to the current design provisions of EC8 [C 1] and ACI 318
[A2]. The main parameters examined in this experimental program and of particular
relevance to current research were the aspect ratio, axial load, and presence of
construction joints; however, the effect of the transverse reinforcement in wall
boundary elements was not studied. As such, it was observed that for structural
walls with limited transverse reinforcement, previous studies do not provide
adequate and conclusive information with respect to the behavior of such type of
medium-rise walls, particularly in the case where the mixed failure modes may
dominate for walls specified by an aspect ratio of 1.625.
The main purpose of the present study is to assess the validity of detailing
relaxation from those required by current design provisions for fully ductile
response. In this study, a description of an experimental investigation of three
flanged structural walls subjected to moderate axial compression, with various
quantities of transverse reinforcement and the presence of construction joints is
presented. Conclusions are drawn concerning the deformation and strength capacity,
the secant stiffness degradation and the energy dissipation characteristics shown by
the walls.
The present study for medium-rise structural walls with limited transverse
1,111
'
,
ill
II'!
\11
III
Iii
I
reinforcement aims to compile the information of economically design structures
which fall between full ductility and elasticity, that is, structures with strengths
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Chapter Three
greater than that required by seismic loading for fully ductile behavior, or less
important structures which do not warrant detailing for full ductility. Moreover, this
comprehensive experimental program can present a better understanding on the
behavior of limited ductile structural walls with various quantities of transverse
reinforcements at wall boundaries and with the presence of construction joints at the
wall base. Finally, the proposed strut-and-tie models can offer insights into the
concept of shear transfer and the contribution of reinforcement in reinforced
concrete squat walls.
3.2 Experimental Program
The three medium-rise shear walls, referred to as Specimens MWI-MW3, were
constructed and tested as isolated cantilever walls with an aspect ratio of 1.625. The
experimental program presented herein aimed at investigating the performance of
medium-rise reinforced concrete walls with limited transverse reinforcement and as
such the effects of reinforcement detailing and construction joints at the wall base
on failure mode, strength, stiffness, and energy dissipation capacity of walls were
investigated.
3.2.1 Material Properties
Ready mixed concrete with 13 mm maximum aggregate specified by a
characteristic strength of 35 MPa was used to cast the specimens. Two types of steel
bars, high yield steel bar (T bar) with the nominal yield strength of 460 MPa and
mild steel bar (R bar) with the nominal yield strength of 250 MPa, were used in all
specimens. Fig. 3.1 displayed the typical stress-strain relationships of the bars.
Among all bars, TI0, RIO, and R6 were used in the walls while T13 and T20 were
applied at the top and base beams.
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Chapter Three
3.2.2 Code Provisions for Confining Reinforcement in Plastic Hinge Regions
Due to the existence of different requirements for the amount of confining
reinforcement in walls in the ACI 318 [A2] and NZS 3101 [N1] codes, design
equations in both design codes to ensure adequate ductility are described as follows.
3.2.2.1 ACI 318 Code Provisions
The required area of hoop reinforcement is given by the larger of
Ash = 0.3s he(~g -IJ fe'eh fyh
and
A'h = 0.09s h fe'e fyh
(3.1)
(3.2)
•
where Ash is the total cross-sectional area of transverse confining reinforcement
within spacing s and perpendicular to dimension he; s is the spacing of
transverse reinforcement measured along the longitudinal axis of the structural
member; he is the cross-sectional dimension of column core measured
center-to-center of confining reinforcement; Ag is the gross area of section; Aeh
is the cross-sectional area of a structural member measured out-to-out of transverse
reinforcement.
3.2.2.2 NZS 3101:1995 Code Provisions
Where the neutral axis depth in the potential yield regions of a wall, computed for
the approximate design forces for the ultimate limit state, exceeds:
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Chapter Three
(3.3)
(3.4)J1 Ag fe' ( C JA = (-+O.l)s h --- --0.07sh 40 h e A f L
e yh w
The following requirements of the transverse reinforcing steel shall be satisfied in
that part of the wall section which is subjected to compression strains due to the
design forces.
relied on in the design; L w is the horizontal length of wall.
moments refer to the base section of wall; J1 is the displacement ductility capacity
overstrength to moment resulting from specified earthquake forces, where both
where Ce is the distance of the critical neutral axis from the compression edge of
the wall section at the ultimate limit state; ¢o is the ratio of moment of resistance at
For medium-rise structural walls located in areas of low to moderate seismicity, it
may be appropriate to require reinforcement ratios which are between the gravity
load design and the seismic design requirements. The overall dimensions and
reinforcement details of the specimens tested are shown in Fig. 3.2. Each test
3.2.3 Details of Test Specimens
75 mm corresponding to 70% of the transverse confining reinforcement required by
mm diameter, giving a reinforcing ratio of 1.4 percent (minimum code requirement
specimen has a web reinforcement ratio of 0.50 percent and 150 mm wide x 300
mm deep boundary elements which are reinforced with eight mild steel bars of 10
is 1.0 percent). Reduced confinement in the boundary elements for specimens
MW1and MW3 is provided by 6.0 mm diameter closed stirrups (hoops) spaced at
NZS 3101 at a limited displacement ductility of 3.0 and 25% of that required by
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Chapter Three
ACI 318 as shown in Eq. (3.4) and Eq. (3.1), respectively. While for specimen
MW2, the vertical spacing of the hoops is meant to be 200 mm which corresponds
to 30% and 10% of the transverse confining reinforcement required by NZS 3101
and ACI 318, respectively for seismic detailing of fully ductile walls. Note that the
hoops enclosing the main flexure bars in wall boundary elements were detailed in
accordance with the requirements stipulated by ACI 318-02: having a bent around
90-deg with a six-diameter extension that engages the longitudinal reinforcement
and projects into the interior of the stirrups.
All three medium-rise shear walls tested had the same cross-sectional dimensions
and wall height and were reinforced with identical longitudinal steel. Each
medium-rise wall specimen in this test was 2000 mm wide, 3000 mm high and 120
mm thick. This provided the value of aspect ratios for all three specimens with
1.625 which was calculated according to the respective wall height of 3250 mm
measured by the vertical distance between the lateral loading point and the wall
base.
The moulds for the three specimens were set up by connecting the standard steel
sheets on which each surface were oiled to make it smooth just prior to the concrete
casting. During casting, the concrete of all specimens was compacted primarily by
means of a standard internal vibrator. The specimens were cast monolithically in the
vertical direction except that for specimens with construction joints at the base,
Specimen MW3, it was kept to stand for three days after the concrete for the base
beam had been poured, vibrated, and leveled off. Just before the upper part of
concrete was poured, the hardened concrete and the reinforcing bars in the
construction joint area were brushed to remove any loose particles. Then the base
beam concrete surface was moistened and the fresh concrete was poured to the
upper part of the moulds. After seven days, the moulds were removed and the
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Chapter Three
specimens were allowed to expose to laboratory environment until just before
testing. In the end, one day prior to testing the outer surface of every specimen was
made extra smooth for drawing crack patterns during testing.
3.2.4 Experimental Set-up and Loading History
The test rig used in this study is shown in Fig. 3.3. It consisted of two main
independent systems: an in-plane loading system and an in-plane base beam
reaction system. The in-plane loading system comprised one horizontal hydraulic
actuator which was fixed to the reaction wall and two vertical actuators connected
to the strong floor. The test units were subjected to in-plane, reversed cyclic loading
from the horizontal double-acting actuator applied at the level of the top steel
transfer beam (transfer beam 2) as shown in Fig. 3.3. The hydraulic actuator with
1000 mm stroke possessed a capacity of 1000 kN in compression and tension. The
axial loading was applied through two vertical actuators, each with a compression
capacity of 1000 kN and 500 mm stroke, attached to the top beam system, as shown
in Fig. 3.3.
The base beam of the specimens was fixed to the laboratory floor with twelve high
strength rods that prevent uplifting of the specimens and horizontal sliding of the
units along the floor during the application of the horizontal loading. Moreover,
every high strength rod was prestressed to efficiently restrain the rotation and
sliding of the specimen during the test. Constant axial load which corresponds to
0.05 of the cylinder compressive strength of the wall cross section that is equal to
fc'Ag
was adopted in the testing program and kept constant during the entire test by
applying load control to the vertical actuator. Moreover, hinged connections at the
tips of both the vertical and the horizontal actuator prevent any substantial restraint
to the rotation of the top of the wall, thus insuring cantilever behavior. After the
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Chapter Three
total constant axial force was applied, the horizontal loading would be introduced
through the top steel beam (transfer beam 2) of the specimens.
In previous tests of ductile members, the displacement loading, in which the level of
displacement increases according to the ductility factor (f.1 t1 = I1j~y)' controls the
subsequent cycles. However, since the present test units were meant to exhibit only
limited ductility, it was thought more valuable to displace the units to deflections
corresponding to selected values of drift due to the fact that the ductility factor
(f.1 t1 =~/~ y) depends heavily upon the definition of ~ y which is not readily
identified. As mentioned in the previous research [M1], a value of ~/ hw = 0.01 is
considered a practical limit on the drift to be realistically expected in low-rise
structural wall buildings. Herein the end of the test was reached at a drift ratio of
1.33% or the strength dropped to less than 80% of the recorded maximum loading.
In this study, the tested specimen was subjected to two cycles at each displacement
level except that only one cycle was applied to the specimen at a drift ratio of
1/2000. Fig. 3.4 demonstrates the detailed loading sequences during the testing.
3.2.5 Instrumentation of Wall Specimens
For measurement of top deflection, flexure deformations, and shear deformations,
two types of Linear Variable Differential Transducers (LVDT), with 100 mm travel
and 50 mm travel, were introduced as shown in Fig. 3.5. One LVDT numbered as
L14 with 100 mm travel was installed at the top of the specimen to monitor the top
lateral displacement. A total of ten LVDTs along the two vertical edges of the
specimens were arranged to measure the flexure deformation. The panel shear
deformations were detected by two LVDTs (L4 and L5) distributed along diagonal
directions of the panels. Two inclined LVDTs (L2 and L3) with one end of the steel
rods at the base beam were used to the measure the sliding deformations of the wall
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Chapter Three
panels. The sliding of base beam was detected by one LVDT named as LI with 50
mm travel positioned at the strong floor.
FLA type 5 mm-gauge length strain gauges with 10m vinyl-insulated lead wires
were used to measure the local strains of the selected reinforcing bars. The strain
gauges were attached to merely one layer of the reinforcing nets such that only the
outer layer for the bars of boundary elements was considered to attach the strain
gauges due to the symmetric configurations of the wall units. During testing, the
strains of the bars were recorded by an automatic datalogger and a strain gauge was
deemed no longer reliable when the strain exceeded 0.02.
3.3 Experimental Results
The global behavior, represented by crack patterns and hysteretic loops, and local
response such as longitudinal and transverse bar strain distribution of the tested
specimens are presented in the following figures. For those concerning the crack
pattern at different drift ratios, the dashed lines in the grid lines, indicating the
spacing of the reinforcement, represent the negative cracks opened during negative
loading while the continuous lines refer to the positive cracks in the positive loading.
The blackish areas as shown in the figures represent the splitting of the concrete.
3.3.1 Experimental Results of Specimen MWI
3.3.1.1 Global Behavior
Fig. 3.6 demonstrates the crack patterns and failure modes of specimen MWI at
drift ratios of 0.25%, 0.5% and 1.0% corresponding to the initial cracking stage,
crack development stage and final failure stage respectively. Fig. 3.6(a) shows the
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Chapter Three
initial flexure cracks at the lower part of the reference wall observed at a lateral
displacement of approximately 8.13 mm corresponding to a drift ratio of 0.25%.
With increasing drift ratios up to 0.5% as shown in Fig. 3.6(b), horizontal flexure
cracks spaced at approximately 250 mm (the web horizontal reinforcement spacing)
appeared and extended up to 70% of the wall height and moreover, shear cracks
propagated from the wall boundaries toward the opposite side and from the bottom
upward. Increasing the lateral displacements up to failure, it was observed that the
web cracks propagated more extensively toward the opposite side in the lower part
of the wall and the concrete in the left wall boundary element was spalling
considerably. This suggested that with the increase in lateral displacements, the
contribution of shear nature to the wall behavior became more significant with the
formation of diagonal shear cracks across the entire web at the final stage of testing.
However, at the attainment of wall failure, the opening of these diagonal shear
cracks remained small and the values of web horizontal bar strains across the
diagonal shear cracks were recorded to be low as shown in Fig. 3.10. This indicated
that although the shear nature contributed more to the wall behavior with respect to
the increasing lateral displacements up to failure, the overall performance of
specimen MWI was still dominated by flexure.
The lateral load versus the top displacement relationship for specimen MW1 is
shown in Fig. 3.7. The ideal strength of ~ =375 kN is obtained from rational
section analysis and exceeded by 13% and 12% for the positive and negative
loadings, respectively. In the positive loading direction, the observed base shear at
the first cracking occurs at a lateral displacement of 8.13 mm is approximately
301.0 kN corresponding to a nominal shear stress of 0.21g, and no yielding of
reinforcements at this time. As the test progressed, the specimen attained its
theoretical flexure strength as the vertical boundary element bars yielded in tension
at the wall base as shown in Fig. 3.7. Also, it can be seen from this figure that with
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Chapter Three
the increase of wall drift ratio to 1%, the maximum base shear as shown in Table
3.1 was achieved as the longitudinal bars in boundary elements yielded in tension
almost throughout the entire height of the wall. At this time, a number of diagonal
struts were formed and spalling of the concrete cover at the left wall boundary near
the base was observed.
3.3.1.2 Local Response
The plots of strains in the outermost longitudinal bar (#A bar) in the boundary
elements are presented in Fig. 3.8 in four different positions along the wall height:
50 mm (#A35), 550 mm (#A37), 1300 mm (#A39), and 2300 mm (#A41) above the
wall base. Longitudinal strains under both positive and negative loading directions
are demonstrated. In the compression boundary element, strains in the selected bar
are significantly less than those in the tension boundaries with respect to same wall
drift ratios and the compressive strain in these longitudinal bars is generally
observed in the region up to a wall height of 500 mm. In the tensile boundary
elements, at a wall drift ratio of approximately 0.30%, the longitudinal bar (#A bar)
experiences first yielding and strain of the whole bar almost reaches yield strain
after attaining a wall drift ratio of 0.67%. Under the negative loading direction, the
strain distribution of other longitudinal bars along three different sections is also
illustrated in Figs. 3.9(a), 3.9(b), and 3.9(c) respectively. The three wall sections,
section 1-1, section 2-2 and section 3-3, as shown in Figs. 3.9(a), 3.9(b), and 3.9(c),
respectively, are located at respective wall heights of 50 mm, 550 mm, and 1300
mm above the wall base. As indicated in these figures, with the increase of wall
drift ratios the neutral axis depth is observed to shift from the section middle to
around 600 mm for the section 3-3 and 400 mm for the section 1-1 calculated from
the left flange. The flexure plane section hypothesis can be applied to three wall
sections up to a drift ratio of 0.25%.
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Chapter Three
The strains along each selected horizontal web bars at each displacement peak are
shown in Figs. 3.IO(a)-3.IO(c). The three selected horizontal web bars, R, T, and W
bars as shown in the figures, are located at respective wall heights of 250 mm, 750
mm, and 1500 mm above the wall base. The figures show that as the test progressed,
the increase of strains along the selected horizontal web bars (R, T, and W bar) are
observed to be concentrated along the main diagonal struts during both loading
directions (positive direction: R7, R8, TI5, W70; negative direction: R5, R6, TI4).
However, strains remained small in web bars lying on or off the main diagonals.
Moreover, the bars in the center of the wall (T and W bar) were observed to be
highly strained since this portion of the wall was situated along the main diagonals
for both loading directions.
3.3.2 Experimental Results of Specimen MW2
For the purpose of investigating wall behavior caused by employing different
quantities of transverse reinforcements, less volumetric ratio of transverse
reinforcements were provided to the boundary elements of specimen MW2 in
contrast with those in specimen MW 1. Hence, the vertical spacing of transverse
reinforcements in wall boundaries was modified from 75 mm, which was applied in
specimen MWI, to 200 mm for specimen MW2.
3.3.2.1 Global Behavior
The plots of crack patterns and failure modes of specimen MW2 are shown in Fig.
3.11 and typical hysteretic loops along with the ductility and drift capacity of the
wall are demonstrated in Fig. 3.12. Fig. 3.11 (a) shows that the initial flexure cracks
located at the lower part of wall boundary elements are observed at a drift ratio of
0.170/0 for a base shear of approximately 300 kN. As test progressed, the outermost
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Chapter Three
flexure reinforcement in left boundary elements experienced yielding at a drift ratio
of 0.25% (8.1 mm) corresponding to a base shear of 340 kN and based on this, a
magnitude of yield displacement of 9.0 mm is obtained as shown in Fig. 3.12.
Displacement by 16.25 mm, the flexure cracks in the boundaries as shown in Fig.
11 (b) become denser and extend up to the top of the wall. For the positive loading
direction, the maximum base shear of41 0.2 kN is achieved at a lateral displacement
of 21.37 mm corresponding to a displacement ductility of 2.4. When the wall lateral
displacements are further increased to failure, it is observed that approximately 40
degree inclined struts formed and the strength degraded by approximately 15% at a
drift ratio of 1.0%.
3.3.2.2 Local Response
Fig. 3.13 plots the strains in the outermost longitudinal bar (#A bar) in the boundary
elements under both negative and positive loading directions. In the compression
boundary element, strains in the selected bar are significantly less than those in the
tension boundaries with respect to same wall drift ratios and do not approach yield
until the wall is displaced at a drift ratio of 0.67%. In the tensile boundary elements,
at a wall drift ratio of approximately 0.30%, the longitudinal bar (#A bar)
experiences first yielding and strain of the whole bar almost reached yield strain
after attaining a wall drift ratio of 0.67% where the maximum wall lateral strength
is attained during testing. Figs. 3.9(a), 3.9(b), and 3.9(c) present the strain
distribution of other longitudinal bars along three different sections under the
negative loading direction. As indicated in these figures, with the increase of wall
drift ratios, the neutral axis depth is observed to shift from the section middle to
around 600 mm (0.3Lw ) for the section 3-3 and 200 mm (O.ILw ) for the section 1-1
calculated from the left flange. For section 2-2 and 3-3 at wall heights of 550 mm
and 1300 mm, respectively the planes remain to be plane till the final stage of the
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Chapter Three
testing. For wall section 1-1, it is observed that at the end of the testing the plane
does remain to be plane.
Figs. 3.15(a)-3.15(b) illustrate the strains along each selected horizontal web bars at
each displacement peak. The two selected horizontal web bars, Rand T bars as
shown in Figs. 3.15(a)-3.15(b), are located at respective wall heights of 250 mm,
750 mm above the wall base. As it was found in Specimen MWl, the strains in the
horizontal bars are generally small and less than half yield strain except for T bar as
shown in Fig. 3.15(b) along the main diagonals (T14, T15 in positive loading
direction and T14 under negative loading direction).
3.3.3 Experimental Results of Specimen MW3
For the purpose of investigating the behavior of walls with existing construction
joints at the base, Specimen MW3 was tested to make a comparison with the
experimental results of Specimen MW 1 which subjected to a same level of axial
loading but with no construction joints at the wall base.
3.3.3.1 Global Behavior
The crack patterns and the final failure modes of the tested wall are shown in Fig.
3.16. At the beginning of the testing, almost horizontal cracks in the lower part of
the wall initially developed at a displacement of 5.6 mm for a base shear of
approximately 291.2 kN. Increasing the displacement to 8.3 mm resulted in the first
yielding of the flexure reinforcements in the boundary elements as shown in Fig.
3.18. As the test progressed, Specimen MW3 developed its maximum base shear of
410.3 kN at 31.74 mm top lateral displacement. At this time, a large portion of the
cracks was formed and several diagonal struts crossed in the middle of the wall web.
I
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Chapter Three
Small interconnected horizontal cracks were observed at the base of the wall during
further cycling the wall to the final testing stage. Fig. 3.17 illustrates the lateral load
versus the top displacement relationship of Specimen MW3.
3.3.3.2 Local Response
The plots of strains in selected longitudinal bars in the boundary elements are
shown in Fig. 3.18. It was shown in the figure that strains of the selected bars (#A
bar) in the compression boundary element are negligible and in the tensile boundary
elements, at a wall drift ratio of approximately 0.25%, the longitudinal bar (#A bar)
experienced first yielding and thereafter yield strain of the bar was concentrated up
to a wall height of 500 mm from the wall base. The strains of other longitudinal bars
along three different sections under the negative loading direction are also presented
in Figs. 3.19(a), 3.19(b), and 3.19(c), respectively. As indicated in these figures,
with the increase of wall drift ratios the neutral axis depth is observed to shift from
the section middle to around 400 mm (0.2Lw ) for the section 3-3 and 100 mm
(0.05L w ) for the section 1-1 calculated from the left flange. For section 2-2 and 3-3
at wall heights of 550 mm and 1300 mm respectively, the planes remain to be plane
till the final stage of the testing. For wall section 1-1, the flexure plane section
hypothesis can not be applied after the attainment of a wall drift ratio of 0.5%.
The strains along each selected horizontal web bar at each displacement peak are
shown in Figs. 3.20(a) and 3.20(b), respectively. From these figures, it can be
noticed that as the test progressed, the increase of strains along the selected web
bars (R and T bars) is observed to be concentrated along the main diagonals during
both loading directions (positive direction: R7, R8, T15; negative direction: R5, R6,
TI4).
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Chapter Three
3.4 Discussion of Experimental Results
In this section, a simple discussion of experimental results was presented in terms of
the observed behavior of each specimen. The strength and deformation
characteristics for each specimen are listed in Table 3.1. Note that different methods
of defining the yield displacements of walls existed and the yield displacement
herein was determined as the displacement when flexure reinforcements at the wall
boundary elements yield.
3.4.1 Crack Patterns and Failure Modes
It is observed that the initial flexure cracks within the length of the boundary
elements located at the lower part of specimens occurred at a drift ratio ranging
from 0.17% to 0.250/0 prior to flexure yielding of vertical boundary element bars. As
test progressed, all three specimens experienced yielding of flexure reinforcements
in the boundaries at a drift ratio varied between 0.25% and 0.33% corresponding to
an average base shear of approximately 340.0 kN. With increasing lateral
displacements until the base shear reaches maximum, the flexure-shear cracks of all
specimens are extended up to the wall top and a number of diagonal struts are
formed to efficiently transfer the lateral loading from the wall top to the bottom.
The recorded strains in vertical boundary element bars of all three specimens are
observed to be not uniformly distributed along the wall height, but significantly
changed with the increase of wall height on the tension side. Note that the rate of
change in tensile force in the boundary reinforcements implies bond forces acting
along the bars and is observed to be common in slender structural walls. This
suggests that the lateral force is mainly transferred by the integrity of
concrete-to-steel bond rather than diagonal compression struts.
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Chapter Three
Moreover, the strains along selected horizontal web bars are recorded to be
concentrated along the main diagonal struts during both the loading directions, but
remain small throughout the test. This gives a further verification of the
above-mentioned behavior that the specimens are dominated by flexure rather than
shear. However, concrete crushing and reinforcement buckling at the wall
boundaries as well as shear sliding at the base for all specimens are also observed,
but it does not significantly influence the flexure behavior of the walls. Accordingly,
it can be concluded that all specimens eventually failed in a flexure - shear mode.
3.4.2 Backbone Envelopes of Load-displacement Curves
The backbone envelopes of load-displacement curves have long been recognized to
be a critical feature in modeling the inelastic behavior of RC walls. Generally, it is
generated with the curve determined from a monotonic test and herein is
constructed by connecting the peaks of recorded lateral load versus top
displacement hysteretic loops for the first cycle at each deformation level of the
tested specimens. Fig. 3.21 shows the backbone envelopes of load-displacement
curves of all specimens tested along with the estimated average flexure and drift
capacities (refer to Table 3.1). From the figure, almost linear elastic behavior prior
to flexure yielding is observed for all specimens and thereafter the response curve
changes rapidly. Moreover, it can be seen that all three specimens are capable of
developing their flexure strength prior to failure, which is a prerequisite of adequate
seismic performance. Meanwhile, a good agreement between the calculated
maximum strength and measured maximum flexure strength is observed which
indicates that the maximum strength for each specimen is governed by the
maximum flexure strength obtained from inelastic section analysis.
In the case of Specimen MW3 with construction joints, almost same maximum
flexure strength is developed by comparing with that of Specimen MWI without
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Chapter Three
III
1'""'---__
construction joints but interestingly, slightly larger drift capacity for the Specimen
MW3 is observed. This observation does clearly indicate that for this specimen
tested, sliding shear does not inhibit the development of strength and deformation
capacities. In the case of Specimen MW2, although same drift capacity is achieved
by comparing with that of Specimen MW 1, the strength degradation for this
specimen is observed to be more severe than that of Specimen MW1 after the
attainment of the maximum flexure strength corresponding to a wall drift ratio of
0.67%. This can be due to the use of more confinement reinforcements in the
specimen which could have a favorable effect on inhibiting severely strength
degradation.
3.4.3 Components of Top Deformation
The top deformation of walls in this testing is mainly caused by three components:
flexure displacement, panel shear displacement, and sliding displacement. Note that
the flexure component of the total deformation also includes the contribution from
bond slip in longitudinal bars at the base of the wall. From figures of strain profiles
such as Figs. 3.9(a) and 3.9(b) etc. for all three tested specimens, it is found that
flexure deflection measured by LVDTs at the left and right side of tested specimens
could be overestimated and as such the effect of internal strains should be
considered to accurately evaluate the flexure deflections. For this purpose, the plane
section of tested specimens is assumed to deflect with respect to the best fit line of
internal strain distributions and in this study nonlinear strain distributions along the
wall sections 1-1 and 2-2 for each tested specimen are represented by linear trend
lines which consider the internal strains mostly. It is found by comparing the results
considering the effects of internal strains that the flexure deformations of tested
specimens by use of this kind of LVDT measure arrangement are overestimated by
a percentage value ranging from 3.78 to 6.41. Fig. 3.22 illustrates the ratios of three
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Chapter Three
displacement components to the total deformation with respect to wall drift ratios
by considering the effect of internal strains on flexure deflection of tested
specimens. It can be seen from the figure that as test progressed, the contributions
of the flexure deformations reduce slightly while the shear and sliding deformations
increase. However, it is recognized that at the final testing stage for all specimens,
the flexure deformation still dominates the response as it accounts for more than
50% of the total displacement. Note that the horizontal sliding displacement at the
base of the wall is negligible at all stages of testing for all three specimens tested, as
only up to approximately 10% of the total displacement was attained for Specimen
MW3 with construction joints at the wall base. This indicates that under axial
compression, the presence of the construction joints at the wall base have a minor
effect with respect to sliding. Meanwhile, the flexure displacement contribution of
Specimen MWI tends to be greater (65% compared with 58% in the positive
loading direction) at ultimate stage of testing than that of Specimen MW2 while the
shear components of total displacements for Specimen MWI becomes slightly less
at that stage (26% compared with 30% of the total displacements). This suggests
that for Specimen MW 1 with more content of the transverse reinforcements in wall
boundaries, the flexure contribution of the total deformation becomes greater. It is,
therefore, considered that the content of transverse reinforcement for flanged
speCImens can have an important effect in achieving a more ductile hysteretic
response.
3.4.4 Curvature Distribution along the Wall Height
Fig. 3.23 shows the average curvature distribution along the wall height, but only
the first cycle at certain drift level is presented there for all three specimens. It can
be observed that for all specimens, the rate of increase of the wall curvatures tends
to rise with respect to the increasing drift ratios of the walls. For Specimen MWl,
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Chapter Three
the curvature is highly concentrated at the bottom region and for wall region higher
than 500 mm from the wall base, the curvature remains constant at a lower level. In
negative and positive loading directions, the bottom curvatures of this specimen
were observed to be close with respect to same drift ratios and increased
significantly after the attainment of the wall drift ratio of approximately 0.33%
which occurred just after the onset of first yielding of this specimen. Similar
observed trend of wall curvature variations along the wall height for Specimen
MW3 are illustrated in Fig. 3.23. However, at a drift ratio of 1.0%, the average
curvature of Specimen MW3 is observed to be approximately 60% higher than that
of Specimen MW 1. In the case of Specimen MW2, unlike the observed behavior of
curvature distribution in Specimens MWI and MW3, the maximum curvature
capacity for this specimen is concentrated at a wall height of approximately 500
mm.
3.4.5 Stiffness Characteristics
Previous research [AI, T2, P5] indicated that the true stiffness of the wall elements
was significantly lower than that corresponding to gross section properties, even at
the serviceability limit state. It is, therefore, essential to evaluate realistic stiffness
properties of wall elements which can lead to more accurate modeling and analysis
of RC buildings with structural walls. The values of initial stiffness for each
specimen in both loading directions defined at the first yielding of longitudinal
reinforcements at wall boundaries are listed in Table 3.1. Fig. 3.24 demonstrates the
detailed stiffness properties of the walls which were evaluated using secant stiffness
at the peak of first cycle at each deformation amplitude. As expected, all specimens
experienced considerable reduction in stiffness as increasing wall deformations. At
the early stage of testing, the stiffness of each specimen rapidly dropped to about
25% of its uncracked stiffness by a drift ratio of approximately 0.15%. With the
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Chapter Three
increase of the wall top drift, the stiffness of each specimen further decreased and at
the final stage in testing, it remained rather low levels with approximately 30% of
its initial stiffness as shown in Table 3.1. Moreover, the degradation ratio of secant
stiffness which evaluated by dividing the values of secant stiffness at the initial
loading stages by those corresponding to the final loading stages was achieved to be
about 85% for all walls tested. This suggested that the content of transverse
reinforcements in wall boundary element and the presence of construction joints at
the wall base had negligible effects on the stiffness characteristics of the tested
walls under such levels of axial compression.
3.4.6 Energy Dissipation
The energy dissipation capacity for each specimen, which is calculated from the
inner area of load-displacement curves, has long been recognized to be of
paramount importance in the evaluation of the seismic performance of RC walls.
Fig. 3.25 shows the energy dissipation capacity of each specimen with respect to its
drift ratios. From this figure, it can be seen that prior to yield, rather small amount
of energy was dissipated, and thereafter the increase rate of energy dissipation for
all specimens tended to rise with increasing top drift ratios. In the case of Specimen
MW1 with more content of transverse reinforcements in the wall boundary
elements, the amount of energy dissipated was larger than that corresponding to
Specimen MW2. This can be due to the favorable effect of the transverse
reinforcements in wall boundaries with regard to its ability of developing more
ductile behavior for the wall tested. For Specimen MW3 with construction joints at
the wall base, although the contribution of sliding components to the total
displacements was observed to be small (less than 10%), lower amount of energy
was dissipated in contrast to that dissipated by Specimen MWI which was observed
to be 20% lower than that of Specimen MW1 at ultimate stage of testing. This could
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Chapter Three
be explained by the presence of construction joints at the wall base which led to
significant pinching of hysteretic loops of specimens at the final stages in testing.
The amount of energy dissipated by tested walls corresponding to the two different
loading cycles is also presented in Fig. 3.25, in which the dashed lines represent the
second cycles of the corresponding specimen. It was observed apparently that the
initial cycles dissipated more energy than the second loading cycles for all three
walls. Meanwhile, for the purpose of comparing the amount of energy dissipated by
individual components for each specimen, the flexure deformations and shear
deformations as well as sliding deformations of all specimens are separated from
their top displacements and are plotted against the lateral loadings as shown in Fig.
3.26. From the figure, it is evident that the energy dissipated by the flexure
deformation is much higher than that by shear or sliding deformations.
3.5 Extrapolation of Test Results
The shear force transfer mechanism of squat structural walls has been investigated
by many researchers [PI-3, WI] and was well outlined by Park et al. [PI] in which
the shear force is transferred to the wall base by a middle strut and a truss in the
triangular region beside the strut. Similarly, a strut-and-tie analytical model as
shown in Fig. 3.27 is proposed to simulate the behavior of Specimen MWI. The
concrete contribution is provided by a direct strut (dashed line) from the loading
point to the base of the wall and is kept constant after the onset of diagonal cracking.
The cross section of the concrete struts converging to the base of the wall is
approximately equal to Ac = 1.4 .c .b, where c is the depth of the compression
zone as shown in Fig. 3.28(a) calculated by the bending theory, and b is the wall
width. The area of the two outer struts, outer strut 1 and 2 as shown in Fig. 3.28(b),
is assumed to be 1/3 and 1/4 the area of the inner struts (Fig. 3.28(a)) respectively,
since the shear force is mainly transferred by the inner diagonal strut.
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Chapter Three
The angle of struts FG and DE is 28.4° which is close to the average angles of
diagonal cracks in the lower part of the wall. The transverse reinforcement within a
distance of 1000 mm, including eight web horizontal bars in the lower part of the
wall, is concentrated in horizontal member GF and DE. The eight longitudinal bars
in the web plus eight longitudinal bars in the flange are clustered in the vertical
member AE in the center of the flange. This consideration can be validated since the
strains of those bars at the lower part of the wall were observed to be beyond yield
strains at the maximum load as shown in Figs. 3.9(a) and 3.9(b). It can also be
observed from Fig. 3.9(c) that in the upper part of the wall, the bars with strains
beyond yield are mainly concentrated in wall flanges rather than the bars in wall
web. This variation agrees well with the reduced tensile force in tie AD compared
with those in tie DF and FH. The failure load of the truss is assumed when the
longitudinal reinforcement yield which is observed during tests.
The concrete contribution for shear in tested specimens can be estimated by the
strength at the onset of the diagonal cracks Vcr which can be detected by the strain
gauge attached to the horizontal reinforcement because significant tensile strain is
developed at this stage. When the horizontal reinforcement is sufficient to resist the
applied shear, the tensile strain of the reinforcement will be stable at a certain level
which is generally less than its yield strain, as shown in Fig. 3.29. The strengths
detected by the gauges attached to the web horizontal bars of tested specimens are
listed in Table 3.2. It can be seen that the detected strengths are close to that
calculated by NZS 3101 code within the range of approximately 10%. Provided that
the direct strut takes the shear force equal to the diagonal crack strength 293.5 kN
listed in Table 3.2, the other member forces can be determined. The member forces
at the maximum negative strength are presented in Fig. 3.27.
By use of this model, the average strain of ties EF and CD is evaluated and depicted
in respective Fig. 3.29(a) and 3.29(b) which also show tested strain history of
horizontal reinforcement against the applied shear force. It can be observed that the
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Chapter Three
average tensile strain predicted by the assumed strut-and-tie model agrees well with
the tensile strain history of web horizontal bars in walls tested. Thus, the proposed
model may provide insights into the force transfer mechanism and contribution of
web reinforcements of medium-rise walls under low axial loadings. This can be
further verified by employing this model to Specimens MW2 and MW3, as shown
in Figs. 3.30 - 3.31, respectively. Figs. 3.32 and 3.33 illustrate the strain history of
the gauge (TI4) along diagonal strut in the web horizontal bar (#T bar) for
Specimens MW2 and MW3, respectively. Also, the horizontal bar strains calculated
by using the proposed strut-and-tie model are presented in Figs. 3.32 and 3.33. It is
observed that the strains predicted by use of the assumed strut-and-tie model in
horizontal web bars of structural walls agree well with the tested data.
3.6 Conclusions
Three isolated cantilever reinforced concrete walls with an aspect ratio of 1.625
have been tested under cyclic loading up to failure. Strength and deformation
capacity characteristics of all specimens tested were summarized in Table 3.1. On
the basis of the experimental results presented herein, it shows that all three
specimens generally behaved in a flexure manner and were capable of developing
their flexure strength prior to failure, which is a prerequisite of adequate seismic
performance. Values of drift at initial cracking range from 0.17% to 0.25%.
Ultimately, the tested medium-rise structural walls designed for low to moderate
seismic areas can generally develop a drift capacity not less than 1%.
The content of transverse reinforcements at the wall boundaries, a reduction to 30%
and 10% that required by NZS 3101 and ACI-318 code corresponding to fully
ductile walls, might be considered as an effective measure for confining the
concrete in the compression zone in terms of the limited ductile performance of
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Chapter Three
walls. Moreover, for the specimens with more content of the transverse
reinforcements in wall boundaries, the flexure contribution of the total deformation
becomes greater. This indicates clearly that seismic performance such as drift,
ductility and energy dissipation capacity can be enhanced by increasing the amount
of the transverse reinforcement at the boundary elements of a wall. It is concluded
that the content of transverse reinforcement in wall boundary element and the
presence of construction joints at the wall base have negligible effects on the
stiffness characteristics of the tested walls under such level of axial compressions.
By decomposing the total lateral deformation into flexure and shear components as
well as sliding components, it can be demonstrated that the bulk of the energy
dissipation is due to flexure. The amount of energy dissipation due to shear
components does not change much under the condition of axial loadings on the
specimen. With regards to the energy dissipation contributed by sliding components,
it is found to increase slightly due to the presence of construction joints at the wall
base, but still remained at a low level up to the final stage in testing. However,
comparing the results considering the effects of internal strains, the flexure
deformations of tested specimens by use of this kind of LVDT measure arrangement
are overestimated by a percentage value ranging from 3.78 to 6.41.
The reasonable strut-and-tie analytical model for medium-rise structural walls,
which accounts for contribution of horizontal and longitudinal web reinforcements,
is developed to accurately reflect the force transfer mechanisms of medium-rise
structural walls under cyclic loadings. The tensile strains in horizontal web bars of
structural walls can be predicted by use of the assumed strut-and-tie model which
agrees well with the tested data. This provides evidence that the assumed
strut-and-tie model is a reasonable model for the flow of forces and contribution of
web reinforcements in walls tested.
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Tab
le3.
1-
Obs
erve
dst
reng
ths
and
duct
ilit
yo
fsp
ecim
ens
test
ed
No.
I(a
)I
(b)
I(c
)I
(d)
I(e
)I
(f)
I(g
)(1
)(2
)(3
)(4
)I
(5)
I(6
)I
C')I
(8)
PV;
.V"
"-,:U
:(o
~'~.
L~,:
a;.)
r;.",.
:l..
p;:
Ip.,
.I
PI:
IP
~IP
.\7
SIP
.K!I(
t;4
)r;
_K
,j1~
0=
D·D
D·
D··
D..
n·C
D."
kNIu
'\10
:..'"n
un
10:..'"
nu
nD
· C.D
.DD
"30
1.0
424.
63"
'5.0
1.13
9.1
41.2
3.5
1.0
~nn
I0.
500.
50lA
O0.
881.
203.
640.
05-2
93.3
-t22
.1·3
"'5.
01.
12·9
.539
.5·3
A·1
.0
I29
4.1
410.
23"
'5.0
1.09
9.0
41.-:
-3.
61.
0M
'Y2
I0.
50I
0.50
I1
AOI
0.33
I1
20
I3.
64I
0.05
·289
.4-t
03
A·3
-:-S
.01.
0-:-
·91
41.2
-3.5
.1.0
'0
I\.l
W3·
1O
.SO
IO
.SO
IlA
OI
0.88
I1.
20I3
.64
IO.O
SI
291.
241
0.3
3"'S
.O1.
0910
.0}
'7"
41
130
0
·286
.3·3
92
A-3
'75.
01.
04-9
.838
.3-t
.2-1
3
Not
e:(a
)H
oriz
onta
lw
ebre
info
rcem
entr
atio
;(b
)\'e
rtic
alw
ebre
info
rcem
entr
atio
;(c
)F
lexu
ralr
einf
orce
men
trat
ioin
boun
dary
elem
ent;
(d)
\i:l
lum
etri
cra
tio
of
tran
s'\'e
rse
rein
forc
emen
tin
bOU
ldar
yel
emet
t:(e
)\i
:llu
met
ric
rati
oo
ftr
ans'
\'ers
ere
info
rcem
ett
requ
ired
by
NZ
Sco
de:
(D\i
:llu
met
ric
rati
oo
ftr
ans\
'ers
ere
info
rcem
entr
equi
red
by
AC
Ico
de:
(g)
A",
iall
oad
rati
o:
(I)
Obs
er\'e
dsh
ear
forc
eat
firs
tcr
acki
ng;
(2)
!\.f
axim
umob
ser\
'ed
stre
ngth
duri
ngth
ete
st:
(3)~la..",imum
idea
lfle
xUIll
Ist
reng
th:
(4)
The
rati
oo
fIn
a."'
imum
ob
sm'e
dsh
ear
stre
ngth
toid
eal
flex
ural
stre
ngth
:(5
)Y
ield
dis
pla
cem
ett
wh
enou
ter
boun
daI!
'lon
gitu
dina
lrei
nfor
cem
etts
yiel
d:(6
)T
hein
itia
lst
iffn
ess
for
the
ith
spec
imen
;C
')M
axim
um
disp
lace
men
tdu
ctil
ity
le\'e
l;(8
)Max
imu
mto
pdr
iftr
atio
achi
e\'e
d.
*de
note
sth
esp
ecim
enw
ith
cons
truc
tion
join
tsat
the
wal
lbas
e.Q ~ ~ :;l ~ II>
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Chapter Three
Table 3.2 - Strengths at onset of diagonal cracks of specimens testedand predicted by analytical models
~ MWI MW2 MW3
310.5 291.6 295.3Vcr (kN)
-293.5 -285.7 -281.4
Vc (kN)274.0 274.0 274.0-274.0 -274.0 -274.0
Vcr /Vc1.13 1.06 1.08
1.07 1.04 1.03
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Chapter Three
R6
.Jl Q _
R10
T13
T20
::..;;...-..------------~ ---- ----- ---------
~------------------------
fI)
~en ----------
300 ~----------------~------------------------------~
400
500
700 ~i--------------------1cu
r:L!.
600
200~-----------------------------------------------------
100-------------------------------------------------------
Strain
0.10.090.080.070.060.050.040.030.020.01
o +,---,-------,------,-----,---------,---,-------,------,-----,--------1
o
Fig. 3.1 - Stress-strain relationship for steel reinforcements
..........
o
EIII I I I I ~TIOH&V
_----4, r I I I I I I It. ---'k--
oo
~ I"·'·'" '1·'1 I----..+-
800
~1700
~800
2RIO
Ill: :W- ~:~752RlO
Fig. 3.2 - Details of Specimen MWI
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e =:11751
i--IIIi-
I1
I
- 101 -
Fig. 3.4 - Applied loading history
Cycle Number
Chapter Three
I
1I I 1 I I 1 I
- - -1- - - -I - - - -+ - - - +- - - - I- - - -1- - - ....1 I I 1 1 I I II I I I 1 I I II I I I 1 I 1 I I I I I
___ 1 -I -l- - __ .J- _ - - I- I ~ .J- - - - I- - - - 1- - - -I - ~
I I I I I I I / 1 1 I II I I 1 I I I I / I I II I I I 1 I 1 I I I I I I I
___ 1 I ..1 1.. 1 1 -.J 1.. L 1 1 ..1 L _ -.J
I I I I I I I I I I 1 1 I 1I I I I I I I I I I I I I II 1 I 1 I I I I I I I I I I I I
___1 1 ...L 1.. 1 1 ..1 .1 L 1 1 ..1 L L I __ -.J
I I I I I I I I I I I I I I I II I I I 1 1 I I I I I 1 I 1 I II I I I I I I I I I I 1 I I / I
Fig. 3.3 - Experimental set-up
I 1
1 : I I 1 I I I 1 : e = 1: /100 I---I----I----r---,- Drift ratio = --r----,---/----I----r---r----,---I--
: : : : : : : : 8 = t/150: : :I I I I I I I I I () =11/200 I I I I
- - -1- - - ~ - - - -r - - - , - - - r- - - -I - - - "1 - - - r- - - - , - - - 1- - - -I - - - -r - - - r- - - ,- -I - -
I I I I I I I 8 = if3001 I I I I I II I I I I 1 1 I I I I I I I II I / I 1 8 =11/4001 I I I I I I I /
---:----:---~-e~!T6-00:----:---~---7---~---:----:-- ~-- ~- ~- :--: O=~ /1000 : : : : : : : : : : : :
- - -1- - - -I - - - -+ - - - + - - - I- - - -I - - - -I - - + - - I- - 1- - I - - - - - 1- - I-- - / - - - - -
8=1/:2000 : : : : : : : : :: : I :
1 I I 1 I I I I I
20
50
30
40
-30
-50
-20
-40
ES 10"E~ 0
~~ -10(5
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Chapter Three
3o~
o~
100
L2
3600
1700
L3
L8
L7
L6
~r--
~ 1600 r~ I~ I ::I:" ~
o~ I L9
~r--
oo..0
Fig. 3.5 - LVDTs support arrangements in specimen walls
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Fig. 3.7 - Lateral load - top displacement relationship of Specimen MW 1
-... '\'
11.0%
(c)At a drift ratio of 1.00/0
10 15 20 25 30 355
Ductility
0.05-0.25% 0.33% 10.5% 10.67%
1
1 1 1 1 1 1- - - r - - "1 - - - 1- - - ., - - -I - - - r - -
1 1 1 1 1 1
1 1 1 1 1 I- - - , - - , - - - 1- - - ,. - - -I - - - r- - -
1 1 1 1 1 1I 1 1 I 1 1
- - - t- - - "1 - - - r - - I - - -I - - - t- - -
Vj=375kN Specimen MW1
--- - , --1---,
111 I 1211 1 1 1---,--,---
-1- -
1
o
(b)At a drift ratio of 0.5%
-r--
Z-~_---I---r----r--
: 12: : I 1 :
Chapter Three
-.;: '_-K::.~;:~.:'
.....~ ~ ...
- 103 -
Displacement (mm)
Fig. 3.6 - Crack patterns of Specimen MWI
II! !I ~ I Drift ratio
--F jI .[--r~~I~kN--,- I I I --,--,--
1 1 I 1 1 11 1 1 1 1 1
- - -r - - - 1- - - 1'" - - -; - - - r- - - -r - -1 I 1 1 1 I1 1 1 1 I
- - - r - - "1 - - - r - - ., - - -I - - - r - -1 I 1 1
1 1 1 I~;..:-~.;;;;.-....;;;-;..:-;;.J.-=-+"-;;.=...;;;;.-....;;;-~-;;L.;-:::;..,;;;;.-4_t~=_F'_T_~- - - t- - - "1 - - - 1- - - I - - -1- - - r - -
1.0%: :0.67°40.5%: 0.33~4 O.25-Q.05% Drift ratio : : : :-600
-35 -30 -25 -20 -15 -10 -5
600
500
400
300
~200
100"'0
co;j
..9 0"';;..;
-100.sco;j
....:l -200
-300
-400
-500
(a)At a drift ratio of 0.25%
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Chapter Three
2000
2500
500
E.Sf<ll
1500 Ji(J)lIJ<ll
CD
~1000 ~
Ol'0;I
pn+>A41 DA39
A37 -A35 _
+
#A bar
---1%
#Abar
Specimen MW1
A41nD~-)A39
A37A35 -
A3"n :V ., It 111 I' ,_ , ~
A351 I v j • ~ I.' I ~ , I .,c. l J I 0
-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008
A411 .... i "I
§.~ A391.Q Ir----+----
~~;:jroo
Strain
Fig. 3.8 - Strain distribution in outermost longitudinal bars of Specimen MWI
Specimen MW1
JO[, . _. -w __ -lA' _-IA
.. -. -
Wall width marked from the left (mm)
1000 1200 1400 1600 1800 2000800600
-- -0.1%
- -lA- -0.25%
- - -0.50%
-_-1.0%
400
Drift ratio =
--0.05%
__0.17%
-0.33%
-0.67%
0.005
0.004
0.003
0.002
0.001;::'a~f/)
-0.001
-0.002
-0.003
-0.004
-0.005
Fig. 3.9(a) - Strain profiles of the vertical bars along section I-I of Specimen MWI
Wall width marked from the left (mm)-0.001
0.005
IJ['Drift ratio =
0.004 ~- -t:.- -0.05% ---0.10%
- -.- -0.17% 0.25%
0.003 1 - -x- -0.33% -0.50%
;::- -- -0.67% ---1.00%
'a 0.002l-;
in0.001
-0.002
Fig. 3.9(b) - Strain profiles of the vertical bars along section 2-2 of Specimen MWI
- 104-
II!I
I~
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
R8
T16
R6 R7
T14 T15
R5
- 'j .f • - -, • = . , - .. • • = ''")i;1 ; "i
T13
,..
RB
T16
Gauge location
Gauge location
Wall width marked from the left (mm)
- 105 -
Chapter Three
- .",- -0.05% ---0.10%
- .... -0.17% 0.25%
- -)1'- - 0.33% - 0.50%
- - -0.67% -1.00%
Drift ratio =
600 800 1000 1200 1400 1600 1800 2000
R7R6
T14 T15
Specimen MW1
R5
T13
10- 'n I I lI'~Specimen MW1
l--- >----
Drift Ratio =l--- ~_ITI'
el¥R7 >----
10-
RBar
+79
t{ 1°/n
~ 7 0.67% ~ ~ ./n<:o
""~ 77 0.5%~~~~ -A~
~ff 0.1%-0.33% 0.1%-0.33%P
11I I
I----
~"' ..Specimen MW1 ~
~'~
~
7~1% Drift Ratio = 1II---- I ..... r~ .. n<4 n!~
/ ~ .1%
1/ ~ T Bar
/h~ 0.67% /~
/# "" ~ / \g "~.5% /0.67% \W /1'---~ / \V Z- t----.~ 0.5% \~
=::::::::: ~
t--0.1%-0.33% 0.1%-0.33%
Fig. 3.1 O(b) - Strain distribution in the horizontal web bar (T bar) ofSpecimen MWI
Fig. 3.10(a) - Strain distribution in the horizontal web bar (R bar) ofSpecimen MWI
0.003
0.0025
0.002
0.0015
;:::';;j 0.001l-;
i/)0.0005
0
-0.0005
-0.001
0.002
0.0018
0.0016
0.0014
0.0012
;::: 0.001'a.ti 0.0008rJ)
0.0006
0.0004
0.0002
-0.0002
0.002
0.0018
0.0016
0.0014
0.0012
;::: 0.001';;jl-; 0.0008i/)
0.0006
0.0004
0.0002
-0.0002
Fig. 3.9(c) - Strain profiles of the vertical bars along section 3-3 of Specimen MWI
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Chapter Three
I--
RI I
1['I--= wo,.
Specimen MW1
- +I--"",,,.. .f~% Drift Ratio = W"\r'7!I
f-~ J 0.670~
f-
E/'"---- ~ WBar
E/ 0.5% --- +
f/ 1%
J/ ~.67%
J/ --- ~~0.33%
------0.1%-0.5% ~
0.002
0.0018
0.0016
0.0014
0.0012
~ 0.001
'a~ 0.0008
~0.0006
0.0004
0.0002
0
-0.0002W21 W70 W22 W21 W70 W22
Gauge location
Fig. 3.1 O(c) - Strain distribution in the horizontal web bar (W bar) ofSpecimen MWI
r;;:"", '.'~N'"
~~
.' I,"
~
"'r 1,:'"'--"'-
~rn"'lr
--..:;::
1'--.- N I,' l'<.J 1/1 I.-~_.-
I~ ....".~n·11 I ~n I ~'I
(a)At a drift ratio of 0.17% (b)At a drift ratio of 0.5% (c)At a drift ratio of 1.0%
Fig. 3.11 - Crack patterns of Specimen MW2
- 106-
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Chapter Three
E.sEIIIQ)
coQ)u:lIIIcoQ)>
..8<t.EOl
'QiI
1500
2500
1000
500
2000
11.0%
#Abar
.AA1pn~DA39 _
A37A35
Drift ratio
5 10 15 20 25 30 35
I1 1 I 1 1
- - ..... - - --I - - -1- - - l- - - --l - - -1- - -1 I 1 1 1 11 I 1 1 1 1
- - ..... - - ....J 1__ - .l- - - --l __ -1 _
I 1 1 1 1 I
1 1 1 1 1 1__ ..1 __ -.J 1 L __ .J 1 _
Vj=375kN : Specimen MW2I 1 1 L __ .J 1 _
Ductility 1 1 1 1
1 1 I 1__ 1 I 1 1.. __ .J 1 _
I I 1 II 1 1 11 1 1 1
0.05-0.25%iO.33%1 0.5%1 0.67%
o-5
--1%
-20 -15 -10
Specimen MW1
Strain
M1A39
A37
A35
- 107 -
Displacement (mm)
-------:0.67°~ 0.5%: 0.3304 0.25-~.05%1 1 1 1 1
: I!!I ! Drift ratio
--:-:.Jo·[ --~~c~ili;;---i"-
: Vj=375kN- - i" - - - --j - - -I - -
1 I 1
I 1 1
- - + - - --1 - - -1- - - t- - - ~ - - -1- -1 1 1 1 1 I
1 I 1 I 1 1- - ..... - - --I - - - 1- - - l- - - -4 - - -I - -
I 1 1 1 1 11 1 1 1 1 1
A37'f----+---+--..--+---f-iII+4rI-+-1~-_+il-~-+_--__+--~
A39r----+---
A351l=:::;::::==;=:::;::::==~~=:=~~I==!~==;~::;:::::!:::j==;::;:::::==:::j==;:::;!:~0-0.008 -0.006 -0.004 -0.002 0.002 0.004 0.006 0.008
A41 f----+---r-::..;.;;.::..;.=;.;;.,.--..-tIIIIt-~t__~t__--+_--_+--__I
-600
-35 -30 -25
Fig. 3.12 - Lateral load - top displacement relationship of Specimen MW2
600
500
400
300
200ZC 100"'0ro
0.9~
-100'""~ro.....:l -200
-300
-400
-500
Fig. 3.13 - Strain distribution in outermost longitudinal bars of Specimen MW2
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
Chapter Three
."".
1000 1200 1400 1600 1800 2000
Wall width marked from the left (mm)
Specimen MW2
D[400
.0- ..
Drift ratio = ._ 0"- ~ '
-0.05% .... ·0.1%
____ 0.17% ~ -..~ - 0.25%
_ 0.33% - -.... - 0.50%
- 0.67% • -0- - 1.0%
0.005
0.004
0.003
0.002
I=l 0.001'§~
-0.001
-0.002
-0.003
-0.004
Fig. 3.14(a) - Strain profiles of the vertical bars along section 1-1 ofSpecimen MW2
1800 20001600
,...1200 1400
!J--i~lF•..-=-: :; 20, r>' ~M' - - 600' - . ""8~0 1000
0.005 _
D['Drift ratio =• -6' -0.05% -0.10%0.004 -I- -II.- -0.17% 0.25%
- -:t. -0.33% ---0.50%0.003 -I• - '0.67% ---1.00%
~ .... - -+
I=l0.
002 1';jSpecimen MW2~
f/)0.001
-0.001Wall width marked from the left (mm)
-0.002
Fig. 3.14(b) - Strain profiles of the vertical bars along section 2-2 ofSpecimen MW2
600 800 1000 1200 1400 1600 1800 2000
Wall width marked from the left (mm)
-yf"'1" u'I7,;u'-,HIPi.
0.25%
-0.10%
---0.50%
-1.00%
Drift ratio =
• -t;. '0.05%
.... ·0.17%
· -x·· 0.33%
• - -0.67%
Specimen MW2
n-0.003
0.0025
0.002
0.0015
I=l';j 0.001l-;
~0.0005
0
-0.0005
-0.001
Fig. 3.14(c) - Strain profiles of the vertical bars along section 3-3 ofSpecimen MW2
- 108 -
\
I
~
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
-
IlI I
II-Specimen MW2
f--- -Drift Ratio ..- r.i !ton -
-RBar +
r--2% + 1%
I 0.67 0:--- O~
~ II ---- 0.5% \~ ///~ \ /
...............~VI/ --~\~~
0.1%-Q.33% 0.1 %-0.33%:r
0.002
0.0018
0.0016
0.0014
0.0012
;::0.001.;
ti 0.0008rJ)
0.0006
0.0004
0.0002
-0.0002R5 R6 R7 R8 R5 R6 R7 R8
Chapter Three
Gauge location
Fig. 3.15(a) - Strain distribution in the horizontal web bar (R bar) ofSpecimen MW2
n I I
ld[f--- Specimen MW2 r---
f--- f--
~ .......... 1%Drift Ratio ..
f---114'r.l nlm
f--
/ ~ +
I .....TBar
/ ./ '0.67% 0.670/~\
1/ ~ / \/// ,...",-~ \
V// / __ 0.33% 0.5% \ ".
~V ~ --~- .~ _.. -0.1%-Q.25% 0.1%-Q.33%
0.002
0.0018
0.0016
0.0014
0.0012
;:: 0.001"C;;... 0.0008
l'l)0.0006
0.0004
0.0002
-Q.0002T13 T14 T15 T16 T13 T14 T15 T16
Gauge location
Fig. 3.15(b) - Strain distribution in the horizontal web bar (T bar) ofSpecimen MW2
........
t--r--.
I I(a)At a drift ratio of 0.25%
'---. ....... :'>(.--
(b)At a drift ratio of 0.5%
""' 'IX: ............ " ,.[~...,
(c)At a drift ratio of 1.33 %
Fig. 3.16 - Crack patterns of Specimen MW3
- 109-
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Chapter Three
Specimen MW3
10 15 20 25 30 35 40 455
Vi=375kN
0.05-0.33% 10.5%1 0.67% 11.0% 11.3%
- - r - -,- - l" - - r - -1- - I - - ,- - -1- -
Ductility I I I I I I, , I I I 1 I
- - I" - -I - - I - - I" - - , - - I - - 1- -
Drift ratio I I I I II I I I I
o
I1 I I I I I I I
- - f- - -1- - -+ - - I- - -1- - -I - - +- - -1- -I I I I I I I II 1 I I I I I I
- - - - r - -I - - ""t - - r - -1- - --j - - r - - 1- -
I I I 1 I I I 1I I I I I I I I
- - r - -1- - l" - - r - -1- - -, - - ,- - -1- --,--I ~ I
""~-1--;- -2i l- - i -1-:1 - i - -
I I 1 I I I I
- T'- - "I -,I - -1- -I' - -, T -1 1-1- Till1:.0%: 0:.67%: O.~% 0~33-0:05%
__ :__ I .! I ~ l I I D'
~t~ Ji I[ !-D~t:~~O- -- _1- _ :I I I I
--:- -i--~--:- -~ ~ ~!~ ~;~ ~ jI I I : : 1-
-600-45 -40 -35 -30 -25 -20 -15 -10 -5
600
500
400
300
Z 200~'- 100"'0~
..9 0Cd~
-100(I)
~~
-200
-300
-400
-500
Displacement (mm)
Fig. 3.17 - Lateral load - top displacement relationship of Specimen MW3
2500
A41
A41 M1n 2000A39 A39 _ E
~ A37 A37 _ .s.~ A35 A35 _ ~~ 1500 (J)
CDC,) #Abar -0.50% #Abar gJ0 A3- -0.67% (0
(I) Specimen MW3CD
~ -1% ~::1
I
1000~
.c<C
0 1:01
+ '03I
A3 500
A31 J I ,.-..... 1 I ~ 1 I I Ii, I 1 J " j j 0
-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008
Strain
Fig. 3.18 - Strain distribution in outermost longitudinal bars of Specimen MW3
- 110 -
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00
1000 1200 1400 1600 1800 2000
• -. -••••• -•• - - - -Jil. - - - - - -' - - - - - - -~.'
800 1000 1200 1400 1600 1800 2000
Chapter Three
- 111 -
...... - .....
.... -. - • - _ ••• ,.. .. + 4 , ..., ••••
800
800 1000 1200 1400 1600
Wall width marked from the left (mm)
Wall width marked from the left (mm)
Wall width marked from the left (mm)
600
600
400
400
I}[Drift ratio =
-0.10% - -.. -0.17%
0.25% - -x- -0.33%
--0.50% • -- ·0.67%
-1.00%Specimen MW3
D[ Drift ratio =
-0.10% - -. -0.17%
0.25% - -~ -0.33%
-0.50% - -+- -0.67%.t
-1.00%Specimen MW3
D[Drift ratio =
--0.10% - -.. -0.17%
0.25% - -,,- -0.33%
--0.50% - -~ -0.67%
--&-1.00%Specimen MW3
200
-0.002
-0.002
0.01
0.008
0.006
0.004I:::::';;.,
~ 0.002
0.01
0.008
0.006
I:::::'; 0.004;.,
~
0.002
0.007
0.006
0.005
0.004I:::::';
0.003.t:lr:/)
0.002
0.001
-0.001
Fig. 3.19(a) - Strain profiles of the vertical bars along section I-I ofSpecimen MW3
Fig. 3.19(c) - Strain profiles of the vertical bars along section 3-3 ofSpecimen MW3
Fig. 3.19(b) - Strain profiles of the vertical bars along section 2-2 ofSpecimen MW3
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I ISpecimen MW3
f----
Il !J[t---
+ Drift Ratio =r----
/I'~%r---
r---- ~.6011
/ ~67'~t"7
r---
r---- R Bar r---
II.~5~ ""
//~~""""1///1~~~ + 1%
-----rill ~ ~ / ~67O ~~~/ ~ ~
........~ 0.5%
0.1%-0.33% 0.1%-0.33%
0.002
0.0018
0.0016
0.0014
0.0012
~0.001
";0.0008.]:j
if)0.0006
0.0004
0.0002
0
-0.0002R5 R6 R7 R8 R5 R6 R7 R8
Chapter Three
Gauge location
Fig. 3.20(a) - Strain distribution in the horizontal web bar (R bar) ofSpecimen MW3
n I I n'I----
""'m ..Specimen MW3 l---
I---- l---Drift Ratio =
f---- II" n~ f!4n:! l---
+
TSar /
+ r---.2:0 lyfo 1\/ ~ .h5%/1\\
/ 0.67Y~~ V //1\\\/ //~o~ // ~\
1/ 1/./ ~ V \~
0.1%-0.33% 0.1%-0.33%
0.002
0.0018
0.0016
0.0014
0.0012
~0.001
"§ 0.0008
ci50.0006
0.0004
0.0002
0
-0.0002T13 T14 T15 T16 T13 T14 T15 T16
Gauge location
Fig. 3.20(b) - Strain distribution in the horizontal web bar (T bar) ofSpecimen MW3
r - - -1- - - l" - - - r - - -1-50E~
1 I I Drift ratio = I
~ - - -: - - -+- - - ~ - -~ -40gI I I I -gI-- - - -1- - - -+ - - - I-- - - 0 -3tleI I I I =~ - - -: - - - ~ - - - ~ - - ~ -weI I I I j~ - - -:- - - ~ - - - ~ - - -:-weI I I I I
-~o -40 -~o -~o -~or - - -1- - - "T - - - r - - -I-weI I I I IL I .1 - - - L - - -1_20I I I I II I I I Ir - - -1- - - -t - - - r - - -I
~_~I I -
1---1---
-025".t'cf" -a:"5"l,r - - 1f1)% - r - - -I
~~~-~- -- I-- - - -1- - - -+ - - - I-- - - -I
I I I I II I I I I- - r - - -1- - - "1 - - - r - - -I
I I I I I~ __ L 1 J. L I
: : Displacement (mm):
1:0 ~ ~o 4> ~O- - - r - - -1- - - "1 - - - r - - -I
I I -+-MW1 I___ L 1_ _ _ _I
I I ..... MW2 I
___ ~ :__ --MW3 __ II
I I 1V, = -375 kN I I I I
I 1---1---1---1Drift ratio = I I I I
___ ~ 1 ..l L I
Fig. 3.21 - Backbone envelopes of load-displacement curves for tested specimens
- 112-
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- 113 -
1
-1--------------1 1-1- -1- - - - - - - - - - --
_ -I-I 1 1 1 1 1
_1- _1 __1__ 1__ 1__ 1__
_ ~ _ ~ _ --Flexure _: __ :__ :__
1 1 --Shear 1 1 I
- ~ - ~ - -:-Slid~ng -:- -;- -:- -
-~~~-I--I--
-,- -1- -1- -1- -1- -1- -I--
1 I I 1 1 1 1-1------ 1
I
Chapter Three
Drift ratio (%)
(c)
-0.88 -<l.75 -<l.63 -<l.5 -<l.38 -<l.25 -<l.13 0 0.13 0.25 0.38 0.5 0.63 0.75 0.88 1
1 1 I 1 1 1 1-I-'I-'I-T-T-'I-I-
1 1 1 1 I 1 I-1-1'-1'-'-'-1-1-_L_L_L_l.._l.._J._-.l _
_L_L_L_l._.1_..L_.l_I 1 1 I 1 1 1
-1--t--t--..L-..L-.J.-4-1 1 1 Specimen MW3 1
-I--t--t--T-,-,-"t-1 1 1 1 I 1 1-,-,-r--,-,-,-,--r-=::~-
1 1 1 1 1 1 1---I-I'-I-T-I-I-
1
o 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1
--- 100~~ 90
I;/J
~ 800
""5 70
.D 60'C'S 500
40C,)
4-;0 30
~ 20.gC,) 10rol-; 0~
-1
Drift ratio (%)
(a)
I 1 1 1 1 1 1 1 1 1 1 1 1 1- T - 1" - --, - -1- - 1- - I - T - - -1- -1- - r - T - 1" - --, - -1- -
1 1 1 1 1 1 I 1 1 1 1 1 I 1- T -1- -1- -1- -1- -1-1- - -1- -1- - 1- - T - 1- -I - -I--
I 1 1 I 1 I I I 1 1 1 1 1 1-T-I--------I-I- -~----~-T--------
_ .J. _ -.J __ 1__ 1_ _ _ _ _ _1 __ 1__ L _ .J. _ -.J __ 1__
1 1 1 1 I 1 I 1 1 1 I 1_l._J._...J __ I__ L_L_.1 1__ 1_ .1_...J __ I__
I I I I Specinen M'N1 1 I 1 -- FIeXlJ"e 1 1 1-+--+---1--1--1--'--+- --1--1- --Shear +---1--1--
_~_~_~__ :__ :__ ~_~ :__ : Siting ~_~ __:__
-,~'~- -~:~-t--r-+~~--I 1 1 1 1 1 1 1 1 1 1 1 1 1
- T - T - --, - -1- - 1- - I - T - - -1- -1- - r - T - T - --, - -1- -
o ~::::==~~=j....-l-I_~~I~~.iI~=+=~==~::::;=~I~~1~-1 -0.88 -0.75 -0.63 -0.5 -038 -0.25 -013
70
30
20
10
60
40
50
90
80
~ 100
e...,
Drift ratio (%)
(b)
Fig. 3.22 - Contribution of various deformation modes to total displacement of walls
Ol---i----i--;---;---;.--r---;---i--i---r-j---i----i--;---;---i-1 -<l.88 -<l.75 -0.63 -0.5 -<l.38 -0.25 -0.13 0 0.13 0.25 0.38 0.5 0.63 0.75 0.88 1
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
Chapter Three
3000 r-------,----.,---...,..-------...__-----,----.,---...,..------,
0.040.030.020.01
I I I I I I1 "il--:--~--~--:--~-- --:--~ -:--, I I I 5 I
I Specimen MW3 I I I 4 II I I I I I I 'I I
--1--4--1---1--4-- --I--~ ; -I--I I I I I I I - II I I I I I I II I I I' 1 I I I I
_ _I __ .1 __ L __ 1__ .1 _ _ _ 1__ ...J __ 1. __ 1 I __
I I I I I I I I I II I I I I I I I I II I I I I I I I I II I I I I 1 I I I I- -,- - I - - I" - -1- - I - - - -1- - I - - "I - - 1- - -I - -
I I I I I I I , I I
Drif\ratio: : : I I::: Drifl~tiO-~--l--I-- I -I--r-~--
I I \ j I I I I, \!
o -1% 0.05;0- 0.33% V.VI 'u "u
.{I.06 -0.05 .{I.04 -0.03 -0.02 .{I.01 0 0.01 0.02 0.03 0.04 0.05 0.00
SOD
2000
1000
1500
Curvature (rad/m)
(c)
3000 r-----r-----,---.,--~-...,.-----. ------r-____r----.,.--~--r--~
o[Jl
c::l..0
~~o>o~oul:l.s[Jla
S 2500
5
(a)
.{I. 01.{I.02.{I.03
II
I I I---r----t----t---
II Specimen MW1I I I
---f----I----I---I I II I II , I
___ L L L __
I I II I II I 1
___ L L L __ ._I I II I II I II I
---1---1--Drift ratio I
I
I!!',
---t--- --
, 5II 4
---1--- 3 --
--~---' --- '---III
_1 1 1 _1
III I
--1---1---I Drift ratioI
o I -1%~ -06z:rn -05tJ;.oM~Obs°(nJo:SC!% 0.6701 '1%:
.{I.04
0.01 0.02 0.03 0.04 0.05 0.06
I I I , 1_:__~-!!I_-I I 5
I I 4
I 3-~- 1 ; --
III
_ ...J __ 1. __ 1__ ...J __, 1 I 1
I I II , I
I I-1- - -I--
II,
---I--
I IDriflratio I
I I
SOD
2000
1000
1500
o[Jl
c::l..0
~~o>o~oul:l.s[Jla
S 2500
5
1
II I I I
- -1- - -t - - t- - -1- -
:Specimen MW2 :I I I I
- -I - - -+ - - I- - -I-I I I II I I, I I
__1__ .1 __ L: v:1
:I 1 II I
- -,-
OJ -1'0 X"/rn",\,u!fYI'Olvl'O-U'(lIUW'PI'U
.{I.06 -0.05 .{I.04 -0.03 -0.02 .{I.01
SOD
2000
1000
1500
3000 _
Curvature (rad/m)
Curvature (rad/m)
- 114-
Fig. 3.23 - Wall curvature distribution of tested walls
(b)
o[Jl
c::l..0
-;~o>o~oul:l.s[Jl
a
S 2500
E.
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Fig. 3.24 - Secant stiffness of tested walls with respect to drift ratios
III..... "II
!i
1.2
1.2
0.80.6
~MW1
-.-MW2
~MW3
0.8
0.2 0.4
Chapter Three
- - - - T - - - -I - - - - I - - - -I - - - - r - - - ,I I I 1 I II I I I I I
___ l ~ L ~ L JI I I I I II I I I 1 I1 I I I I I
-+---~----~---~----~---~
I 1 I I I II I I I I II I I I I I
T - - - -I - - - - I - - - -, - - - - r - - - ~
: : ~MW1: :____ ~___ L ~
: - -~- - MW2 : :I I'
---:--- -MW3 ~---~
I I II I I
-------------~---I
, I
1 I I
- - -I - - - - ~ - - - -lI I
II
---I
1
1
I----T
I1
I----i--II I
- - - - ~ - - - -I - - -I II 1I I I I
----T---~----,---~----
I I I I II I I I I
0.6
o
Drift ratio (%)
0.40.2
-0.8 -0.6 -0.4 -0.2
oo
-1
8000
Drift ratio (0/0)
12000
•10000
- 115 -
r---~----r---~----r---T---
I I I I I II I I I 1 IL J 1 J 1 1 __I I I I I II I I I I II 1 I I I I
~---~----~---~----~---+-
1 1 I I I II I I I I I1 I I 1 Ir---~--- ---~----r---T
I I I I 1I I I I I IL ~ L ~ L _I 1 I I II I I I II I I I 1r---~----r---~----r--
I I I 1 II I I 1 II I I I I~---~----~---~----
I I I II I I I
~ - - - -l - - - - 1- - -I III I 1 I Ir---~----r---~----r---T---
I I I I I I1 I I I 1 1
o-1.2
Fig. 3.25 - Energy dissipation capacity of each specimen with respect to thedrift ratios
90
80
70
S 60SZ 50CCflCfl~ 40~~1;; 30"EC':l<:,)
20~r./)
10
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Chapter Three
r - -I - - T - - r.::. , -509 T - -1- - ""1 - - r - -1- - l
I I 1 I ~ Ir - -I - - i- - - r :; .., -400I I I I ~ II- - -I - - + - - 1- ::; --I -3091 , I 1:Ii 1
~ - -: - - ~ - - :- 3 ~ ~oo
l __ I __ 1 - - 1- - -I 409I I I I II I I I
$ 10 1p 20 2p- Flexural displacement (mm)l
, , , , 1
--I---j--j---I--"1
I 1 1 1 I
--I---l--j---I---1
I Specimen MW1 I__ I__ ....l __ L __ I__ J
1 I I 1 I__ 1__ J __ L __ 1__ J
I - - I - - 1- - -1- ::.. -1- - - -I - - , - - ""1 - - ""1
I I I I ~ I I I I I1---1---1---1-:;-1- ---1--'--'--1
l __ 1 1 1_ ~ _1_4
1 J _ JI I I 1 = I 1 IL - - L - -1- - -1- !ii -1- _ _ _ _ .JI I I I"; I I II- - - I- - - 1- - -1- ...I -1-29 - -t - - ..,I 1 1 I I I I 1r - - r - -1- - -1- - -1-1 - -, - - ""1 - - I
I 1 I I 1 I I
__ 3 __ ~ __ .§ __ j
Shear displacement (mm) 1- . ~
I , I 1---I--..,---t--..,
I I 1 I- - -I - - ..., - - ..., - - I
I Specimen MW1 I- - -I - - , - - , - - 1___I __ J __ J __ J
r - - 1- - - 1- - -,- - -;....--"300
~ - -:- - -:- - -:- -. !500
~ - -:- - -:- - -: - - . ~ ~OOL - - 1- - - 1- - -1- - . !ii 300I I I 1 1;;r - - 1- - -1- - -1- - . ~ 200I 1 I I Ir - - 1- - - 1- - -I - - -1-1
I I I I
-~==-~ ~11.~~ _-XIr-I I I 1 II - - 1- - -1- - -I - - -I -500
l - - 1- - - 1- - _I I -600
__ Ql5__ 1 __ 115__ ~
Sliding displacement (mrn)--i--i--,--j---t--i---+-_
i1 1 I I--,--,---,..--,1 Specimen MWl I
--l--j--T--j--J_-l __ l __ J
(a)Flexure displacement of
Specimen MW 1
(a)Shear displacement of
Specimen MW 1
(a)Sliding displacement of
Specimen MW 1
1 115 ~ 215 J
- ~sliding displacement(mm)
9-t-i- - -t - -I - - 1- - t- - 1I I I I I I
& - + - -4 - -I - - 1- - "" - ~
I I Specimen MWl I_ .1 _ .J __, __ ,__ L _ J
1 1 I I I I_ 1 _ J __,__ 1__ l _ J
-~ __~~.5_; _ ~II Ir-+---t
I I I1-- --+1
L _ .1_
I I I I I I
l - 1.. - ..!. - _1- -1- -.!so&
r-T-"1-'--I-~r509
I I I I 1 ~ 1r - t- - -t - -I - -1-:; ~oo
I I I 1 I ~ IL - +- - -+ - -I - -1-:;; Iaoe1 I I 1 1:Ii I
~ -t - ~ - -: - -:- ~ ~I I I I I II-I -I - -1- -1- - 1
1 I I I I
.. ~ $ 110 112
~ - Shear displacement (mm):
- -t - -1- - t- - -1- -I- - -j
I 1 I I 1 I- -+ - -1- - +- - -1- -I- - -I
1 I Specimen MW2 1_ .1 __ 1__ L _ -l __ ,__ J
1 1 I I I I_ ..!. __ 1__ 1.. __I __ 1__ J
r - , - -1- - T :::"-1 SOlt-r - , - -1- - T - -I - - 1- - l1 1 I 1 ~ I
r - -t - -1- - +- :; -i40lt-I I I I ~ ,1---4--I--+- ::;-I30lt-I I 1 1:Ii I
~ - ~ --:- -t 3-: 209-
I 1 I I II-I - -1- -I - -I 1 I
r - - r - - 1- - -1- ~-1-590-, - - -I - - , - - ""1 - - l
I I I 1 ~ Ir - - r- - -1- - -1- :; -1-400
1 1 1 I ~ II- - -I- - -1- - -1- =-1-300
1 I I I ~ 1
~ - - ~ - -:- - -:- '::-:-200
l_ -1- - _1- __ 1__ -1-1001 I 1 I II I I I
-a5 -aD ~ 10 1p 210I - - 1- - 1- Flexural displacement (mm) II I , , , Ir---r- --"'--"'---j--"1, 1 1 I I I1--- ----1---l---l---11 1 Specimen MW2 IL__ __~ __ ....l __ ....l __ J
I I I I I I I I IL __ 1 1 1 1_
590I __ .J __ .J __ J
(b)Flexure displacement of
Specimen MW2
(b)Shear displacement of
Specimen MW2
(b)Sliding displacement of
Specimen MW2
r -1- -1- -,- ,-.::. - -!p00I I 1 I 1 ~ Ir -1- -1- -1- .., -:; - "'09I I 1 I I ~ II- - 1- -1- -I - --I - :;; - 4091 I I I I :Ii 1
~ -:- -:- -:- ~ - ~ - ~oo
~ -:- -:- -: - -: - ~ - i ooI I I I I 1
-35 -3D -a5r -I--
I I 1r - -I
1 I I-I-
I IL _I__
I I 1 I I I IL _1__ 1__1__I _ .J -..500
- r - I -1- -1- -1-""1 - l
~ 10 115 210 215 ~ 315- Flexural displacement (mm) I
, , , , , , 1
- r - r - 1- -I - -I - -j - "1
I I 1 I I I I- +- - I- -1- -I - --I - -l - -1
I I Specimen MW3 I_ L _ L _ 1__I __I _ ....l _ J
I I I 1 1 I 1_ L _ L _ 1__I__I _ .J _ J
r - - , - - -1- - - r ~oo
I 1 1 ~r---t---I--- :;400I I I ~
L - - -+ - - -1- - - =3001 I I 5L __ .1 - - _1- - - .,; ~ooI 1 I ...I
~ - - ~ - - -:- - - ~ -1
: I 1 1-6 -6 -14r - - "1 - - -I-I Ir---t-
I I
L-IL_I I I IL.. __ J 1 L.. --500
:- - - T - - , - - -1 I 1
- - --j - I) --:---j - --I
v-t--~--_:1- _ I I I: _: __ J
I:
I 1
2 .. ~ II
- - .Shear displacement (mm) :
- - - 1- - - +- - - .., - - - 1
I I I I- - - 1- - - +- - - ---j - - - I
1 I___ 1 Specimen MW3 _ 1
I I I I___ 1 1.. __ J I
r - - , - - -,- - - r ~~
~--~---:---~~!l __ J 1 1.. --. ~
I I I I =L - - -+ - - -1- - - L -; !iiI I I I 1;;f-- - - -+ - - -1- - - +- ---'~::
I I I Ir - - , - - -1- - - T -1901 I 1
IL __
1
f--
I
~ ==j===:= ==t:
- - -1- - - T - - ""1 - --I
I I I 1
- - -1- - - T - - -1- --,
I ~I__ J II I
_J- __ ....l 1
1 I I
_;d= =~ ==~===:1 I 1
1 2 :J 4
==;1~i:9 ~i~l:c~m~~(~;):I I I I
- - -1- - - +- - - -l- --II I 1 I
- - - 1- - - .,.. - - ..., - - -ISpecimen MWJ I
-- ---1---1
___ 1_ _ 1 __ J - - _I
(c)Flexure displacement of
Specimen MW3
(c)Shear displacement of
Specimen MW3
(c)Sliding displacement of
Specimen MW3
Fig. 3.26 - Flexure, shear and sliding displacements of Specimens MWI to MW3
- 116 -
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Tie
Chapter Three
Tie
Tie
Inner strut
Outer strut 1
Outer strut 2
V
---.!'---N_2 ------=:h-=--- N_l_t
Fig. 3.28(b) - Forces acting at basesection of strut-and-tie model
294kN
P(-)=422.1 kN
0trl
//
c,c,}// trl
00\", 0 0
" M 0
128.6 :3
D
0 000 0r- 0M
F
trl
r-..: 0"'1" 0"'1" :3
H
FI
294kN
BI
ir---------""O:~1
01~I0"11Nj,
I
C I"llI-II.......--~...;;..;...;;t--a.;;..-......I----±.. J
----. VMU
h
L
- 117 -
Fig. 3.27 - Strut-and-tie model of Specimen MWI
The area ofconcrete struts
A,
F2
Fig. 3.28(a) - Forces acting at wall basesection
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Chapter Three
500 ,Strut-and-Tie predictions nTl3 m Gauge #T14
Tli T1~
Z :,(~c SpedmenMW1VC)l-<
c2-<:f8g~III\~
0.0011 0.0015 0.0019l-<
StrainroV
.J:lf/)
-300
-400
-500
(a) Horizontal gauge #T14
500mStrut-and-Tie predictions
R'400 fIj) ...-_ ..300 • •••• •
\JO'V1O Gauge tlWfS
~200
100 Specimen MW1VC)l-<
c2~·~~l! o(~~~
0.0011 0.0015 0.0019l-< StrainroV
-200.J:lf/)
-300
-400
-500
(b) Horizontal gauge #W69
Fig. 3.29 - Strain history of gauges in horizontal bars of Specimen MWI
P(-)=403.4 kN
294 kN
B
l. ..<,lC
D
'C
~
<'l
~
H
ccc
ccc
ccc
l..8..5..O. 1850
Fig. 3.30 - Strut-and-tie model ofSpecimen MW2
Fig. 3.31 - Strut-and-tie model ofSpecimen MW3
- 118 -
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Chapter Three
0.0019
Strain
Gauge #114
Gauge #T14
TJ3 T16
mT15
Strain
0.0015
Tl.3 T16
Tl~ m
~-)
Specimen MW2
Specimen MW3
0.0011 0.0015 0.0019
0.0011
Strut-and-Tie predictions
Strut-and-Tie predictions
- 119 -
500
400
30
-400
-500
500
400
300
Z 200
C 100~
u~
~~ -0.0001~ -100~
...dr./)
-200
-300 ........
-400
-500
Fig. 3.32 - Strain history of the gauge T14 in selected horizontal bar ofSpecimen MW2
Fig. 3.33 - Strain history of the gauge T14 in selected horizontal bar ofSpecimen MW3
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Chapter Three
REFERENCES
[AI] Aktan, A. E., and Bertero, V. v., "RC Structural walls: Seismic Design for
Shear," Journal of Structural Engineering, ASCE, V. 111, No.8, Aug. 1985,
pp. 1775-1791.
[A2] ACI Committee 318, "Building Code Requirements for Structural Concrete
(ACI 318-02) and Commentary (318R-02)," American Concrete Institute,
Fannington Hills, Mich., 2002, 391 pp.
[Cl] CEN Technical Committee 250/SC8, "Eurocode 8: Earthquake Resistant
Design of Structures - Part 1: General Rules (ENV 1998 1-1, 1-2, and 1-3),
CEN, Brussels, 1995.
[Fl] Fintel M., "Shearwalls - An Answer for Seismic Resistance?", Concrete
International, Vol. 13, No.7, pp.48-53.
[Ll] Lefas, L.D., Kotsovos, M.D. and Ambraseys, N.N., "Behavior of Reinforced
Concrete Structural Walls: Strength, Defonnation Characteristic, and Failure
Mechanism", ACI Structural Journal, V.87, No.1, Jan-Feb 1990, pp.23-31.
[M1] Mestyanek 1. M. "The Earthquake of Resistance of Reinforced Concrete
Structural Walls of Limited Ductiltiy" Master thesis, University of
Canterbury, Christchurch, New Zealand, 1986.
[M2] Maier J. and Thurlimann B., "Shear Wall Tests" The Swiss Federal Institute
of Technology, Zurich, Switzerland, 1985, 130pp.
[Nl] New Zealand Standard Code of Practice for the Design of Concrete
Structures, NZS 3101: Part 1, 185 p.; Commentary NZS 3101: Part 2,247
p.; Standard Association of New Zealand, Wellington, New Zealand.
[PI] Park, R., and Paulay, T., "Reinforced Concrete Structures", John Wiley &
Sons, New York, 1975,769 pp.
[P2] Paulay, T., and Priestly, M. J. N., "Seismic Design of Reinforced Concrete
and Masonry Buildings", John Wiley & Sons, New York, 1992,744 pp.
- 120-
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Chapter Three
[P3] Paulay, T., Priestley, M. J. N., and Synge, A. J., "Ductility in Earthquake
Resisting Squat Shearwalls", American Concrete Institute, Detroit,
July-August, 1982, pp. 257-269.
[P4] Pilakoutas, K. and Elnashai, A., "Cyclic Behavior of Reinforced Concrete
Cantilever Walls, Part I: Experimental Results" ACI Material Journal, V.92,
No.3, May-June, 1995, pp. 271-281.
[P5] Pilakoutas, K. and Elnashai, A., "Cyclic Behavior of Reinforced Concrete
Cantilever Walls, Part II: Discussions and Theoretical Comparisons" ACI
Material Journal, V.91, No.2, May-June, 1995, pp. 1-11.
[P6] Penelis, G. G., and Kappos, A. J., "Earthquake-Resistant Concrete
Structures," E&FN Spon, London, UK, 1997.
[Tl] Thomas N. Salonikios, Andreas J. Kappos, Ioannis A. Tegos, and Georgios
G. Penelis, "Cyclic Load Behavior of Low-Slenderness Reinforced Concrete
Walls: Design Basis and Test Results" ACI Structural Journal, V 96, No.4,
July-August 1999, pp. 649-660.
[T2] Thomas N. Salonikios, Andreas J. Kappos, Ioannis A. Tegos, and Georgios
G. Penelis, "Cyclic Load Behavior of Low-Slenderness Reinforced Concrete
Walls: Failure Modes, Strength and Deformation Analysis, and Design
Implications" ACI Structural Journal, V97, No.1, Jan-Feb 2000, pp.
132-142.
[WI] Wood, S.L., "Shear Strength of Low-Rise Reinforced Concrete Walls" ACI
Journal, V87, No.1, Jan-Feb 1990, pp.99-107.
- 121 -
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Ag
Aeh
Ash
Ce
fe'
f y
he
Ki
Lw
Ver
Vrnax(feSf)
Vrnax(teSf)
fJ
fJ~max
L1 y
s
¢o
Chapter Three
NOTATIONS
Gross area of section
Cross-sectional area of a structural member measured out-to-out of
transverse reinforcement
Total cross-sectional area of transverse confining reinforcement
within spacing s
Distance of the critical neutral axis from the compression edge of the
wall section
Cylinder strength of concrete
Yielding stress of reinforcing steel bar
Cross-sectional dimension of column core measured center-to-center
of confining reinforcement
The initial stiffness for the ith specimen
Horizontal length of wall
Observed shear force at first cracking
Maximum observed strength during the test
Maximum ideal flexure strength
Displacement ductility factor
Maximum displacement ductility
Yield displacement of the walls
Spacing of transverse reinforcement
Ratio of moment of resistance at overstrength to moment
- 122 -
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Chapter Three
Horizontal web reinforcement ratio
Vertical web reinforcement ratio
Flexure reinforcement ratio in boundary element
Volumetric ratio of transverse reinforcement in boundary element
Maximum top drift ratio achieved
- 123 -
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Chae.ter Four
CHAPTER FOUR
STIFFNESS CHARACTERISTICS OF STRUCTURAL
WALLS WITH LIMITED TRANSVERSE
REINFORCEMENT
Abstract
This study intends to investigate the stiffness characteristics of reinforced concrete eRC)
squat structural walls with limited transverse reinforcement. An analytical approach,
combining the inelastic flexure and shear components of deformation, is proposed to
properly evaluate the initial stiffness of RC walls tested. In this approach, the flexure
deformation is calculated by use of a standard moment-curvature analysis and the shear
deformation is determined by applying the truss mechanism to the RC walls studied.
Based on this proposed analytical approach, a comprehensive parametric study
including a total of 180 combinations is carried out and a simple expression according
to this study is proposed to determine the initial stiffness of RC walls studied as a
function of three factors: yield strength of the outermost longitudinal reinforcement,
applied axial compression and wall aspect ratios. Finally, the proposed stiffness
formulae are validated with experimental results and it is found, by comparison with
other stiffness predictions, to be more effective.
Keywords: RC structural walls; Limited transverse reinforcement; Stiffness
characteristic; Elastic un-cracked stiffness; Cracked analytical stiffness; Initial stiffness
- 124 -
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Chapter Four
4.1 Introduction and Background
Stiffness properties of all elements of RC wall structures can affect the estimation of
the fundamental period, displacements and distributions of internal force response
between walls. The magnitude of the initial stiffness, £1e depends on the intensity
and distribution of stress on a wall cross-section as well as the extent of flexure
cracking. Flexure cracking causes reduction in net cross sectional area and moment of
inertia, and hence reduction in initial flexure rigidity of the wall section. This leads to
the accurate prediction of initial stiffness of RC members becoming increasingly
difficult. Thus, rough estimates of the stiffness are employed in the analysis of RC
members under lateral loads. In practice, the value of 0.35 and 0.70 the gross moment
of inertia for cracked and un-cracked walls, respectively is widely employed in the
first-order analysis. However, this simplification may not be appropriate in many
practical cases as the recommended moment of inertia for walls is independent of the
reinforcement content and axial load level. To refine this over-simplified stiffness
consideration in beams and columns, several equations [S 1, M 1] have been presented
to properly estimate the initial stiffness of beams and columns up to date and different
parameters such as axial loads, concrete compressive strength and shear deformations
are considered in these equations. Meanwhile, future recommendations for frame
analysis have also been presented to apply them to current design codes.
In the case of RC walls under lateral loads, Priestley et al [P2-4] indicated that the NZS
3101 [N1] design code recommendations for estimating the stiffness of RC walls fail to
recognize the influence of two important factors: the member strength and the
reinforcement grade. For stiffness evaluation of RC walls, it is found that the curvature
does not vary significantly with reinforcement grade or axial load level. However, the
- 125 -
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Chae.ter Four
wall strength has well been recognized to depend significantly on the axial load and
reinforcement content. Thus the recommendations in NZS 3101 design code [N1] for
stiffness estimation are found to be over-simplified and may lead to an inaccurate
assessment of the wall stiffness. To obtain a more accurate evaluation of wall stiffness
as determined from the curvature at first yield and the flexure strength, several
proposals on how stiffness values can be more realistically assessed were presented
[P3-4]. However, Fenwick and Bull [Fl] found that these proposals also proved to be
over-simplified. To further improve the stiffness predictions, Fenwick and Bull [Fl]
analytically investigated a range of slender rectangular walls with uniformly spaced
reinforcement and proposed an expression for predicting the initial wall stiffness as a
function of concrete strength, reinforcement grade and axial loads. As indicated from
the research [Fl], the expression proposed for accurately predicting the wall stiffness is
only justified for slender rectangular walls for which the response is dominated by
flexure and thus the proposals cannot be applied to RC walls with low aspect ratios as
their response may be controlled by shear deformations.
This study strives to establish more consistency and accuracy in predicting initial
stiffness of these low-rise structural walls with limited transverse reinforcement. Firstly,
a brief review of current ACI 318-02 [AI], NZS 3101-1995 [Nl] and FEMA 356
provisions [F2] concerning the initial stiffness, Ele is presented and the need for
modifications to the current code provisions is explained. Secondly, to better
understand the initial wall stiffness, this study makes a comparison among several
stiffness characteristics: un-cracked elastic stiffness, fully cracked wall stiffness and the
initial wall stiffness. Thirdly, this study proposes an analytical approach to properly
assess the initial flexure and shear stiffness of squat RC walls with limited transverse
reinforcement. In this approach, an extensive parametric study including a total of 180
- 126-
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Chapter Four
combinations are carried out and the influences of several important parameters on the
wall initial stiffness are pointed out. Finally, based on the parametric study, a simple yet
accurate formula combining both shear and flexure deformations is presented to predict
the initial stiffness of squat RC walls. The obtained analytical results by use of this
simple formula is verified against experimental results and compared with alternative
recommendations from several existing codes and standards.
4.2 Previous Research in Evaluating Initial Wall Stiffness
In the following sections, the effective moment of inertia, Ie' is defined as the moment
of inertia that a uniform elastically responding wall would have, such that when it is
subjected to the lateral force that causes first yield, or a strain of 0.002 in the concrete,
it sustains the same deflection. The initial stiffness of walls defined as explained above
is utilized in several previous researches and current design codes. They are briefly
reviewed in the subsequent sections.
4.2.1 Research Conducted by Fenwick and Bull [Fl]
The member initial stiffness can be obtained by the integration of curvatures over the
cracked and un-cracked sections along the member. The standard beam theory
describes the behavior of idealized cracked and un-cracked sections and can be used to
determine the initial stiffness theoretically. However, the real flexure response is
complicated by bond slip and tension stiffening. Moreover, the presence of axial
compression further stiffens the member by delaying the onset of cracking. After
considering these, Fenwick and Bull [F 1] conducted a parametric study considering
three main parameters: axial loads (N), yield strength of longitudinal reinforcement
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Chal!..ter Four
(MPa) and concrete compressive strength (MPa). The authors went on to propose the
following expression as shown in Eq. (4.1) by relating the effective moment of inertia
to the moment of inertia of un-cracked concrete section for cantilever walls subjected
to flexure deformations predominantly.
P 190 'Ie =0.267(1+4.4-,U-)(0.62+-)(0.76+0.005! )1
! A legc g Y
(4.1)
The authors suggested that the expression could be used for slender walls whose
behavior is dominated by flexure. Meanwhile, it was assumed that the longitudinal
reinforcement was spread uniformly along the walls with rectangular cross sections. In
the downside, the expression could not be applied on walls with low aspect ratios as
their behavior is normally dominated by deformation associated with shear, also not for
walls with other types of reinforcement arrangements and other shapes of wall cross
section.
4.2.2 Research Conducted by Paulay and Priestley [P2]
Paulay and Priestley [P2] proposed the following equation as shown in Eq. (4.2) by
relating the equivalent moment of inertia of the wall cross section at first yield in the
extreme fiber, Ie (mm4) to the moment of inertia of the un-cracked gross concrete
section, I g (mm4) to determine the initial stiffness of RC structural walls which
respond in a flexure manner.
100 I:)II =(-+~t- g
e I y Ie Ag
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(4.2)
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Chapter Four
It can be seen from Eq. (4.2) that two important parameters: yield strength of
longitudinal reinforcement, f y (MPa) and axial load ratio, Pu/ fe' Ag
are considered
to reflect the initial stiffness of flexure walls. Moreover, to account for shear
deformation contributions to the wall stiffness, the authors [P2] presented the following
expressions as shown in Eqs. (4.3a) and (4.3b) to evaluate the initial stiffness for
structural walls with aspect ratios less than 4.0:
I = Ie (4.3a)w 1.2 +F
where
F=30Ie
(4.3b)h 2b I
w w w
In Eqs. (4.3a) and (4.3b), the shear deformation contribution to initial wall stiffness is
reflected by employing three overall dimensions: wall height, hw ' wall thickness, bw
and wall length, Iw •
4.2.3 ACI 318-02 [AI]
ACI 318-02 [AI] recommends the application of the following effective stiffness
coefficients for walls in the structure. The initial stiffness, EIe
taken as 0.875 of those
in the work of MacGregor and Hage [HI], is proposed to be 0.70Ee Ig
and 0.35Ee Ig
for un-cracked and cracked walls, respectively.
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Chal2.ter Four
4.2.4 NZS 3101: 1995 [Nl]
Table 4.1 lists the stiffness coefficients recommended by the New Zealand Concrete
Structures Standard (1995) [N1] for the elastic seismic analysis of walls. As shown in
Table 4.1, the stiffness coefficients are dependent on the axial load, N and the
expected inelastic ductility demand, f1. Under the ultimate limit state, three cases
designed for ductility f1 > 1, that is J..l = 1.25, 3.0 and 6.0, are all assumed to have
passed their yield point. Therefore, the lower bound stiffness values (for example,
0.15/g to 0.45/g' 0.30Ag to 0.80Ag for walls under the axial load ratios,
N / fc'A g from -0.1 to 0.2) should be used in the analysis. However, under service
level state, larger stiffness coefficients can be used for the limited ductile cases, that is
f1 = 1.25 or 3 since there would be less yielding than compared to walls designed with
ductility demands of f1 = 6 . Note that in the case of f1 = 1.25, the walls can be
analyzed using their fully un-cracked stiffness under the service level earthquake.
4.2.5 FEMA356 (FEMA2000) [F2]
FEMA 356, Prestandard and Commentary for the Rehabilitation of Buildings (FEMA
2000) [F2] suggests that RC structural walls respond in three different levels
corresponding to their aspect ratios. It recognizes that RC walls with an aspect ratio of
less than 1.5 and greater than 3.0 are dominated by shear and flexure, respectively, and
the walls with intennediate aspect ratio are controlled by both flexure and shear. In
FEMA 356, the use of wall initial stiffness values as listed in Table 4.2 that correspond
to the secant value at the yield point of the wall is recommended for linear and
nonlinear analysis.
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Chapter Four
4.3 Stiffness Characteristics
There are two common methods used to define the initial wall stiffness as shown in Fig.
4.1. The structure in the first method is cyclic loaded to ± 75 percent of the nominal
strength, with the recorded displacement of the structure being extrapolated to the
position at the nominal strength level. This defines the ductility displacements, ~ yl as
shown in Fig. 4.1. In the second method, the structure is loaded until either the first
yield occurs in the longitudinal reinforcement or the maximum compressive strain of
concrete reaches 0.002 at the critical section. The displacement at this point, which can
be found by integrating the curvatures over the height of the wall, is then extrapolated
to the level of the nominal strength and this determines the ductility displacement, ~ y2 •
Generally, the two methods present similar values and in this study, the second
approach is adopted.
For RC structural walls subjected to a specified loading, the final deflection can be
modeled through a secant stiffness determined by calibration to tests or detailed
analytical models. However, accurate establishment of the secant stiffness for low-rise
structural walls is complicated by the interaction of shear, flexure, and axial loading.
Considering the interaction of shear, flexure and axial loading, two analytical stiffness
of low-rise structural walls, elastic un-cracked stiffness and cracked analytical stiffness,
are presented as follows and compared with experimental pre-cracking stiffness and
initial stiffness of tested walls. The RC structural walls tested in this study and reported
in previous chapters are used for this purpose.
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Chal!.ter Four
4.3.1 Elastic Un-cracked Stiffness
The elastic un-cracked stiffness, K e of a RC wall can be readily obtained in a
closed-form solution by employing the principle of virtual work. The total deformation
of an un-cracked RC wall is separated into components due to flexure, L\ jl , and shear
deformations, L\ sh. In the case of cantilever walls which were tested in this program,
the total deformation is given by Eq. (4.4)
VH 3 VHL\ - --+k-···-L\ =L\ jl + sh - 3E I GcA
gt c g
(4.4)
Thus the elastic un-cracked lateral stiffness, K e can be determined by the following
equation
K _ _V _e - L\ - H 3 kH
t __ +__3EcI g GcAg
(4.5)
where the shear modulus of the concrete, Gc = E c /2(1 + v) and coefficient, k IS
related to the cross section shape and defined herein as the ratio of Ag / A w •
Table 4.3 presents the elastic un-cracked stiffness, K e for all specimens tested,
calculated according to Eq. (4.5). The experimental pre-cracking stiffness, K pc' also
shown in Table 4.3 is obtained at a drift ratio of 0.05%, corresponding to the
pre-cracking stage during the testing. As listed in Table 4.3, the value of elastic
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Chapter Four
un-cracked stiffness is much higher than the experimental pre-cracking stiffness
obtained and a ratio between them, K e / K pc is computed to be approximately 3.0: 1.0
for medium-rise RC walls. While, for low-rise RC walls with and without axial
loadings, the ratio is observed to be 4.5: 1.0 and 3.2: 1.0, respectively. Accordingly, it
can be concluded that even at early stages of testing, the elastic un-cracked stiffness
does not represent the true wall stiffness and as such, linear elastic analysis based on
the calculated un-cracked stiffness cannot accurately predict the structural forces. This
deviation between the elastic un-cracked stiffness and experimental pre-cracking
stiffness is likely to be affected not only by the loading history, but also by the material
characteristics, dimensions, and curing conditions.
4.3.2 Analytical Cracked Stiffness
The analytical cracked member stiffness, K c also shown in Table 4.3, is determined
using a well-known nonlinear cyclic section analysis computer program:
RESPONSE-2000 [Bl], without taking the shear deformation into consideration. The
cracked section stiffness, E c1e at the first yield of the outermost bars in tension flange
is calculated by employing the following expression:
(4.6)
Thus the analytical cracked member stiffness, K c , for a cantilever wall at the stage of
the first yield of outermost bars can be expressed as:
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Chapter Four
(4.7)
As shown in Table 4.3, the ratio between the analytical cracked member stiffness, K c '
and experimental pre-cracking stiffness, K pc' is achieved to be approximately 2.5: 1.0
and 1.7: 1.0 for the low-rise and medium-rise RC structural walls, respectively. It
indicates that the analytical cracked member stiffness, K c ' is much closer to
experimental pre-cracking stiffness, K pc' than the elastic un-cracked stiffness, K e •
Moreover, the value of the analytical cracked member stiffness, K c ' could be further
decreased if shear displacements were included in its analysis, and thus the previous
comparison would improve. Therefore, it is advisable to use the member analytical
cracked stiffness, K c' rather than elastic stiffuess, K e' as the wall pre-cracking
stiffness for elastic analysis.
The experimental initial stiffness, K;, of the tested walls which is obtained by the
previous mentioned approach is also shown in Table 3.4. The ratio, K c / Ki' between
the analytical cracked stiffness, K c ' and experimental initial stiffness, K;, is observed
to be around 3.0:1.0 and 2.2:1.0 for low-rise and medium-rise RC walls, respectively as
shown in Table 4.3. It indicates that without considering the contribution of the shear
defonnation to initial stiffness of RC walls with low aspect ratios, rather large gap can
occur between the analytical cracked member stiffuess and experimental initial
stiffuess. Both stiffuesses are obtained at the first yielding of outennost bars in tension
flange. It is, therefore, considered to be necessary to account for the contributions of
the defonnations from both flexure and shear to initial stiffuess of walls with low
aspect ratios. In the following analysis, a method, considering both flexure and shear
defonnation, is proposed to properly evaluate the initial stiffuess of squat RC structural
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Chapter Four
walls.
4.3.3 Initial Stiffness
4.3.3.1 Flexure Deformation Determination
To determine the total flexure deformation at the top of the walls, the cantilever wall is
divided into five segments along the height of the wall. The curvatures of the five
sections along the height of the wall as listed in the column (5) of Table 4.4 can be
obtained from their bending moments by the use of moment curvature analysis for each
section, when the base section of the wall attains yield moment, My. The flexure
deformations at the top of the wall, induced by each wall section, are presented in
column (6) of Table 4.4. Thus, the total flexure deformation at the top of the wall, ~Yf'
can be obtained by cumulating the flexure deformations caused by every segment.
4.3.3.2 Shear Deformation Determination
The concept of modeling cracked RC members as a truss has been around for many
years since it provides a more promising way to treat shear. The truss analogy not only
presents a clear concept of how a RC wall resists shear, but also properly manages the
interaction of flexure, shear and axial load. In truss analogies, longitudinal
reinforcement is represented by the longitudinal chords of a truss while transverse steel
is represented by transverse tensile ties. The concrete in compression is considered to
be the compression chord members. The longitudinal chords and the transverse ties are
internally stabilized by the diagonal struts which model the concrete compressive stress
field. For simplicity, the longitudinal chords, transverse tensile ties, and diagonal struts
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Chae.ter Four
are assumed to be joined together through rigid nodes.
The contribution of the shear deformation to the total deformation in low-rise structural
walls under large shear forces can be significant since diagonal cracks must be
expected at the service limit state. Hence, in the case of low-rise walls under service
conditions, the designer also needs to be able to assess the order of expected shear
deformations. Moreover, it is well-recognized that generally, a greater proportion of the
load in low-rise structural walls is to be carried by truss action and hence the
deformation induced by truss mechanism should be emphasized in this study. Based on
this, Park and Paulay [PI] in their study proposed a method to calculate the shear
stiffness of short or deep rectangular beams of unit length by using the model of the
analogous truss. The shear stiffness is defined as the magnitude of the shear force that
when applied to a beam of unit length, will cause unit shear displacement at one end of
the beam relative to the other. The shear deformation of these unit length members
under certain shear force can be assessed by using the shear stiffness. Fig. 4.2 presents
the shear distortions according to the analogy truss mechanism for structural walls
studied.
As shown in Fig. 4.2, the horizontal reinforcements and concrete act as tension and
strut members, respectively, while the vertical reinforcements form the left and right
chords. The total shear distortion includes two components: elongation of the
horizontal reinforcements, ~s, and the shortening of the compression strut, ~c. As
indicated in Fig. 4.2, the shear distortion, ~ v can be defined by
L\ v = L\s + L\ R = L\s + L\ c / sin a
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(4.8)
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Chapter Four
Assuming that the shear force taken by the wall panel is Vs ' the stress of horizontal
where a is the inclination of compression strut.
reinforcement can be expressed as
fs= vs·sdcota· Ah
(4.9)
where d is the length of wall panel, s is the distance between horizontal
reinforcements, and Ah is the area of horizontal reinforcement spaced at a distance s.
Hence the elongation of the horizontal reinforcement becomes
(4.10)
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Hence the shortening of the concrete strut is
(4.11)
(4.12)
VsfCd=bL·
w cs sIn a
The concrete compression stress is obtained
where bw is the depth of wall panel and Lcs is effective depth of the compression
strut as shown in Fig. 4.3.
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Chap,.ter Four
where ho is the height of the wall panel. By making the appropriate substitution for
web horizontal steel content, Ph = Ah/ sbw , and modular ratio, n =E s / E e , the shear
distortion in the wall panel can be expressed as
() = ~v = ~s + ~R = Vs ( 1 + hon Jv ho ho hoEsbw cot a· Ph Les sin 2 acos a
(4.13)
when (}v =1 and Les =d . cos a, the shear stiffness of the wall panel can be defined
by the following expression:
. 2 2_ Ph sIn a· cos a b d
K v - . 4 E s w
sIn a+nph(4.14)
Eq. (4.14) indicates that the unit shear stiffness of the wall panel is mainly dependent
on the extent of the crack angles. To accurately estimate the theoretical crack angle,
Kim and Mander [K1] derived the following equation by considering the energy
minimization on the virtual work done by the shear and flexure components.
a = tan- l
1
Ph n +1.57 Ph x~ \4Pv Ag
1+ Phn(4.15)
This equation was first employed by Kim and Mander [K1] to analyze the reinforced
concrete beam-columns as a truss consisting of a finite number of differential truss
elements. Later, Matthew [M2] applied the proposed equation to both shear and flexure
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Chapter Four
critical structural walls. It was shown in their research that the theoretical crack angles
agreed well with the experimentally observed crack angles reported by previous studies.
As such, Eq. (4.15) is also employed in this study to conduct the following validation
and parametric studies.
Hence the shear displacement caused by the yield lateral force Fy
would be
The proposed approach IS validated in the following sections by companng the
After the flexure and shear deformation at the top of wall under yield lateral load are
obtained, the initial stiffness of walls can be determined as:
4.3.3.3 Combination of Shear and Flexure Response
F yK.=----1 ~ +~
yf yv
Hence the effective moment of inertia for a cantilever wall can be expressed as:
K.h 3Ie =_l_W_
3Ec
4.3.4 Validation of the Proposed Approach
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(4.16)
(4.17)
(4.18)
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Chae.ter Four
theoretical estimations with experimental observations. The experimental results of the
eight tested walls reported in the previous chapters are considered here. The theoretical
results of the initial stiffness, K i(a)' of the eight specimens, together with the
experimental results, Ki(e)' are listed in Table 4.5. To compare the accuracy of the
experimental results with the numerical ones, the error value, r, of numerical results
to the experimental ones as shown in Eq. (4.19) is also provided in Table 4.5.
IK;(a) -K;(e)lxlOO
%
Y= Ki(e)
(4.19)
Table 4.5 shows that the analytical initial stiffness of eight RC walls tested agrees well
with the experimental results, as the average error value is observed to be 8.6%. This
indicates that the proposed approach can assess initial stiffness of squat RC structural
walls with or without axial loads with a satisfactory level of accuracy.
4.4 Parametric Study for Initial Stiffness of Squat Structural Walls
A parametric study based on the proposed approach is carried out to investigate the
effect of various parameters on the initial stiffness of low-rise structural walls with
limited transverse reinforcement. The primary parameters investigated are: yield
strength of longitudinal reinforcement in wall boundaries, f y ' longitudinal
reinforcement content in wall boundaries, Ph , the axial load ratio, N/ifc'Ag ) and
aspect ratios, hw/ lw . The investigated range of the parameters is listed in Table 4.6. It
should be noted that the ranges investigated are typical cases for almost all practical
low-rise structural walls in low to moderate earthquake regions. All specimens studied
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Chapter Four
have three common characteristics, namely, 1) aspect ratios less than 2.0, 2) specimens
are isolated walls, and 3) solid with symmetrical wall cross section. The tested wall,
Specimen LW1, with an aspect ratio of 1.125 as depicted in chapter two is considered
to be the reference wall. The longitudinal and transverse reinforcement content in the
wall web is kept at a low level of around 0.48 percent and remains to be the same for
all specimens studied. The concrete compressive strength of all other specimens studied
is the same as that of Specimen LWl.
In the parametric study, the effect of investigated parameters on the initial stiffness of
walls is to be presented by the dimensionless stiffness ratio k defined as follows
(4.20)
Tables 4.7 - 4.9 present the analytical results of the stiffness ratio with the variation of
the investigated parameters for longitudinal reinforcement content in wall boundaries
of 1.4%, 2.8% and 4.2%, respectively. In the mean time, influence of different
parameters investigated on stiffness ratios is also plotted in Figs. 4.4 - 4.6 and is
discussed below.
4.4.1 Influence ofAspect Ratio
As listed in Tables 4.7 - 4.9, the stiffness ratio generally increases with the addition of
aspect ratios for walls with same reinforcement detailing and under the same level of
axial loads. With the increase of aspect ratios from 1.125 to 1.625 and 1.925, stiffness
ratios for walls without axial loads rise by approximately 70% and 1080/0, respectively.
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Cha12.ter Four
The rate of increase of stiffness ratios becomes more significant for RC walls under an
axial load level of 0.1 as it rises by approximately 85% and 130%, respectively.
This variation of stiffness ratios with the change in aspect ratios can also be observed in
Figs. 4.4(a) - 4.4(c) corresponding to three different longitudinal reinforcement ratios
in wall boundaries: 1.4%, 2.8% and 4.2%, respectively. As wall aspect ratios rise by
around 44% and 70%, an increase in stiffness ratios of approximately 80% and 124%,
respectively is obtained for walls, subjected to an axial load ratio of 0.05. However, for
walls with different longitudinal reinforcement content in wall boundaries, the variation
of the stiffness ratios with the change of aspect ratios remain almost the same, which
indicates that the longitudinal reinforcement content in wall boundaries plays a minor
role in contributing to the initial stiffness of squat RC walls.
4.4.2 Influence of Axial Load
It is generally recognized that the initial stiffness of RC walls should increase with
added axial loads due to the decrease in depth of flexure cracks. Although the yield
curvature does not vary significantly for walls with or without axial loads, the presence
of axial load can effectively increase its strength and thus lead to larger initial flexure
stiffness. Analysis shows, in Tables 4.7 - 4.9, that with the added axial loads the
stiffness ratio generally increases and approximately 6% and 10% stiffness ratio rise is
observed for walls with an aspect ratio of 1.125 when the level of axial load ratio
increases from 0.0 to 0.05 and 0.1, respectively.
Figs. 4.5(a) - 4.5(c) clearly illustrate the increase of stiffness ratio with the added axial
loads corresponding to three different levels of longitudinal reinforcement content in
wall boundaries: 1.4%,2.8% and 4.2%, respectively. As shown in Figs. 4.5(a) - 4.5(c),
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Chapter Four
the influence of axial load on the stiffness ratio can be more significant with increase of
aspect ratios. It is observed that stiffness ratios for walls with an aspect ratio of 1.625
rise by approximately 13% and 19% with the change of axial load ratio from 0.0 to
0.05 and 0.1, respectively, while for walls with an aspect ratio of 1.925, approximately
160/0 and 26% rise of stiffness ratios is obtained for walls with the increase of axial load
from 0.0 to 0.05 and 0.1, respectively.
4.4.3 Influence of Longitudinal Reinforcement Content in Wall Boundaries
Tables 4.7 - 4.9 shows that with the addition of longitudinal reinforcement content in
wall boundaries from 1.4% to 2.8% and 4.2%, stiffness ratios rise slightly by
approximately 2% and 5%, respectively for walls without axial loads. The stiffness
ratios increase by approximately 2% and 3%, respectively for walls under axial loads of
0.05 with the increase of longitudinal reinforcement content from 1.4% to 2.8% and
4.2%, while this increase reduces to approximately 1.0% and 3.0%, respectively for
walls under the axial load ratio of 0.1.
The influence of longitudinal reinforcement ratios in wall boundaries on stiffness ratios
is also presented in Figs. 4.6(a) - 4.6(c) for three different aspect ratios: 1.125, 1.625
and 1.925, respectively. From Fig. 4.6, stiffness ratios for walls without axial loads are
observed to rise slightly for walls corresponding to all aspect ratios, while for walls
under the axial load ratio of 0.1, the stiffness ratios for walls with all aspect ratios
studied almost remain the same. This suggests that the influence of longitudinal
reinforcement ratios in wall boundaries on stiffness ratios is not significant. For
simplicity, the effect of longitudinal reinforcement content in wall boundaries on initial
stiffness of RC structural walls with low aspect ratios can be neglected.
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Chal2.ter Four
4.4.4 Influence of Yield Tensile Strength of Longitudinal Bars in Wall Boundaries
As presented in Tables 4.7 - 4.9, the stiffness ratios generally decrease with the
increase of yield strength of outennost longitudinal bars. With the increase of yield
strength from 308 MPa to 458 MPa, stiffness ratios of walls without axial loads
decrease by approximately 6% for walls with an aspect ratio of 1.125, while there is
approximately 10% and 14% reduction in stiffness ratios is observed for walls with an
reinforcement content in wall boundaries on wall initial stiffness is conservatively
axial load ratios of 0.05 and 0.1, respectively.
stiffness of squat structural walls. For simplicity, the influence of longitudinal
(4.21)Ie [100 N J( hw hw
2 J-=0.19 -+-,- 0.53+0.37-+0.31-2
I g I y Ie Ag Lw Lw
aspect ratio of 1.625 and 1.925, respectively. The influence of yield strength of
longitudinal bars on stiffness ratios becomes insignificant for walls with aspect ratios of
1.125 since just 2.5% and 3% reduction in stiffness ratios is observed for walls with
Based on the previous parametric study, Eq. (4.21) which considers three
4.5 Proposed Equation for Moment of Inertia of Structural Walls
disregarded.
above-mentioned parameters investigated: yield tensile strength of steel bars in wall
boundaries, axial loads, and aspect ratios is proposed to properly evaluate the initial
Where Effective and gross moment of inertia, Ie and I g : mm4; Axial force, N: N;
Reinforcement yield strength, Iy and concrete compressive strength, Ie': MPa; Height and
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Chapter Four
Length of the wall panel, hw and Lw : mm; Gross cross section area, Ag : mm2.
The analytical results of stiffness ratios for all studied walls (column 1) along with
those calculated by employing Eq. (4.21) and Eq. (4.3) are listed in Table 1 at Appendix
A. In Appendix A, Column 3 of Table 1 presents the comparisons between the proposed
equation (Eq. (4.21)) and the results from the parametric study (Column 1 of Table 1).
It is found that the predictions by Eq. (4.21) is quite reasonable since the mean
predicted/analytical stiffness ratio, EIe / EIg is obtained to be 1.00 with a standard
deviation of 0.17 which is the least values compared with those achieved by Eq. (4.3)
and current design provisions. Columns 7, 9 and 11 in Table 1 at Appendix A
corresponding to those proposed by ACI 318, NZS 3101, and FEMA 356, respectively
show that the mean recommended/analytical EIe / EIg ratio is found to be 3.37, 2.87,
and 4.81 with a standard deviation of 1.20, 1.05, and 1.71, respectively.
Fig. 4.7 illustrates the comparisons of stiffness ratios proposed by the parametric
studies, proposed Eq. (4.21) and previous Eq. (4.3) at different aspect ratios of walls
studied. In the mean time, the trend lines of the obtained stiffness ratios are also
presented in the figure to indicate the variation of the stiffness ratios with the change of
axial loads. As can be seen in Fig. 4.7, the prediction by Eq. (4.21) compares better
than results calculated by Eq. (4.3) since the mean predicted/analytical stiffness ratio,
as shown in Table 1 at Appendix A, for Eq. (4.3) is 1.63 with a standard deviation of
0.34. However, proposed Eq. (4.21) generally presents higher value of stiffness ratios
than that of analytical results for walls with an aspect ratio of 1.125.
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i
1
----- . _.._.._-------
Chae.ter Four
4.6 Comparisons of Analytical Stiffness Ratios with Tested Results
Results from RC low-rise structural walls tests as listed in Table 4.10 are compared
with analytical results using the proposed model, Eq. (4.21) and other provisions
previously reviewed. Experimental stiffness values, E1e from the tests are back
calculated by dividing the displacements at the yield point by the tested yield strength
with an elastic model. All tested walls have aspect ratios not larger than two and axial
load ratios which range from zero to 0.2, which covers almost all conditions likely to
be encountered in practice. Yield strengths of outermost longitudinal bars for all
specimens range from 300 MPa to 585 MPa. It is believed that the proposed stiffness
model is applicable for all values of yield strengths studied. The longitudinal and
transverse reinforcement content in the wall web is limited and remains at a low level
for all walls selected.
Table 4.10 and Fig. 4.8 presents the comparison between the experimental and
calculated stiffness (E1e / E1g) for the proposed model and that presented by other
proposals. The tested initial stiffness value is reported at column 1 in Table 4.10.
Column 3 in Table 4.10 presents the ratio between the stiffness proposed by Eq. (4.21)
and the tested results. As shown in Table 4.10, the initial stiffness predicted by Eq.
(4.21) agrees well with that from the tested results as a mean stiffness ratio and
standard deviation is obtained to be 0.91 and 0.41, respectively. For comparison, Table
4.10 also lists the initial stiffness calculated by Eq. (4.3) as shown at column 4 which a
mean stiffness ratio and standard deviation is observed to be 1.48 and 0.54,
respectively.
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Chapter Four
Fig. 4.8 illustrates the comparisons of stiffness ratios, E1e / E1g computed by the
currently proposed Eq. (4.21), Eq. (4.3) proposed by Paulay et al. and NZS 3101 design
codes. Tested-to-calculated values of stiffness ratios in Fig. 4.8 indicate that NZS code
significantly over-estimates the initial stiffness of low-rise structural walls. Of the two
equations proposed, the currently proposed Eq. (4.21) with a standard deviation of 0.41
appears to be more accurate in initial stiffness evaluation than Eq. (4.3) as a standard
deviation of 0.54 was obtained.
4.7 Conclusions
This research focuses primarily on RC flanged structural walls with aspect ratios less
than two and with limited transverse reinforcement in the wall web. It is believed that
this analysis can be extended to walls with a rectangular cross-section since
longitudinal reinforcement content in wall boundaries relating to the boundary area is
found to have a minor effect on initial stiffness evaluations.
This study develops an effective method to evaluate the initial stiffness of RC low-rise
structural walls with limited transverse reinforcement by incorporating both the flexure
and shear deformations. The method uses the results of moment curvature analysis to
calculate the flexure deformation, and applies the truss mechanism to evaluate the shear
deformation. This analytical method of stiffness evaluation is found to be reasonable
when compared with the results from the current test program.
Parametric study shows that three critical parameters: yield strength of outermost
longitudinal reinforcement, applied axial load, and wall aspect ratios, influence the
initial stiffness of low-rise structural walls most. A simple expression to evaluate the
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Cha12ter Four
wall initial stiffness accounting for both flexure and shear defonnations is proposed and
validated with the tested data based on this parametric study. The results obtained are
found to be in good agreement with experimental work.
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Chapter Four
Table 4.1 - Effective section properties by New Zealand Standard (NZS 1995)
Checks at Checks at serviceability limit stateType of member ultimate
limit state /l =1.25 /l =3.00 /l = 6.00
NI{A.=0.20.451 g I g 0.701. 0.45Ig
0.80Ag Ag 0.90Ag 0.80Ag
N I fAg =0.00.251. I. 0.50Ig 0.25Ig
Walls 0.50Ag A 0.75Ag 0.50Agg
N I fAg =-0.10.15Ig I g 0.40Ig 0.151.
0.30Ag Ag 0.65Ag 0.30Ag
Table 4.2 - Initial stiffness coefficients for linear analysis of walls in FEMA 356
Type of member Flexure Rigidity Shear Rigidity Axial Rigidity
Walls (uncracked) 0.8E,1g 0.4E,A w E,A.
Walls (cracked) 0.5E,1. O.4E,A w E,A.
Table 4.3 - Stiffness evaluation of all tested walls
K e K pc K c K i K e / K p , K, / K pc K, / K,No.kNlmm kNlmm kN/mm kNlmm
LW1 532 120.2 195.0 60.6 4.4 2.5 3.22
LW2 532 174.0 256.6 84.7 3.1 2.4 3.03
LW3 532 150.4 256.6 73.5 3.5 2.8 3.49
LW4 532 119.3 195.0 58.4 4.5 2.5 3.34
LW5 532 168.2 256.6 76.9 3.2 2.5 3.34
MW1 233 83.12 85.1 41.2 2.8 1.7 2.07
MW2 233 81.41 85.1 41.7 2.9 1.7 2.04
MW3 233 75.76 85.1 37.5 3.1 1.8 2.27
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Chal!.ter Four
Table 4.4 - Flexure deformation determination
(1) (2) (3) (4) (5) (6)
Section Position Segment Moment Curvature Deformation
(i) (mm) width(mm) (kNm) (rad/m) (mm)
0 0 0 0 0
1 550 500 180.0 0.05X 10-3 0.04
2 1300 500 425.2 0.10X 10-3 0.10
3 1550 500 507.2 0.25X 10-3 0.19
4 1975 350 649.0 0.75X 10-3 0.52
5 2200 100 723.6 0.97X 10-3 0.21
Specimens LWI and LW4: Total flexure deformation = 1.00
0 0 0 0 0
1 550 500 285 0.08X 10-3 0.02
2 1300 500 675 0.20X 10-3 0.13
3 1550 500 805 0.32X 10-3 0.25
4 1975 350 1026 0.80X 10-3 0.55
5 2200 100 1143 1.10X 10-3 0.24
Specimens LW2, LW3 and LW5: Total flexure deformation = 1.20
0 0 0 0 0
1 675 750 243 0.08X 10-3 0.04
2 1425 750 512 0.20X 10-3 0.21
3 2175 750 782 0.32X 10-3 0.52
4 2850 600 1025 0.80X 10-3 1.37
5 3200 100 1151 1.1 OX 10-3 0.35
Specimens MW1, MW2 and MW3: Total flexure deformation = 2.50
(3)i = (2)i / 2250x My
(6)i = (5)i X (3)i X 2000 (2000 mm is the wall length)
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Table 4.5 - Experimental and analytical results for initial stiffness
of the eight specimens tested
Chapter Four
Specimens Experiment Analysis Error value
No. (kN/mm) (kN/mm) r (%)
LWI 60.6 64.4 6.3
LW2 84.7 67.6 20.2
LW3 73.5 67.6 8.0
LW4 58.4 64.4 10.3
LW5 76.9 67.6 12.1
MWI 41.2 40.7 1.2
MW2 41.7 40.7 2.4
MW3 37.5 40.7 8.5
Average error value = 8.6
Table 4.6 - Parameters investigated
No. Name Description Range Investigated
1 I y (MPa)Yield strength of longitudinal
308,358,408,458reinforcement in wall boundaries
2 Ph (%)Longitudinal reinforcement content
1.4, 2.8, 4.2in wall boundaries
3 N/(j'Ag ) Axial load ratio 0.00,0.025,0.05,0.075,0.10
4 hw/lw Aspect ratio 1.125, 1.625, 1.925
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Cha12ter Four
Table 4.7 - Stiffness ratio, E1e / E1g (%) for walls with Ph = 1.4%
Item I hw/lw
N 1.125 1.625 I 1.925
fc'Ag
f y (MPa)
308 358 408 458 308 358 408 458 I 308 I 358 I 408 I 458
0.0 6.68 6.51 6.43 6.29 11.46 10.92 10.68 10.32113.89 I 13.10 I 12.75 I 12.23
0.025 6.90 6.82 6.70 6.53 12.09 11.84 11.47 I 10.95 I 14.90 I 14.50 I 13.94 I 13.12
0.05 7.01 6.96 6.93 6.84 12.58 12.31 I 12.17 I 11.89 I 15.69 I 15.26 I 15.04 I 14.61
0.075 7.19 7.15 7.06 7.00 13.04 12.88 I 12.60 I 12.43 I 16.46 I 16.20 I 15.74 I 15.48-0.1 I 7.28 I 7.20 I 7.15 I 7.07 I 13.35 I 13.06 I 12.91 I 12.68 I 16.99 I 16.49 I 16.24 I 15.89
Table 4.8 - Stiffness ratio, E1e / E1g (%) for walls with Ph =2.8%
Item I hw/lw
N 1.125 I 1.625 I 1.925
fc'Ag
f y (MPa)
308 358 408 458 308 358 408 458 I 308 I 358 I 408 I 458
0.0 6.82 6.73 6.60 6.42 11.88 11.62 11.19 11.02 114.61 I 14.22 I 13.53 I 13.34-0.025 11.95 I 11.82 I 15.68 I 15.17 I 14.74 I 14.55
0.05 12.40 I 12.28 I 16.26 I 15.85 I 15.45 I 15.26--0.075 I 7.25 I 7.19 I 7.10 I 7.04 I 13.27 I 13.04 I 12.75 I 12.56 I 16.85 I 16.48 I 16.01 I 15.70-0.1 17.31 17.25 I 7.18 17.12 I 13.42 I 13.23 I 13.02 I 12.80 117.13 I 16.78 I 16.44 I 16.08
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Table 4.9 - Stiffness ratio, Ele / EIg (%) for walls with Ph =4.2%
Chapter Four
Item hw/lw
N 1.125 1.625 1.925
fc'Ag f y (MPa)
308 358 408 458 308 358 408 458 308 358 408 458
0.0 6.99 6.94 6.91 6.87 12.42 12.26 12.17 12.05 15.49 15.24 15.10 14.91
0.025 7.15 7.07 7.02 6.97 12.92 12.67 12.50 12.37 16.29 15.90 15.62 15.40
0.05 7.24 7.17 7.12 7.07 13.20 12.98 12.82 12.68 16.74 16.39 16.12 15.91
0.075 7.31 7.24 7.19 7.13 13.45 13.22 13.05 12.86 17.16 16.78 16.52 16.20
0.1 7.35 7.30 7.23 7.19 13.58 13.40 13.20 13.05 17.37 17.08 16.75 16.50
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Chae.ter Four
Table 4.10 - Comparison of tested versus predicted stiffness
!v hw N Ele / EIgUnits
lw fc'Ag
Tests Eg. (4.18) Ratio Eg. (4.3) Ratio(MPa) [1] [2] [3 ]=[2]/[ 1] [4] [5]=[4]/[ 1]
Specimens tested by SYnge et al. [S 1]
Wall 1 300 0.5 0.00 0.047 0.050 1.06 0.097 2.06
Wall 3 315 0.5 0.00 0.024 0.048 1.98 0.065 2.71
Specimens tested by Mestyanek [M 1]
Unit 1.0 520 1.0 0.00 0.055 0.044 0.80 0.107 1.95
Unit 1.5 530 1.5 0.00 0.099 0.064 0.64 0.126 1.27
Unit 2.0 500 2.0 0.00 0.138 0.095 0.69 0.143 1.04
Specimens tested by Lefas et al. [L1]
SWll 470 1.0 0.00 0.090 0.049 0.54 0.123 1.37
SW12 470 1.0 0.10 0.137 0.072 0.52 0.158 1.15
SW13 470 1.0 0.20 0.124 0.095 0.76 0.185 1.49
SW21 470 2.0 0.00 0.226 0.101 0.45 0.160 0.71
SW22 470 2.0 0.10 0.367 0.149 0.41 0.224 0.61
SW23 470 2.0 0.20 0.368 0.196 0.53 0.283 0.77
Specimens tested by Salonikios et al. [T 1]
LSWI 585 1.0 0.00 0.140 0.039 0.28 0.105 0.75
LSW2 585 1.0 0.00 0.093 0.039 0.42 0.105 1.13
LSW3 585 1.0 0.07 0.124 0.055 0.45 0.133 1.07
MSWI 585 1.5 0.00 0.130 0.058 0.44 0.105 0.81
MSW2 585 1.5 0.00 0.048 0.058 1.20 0.105 2.19
MSW3 585 1.5 0.07 0.147 0.081 0.55 0.133 0.90
Specimens tested by Young et al. [Yl]
WR-20 449 2.0 0.10 0.149 0.153 1.03 0.230 1.54
WR-I0 449 2.0 0.10 0.178 0.153 0.86 0.230 1.29
WR-O 449 2.0 0.10 0.117 0.153 1.31 0.230 1.97
WB 342 2.0 0.10 0.138 0.187 1.35 0.245 1.78
Current tested specimens
LWI 382 1.125 0.00 0.061 0.082 1.35 0.130 2.13
LW2 382 1.125 0.05 0.082 0.095 1.16 0.142 1.73
LW3 382 1.125 0.05 0.072 0.095 1.32 0.136 1.89
LW4 382 1.125 0.00 0.060 0.082 1.37 0.127 2.12
LW5 382 1.125 0.05 0.073 0.095 1.30 0.140 1.92
MWI 382 1.625 0.05 0.124 0.138 1.12 0.190 1.53
MW2 382 1.625 0.05 0.125 0.138 1.11 0.191 1.53
MW3 382 1.625 0.05 0.113 0.138 1.22 0.186 1.65
Mean= 0.91 1.48
Stdev= 0.41 0.54
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Chapter Four
Reinforcement first reach
yield strain or concrete strain
reach 0.002
Displacement
~Yl ~Y2
Fig. 4.1 - Initial stiffness determination [FI]
~v
~R~c
\
\
Steel tension member
a\
w~s
Fig. 4.2 - Shear distortion of wall panel using analogous truss [P 1]
\
~
\
\
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Concrete compression strut
Chal!.ter Four
- 156 -
Lcs
a
fix As
CD
Fig. 4.3 - Compression strut of wall panel
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Chapter Four
20 -,---------------, r----,
f, =
308MPa
358MPa
408MPa
458MPa
---+--- Aspect ratio: 1.125...•... Aspect ratio: 1.625
--- Aspect ratio: 1.925
.9 8 j "",--~-~-~--1;:=-""-""'--...-...----------..- --------1<;; 1=...VJ
~ 4~::::c/l
0.100750.050.025
0+----,--------,--------1
oAxial load ratio
(a)
20 :-------1,-------,Pb=2.8% r= f,=
308MPa
358MPa
408MPa
458MPa
20 -,-------------, ,..__...,
---+--- Aspect ratio: 1.125
_. -.- .. Aspect ratio: 1.625
---A- Aspect ratio: 1.925
---+--- Aspect ratio: 1.125
.. -.-. - Aspect ratio: 1.625
---A- Aspect ratio: 1.925
.9 8 j "",-",,--"",-"",-..- _--_-_-_-...- ---=-=-~-~--=-=---'1-<;; ~...VJ
~ 4sE::::c/l
.g 8 .....- _--_-_-.-_-_--_-_-..-_-_-_-----4-...-=-=--=-1~VJVJtl) 4
sE::::c/l
0.10.0750.050.025
o+---,-------,------,-----1
o0.10.05 0.0750.025
O+---r------,-----,-------j
oAxial load ratio Axial load ratio
(b) (c)
Fig. 4.4 - Influence of wall aspect ratios on stiffness ratios
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Cha12.ter Four
f r =308MPa
358MPa
408MPa
458MPa
1.925
1.8751.625
(c)
Aspect ratio
1.375
-+--- Axial load ratio: 0.0"" "."." Axial load ratio: 0.05 ---+- Axial load ratio: 0.1
f y =308MPa
358MPa
408MPa
458MPa
Pb = 4.2%
1.875
18 -,1---------------, ,....--...,
~ 16'--
~~ 14
"~~ 12
.8 10~:.-.VJ 8VJ(1)
~~ 6~
4
1.125
1.625
(a)
Aspect ratio
----+- Axialload ratio: 0.0
"" ".""" Axialload ratio: 0.05-.-Axialload ratio: 0.1 I 1.925
1.375
II' =308MPa
358MPa
408MPa
458MPa
1.925
Pb=1.4%
1.875
18
~ 16'-
~~. 14
"~~ 12
.8 10~:.-.VJ 8VJ(1)
~~ 6~
4
1.125
1.625
Fig. 4.5 - Influence of axial load on wall stiffness ratios
(b)
Aspect ratio
----+- Axial load ratio: 0.0"" ".""" Axial load ratio: 0.05-.- Axial load ratio: 0.1
1.375
Pb = 2.8%18
~ 16t..
~~ 14........
~~ 12
.:i 10~:.-.VJ 8VJ(1)
~6~
~
4
1.125
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Chapter Four
8i-------Ir-----,
Aspect ratio: 1.125 f, =; 308MPa
358MPa~-+~.I
I:§~si~~~~~Ef#-~1408 MPa7 t 458MPa
~~
,...," 7.5~"-
ki'
.9-;;;:;; -+- Pb = 1.4%<Jl<Ll~ ...•... Pb =2.8%
E 6 JI-----,----~--.----------'~, ---.- Pb =4.2%!
o 0.025 0.05 0.075 0.1
Axial load ratio
(a)
14 -,-------------i.r----,
f, =308MPa
358MPa
408MPa
458MPa
0.10.075
---+-Pb =1.4%.. ·... ·Pb=2.8%
---'-P b = 4.2%
0.050.025
Aspect ratio: 1.925
18 -,-------------- .--_---,
12 +------,-----,---,-------1
o
f, =308MPa
358MPa
408MPa
458MPa
Aspect ratio: 1.6250-~
ki" 13"-
ki'
.912
-;;;....<Jl .';-
---+-Pb = 1.4%<Jl<Ll II§ .. ·•.. ·Pb=2.8%~
---'-Pb =4.2%10
0 0.025 0.05 0.075 0.1
Axial load ratio Axial load ratio
(b) (c)
Fig. 4.6 - Influence of longitudinal reinforcement ratios in wall boundarieson wall stiffness ratios
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Chae.ter Four
Parametric
study
Eq. (4.18)
Eq. (4.3)
• Parametric study
Eq. (4.18)
• Eq. (4.3)
Parametric
study
Eq. (4.18)
Eq. (4.3)
Eq. (4.18)
• Eq. (4.3)
• Parametric study
(a)
(b)
Trendhne type:
0.1
0.1
0.075
0.075
ACI318
0.05
Axial load ratio
ACI318
0.05
Axial load ratio
Aspect ratb: 1.125
0.025
0.025
~----f---------l---------!--------
Aspect ratb: 1.625
-: =-- : :..- .- .:.t.- .:..-.--=--.--=--f...;.-·-......~~--t ... -- ...!' -~ -= :-::
-------~-- . .. J. - --::..-.-~-~.::..-.-~~--- ...... ---=-" _.-. . . -----
---------~------- . .......:.. -.---------.. ---------
-------------~~~~--------
1--------------------~==::::::::::::-='""1.. Trendhne type:-4
40
35
~30
......'"25k:l
"......"kl 20
6';g 15~
rJ':J10rJ':J
(l)
~~~
0
0
40
35
~ 30......bC
k:l25"
~20
.9~ 15~
rJ':JrJ':J 10(l)
~~~
0
0
40 -,-----------------------------,
35 9 •ACI318
• Parametric study
Eq. (4.18)
• Eq. (4.3)
Trendline type:
Parametric
study
Eq. (4.18)
Eq. (4.3)
Aspect ratb: 1.925
NZS 3101
10
~ 30~"
"" 25......k:l6 20 f- --------:-- : -----~-------.- . -' -' -' --~-~-'-' -' - ..... - . -'-~ 15 : : :: : : : : J-. -. -.-.-.-.-.--f~.-,-:-- -~;--------@ ~ ~--------l
.....~
(c)
0.10.0750.05
Axial load ratio
0.025
O-t----------,------r---------r----------j
o
Fig. 4.7 - Comparisons of stiffness ratios proposed bythe parametric study and equations
- 160 -
•
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Chapter Four
•
• NZS 3101 with a standarddeviation of 1.81
t::.• Eq. (4.21) proposed by current
research with a standarddeviation of 0.40
1:, Paulay's Eq. (4.3) with astandard deviation of 0.54
0-1""--------,-----,------r-------,-----------'
0.5
•0.4 ... •••
~ 0.3~........ ••;j
~ 0.2"~
0.1
o 0.1 0.2 0.3 0.4 0.5
£1 tested / £1 g
Fig. 4.8 - Comparison of initial stiffness between theanalytical results and tested data
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Cha12.ter Four
REFERENCES
[A 1] ACI Committee 318, "Building Code Requirements for Structural Concrete
(ACI 318-02) and Commentary (318R-02)," American Concrete Institute,
Farmington Hills, Mich., 2002, 391 pp.
[B 1] Bentz, E. C., and Collins, M. P., "Reinforced Concrete Sectional Analysis
Using the Modified Compression Field Theory," Response-2000, Version
1.0.5.
[F1] Fenwick, R., and Bull, D., "What is the Stiffness of Reinforced Concrete
Walls," SESOC Journal Vol.13, No.2, Sept. 2000.
[F2] FEMA 356, "Prestandard and Commentary for the Rehabilitation of
Buildings," Federal Emergency Management Agency, Washington, D.C.,
2000.
[HI] Macgregor, J. G. and Hage, S. E., "Stability Analysis and Design Concrete,"
Proceedings, ASCE, V. 103, No. ST 10, Oct. 1977.
[K1] Kim, J. H., and Mander, J., "Truss Modelling of Reinforced Concrete
Shear-Flexure Behavior," Technical Report MCEER-99-0005, University at
Buffalo, New York, 1999.
[M1] Madhu, k. and Ghosh, S. K., "Flexure Stiffness of Reinforced Concrete
Columns and Beams: Analytical Approach," ACI Structural Journal, V. 101,
No.3, May-June 2004, pp. 351-363.
[M2] Matthew, R. E. L., "Analytical Modelling of Reinforced Concrete Wall
Behavior Under Seismic Loading," Master thesis, University of Canterbury,
Christchurch, New Zealand, Feb. 2001, 174 pp.
[N1] New Zealand Standard Code of Practice for the Design of Concrete
Structures, "NZS 3101: Part 1,185 p.; Commentary NZS 3101: Part 2,247
p.;" Standard Association of New Zealand, Wellington, New Zealand.
- 162 -
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Chapter Four
[PI] Park, R., and Paulay, T., "Reinforced Concrete Structures," John Wiley &
Sons, New York, 1975, 769 pp.
[P2] Paulay, T., and Priestley, M. 1. N., "Seismic Design of Reinforced Concrete
and Masonry Buildings," John Wiley & Sons, New York, 1992, 744 pp.
[P3] Priestley, M. J. N. and Kowalsky, M. J., "Aspects of Drift and Ductile
Capacity of Rectangular Cantilever Structural Walls," Bulletin of NZSEE,
Vol. 31, No.2, June 1998, pp. 73-85.
[P4] Priestley, M. J. N., "Brief Comments on Elastic Flexibility of Reinforced
Concrete Frames and Significance to Seismic Design," Bulletin of NZSEE,
Vol. 31, No.4, Dec. 1998, pp. 246-259.
[SI] Sameh, S. F. M. and Horoshi, K. and Gregory G. D., "Stiffness Modeling of
Reinforced Concrete Beam-Columns for Frame Analysis," ACI Structural
Journal, V. 98, No.2, March-April 2001, pp. 215-225.
- 163 -
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NOTATIONS
Ah Area of horizontal reinforcement spaced at a distance s
Ag Gross cross section area
Av Net cross section area
bw Depth of wall panel
d Length of wall panel
Ie ' Cylinder strength of concrete
Iy
Yielding strength of reinforcement
Fy Yield lateral force
ho Height of the wall panel
hw Height of the cantilever wall
k Dimensionless stiffness ratio
K c Analytical cracked member stiffness
K e Elastic un-cracked stiffness
K. The wall initial stiffnessI
K i ( aJ The theoretical initial stiffness
Ki(eJ The experimental initial stiffness
K pc Pre-cracking stiffness
K v Shear stiffness
Les Effective depth of the compression strut
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Chal!.ter Four
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n
s
~Yf
~YV
~s
~sh
r
Length of the cantilever wall
Yield moment
Axial loads
Modular ratio
Spacing of transverse reinforcement
Shear force taken by the wall panel
Inclination of compression strut
Longitudinal reinforcement content in wall boundaries
Horizontal steel content
Longitudinal steel content
Shortening of the compression strut
Flexure deformation
Flexure displacement caused by the yield lateral force Fy
Shear displacement caused by the yield lateral force Fy
Elongation of horizontal reinforcements
Shear deformation
Yield displacement of the walls
Shear distortion
Yield curvature
The error value
- 165 -
Chapter Four
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Cha/2.ter Four
Appendix A
Table 1 - Influence of various parameters on stiffness ratio, Ele / Elg (%) of
structural walls with low aspect ratios
No. Anal Eg. Ratio Eg. Ratio ACI Ratio NZS Ratio FEMA Ratio
-ysis (4.18) [3] = (4.3) [5] = 318 [7] = 3101 [9] = 356 [11] =
[1] [2] [2]/[1] [4] [4]/[ 1] [6] [6]/[1 ] [8] [8]/[ 1] [10] [10]/[1]
1 6.68 8.23 1.23 14.67 2.20 35.0 5.24 25.0 3.74 50.0 7.49
2 6.51 7.08 1.09 13.49 2.07 35.0 5.38 25.0 3.84 50.0 7.68
3 6.43 6.21 0.97 12.48 1.94 35.0 5.44 25.0 3.89 50.0 7.78
4 6.29 5.53 0.88 11.61 1.85 35.0 5.56 25.0 3.97 50.0 7.95
5 6.82 8.23 1.21 14.67 2.15 35.0 5.13 25.0 3.67 50.0 7.33
6 6.73 7.08 1.05 13.49 2.00 35.0 5.20 25.0 3.71 50.0 7.43
7 6.6 6.21 0.94 12.48 1.89 35.0 5.30 25.0 3.79 50.0 7.58
8 6.42 5.53 0.86 11.61 1.81 35.0 5.45 25.0 3.89 50.0 7.79
9 6.99 8.23 1.18 14.67 2.10 35.0 5.01 25.0 3.58 50.0 7.15
10 6.94 7.08 1.02 13.49 1.94 35.0 5.04 25.0 3.60 50.0 7.20
11 6.91 6.21 0.90 12.48 1.81 35.0 5.07 25.0 3.62 50.0 7.24
12 6.87 5.53 0.81 11.61 1.69 35.0 5.09 25.0 3.64 50.0 7.28
13 6.9 8.86 1.28 15.27 2.21 35.0 5.07 27.5 3.99 50.0 7.25
14 7.01 9.50 1.35 15.82 2.26 35.0 4.99 30.0 4.28 50.0 7.13
15 7.19 10.13 1.41 16.34 2.27 35.0 4.87 32.5 4.52 50.0 6.95
16 7.28 10.77 1.48 16.82 2.31 35.0 4.81 35.0 4.81 50.0 6.87
17 6.82 7.71 1.13 14.16 2.08 35.0 5.13 27.5 4.03 50.0 7.33
18 6.7 6.85 1.02 13.23 1.97 35.0 5.22 27.5 4.10 50.0 7.46
19 6.53 6.17 0.94 12.42 1.90 35.0 5.36 27.5 4.21 50.0 7.66
20 6.96 8.35 1.20 14.79 2.13 35.0 5.03 30.0 4.31 50.0 7.18
21 6.93 7.48 1.08 13.92 2.01 35.0 5.05 30.0 4.33 50.0 7.22
22 6.84 6.80 0.99 13.17 1.93 35.0 5.12 30.0 4.39 50.0 7.31
23 7.15 8.98 1.26 15.37 2.15 35.0 4.90 32.5 4.55 50.0 6.99
24 7.06 8.11 1.15 14.56 2.06 35.0 4.96 32.5 4.60 50.0 7.08
25 7 7.44 1.06 13.87 1.98 35.0 5.00 32.5 4.64 50.0 7.14
26 7.2 9.62 1.34 15.92 2.21 35.0 4.86 35.0 4.86 50.0 6.94
27 7.15 8.75 1.22 15.16 2.12 35.0 4.90 35.0 4.90 50.0 6.99
28 7.07 8.07 1.14 14.52 2.05 35.0 4.95 35.0 4.95 50.0 7.07
29 7.03 8.86 1.26 15.27 2.17 35.0 4.98 27.5 3.91 50.0 7.11
30 6.94 7.71 1.11 14.16 2.04 35.0 5.04 27.5 3.96 50.0 7.20
31 6.85 6.85 1.00 13.23 1.93 35.0 5.11 27.5 4.01 50.0 7.30
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Chapter Four
32 6.79 6.17 0.91 12.42 1.83 35.0 5.15 27.5 4.05 50.0 7.36
33 7.15 9.50 1.33 15.82 2.21 35.0 4.90 30.0 4.20 50.0 6.99
34 7.07 8.35 1.18 14.79 2.09 35.0 4.95 30.0 4.24 50.0 7.07
35 6.99 7.48 1.07 13.92 1.99 35.0 5.01 30.0 4.29 50.0 7.15
36 6.95 6.80 0.98 13.17 1.89 35.0 5.04 30.0 4.32 50.0 7.19
37 7.25 10.13 1.40 16.34 2.25 35.0 4.83 32.5 4.48 50.0 6.90
38 7.19 8.98 1.25 15.37 2.14 35.0 4.87 32.5 4.52 50.0 6.95
39 7.1 8.11 1.14 14.56 2.05 35.0 4.93 32.5 4.58 50.0 7.04
40 7.04 7.44 1.06 13.87 1.97 35.0 4.97 32.5 4.62 50.0 7.10
41 7.31 10.77 1.47 16.82 2.30 35.0 4.79 35.0 4.79 50.0 6.84
42 7.25 9.62 1.33 15.92 2.20 35.0 4.83 35.0 4.83 50.0 6.90
43 7.18 8.75 1.22 15.16 2.11 35.0 4.87 35.0 4.87 50.0 6.96
44 7.12 8.07 1.13 14.52 2.04 35.0 4.92 35.0 4.92 50.0 7.02
45 7.15 8.86 1.24 15.27 2.14 35.0 4.90 27.5 3.85 50.0 6.99
46 7.07 7.71 1.09 14.16 2.00 35.0 4.95 27.5 3.89 50.0 7.07
47 7.02 6.85 0.98 13.23 1.88 35.0 4.99 27.5 3.92 50.0 7.12
48 6.97 6.17 0.89 12.42 1.78 35.0 5.02 27.5 3.95 50.0 7.17
49 7.24 9.50 1.31 15.82 2.19 35.0 4.83 30.0 4.14 50.0 6.91
50 7.17 8.35 1.16 14.79 2.06 35.0 4.88 30.0 4.18 50.0 6.97
51 7.12 7.48 1.05 13.92 1.96 35.0 4.92 30.0 4.21 50.0 7.02
52 7.07 6.80 0.96 13.17 1.86 35.0 4.95 30.0 4.24 50.0 7.07
53 7.31 10.13 1.39 16.34 2.24 35.0 4.79 32.5 4.45 50.0 6.84
54 7.24 8.98 1.24 15.37 2.12 35.0 4.83 32.5 4.49 50.0 6.91
55 7.19 8.11 1.13 14.56 2.03 35.0 4.87 32.5 4.52 50.0 6.95
56 7.13 7.44 1.04 13.87 1.95 35.0 4.91 32.5 4.56 50.0 7.01
57 7.35 10.77 1.46 16.82 2.29 35.0 4.76 35.0 4.76 50.0 6.80
58 7.3 9.62 1.32 15.92 2.18 35.0 4.79 35.0 4.79 50.0 6.85
59 7.23 8.75 1.21 15.16 2.10 35.0 4.84 35.0 4.84 50.0 6.92
60 7.19 8.07 1.12 14.52 2.02 35.0 4.87 35.0 4.87 50.0 6.95
61 11.46 11.99 1.05 19.27 1.68 35.0 3.05 25.0 2.18 50.0 4.36
62 10.92 10.32 0.94 17.27 1.58 35.0 3.21 25.0 2.29 50.0 4.58
63 10.68 9.05 0.85 15.65 1.47 35.0 3.28 25.0 2.34 50.0 4.68
64 10.32 8.06 0.78 14.30 1.39 35.0 3.39 25.0 2.42 50.0 4.84
65 11.88 11.99 1.01 19.27 1.62 35.0 2.95 25.0 2.10 50.0 4.21
66 11.62 10.32 0.89 17.27 1.49 35.0 3.01 25.0 2.15 50.0 4.30
67 11.19 9.05 0.81 15.65 1.40 35.0 3.13 25.0 2.23 50.0 4.47
68 11.02 8.06 0.73 14.30 1.30 35.0 3.18 25.0 2.27 50.0 4.54
69 12.42 11.99 0.97 19.27 1.55 35.0 2.82 25.0 2.01 50.0 4.03
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Chal!.ter Four
70 12.26 10.32 0.84 17.27 1.41 35.0 2.85 25.0 2.04 50.0 4.08
71 12.17 9.05 0.74 15.65 1.29 35.0 2.88 25.0 2.05 50.0 4.11
72 12.05 8.06 0.67 14.30 1.19 35.0 2.90 25.0 2.07 50.0 4.15
73 12.09 12.91 1.07 20.30 1.68 35.0 2.89 27.5 2.27 50.0 4.14
74 12.58 13.84 1.10 21.29 1.69 35.0 2.78 30.0 2.38 50.0 3.97
75 13.04 14.76 1.13 22.24 1.71 35.0 2.68 32.5 2.49 50.0 3.83
76 13.35 12.36 0.93 23.15 1.73 35.0 2.62 35.0 2.62 50.0 3.75
77 11.84 11.24 0.95 18.39 1.55 35.0 2.96 27.5 2.32 50.0 4.22
78 11.47 9.97 0.87 16.84 1.47 35.0 3.05 27.5 2.40 50.0 4.36
79 10.95 8.99 0.82 15.56 1.42 35.0 3.20 27.5 2.51 50.0 4.57
80 12.31 12.16 0.99 19.46 1.58 35.0 2.84 30.0 2.44 50.0 4.06
81 12.17 10.90 0.90 17.98 1.48 35.0 2.88 30.0 2.47 50.0 4.11
82 11.89 9.91 0.83 16.76 1.41 35.0 2.94 30.0 2.52 50.0 4.21
83 12.88 13.08 1.02 20.49 1.59 35.0 2.72 32.5 2.52 50.0 3.88
84 12.6 11.82 0.94 19.07 1.51 35.0 2.78 32.5 2.58 50.0 3.97
85 12.43 10.83 0.87 17.90 1.44 35.0 2.82 32.5 2.61 50.0 4.02
86 13.06 14.01 1.07 21.47 1.64 35.0 2.68 35.0 2.68 50.0 3.83
87 12.91 12.74 0.99 20.11 1.56 35.0 2.71 35.0 2.71 50.0 3.87
88 12.68 11.76 0.93 19.00 1.50 35.0 2.76 35.0 2.76 50.0 3.94
89 12.55 12.91 1.03 20.30 1.62 35.0 2.79 27.5 2.19 50.0 3.98
90 12.23 11.24 0.92 18.39 1.50 35.0 2.86 27.5 2.25 50.0 4.09
91 11.95 9.97 0.83 16.84 1.41 35.0 2.93 27.5 2.30 50.0 4.18
92 11.82 8.99 0.76 15.56 1.32 35.0 2.96 27.5 2.33 50.0 4.23
93 12.9 13.84 1.07 21.29 1.65 35.0 2.71 30.0 2.33 50.0 3.88
94 12.65 12.16 0.96 19.46 1.54 35.0 2.77 30.0 2.37 50.0 3.95
95 12.4 10.90 0.88 17.98 1.45 35.0 2.82 30.0 2.42 50.0 4.03
96 12.28 9.91 0.81 16.76 1.36 35.0 2.85 30.0 2.44 50.0 4.07
97 13.27 14.76 1.11 22.24 1.68 35.0 2.64 32.5 2.45 50.0 3.77
98 13.04 13.08 1.00 20.49 1.57 35.0 2.68 32.5 2.49 50.0 3.83
99 12.75 11.82 0.93 19.07 1.50 35.0 2.75 32.5 2.55 50.0 3.92
100 12.56 10.83 0.86 17.90 1.43 35.0 2.79 32.5 2.59 50.0 3.98
101 13.43 15.68 1.17 23.15 1.72 35.0 2.61 35.0 2.61 50.0 3.72
102 13.23 14.01 1.06 21.47 1.62 35.0 2.65 35.0 2.65 50.0 3.78
103 13.02 12.74 0.98 20.11 1.54 35.0 2.69 35.0 2.69 50.0 3.84
104 12.8 11.76 0.92 19.00 1.48 35.0 2.73 35.0 2.73 50.0 3.91
105 12.92 12.91 1.00 20.30 1.57 35.0 2.71 27.5 2.13 50.0 3.87
106 12.67 11.24 0.89 18.39 1.45 35.0 2.76 27.5 2.17 50.0 3.95
107 12.5 9.97 0.80 16.84 1.35 35.0 2.80 27.5 2.20 50.0 4.00
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Chapter Four
108 12.37 8.99 0.73 15.56 1.26 35.0 2.83 27.5 2.22 50.0 4.04
109 13.2 13.84 1.05 21.29 1.61 35.0 2.65 30.0 2.27 50.0 3.79
110 12.98 12.16 0.94 19.46 1.50 35.0 2.70 30.0 2.31 50.0 3.85
III 12.82 10.90 0.85 17.98 lAO 35.0 2.73 30.0 2.34 50.0 3.90
112 12.68 9.91 0.78 16.76 1.32 35.0 2.76 30.0 2.37 50.0 3.94
113 13.45 14.76 1.10 22.24 1.65 35.0 2.60 32.5 2042 50.0 3.72
114 13.22 13.08 0.99 20.49 1.55 35.0 2.65 32.5 2.46 50.0 3.78
115 13.05 11.82 0.91 19.07 1.46 35.0 2.68 32.5 2.49 50.0 3.83
116 12.86 10.83 0.84 17.90 1.39 35.0 2.72 32.5 2.53 50.0 3.89
117 13.58 15.68 1.15 23.15 1.70 35.0 2.58 35.0 2.58 50.0 3.68
118 13.4 14.01 1.05 21.47 1.60 35.0 2.61 35.0 2.61 50.0 3.73
119 13.2 12.74 0.97 20.11 1.52 35.0 2.65 35.0 2.65 50.0 3.79
120 13.05 11.76 0.90 19.00 1.46 35.0 2.68 35.0 2.68 50.0 3.83
121 13.89 14.70 1.06 21.00 1.51 35.0 2.52 25.0 1.80 50.0 3.60
122 13.1 12.65 0.97 18.65 1042 35.0 2.67 25.0 1.91 50.0 3.82
123 12.75 11.10 0.87 16.78 1.32 35.0 2.75 25.0 1.96 50.0 3.92
124 12.23 9.89 0.81 15.24 1.25 35.0 2.86 25.0 2.04 50.0 4.09
125 14.61 14.70 1.01 21.00 1.44 35.0 2.40 25.0 1.71 50.0 3.42
126 14.22 12.65 0.89 18.65 1.31 35.0 2.46 25.0 1.76 50.0 3.52
127 13.53 11.10 0.82 16.78 1.24 35.0 2.59 25.0 1.85 50.0 3.70
128 13.34 9.89 0.74 15.24 1.14 35.0 2.62 25.0 1.87 50.0 3.75
129 15.49 14.70 0.95 21.00 1.36 35.0 2.26 25.0 1.61 50.0 3.23
130 15.24 12.65 0.83 18.65 1.22 35.0 2.30 25.0 1.64 50.0 3.28
131 15.1 11.10 0.74 16.78 1.11 35.0 2.32 25.0 1.66 50.0 3.31
132 14.91 9.89 0.66 15.24 1.02 35.0 2.35 25.0 1.68 50.0 3.35
133 14.9 15.83 1.06 22.24 1.49 35.0 2.35 27.5 1.85 50.0 3.36
134 15.69 16.97 1.08 23.43 1.49 35.0 2.23 30.0 1.91 50.0 3.19
135 16.46 18.10 1.10 24.58 1.49 35.0 2.13 32.5 1.97 50.0 3.04
136 16.99 15.16 0.89 25.70 1.51 35.0 2.06 35.0 2.06 50.0 2.94
137 14.5 13.78 0.95 19.97 1.38 35.0 2.41 27.5 1.90 50.0 3.45
138 13.94 12.23 0.88 18.16 1.30 35.0 2.51 27.5 1.97 50.0 3.59
139 13.12 11.02 0.84 16.68 1.27 35.0 2.67 27.5 2.10 50.0 3.81
140 15.26 14.91 0.98 21.24 1.39 35.0 2.29 30.0 1.97 50.0 3.28
141 15.04 13.36 0.89 19.49 1.30 35.0 2.33 30.0 1.99 50.0 3.32
142 14.61 12.15 0.83 18.06 1.24 35.0 2.40 30.0 2.05 50.0 3.42
143 16.2 16.05 0.99 22.46 1.39 35.0 2.16 32.5 2.01 50.0 3.09
144 15.74 14.50 0.92 20.77 1.32 35.0 2.22 32.5 2.06 50.0 3.18
145 15.48 13.28 0.86 19.40 1.25 35.0 2.26 32.5 2.10 50.0 3.23
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Chal!.ter Four
146 16.49 17.18 1.04 23.65 1.43 35.0 2.12 35.0 2.12 50.0 3.03
147 16.24 15.63 0.96 22.01 1.36 35.0 2.16 35.0 2.16 50.0 3.08
148 15.89 14.42 0.91 20.68 1.30 35.0 2.20 35.0 2.20 50.0 3.15
149 15.68 15.83 1.01 22.24 1.42 35.0 2.23 27.5 1.75 50.0 3.19
150 15.17 13.78 0.91 19.97 1.32 35.0 2.31 27.5 1.81 50.0 3.30
151 14.74 12.23 0.83 18.16 1.23 35.0 2.37 27.5 1.87 50.0 3.39
152 14.55 11.02 0.76 16.68 1.15 35.0 2.41 27.5 1.89 50.0 3.44
153 16.26 16.97 1.04 23.43 1.44 35.0 2.15 30.0 1.85 50.0 3.08
154 15.85 14.91 0.94 21.24 1.34 35.0 2.21 30.0 1.89 50.0 3.15
155 15.45 13.36 0.86 19.49 1.26 35.0 2.27 30.0 1.94 50.0 3.24
156 15.26 12.15 0.80 18.06 1.18 35.0 2.29 30.0 1.97 50.0 3.28
157 16.85 18.10 1.07 24.58 1.46 35.0 2.08 32.5 1.93 50.0 2.97
158 16.48 16.05 0.97 22.46 1.36 35.0 2.12 32.5 1.97 50.0 3.03
159 16.01 14.50 0.91 20.77 1.30 35.0 2.19 32.5 2.03 50.0 3.12
160 15.7 13.28 0.85 19.40 1.24 35.0 2.23 32.5 2.07 50.0 3.18
161 17.13 19.23 1.12 25.70 1.50 35.0 2.04 35.0 2.04 50.0 2.92
162 16.78 17.18 1.02 23.65 1.41 35.0 2.09 35.0 2.09 50.0 2.98
163 16.44 15.63 0.95 22.01 1.34 35.0 2.13 35.0 2.13 50.0 3.04
164 16.08 14.42 0.90 20.68 1.29 35.0 2.18 35.0 2.18 50.0 3.11
165 16.29 15.83 0.97 22.24 1.37 35.0 2.15 27.5 1.69 50.0 3.07
166 15.9 13.78 0.87 19.97 1.26 35.0 2.20 27.5 1.73 50.0 3.14
167 15.62 12.23 0.78 18.16 1.16 35.0 2.24 27.5 1.76 50.0 3.20
168 15.4 11.02 0.72 16.68 1.08 35.0 2.27 27.5 1.79 50.0 3.25
169 16.74 16.97 1.01 23.43 1.40 35.0 2.09 30.0 1.79 50.0 2.99
170 16.39 14.91 0.91 21.24 1.30 35.0 2.14 30.0 1.83 50.0 3.05
171 16.12 13.36 0.83 19.49 1.21 35.0 2.17 30.0 1.86 50.0 3.10
172 15.91 12.15 0.76 18.06 1.14 35.0 2.20 30.0 1.89 50.0 3.14
173 17.16 18.10 1.05 24.58 1.43 35.0 2.04 32.5 1.89 50.0 2.91
174 16.78 16.05 0.96 22.46 1.34 35.0 2.09 32.5 1.94 50.0 2.98
175 16.52 14.50 0.88 20.77 1.26 35.0 2.12 32.5 1.97 50.0 3.03
176 16.2 13.28 0.82 19.40 1.20 35.0 2.16 32.5 2.01 50.0 3.09
177 17.37 19.23 1.11 25.70 1.48 35.0 2.01 35.0 2.01 50.0 2.88
178 17.08 17.18 1.01 23.65 1.38 35.0 2.05 35.0 2.05 50.0 2.93
179 16.75 15.63 0.93 22.01 1.31 35.0 2.09 35.0 2.09 50.0 2.99
180 16.5 14.42 0.87 20.68 1.25 35.0 2.12 35.0 2.12 50.0 3.03
Mean = 1.00 1.63 3.37 2.87 4.81
Stdev = 0.17 0.34 1.20 1.05 1.74
Note: Ratio means ratio of proposed results to analytical ones
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Chapter Five
CHAPTER FIVE
FINITE ELEMENT PARAMETRIC STUDY OF THE
BEHAVIOR OF STRUCTURAL WALLS WITH
LIMITED TRANSVERSE REINFORCEMENT
Abstract
The main objective of this study is to provide an analytical approach to better
understand the global responses of eight RC structural walls tested and described in the
companion study. For this purpose, a nonlinear finite element analytical procedure
incorporating microscopic material models for these RC walls is used in this study. The
simulated global responses such as strength capacity, stiffness characteristics and
energy dissipation capacity of studied RC walls under reversed seismic loadings are
found to correlate well with the experimental ones by employing this analytical
procedure. A comprehensive parametric study is carried out to report the influence of
several paramount parameters: axial loads, longitudinal reinforcements in the wall
boundary elements, aspect ratio, area ratio of boundary columns and the presence of
construction joints at the wall base on the global behavior of RC walls. Conclusions are
drawn concerning the effects of these parameters on global responses in terms of the
strength capacity, secant stiffness characteristic, energy dissipation capacity and
equivalent damping of the RC walls studied.
Keywords: RC structural walls; Nonlinear finite element; Global response; Energy
dissipation capacity; Equivalent damping; Parametric study
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Cha12.ter Five
5.1 Introduction and Background
Singapore and Peninsular Malaysia are in a seismic risk area with an active earthquake
belt comprising the Sumatra Fault and the subduction zone at about 350 km away from
the closest point. Although there has never been any earthquake damage to Singapore
and Peninsular Malaysia, ground tremors have been felt in these areas many times, and
the incidents have increased significantly in number over the last three decades. Strong
tremors were felt in buildings of Singapore due to the recent north Sumatra Earthquake.
However, in these low to moderate seismic regions like Singapore and Malaysia, RC
structural walls are normally designed in accordance with the British Standard: BS
8110 which excludes the influence of seismic loading. Hence, these RC structural walls
are usually non-seismically or limited seismically detailed which would imply a lack of
confinement reinforcement in wall boundaries. In such non-seismically or limited
seismically detailed structural walls, extensive damages may occur under the
earthquake excitation as a result of excessive shear deformation and severe strength
degradation. Therefore, it is of great concern that the structural performance of these
walls may not be adequate to sustain earthquake-induced loads in regions of low to
moderate seismicity like Singapore and Malaysia. As such, experimental and numerical
studies are needed to investigate the seismic performance of non-seismically detailed
structural walls in terms of their strength and deformation characteristics as well as
energy dissipation capacity.
Up to date, many experimental and analytical studies have been performed [D2, El, Fl,
KI-K6, P4-5, TI-2, W2] to predict the nonlinear behavior of isolated structural walls
subjected to load reversals. Although reliable information on the behavior of RC
structural walls can be obtained through experimental studies as presented in the
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Chapter Five
companIon studies, it is a time consumIng process. As such, in the past decades,
numerous analytical models were developed in the modeling of RC structural walls.
Most of them developed were macroscopic models [VI, Ll, K4-5, 01] for RC
structural walls due to their easy application and in these studies great success has been
achieved at the element level [S 1, XI]. However, these analytical results are usually
only valid for the specific conditions upon which the derivation of the model is based
upon [V1]. Moreover, Sittipunt [S 1] indicated that most of the previous works in the
finite element analysis of the behavior of RC walls are concentrated on the
development of the material models that could reproduce experimental results and few
research studies have used the finite element method to investigate behavior of RC
walls other than that of the specimens tested in the laboratory. Therefore,
general-purpose microscopic models are needed to be developed to describe the
detailed local behavior of RC structural walls. With the development of the nonlinear
structural analysis and nonlinear constitutive laws of material, the finite element
method (FEM) is now a powerful means to yield detailed information on the behavior
of RC structural walls, such as stress-strain relationships in concrete and reinforcing
bars, deflected shapes, and crack patterns, which cannot be obtained from other
analytical methods such as truss models and macroscopic models.
At present, numerous numerical modeling works for nonlinear finite element analyses
of RC structural walls have been carried out but most of them were concentrated on its
behavior under monotonic loading [KI-2, PI, Nl] due to the complexities in the cyclic
modeling of RC composite material after cracking of concrete. Several relevant studies
have been carried out [El, K3, K6, SI-2] in the analysis of actual RC structural walls
regards microscopic models describing the behavior of RC under cyclic loading.
Among them, the fixed crack model [E1, K6, S2] is popularly adopted for the modeling
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Chap.ter Five
of concrete because of its computational convenience and its capability of representing
the physical nature of cracking in RC walls. However, it should be noted that the fixed
crack model has a limitation in describing the rotation of cracks induced by the
interface shear along the cracking surface when a RC structural wall is subjected to
cyclic loadings [K3]. In this study, the rotating crack approach based on total strain is
applied to the constitutive modeling of tensile and compressive behavior of concrete for
the purpose of better understanding of shear effects of RC structural walls. In addition,
although several experimental and analytical studies [M2-3, P3, T1] on the structural
components with construction joints which may cause sliding shear failure have been
investigated, research related to numerical modeling for such structural components,
especially shear walls, is rather limited. In this study, the nonlinear structural interface
element based on Mohr-coulomb friction criterion is adopted to aid in better
understanding of the sliding behavior of such walls.
This study presents finite element analytical models for RC structural walls subjected
to in-plane cyclic loadings. The proposed models are validated through comparison of
the analytical results with extensive experimental data at global and local response
levels with the aid of the finite element computer program DIANA 8.1 [D1]. By
comparisons, this study shows that with proper calibration of material parameters and
implementation of cyclic constitutive relationships, the proposed finite element models
are effective in predicting the nonlinear behavior of RC structural walls and will play
an important role in the ongoing research. These analytical models of the finite element
method are then further used to explore in detail the influence of several critical
parameters: axial loads, longitudinal reinforcements in the wall boundary elements,
aspect ratio, area of boundary columns and the presence of construction joints at the
wall base on the seismic behavior of walls.
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Chapter Five
5.2 Description of Finite Element Models
The computer program DIANA 8.1 (Displacement method Analyser, version 8.1) is a
general purpose finite element code including nonlinear structural analysis of RC
structures, based on the displacement method. The program has been developed at the
TNO Building and Construction Research in Netherlands [D1] since 1972 with
appealing capabilities in the field of concrete where excellent material models are
available.
Reinforced concrete (RC) behaves as a composite structure under load and after
concrete cracking, it shows complicated nonlinear behavior such as bond action
between reinforcement and concrete, aggregate interlock along the crack interface and
compressive deterioration of cracked concrete. To effectively describe the complicated
nonlinear behavior of RC structures, the accurate modeling of the material properties
becomes essential. In this finite element analysis computer program DIANA 8.1, four
types of nonlinear material properties including the concrete, reinforcement and
bondages between them as well as structural interface are to be provided.
5.2.1 Constitutive Models for Concrete
Since concrete is weak in tension, tensile cracking, which is one of the most important
reasons for nonlinearities in RC, can have a significant effect on the behavior of most
RC members, even at an early stage of loading. As a result, proper crack modeling is
crucial to the success of the concrete model that generally includes the reasonable
definitions of three basic components: crack representation, crack initiation and
propagation, and the constitutive relationship for cracked concrete. In the following
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Chap.ter Five
sections, different approaches that have been used to define the three basic components
would be discussed with the emphasis on those adopted in this study.
1) Crack representation
In the finite element analysis, stress and strain are assumed to be continuous within one
finite element. However, when concrete cracks, discontinuities in stress and strain
occur in the concrete matrix. To incorporate these discontinuities into the concrete
model, there are two commonly used approaches to represent cracks: discrete crack
model and smeared crack model. The discrete crack model represents cracks as a
separation of nodes along element boundaries. The post-cracking behavior, such as
tension stiffening, aggregate interlock, and dowel action, can be incorporated into the
model by using linkage elements to connect the separated nodes. Although this model
realistically represents the discontinuities in stress and strain across cracks, cracking
can occur only along element boundaries that introduces bias into the finite element
solution. Moreover, once the separation of the nodes has occurred, crack closing and
reopening needs to be considered as a contact problem. This significantly complicates
the finite element procedure, especially in the problems that involve cyclic loading
[Sl].
In the smeared crack model, the stress-strain discontinuities across the cracks are
averaged over the element in the vicinity of the cracks and thus the stress-strain
relationship of cracked concrete can still be described in a continuous manner. Since
cracks at each integration point are considered separately, cracking can occur in any
direction and multiple cracks are also allowed at each integration point. This model is
found to be suitable for modeling RC members with distributed crack patterns because
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Chapter Five
it represents cracks as being finely spaced. However, for RC members in which one or
few large cracks dominate the response, the discrete crack model might be more
appropriate.
For RC structural walls, behavior of cracks are rather more important than the
development of individual crack. For this purpose, the smeared crack model, in which a
finite region containing several cracks and reinforcing bars are considered to be a
continuum, is quite adequate to describe a RC element. On the other hand, reality is
that local discontinuities, such as sliding along the joint plane, can take place due to the
presence of construction joints at the wall base. To take this effect into consideration,
the introduction of the discrete crack model becomes necessary and would be discussed
in the following constitutive relationship of structural interface.
2) Crack initiation and propagation
This study adopts a strength criterion for crack initiation that concrete cracking occurs
as the principal tensile stress violates the maximum stress condition. After a crack has
formed, the fracture mechanics criterion for the smeared crack model is applied in this
investigation, to determine crack propagation. In this approach, each crack is modeled
by a one-element wide band of concrete elements. The cracks propagate to the next
element at the tip of the crack band when the calculated energy release rate of the crack
band exceeds the critical value, which depends on the fracture energy, Gf of concrete.
3) Constitutive modeling of cracked concrete
The finite element computer program, DIANA 8.1, offers a wide range of material
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Chap.ter Five
models for analysis of nonlinear concrete behavior. The models usually comprise
tension softening, compression softening, and crack closing and reopening which are
the major aspects of the inelastic behavior in concrete. In this study, total strain based
crack models, which describe the tensile and compressive behavior of concrete with a
single stress-strain relationship, is applied to simulate the nonlinear behavior of the
concrete. This particular crack model can be used by either a rotating or fixed crack
situation according to the method of determining a crack direction. The rotating crack
model evaluates the stress-strain relationships in the principal directions of strain vector
and no shear transfer model is needed in this approach since no shear stress appears in
continually updated principal planes. Whereas the fixed crack model assumes that the
crack is fixed once it is generated, in this model the shear transfer model is required.
The fixed crack model is popularly adopted in numerical modeling of concrete cracking
due to its computational convenience and more clearly defines the physical nature of
concrete cracking. However, it also should be noted that the fixed crack model has a
limitation in describing the rotation of cracks induced by the interface shear along the
cracking surface when a RC structural wall is subjected to cyclic loadings [K3]. Thus,
in this study, the rotating crack approach based on total strain is applied to the
constitutive modeling of tensile and compressive behavior of concrete.
To simulate the tension softening effect of concrete after its cracking, the finite element
software, DIANA 8.1, offers a range of softening models, such as linear tension
softening, multilinear tension softening and nonlinear tension softening models. In this
study, the nonlinear tension softening model proposed by Hordijk, Cornelissen and
Reinhardt [D 1] would be adopted as shown in Fig. 5.1. This model proposed an
expression for the softening behavior of concrete which results in a crack stress equal
to zero at a ultimate crack strain, £~:.ult as expressed in Eq. (5.1).
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Ger =5.136-f-
Cnn.ult h· h
Chapter Five
(5.1 )
The tensile strength, h (N / mm 2), tensile fracture energy, Gf (N . mm / mm 2
) and
crack band width, h (mm) as shown in Fig. 5.1 are calculated in terms of the
European CEB-FIP Model Code 1990 [C1]:
it =1.4({~ )~ (5.2)
(5.3)
The value of GfO (N· mm / mm 2) relates to the maximum aggregate size as listed in
Table 5.1. Crack band width, h (mm) is the square root of the area of the element.
Concrete in compression is simulated with commonly used parabolic compression
model as shown in Fig. 5.1 due to its ability to consider both the hardening and
softening behavior of concrete. As indicated in Fig. 5.1, the parabolic compression
model is in connection with the concrete compressive strength, fe', crack band width,
h and compressive fracture energy, Ge which the value is usually defined by 150Gf
[C1]. Moreover, it should be mentioned that concrete subjected to compressive stresses
shows a pressure-dependent behavior, i.e., strength and ductility increase with
increasing isotropic stress. In such a case, the compressive stress-strain relationship
should be modified to incorporate the effects of the lateral confinement. In order to
simulate this behavior, the parameters of the compressive stress-strain function in the
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Chal2.ter Five
computer program DIANA 8.1 are determined with a failure function which gives the
compressive stress as a function of the confining stresses in the lateral directions. The
effect of lateral confinement on the strength and ductility of the concrete is clearly
described in the literature [D 1] by means of considering the corresponding parameters.
In the current implementation of the computer program DIANA 8.1, secant reload and
unload responses is applied to concrete in tension, while linear reload and unload
responses with an initial stiffness same as that concrete in tension is used for concrete
in compression.
5.2.2 Constitutive Model for Reinforcing Bars in Concrete
When modeling the reinforcing steel, bar yielding and strain-hardening effects must be
considered as these effects are the major sources of energy dissipation in RC structures.
In the finite element analytical program, DIANA 8.1, two methods for modeling
reinforcing bars in concrete are available. The first method is called embedded
reinforcement which as its name suggests, denotes that the reinforcement is embedded
within the concrete. The strains of such reinforcement are obtained from the
displacement field of concrete and so a perfect bond is assumed between the
reinforcement and concrete. The second method applies discrete elements such as truss
elements to model reinforcing bars in concrete which allows for the consideration of
bond-slip behaviors between the reinforcement and surrounding concrete by adding
interface elements between the reinforcement and the concrete. In this study, the
embedded reinforcement, which allows the lines of the reinforcement to deviate from
the lines of the mesh, are used to model the transverse reinforcement in wall boundary
elements. Whereas, truss elements are applied to model other reinforcing bars in
concrete with perfect bond as the effect of bending stiffness and dowel action of a
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Chapter Five
reinforcing bar on the behavior of low-rise structural walls is not very large. The
constitutive behavior of the reinforcing bars herein is simulated by an elasto-plastic
material model with strain hardening by employing the Von Mises yield criterion [Dl].
5.2.3 Constitutive Model for Structural Interface
The computer program, DIANA 8.1, supplies several structural interface models by
setting a nonlinear relation between tractions (normal/shear) and relative displacements
across the interface to simulate the interface behavior such as bond-slip, crack dilatancy
and friction. The general constitutive relation is assumed to be incrementally linear
t=D·~u (5.4)
where t is the traction vector, ~u is the vector with the relative displacements, and
D is the tangential stiffness matrix.
To simulate the behavior of existing construction joints between two parts of the
structural walls, shear friction theory as shown in Fig. 5.2 is adopted herein as its
suitable application for this case [D 1]. In the shear friction hypothesis as shown in Fig.
5.2, the roughness is visualized as a series of frictionless fine saw-tooth ramps having a
slope of tan t/J. Assuming sliding along the failure plane m-m, and simple Coulomb
friction, the shear force, V, required to produce sliding is equal to J1P, where f.l is
the friction coefficient between the two elements and P is the clamping force
perpendicular to the sliding plane. The roughness of crack m-m will create a separation
t5 between the two halves as shown in Fig. 5.2(b). If reinforcement is placed across
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where c denotes the cohesion, lfI is dilatancy angle which accounts for crack
dilatancy effect and here equals to friction angle, ¢ for the consideration of associated
plasticity [D1]. The detailed constitutive law for the model is presented in the DIANA
manual [D 1].
(5.5)
(5.6)g=R+tn tanlfl
f=R +tn tan¢(k)-c(k) =0
and the plastic potential surface:
In the simulation of the structural interface, the Coulomb friction criterion is extended
with a gap criterion: assuming that a gap arises if the tensile traction, tn normal to the
interface exceeds a certain value which corresponds to a concrete tensile strength, It.
After the gap formation, the tensile traction, tn is reduced to zero immediately and
the interface, the separation will develop tension T in the reinforcement. The tension
provides an external clamping force on the concrete resulting in compression across the
interface of equal magnitude. Tests revealed that the separation is usually sufficient to
yield the reinforcement crossing the crack. In this study the Coulomb friction model as
shown in Fig. 5.3, suitable application for the shear friction theory, is adopted in the
modeling of existing construction joints between two parts of the structural walls. As
shown in Fig. 5.3, the Coulomb friction model based on the shear friction hypothesis
presented is basically given by the yield surface
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Chapter Five
shear retention according to the aggregate interlock relation of Walraven and Reinhardt
[WI] is adopted to simulate the shear transfer mechanism along the structural interface.
The response diagram for this model is shown in Fig. 5.4.
In accordance with ACI 318 code provisions [AI], which suggests the use of the value
1.4 as the friction coefficient for a crack in monolithic concrete, 1.0 for interfaces
intentionally roughened and 0.6 for the interfaces which are not intentionally
roughened. The specimens tested in this study are representative of poor construction
practices since no specific measures were taken regarding curing at the construction
joints. For this case the friction coefficient of 0.6 which takes the effect of shear
resistance due to dowel action of the reinforcement into account is considered to be
appropriate [AI] in the analysis of construction joints not intentionally roughened at the
surface of the normal weight concrete.
5.3 Applications of the Finite Element Models
In the analysis, the developed concrete constitutive law is a total strain rotating crack
model with nonlinear tension softening according to Hordijk et al [D1] and the
parabolic compression as shown in Fig. 5.1. Eight-node quadrilateral isoparametric
plane stress elements, QU8 (CQ16m), based on quadratic interpolation and 3x 3 Gauss
integration scheme are used to model concrete. The reinforcing bar is simulated as Von
Mises plastic material and the strain hardening rule, developed according to the
uniaxial test of bare bars, is applied. Embedded reinforcement is used to model the
transverse reinforcements in wall boundary elements, whereas other bars is assumed to
be a truss element with two nodes and perfect bond between reinforcement and
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concrete is assumed. Six-node interface element between two lines in a
two-dimensional configuration is used to simulate the interface behavior. The element
is based on quadratic interpolation and the default 4-point Newton-Cotes integration
scheme is applied. The mesh of structural walls is generally divided into four zones: the
wall web, wall flanges, top and base beams. The top and base beam of the wall is
considered to be elastic and rigid during the modeling. The base of the wall is assumed
to be fully anchored against horizontal and vertical movements.
The horizontal cyclic loadings are assumed to act at the center location of the top beam
which is considered to be rigid to uniformly distribute the lateral load to the entire
cross-section of RC structural walls. The loading history of each specimen is similar to
its experiment equivalent, but only one cycle is applied at each pre-defined drift ratio.
In the current implementation of the computer program, DIANA 8.1, the behavior of
unloading and reloading is modeled differently with secant unloading as shown in Fig.
5.1 [Dl]. Moreover, in this nonlinear finite element analysis, regular Newton-Raphson
iteration method in which the tangential stiffness matrix is evaluated for every iteration
is chosen as an iterative procedure since a few iterations to converge to the final
solution is needed.
5.3.1 Verification of the Finite Element Models
Before proceeding with an extensive parametric study using the nonlinear finite
element analysis, it is necessary to verify the reliability of the proposed finite element
models. For this purpose, two RC low-rise structural walls previously tested are
selected. The first RC low-rise structural wall, Unit 1.0 with an aspect ratio of 1.0, was
tested by Mestyanek [M1]. The tested wall eventually failed in shear due to diagonal
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Chapter Five
tension and thus is selected as a typical model to demonstrate shear effects. The second
selected structural wall to validate the finite element models is Specimen S-F1 with an
aspect ratio of 1.3 tested by Wu [W2]. The contribution of the flexure component to
total displacement for this specimen is higher than 70% at the ultimate stage and it is,
therefore, natural to conclude that the wall is mainly dominated by flexure. Table 5.2
lists the material properties of concrete and reinforcing bars according to the measured
values as well as reinforcement ratios of the two selected specimens. Figs. 5.5 and 5.6
depict the overall dimensions and reinforcement layout of the tested specimens, Unit
1.0 and Specimen S-F1, respectively.
With exactly the same geometrical configuration and dimensions as the tested
specimens, the finite element idealizations of Unit 1.0 and Specimen S-F1 are shown in
Figs. 5.7 and 5.8, respectively. In these figures, deformed shapes of both specimens at a
drift ratio of 1.0% are also presented to illustrate the effect of shear/flexure
deformations on the wall performance. Meanwhile, the experimental and analytical
lateral load - displacement relationships of Unit 1.0 and Specimen S-F1 are illustrated
in Figs. 5.9 and 5.10, respectively. From Figs. 5.7 and 5.9, it can be seen clearly that
significant tensile and shear straining has occurred in the wall web due to the
insufficient provisions of shear reinforcements for Unit 1.0, while from Figs. 5.8 and
5.10, it is observed that the response of Specimen S-F1 is dominated by flexure. It can
thus be naturally concluded that the observed analytical responses for both studied
specimens agree well with the respective test results.
Table 5.3 presents the comparisons of finite element predictions and test results in
terms of peak loads obtained and its associated displacements for both specimens. As
observed at Columns (6) and (7) in Table 5.3, for Unit 1.0 and Specimen S-FI, the ratio
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of tested-to-predicted peak loads, Fp,f / Fp,p is obtained to be 0.96 and 1.11,
respectively, while the ratio of tested-to-predicted associated displacements, ~ p,t / ~ p.p
is 0.96 and 1.01, respectively. This comparison indicates that the finite element
analytical results correlate well with the experimental ones in terms of strength and
deformation capacities for both specimens. However, for Specimen S-Fl the pinching
degree of the analytical results as shown in Fig. 5.10 is slightly overestimated which
may be the production of the assumed perfect-bond between the reinforcement and
concrete. In practice, the presence of a bond slip may have a pronounced effect on the
pinching of the specimen. In general, it can be concluded that the proposed finite
element models implemented in the finite element analysis can simulate satisfactorily
the behavior of RC low-rise structural walls under reversed cyclic loadings.
5.3.2 Numerical Investigations of Specimens Tested
Previous chapters presented the experimental results of eight RC structural walls,
including both global responses, such as load-displacement relationships, and local
responses, such as rebar strain history. Based on those results, the following numerical
investigations are carried out to fulfill two main purposes. Firstly, the investigations are
intended to verify the reliability of the proposed finite element models under different
design parameters such as axial loads and aspect ratios, especially the structural
interface model for the tested walls. Secondly, the investigations are extended by
comparing the numerical local response of the structural walls tested with the
experimental ones for which offered more insights on the local behavior of all tested
speCImens.
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Chapter Five
The concrete and steel material properties adopted for these nonlinear finite element
investigations of the total eight specimens are same as those used in the testing. The
detailed concrete properties are listed in Table 5.4 and steel properties as displayed in
Fig. 2.1 at chapter two are employed by these investigations. In the case of structural
walls with construction joints at the wall base, Coulomb friction model, with a gap
criterion and shear retention according to the aggregate interlock relation of Walraven
and Reinhardt [W1], is adopted to simulate the shear transfer mechanism along the
structural interface. In the following investigations a friction coefficient of 0.6, which
takes the effect of shear resistance due to dowel action of the reinforcement into
account, is considered to be appropriate [AI] in the analysis of construction joints not
intentionally roughened at the surface of normal weight concrete.
In the following sections the numerical results, including the respective global and
local responses of selected specimens, according to the proposed finite element models
are compared with experimental ones. After comparison with experiment results, a
parametric study is carried out in this research for RC structural walls to establish the
significance of several critical parameters, such as axial loads, longitudinal
reinforcements in the wall boundary elements, aspect ratio, area of boundary columns
and the presence of construction joints at the wall base. The effect of these critical
parameters on structural performances such as strength and deformation capacity,
stiffness characteristic, energy dissipation and equivalent damping of RC structural
walls is investigated as follows.
5.3.2.1 Predicted Global Response of Specimens Tested
The idealized finite element meshes for total eight specimens are illustrated in Fig. 5.11
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Chal2.ter Five
and have exactly the same geometry configuration and dimensions as the respective
tested specimens. The overstriking black continuous line at the wall base as displayed
in Figs. 5.ll(c) and 5.ll(£) indicates the application of structural interface models to
associated specimens. Embedded reinforcement which allows the lines of the
reinforcement to deviate from the lines of the mesh are used to model the transverse
reinforcements in boundary elements for all specimens tested. Whereas, truss elements
located along the sides of element meshes are applied to model other reinforcing bars in
concrete with perfect bond.
Global behavior is presented in terms of hysteretic loops: lateral load - displacement
relationships of specimens studied. Fig. 5.12 shows the comparison results between the
experimental and analytical lateral load - top displacement relationships for all eight
specimens. In general, analytical results show a good correlation with the experimental
ones in terms of strength capacity, deformation characteristics and energy dissipation
for all specimens. This firstly suggests that the structural interface model incorporated
in the proposed finite element models during numerical investigations is effective in
predicting the behavior of RC structural walls with construction joints at the base.
Secondly, it can be concluded that the proposed finite element models are reliable for
RC structural walls under different axial loading levels and aspect ratios. However, by
comparing experimental initial stiffness, the analytical initial stiffness of specimens
studied is found to be overestimated which may due to the assumptions made as fully
fixed boundary conditions, perfect bond between the reinforcement and concrete, and
loads applied exactly at the wall center line during the analysis.
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Chapter Five
5.3.2.2 Predicted Local Response of Specimens Tested
Numerical investigations of the wall local responses are carried out by outputting the
strain distributions along the selected bars for Specimens LW1, LW2 and MWl. In this
study strain distributions along two types of selected bars, longitudinal and horizontal
bars, are investigated respectively for the selected specimens. Moreover, for the
purpose of better understanding the strain development tendency as tests progressed,
four levels of drift, 0.1 0/0, 0.25%, 0.5% and 1.0% corresponding to the stages of the
pre-cracking, first-yield, post-yield and ultimate respectively, are considered for each
selected specimen. At each drift level, both the experimental and numerical strains in
selected longitudinal/horizontal bars of the studied specimens are presented to calibrate
their correlation.
Fig. 5.13 depicts the experimental and numerical longitudinal strains In selected
vertical bars along wall length of the specified specimens. As shown in Fig. 5.13, for
each specimen under the negative loading, the strain distributions along the wall length
is obtained at two different wall heights: 50 mm and 550 mm measured from the wall
base. These wall heights correspond to exactly the same position of strain gauges
attached to the selected longitudinal bars in the tests and at each wall height, eight
longitudinal strains obtained from eight different selected vertical bars are presented to
indicate the strain distributions along the whole wall length. It is found from Fig. 5.13
that at low drift levels the experimental and numerical longitudinal strains in selected
vertical bars agree well and distribute almost linearly along the wall length which
indicates that the Navier-Bemoulli assumption of plane-sections remaining plane can
be applied to all three selected specimens up to a drift ratio of 0.5%. However, as
displacing the specimens to a drift level of 1.0%, a significant increase of the values of
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longitudinal strains is observed in selected vertical bars and the plane sections do not
remain plane at the wall base. At this drift level the numerical strains in longitudinal
bars for selected specimens are observed to be slightly higher than those in the tests,
but in general good correlation between the numerical longitudinal strains and the
experimental ones in vertical bars is observed at all four drift levels.
Fig. 5.14 depicts the experimental and numerical strain distributions along horizontal
bars of the three selected specimens under both positive and negative loading directions.
At each specimen two horizontal bars, R bar and T bar, located at two different levels
of the wall height: 250 mm and 750 mm measured from the wall base, respectively are
selected to distribute the strains at four different locations along the selected horizontal
bar as displayed in Fig. 5.14. The strain locations exactly correspond to the locations of
strain gauges attached to the specified horizontal bars in the tests. In general, larger
values of horizontal strains along the diagonal struts than those in other positions are
observed as shown in Fig. 5.14 for both experimental and numerical strains in the two
specified horizontal bars for selected specimens. The observed good correlation
between the experimental and numerical results further provides the verification of the
effectiveness of the proposed finite element models in predicting the local responses of
walls studied.
5.3.3 Parametric Study of Squat Structural Walls
In the past decades, most of the parametric works in the finite element analysis of RC
structural walls were concentrated on the development of the material models that
could reproduce experimental results and few researchers used the finite element
method to investigate behavior of RC walls other than that of the specimens tested in
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Chapter Five
the laboratory [81]. According to currently available information at hand, two main
parametric studies using the finite element method were carried out up to date to study
the effects of different design parameters on the response of RC structural walls. The
first study was conducted in Japan by Mikame et al [M4] who investigated the effects
of several design parameters such as different reinforcement ratios, vertical axial stress,
compressIve strength of concrete, confinement of column, and openings on the
behavior of heavily reinforced structural walls. The second was undertaken at
University of Ottawa, Canada by Nasir N. [N3] who investigated the sensitivity of
selected design parameters, such as axial loads and aspect ratios etc, on strength and
ductility of eighteen RC structural walls. However, both studies were carried out for
RC low-rise structural walls subjected to static monotonic loading. This leads to
difficulties in the evaluation of several critical seismic performances, such as energy
dissipation and equivalent damping, which can only be assessed by means of modeling
the structural walls under cyclic loadings.
Accordingly, for the purpose of evaluating the effect of different design parameters on
the wall seismic performances, it is necessary to carry out an extensive parametric
study for RC low-rise walls under cyclic loadings. In this study, five critical design
parameters with their description and investigated ranges are listed in Table 5.4. The
effects of these investigated parameters on the wall seismic performance such as the
secant stiffness, K eq' energy dissipation capacity, Ah and equivalent damping factor,
heq as shown in Fig. 5.15 are investigated. Note that the experimental and analytical
equivalent damping factor, heq for walls studied as displayed in Fig. 5.15 is
determined by
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same as those in the tests.
chapters are selected as reference walls. The overall dimension and reinforcement
(5.7)
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1 ~heq =2,. . F
m!1
m
In the following parametric studies, the tested walls as presented in the previous
where Ah is the energy dissipation calculated from the area enclosed by the
load-displacement loops as shown in Fig. 5.15, Fm , ~ m is the peak lateral load and
top displacement of the loop, respectively.
for all specimens studied. The steel properties used in the parametric studies are the
to fulfill this purpose. The final analysis results will be presented together with the
strength of 3.15 MPa was assumed and a constant Poisson's ratio of 0.2 was selected
experimental ones to aid in better understanding of the effect of different design
the axial load ratio, N /( fe' Ag ) from 0.00 to 0.15. A series of two RC walls with
In the following analysis, the verified finite element material models would be adopted
details of the studied RC walls are kept the same as those tested except those specified
in Table 5.4. The concrete uniaxial compressive strength, fe' of 40 MPa was selected,
with an initial tangent modulus of elasticity equal to 29,900 MPa. A concrete tensile
parameters on the seismic performances of the structural walls studied.
5.3.3.1 Effect ofAxial Loading
The investigations on effects of axial load are carried out for eight cases by changing
:II!
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Chapter Five
aspect ratios fixed to be 1.125 and 1.625, respectively are investigated by selecting the
tested Specimens LW1 and MW1 as reference walls in order to compare the analytical
results with experimental ones in terms of seismic performance. The reinforcement
detail and overall dimensions of the walls studied are kept to be the same as those for
Specimens LW1 and MW1.
5.3.3.1.1 Effect of axial loading on wall strength
Figs. 5.16(a) and 5.16(b) demonstrate the analytical and experimental backbone curves
of the lateral load - top displacement for two series of walls with aspect ratios of 1.125
and 1.625, respectively. In general, the presence of axial loading significantly increases
the wall strength capacity with the augment of axial loads up to a ratio of 0.15 for both
two series of RC walls. Moreover, as shown in Fig. 5.16, the lateral load - top
displacement relationship show tri-linear for both two series of RC walls subjected to
axial loads up to a ratio of 0.05 and a drift capacity of around 1.0% can be achieved for
all walls. When axial load ratio reaches 0.10 or above, an arc-shaped relationship is
observed and the wall drift capacity tends to decrease with the increase of axial loads.
The contribution of axial load ratios to the increase of wall strength is illustrated in Fig.
5.17 where P;,max and PO,max is the maximum strength for RC walls with different
levels of axial loads and without axial loads, respectively. In this figure, the analytical
results corresponding to the two series of RC walls with aspect ratios of 1.125 and
1.625, respectively are outputted together with experimental ones. It can be seen from
the figure that the analytical rate of increase of the wall strength with the augment of
axial load ratio from 0.00 to 0.05 correlates well with experimental results, and is
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obtained to be 0.390 for RC walls with an aspect ratio of 1.125. However, with the
increase of the wall axial load ratio from 0.10 to 0.15, the rate of increase of wall
strength tends to be less, and is observed to be 0.181. Such variation trends can also be
observed for RC walls with an aspect ratio of 1.625 for which the axial loads
contribution to the wall strength decreases from 0.278 to 0.153 when the RC walls
subjected to a higher level of axial loads. This indicates that the effect of axial load on
the wall strength is more significant for RC walls under lower level of axial load (from
0.0 to 0.05) than higher level of axial load (from 0.10 to 0.15).
Moreover, the rate of increase of the wall strength, 0.390 with the augment of axial load
ratio from 0.00 to 0.05 for RC walls with an aspect ratio of 1.125 is larger than that for
RC walls with an aspect ratio of 1.625 as the value is observed to be 0.278. Similar
variation observation can be extended to RC walls subjected to added axial load ratios
from 0.10 to 0.15 as the rate of increase of wall strength for RC walls with aspect ratios
of 1.125 and 1.625, is obtained to be 0.181 and 0.153, respectively. This suggests that
the contribution of axial load to the wall strength for RC walls with a lower aspect ratio
is more significant.
5.3.3.1.2 Effect of axial loading on secant stiffness
The representation of secant stiffness for RC walls under cyclic loading is shown in Fig.
5.15. Based on this representation, the variation of the experimental and analytical
secant stiffness with drift ratios of RC walls subjected to four different levels of axial
loads are presented in Fig. 5.18. In general, experimental secant stiffness agrees well
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Chapter Five
with analytical ones and both experimental and analytical wall secant stiffness degrades
significantly with the increase of wall drift ratios. However, under an axial load ratio of
0.05, the analytical secant stiffness is observed to be overestimated which becomes
more significant for both two series of walls at a lower wall drift ratios of up to 0.33%.
This may be due to the assumptions of fully fixed boundary conditions, perfect bond
between the reinforcement and concrete, and loads applied exactly at the wall center
line during the analysis. Moreover, it can also be seen from Fig. 5.18 that the secant
stiffness of RC walls with same drift ratios generally increases with the addition of
axial loads at each drift level. However, this axial load contribution to the increase of
secant stiffness reduces with the movement of top wall drift from a lower to higher
level. At a lower wall drift ratio, the analytical secant stiffness of RC walls increases
significantly with the added axial loads and a large gap between the secant stiffness of
RC walls under different levels of axial loads is observed at same drift levels. However,
at a higher wall drift ratio, the secant stiffness of a wall under different axial loadings
becomes closer.
The contribution of axial loads to the wall secant stiffness at three different drift ratios,
0.1%,0.33% and 1.0%, are shown in Fig. 5.19 where Ki,a and KO,a is the analytical
secant stiffness with different levels of axial loads and without axial loads, respectively.
In the figure, the rate of increase of secant stiffness is determined by
(Ki,a - KO,a) / KO,a' It can be observed that the rate of increase of secant stiffness
generally decreases with the added drift ratios of RC walls studied. At a drift ratio of
0.1 %, the increase rate of secant stiffness for RC wall with an aspect ratio of 1.125,
increases from 109% to 246% with the augment of axial loads from 0.05 to 0.15, while
at a drift ratio of 0.33%, the corresponding values reduces to be 77% and 188%.
Moreover, the contribution of axial loads to wall secant stiffness is more significant for
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RC walls with axial loads increasing from 0.0 to 0.05 than that for RC walls with axial
loads increasing from 0.10 to 0.15. With added axial loads from 0.0 to 0.05, the
contribution of axial loads is 109% and 99% for RC walls with aspect ratios of 1.125
and 1.625 respectively, while with the augment of axial loads from 0.10 to 0.15, 60%
and 21 % of this axial load contribution to secant stiffness is observed for all studied RC
walls at a drift ratio of 0.1 %. Similar decreasing trend of the contribution of axial loads
to secant stiffness with the augment of axial loads can also be observed for all studied
RC walls at a drift ratio of 0.33%.
5.3.3.1.3 Effect of axial loading on energy dissipation
Fig. 5.20 displays the variation of energy dissipation capacity with the drift ratios for
both two series of walls under four different levels of axial loads. In general, the energy
dissipation capacity increases with the added drift ratios for RC walls under all levels
of axial loads. However, the effect of axial load on energy dissipation capacity is
different for walls at a low drift ratio as compared to those at a high drift ratio. For wall
drift ratios of up to 0.33%, the energy dissipation almost remains to be same for all RC
walls under four different axial loads levels, however, with the increase of drift ratios
from 0.33% to 1.0%, it is observed to increase almost linearly as shown in Fig. 5.20.
This variation trend can also be clearly seen in Fig. 5.21 which shows the contribution
of axial load ratios to energy dissipation for all RC walls studied. In the figure, the
continuous and dashed lines represent for RC walls with aspect ratios of 1.125 and
1.625, respectively. In the figure, the percentage increase of energy dissipation for RC
walls with drift ratios of 0.50% or above is obtained by (Ah,i - Ah,o ) / Ah,o where Ah,i
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--.,'...", ..,....."".,.,1
Chapter Five
and Ah,o are the dissipated energy corresponding to RC walls under axial load ratios
of 0.05 and 0.00, respectively. As displayed in this figure, with the added axial loads
from 0.00 to 0.05, the rate of increase of energy dissipation for RC walls at a drift ratio
of 0.50% is obtained to be 13% and 21 %, while at a drift ratio of 1.0% this rate
increases to 56% and 31 % for RC walls with aspect ratios of 1.125 and 1.625,
respectively. This suggests that the presence of axial loads plays a beneficial effect on
the wall energy dissipation capacity at a high drift ratio, while at a low drift ratio (up to
0.33%) this effect is rather negligible.
5.3.3.1.4 Effect of axial loading on equivalent damping
The effect of axial load on equivalent damping of walls studied is shown in Fig. 5.22.
As indicated in the figure, the equivalent damping of RC walls studied increases with
the added drift ratios in general. At a low drift ratio (up to 0.33%), analytical equivalent
damping for RC walls under all four levels of axial loads, increases slightly but remains
to be low, whereas after that, the equivalent damping is observed to increase
significantly for RC walls under axial load ratios from 0.00 to 0.05.
The contribution of axial loads to the equivalent damping for RC walls studied are also
presented and illustrated in Fig. 5.23. In the figure, the continuous and dashed lines
represent for RC walls with aspect ratios of 1.125 and 1.625, respectively. The
percentage decrease of equivalent damping for RC walls with drift ratios of 0.50% or
above is obtained by (heq,i - heq,o) / heq,o where heq,i and heq,o are the equivalent
damping corresponding to RC walls under axial load ratios of 0.05 and 0.00,
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Cha12.ter Five
respectively. In general, the equivalent damping for both two groups of walls studied
decreases with the added axial loading as shown in Fig. 5.23. With the added axial
loads from 0.00 to 0.05, the percentage decrease of equivalent damping for RC walls at
a drift ratio of 0.50% is obtained to be 42% and 35%, while at a drift ratio of 1.0% this
decrease rate is observed to be 1.4% and 22% for RC walls with aspect ratios of 1.125
and 1.625, respectively. This can be due to the decrease of degree of loop pinching with
the added axial load ratio from 0.0 to 0.15 which indicates its beneficial effect on the
control of pinching behavior.
5.3.3.2 Effect of Longitudinal Reinforcement Content in Boundary Element
The effect of longitudinal reinforcement content in boundary element, Ph on wall
strength is investigated for four cases by changing this content, from 0.7% to 4.2% as
listed in Table 5.5. The tested RC wall, Specimen LWI with an aspect ratio of 1.125, is
selected as reference wall in order to compare the analytical results with experimental
ones in terms of seismic performance. The overall dimensions of the walls studied are
kept to be the same as that of Specimens LWI.
5.3.3.2.1 Effect of longitudinal reinforcement content on wall strength
Fig. 5.24 demonstrates the effect of longitudinal reinforcement content on the backbone
curves of load-displacement loops. As shown in Fig. 5.24, the wall strength capacity
increases significantly with the augment of longitudinal reinforcement content in wall
boundaries and a drift capacity of around 1.0% can be achieved for all walls studied.
Moreover, as shown in Fig. 5.24, the lateral load - top displacement relationship show
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Chapter Five
tri-linear for RC walls having longitudinal reinforcement content of 2.8% or above,
while bilinear backbone curves is observed for RC walls with longitudinal
reinforcement content less than 1.40/0.
The contribution of longitudinal reinforcement content to the wall strength capacity is
shown in Fig. 5.25 where Pa,max and ~,max is the maximum strength for RC walls
with longitudinal reinforcement content of 0.7% and one of the other three different
content levels, respectively. As shown in the figure, with the added longitudinal
reinforcement content in boundary elements from 0.7% to 4.2%, the analytical
maximum strength increases almost linearly and the percentage increase of the
maximum strength is observed to be 140% around.
5.3.3.2.2 Effect of longitudinal reinforcement content on secant stiffness
The variation of the experimental and analytical secant stiffness with the drift ratios of
RC walls having four different ratios of longitudinal reinforcement in boundary
element are shown in Fig. 5.26. In general, the experimental secant stiffness agrees well
with the analytical ones and both experimental and analytical secant stiffness degrade
significantly with the increase of the wall drift ratios. With the increase of wall drift
ratios from 0.1 % to 1.0%, the secant stiffness for RC wall with longitudinal
reinforcement ratio of 0.7%, reduces from 93 kN/mm to 12 kN/mm (around 87%
degradation) in the positive loading direction. For RC walls with same drift ratios, the
secant stiffness increases with the added longitudinal reinforcement content in wall
boundaries. With the augment of this reinforcement content from 0.7% to 4.2%, the
secant stiffness for RC walls at a drift ratio of 0.1 % increases by around 52% from 91
kN/mm to 139 kN/mm.
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Cha12.ter Five
The contribution of longitudinal reinforcement content to the wall secant stiffness at
three different drift ratios, 0.1 %, 0.33% and 1.0%, are shown in Fig. 5.27 where KO,a
and K. is the analytical secant stiffness corresponding to RC walls with longitudinall,a
reinforcement content of 0.7% and one of the other three different reinforcement ratios,
respectively. As shown in the figure, the rate of increase of secant stiffness increases
more rapidly after the wall attained a drift ratio of 0.33% for all four RC walls studied.
At a drift ratio of 0.1 % the increase rate of secant stiffness for RC walls with
longitudinal reinforcement ratio of 1.4% and 4.2%, increases by 1.5% and 52%,
respectively. While at a drift ratio of 0.33% the increase rate of secant stiffness is
observed to be 22% and 133% and at the final stage, it is obtained to be 26% and 144%
for RC walls with longitudinal reinforcement content of 1.4% and 4.2%, respectively.
Moreover, the contribution of longitudinal reinforcement ratio to wall secant stiffness is
observed to be almost the same for RC walls with reinforcement ratios increasing from
1.4% to 2.8% as that for RC walls with reinforcement ratios ranging from 2.8% to
4.2% at/after the attainment of a wall drift ratio of 0.33%. With added longitudinal
reinforcement content from 1.4% to 2.8%, the contribution of this reinforcement
content, represented by a slope between them, is 0.43 and 0.41 for RC walls at drift
ratios of 0.33% and 1.0% respectively, while 0.36 and 0.43 of this reinforcement
contribution to secant stiffness is observed for RC walls with longitudinal
reinforcement content ranging from 2.8% to 4.2%. However, at a drift ratio of 0.1 %,
the contribution of this reinforcement content ranging from 1.4% to 2.8% and from
2.8% to 4.2% is obtained only to be 0.23 and 0.14, respectively which is less than that
for RC walls at higher drift ratios. This indicates that the contribution of longitudinal
reinforcement to the wall secant stiffness is more effective for walls at higher drift
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Chapter Five
ratios.
5.3.3.2.3 Effect of longitudinal reinforcement content on energy dissipation
Fig. 5.28 displays the variation of energy dissipation capacity with the drift ratios for
RC walls under four different ratios of longitudinal reinforcement in boundary elements.
In general, the energy dissipation capacity increases with the added drift ratios for all
RC walls studied. At a drift ratio of 0.1 %, the energy dissipated for RC walls with
longitudinal reinforcement content of 0.7% is obtained to be 0.38 kNm while at drift
ratios of 0.33% and 1.0%, it is approximately 0.66 kNm and 8.12 kNm, respectively.
This indicates that the effect of longitudinal reinforcement on energy dissipation
capacity is different for walls at a low drift ratio (up to 0.33%) as compared with that at
a high drift ratio. For wall drift up to a ratio of 0.33%, the energy dissipation almost
remains to be same for all RC walls under all four different longitudinal reinforcement
contents, however, with the increase of drift ratios from 0.33% to 1.0%, it is observed
to increase significantly as shown in Fig. 5.28.
This variation trend can also be clearly seen in Fig. 5.29 which shows the contribution
of longitudinal reinforcement to energy dissipation for all the RC walls studied. In the
figure, the percentage increase of energy dissipation for RC walls with drift ratios of
0.50% or above is obtained by (Ah,j - Ah,o) / Ah,o where Ah,o and Ah,i are the energy
dissipated corresponding to RC walls with longitudinal reinforcement content of 0.7%
and one of other three contents (1.4%, 2.8% and 4.2%), respectively. As displayed in
this figure, with the added longitudinal reinforcement contents from 0.7% to 4.2%, the
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Chal!.ter Five
rate of increase of energy dissipation for RC walls at a drift ratio of 0.50% is obtained
to be 56%, while at drift ratios of 0.67% and 1.0%, this rate increases to 32% and 51 %,
respectively. This suggests that at a high drift ratio (from 0.5% to 1.0%), the increase of
longitudinal reinforcement in boundary elements have a significant effect on the wall
energy dissipation capacity, while at a low drift ratio (up to 0.33%) this effect is rather
negligible.
5.3.3.2.4 Effect of longitudinal reinforcement content on equivalent damping
The effect of longitudinal reinforcement on equivalent damping of walls studied is
shown in Fig. 5.30. As indicated in the figure, the equivalent damping of RC walls
studied increases with the added drift ratios in general. At a low drift up to ratio of
0.25% (prior to reinforcement yielding), analytical equivalent damping for RC walls
under four different longitudinal reinforcement contents, decreases significantly and
minimum equivalent damping (below 4%) is observed at a drift ratio of 0.25% except
for RC wall with reinforcement content of 0.7% which it is observed to be above 4% at
a drift ratio of 0.33%. Whereas after that, the equivalent damping is observed to
increase significantly to the final stage (above 12%) for all RC walls studied.
The contribution of longitudinal reinforcement to the equivalent damping for RC walls
studied are also presented and illustrated in Fig. 5.31. The percentage decrease of
equivalent damping for RC walls with drift ratios of 0.50% or above is obtained by
(heq,i - heq,o ) / heq,o where heq,o and heq,i are the equivalent damping corresponding
to RC walls with longitudinal reinforcement content of 0.7% and one of other three
contents (1.4%, 2.8% and 4.2%), respectively. In general, the equivalent damping for
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Chapter Five
RC walls studied decreases with the added longitudinal reinforcements as shown in Fig.
5.31. With the added longitudinal reinforcements content from 0.7% to 1.4% and 4.2%,
the percentage decrease of equivalent damping for RC walls at a drift ratio of 0.50% is
obtained to be 21 % and 44% respectively, while at a drift ratio of 1.0% this decrease
rate is observed to be 7% and 36%, respectively.
5.3.3.3 Effect of Boundary Columns
Previous research [P2, W2] indicates that sliding shear failure of boundary columns in
walls can greatly impair their energy dissipation capacities. Also it is observed [W2]
that RC walls with boundary columns can achieve higher strength, stiffness, and
maintain similar drift levels when subjected to reversed cyclic loading. To better
understand and clarify the influence of wall boundary columns on the global behavior
of walls with cyclic loadings, a parametric study is carried out by varying the cross
sections of boundary elements in four cases. This is achieved by changing the thickness
of boundary columns, tf from 120 mm (rectangular) to 1000 mm which leads to the
ratio of the sectional area of columns to the total sectional area, Ac / At varying from
0.15 to 0.60 as listed in Table 5.5 where Ac and At are the cross section areas of
boundary columns and the entire wall, respectively. The tested RC wall, Specimen
LW1 with an aspect ratio of 1.125, is selected as a reference wall. The overall
dimensions and the quantity of main bars in boundary columns of the RC studied walls
are kept to be the same as that of Specimens LW1.
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5.3.3.3.1 Effect of boundary columns on wall strength
Fig. 5.32 demonstrates the effect of boundary element shapes on the backbone curves
of load-displacement loops. As shown in Fig. 5.32, the wall strength increases
significantly with an increase of boundary column area ratios and a drift capacity of
around 1.0% can be achieved for all walls. This contribution of boundary columns to
the wall strength is also shown in Fig. 5.33 where PO,max and ~,max is the maximum
strength for RC walls with a column area ratio of 0.15 and one of the other three
different ratios, respectively. As shown in the figure, the percentage increase of the
analytical maximum strength is observed to be 14.2% with the increase of area ratio
from 0.15 to 0.60. Moreover, the trend of strength increases is observed to be more
significant as column area ratios increase. With the change of area ratios from 0.15 to
0.30, the analytical maximum strength increases relatively to be 0.11, while the
increasing values are obtained to be 0.21 and 0.63 corresponding to the area ratios
ranging from 0.30 to 0.45 and 0.45 to 0.60, respectively.
5.3.3.3.2 Effect of boundary columns on secant stiffness
The variation of the experimental and analytical secant stiffness with the drift ratios of
RC walls having four different column area ratios are shown in Fig. 5.34. In general,
the experimental secant stiffness agrees well with the analytical ones. The wall secant
stiffness degrades significantly with the increase of the wall drift ratios. With added
wall drift ratios from 0.1 % to 1.0%, the secant stiffness for RC wall with an area ratio
of 0.15, reduces from 90 kN/mm to 15 kN/mm (around 83% degradation) in the
positive loading direction. It can also be seen from the figure that the secant stiffness of
RC walls with same drift ratios increases with the higher column area ratios. With the
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Chapter Five
increase of this ratio from 0.15 to 0.60, the secant stiffness for RC walls at a drift ratio
of 0.1% increases by around 58% from 90 kN/mm to 142 kN/mm.
The contribution of boundary columns to the wall secant stiffness at three different drift
ratios, 0.1 %, 0.33% and 1.0%, are shown in Fig. 5.35 where KO,a and Ki,a is the
analytical secant stiffness corresponding to RC walls with an area ratio of 0.15 and one
of other three different area ratios, respectively. As shown in the figure, the rate of
increase of secant stiffness with the augment of column area ratios, decreases more
rapidly after a wall drift ratio of 0.10% for all four studied RC walls. At a drift ratio of
0.1 %, the increase rate of secant stiffness, for RC wall with area ratios of 0.45 and 0.60,
increases by 29% and 64% respectively. While at a drift ratio of 0.33%, the increase
rate of secant stiffness is observed to be 7% and 16% and at the final stage, it is
observed to be 10% and 17% for RC walls with area ratios of 0.45 and 0.60,
respectively.
5.3.3.3.3 Effect of boundary columns on energy dissipation
Fig. 5.36 displays the variation of the energy dissipation capacity with drift ratios for
RC walls under four area ratios of boundary elements. In general, the energy
dissipation capacity increases with the higher area ratios of boundary columns. At a
drift ratio of 0.1 %, the energy dissipated for RC walls with area ratio of 0.15 is
obtained to be 0.20 kNm around while at an area ratio of 0.60, it is approximately 1.2
kNm. This variation trend can also be clearly seen in Fig. 5.37 which shows the
contribution of boundary area ratios to energy dissipation for all RC walls studied. In
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the figure, the percentage increase of energy dissipation for RC walls with drift ratios
of 0.50% or above is obtained by (Ah i - Ah 0) / Ah 0 where Ah 0 and Ah i are the, J, , ,
dissipated energy corresponding to RC walls with boundary area ratio of 0.15 and one
of other three area ratios (0.30, 0.45 and 0.60), respectively. As displayed in this figure,
with the added area ratios from 0.15 to 0.30, the rate of increase of energy dissipation
for RC walls at a drift ratio of 0.5% and 1.0% is obtained to be 0.0% and 6.0%,
respectively. With boundary area ratios increasing from 0.30 to 0.45, the rate of
increase of energy dissipation at drift ratios of 0.5% and 1.0% is 32% and 14%,
respectively. This suggests that at a high area ratio, the increase of area ratios of
boundary columns have a significant effect on the wall energy dissipation capacity,
while at low area ratios (from 0.15 to 0.30), this effect is rather negligible.
5.3.3.3.4 Effect of boundary columns on equivalent damping
The effect of boundary columns on equivalent damping of walls studied is shown in
Fig. 5.38. In general, the equivalent damping of walls increases with higher area ratios
of boundary columns. At a low drift of up to a ratio of 0.25% (prior to reinforcement
yielding), analytical equivalent damping for RC walls under four different area ratios,
decreases significantly and minimum equivalent damping (below 10.0%) is observed at
a drift ratio of 0.25% except for the RC wall with an area ratio of 0.60 which it is
observed to be above 15% at a drift ratio of 0.33%. Whereas after that, the equivalent
damping is observed to increase significantly till the final stage (above 15%) for all RC
walls studied.
The contribution of boundary columns to the equivalent damping for RC walls studied
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is presented in Fig. 5.39. The percentage increase of equivalent damping for RC walls
with drift ratios of 0.67% and 1.0% is obtained by (heq,i - heq,i-l) / heq,i-l where heq,i
and heq,i-l are equivalent damping corresponding to current and previous area ratios,
respectively. At drift ratios of 0.67%, with the added area ratios from 0.15 to 0.30, from
0.30 to 0.45, and from 0.45 to 0.60, the percentage increase of equivalent damping for
RC walls is obtained to be 9.0%, 15.0%, and 38.0%, respectively. This suggests that the
rate of increase of the equivalent damping becomes more significant with the added
boundary area ratios with the observations that the largest rate obtained with the area
ratio changing from 0.45 to 0.60 while the least value of increase rate for area ratios
ranging from 0.15 to 0.30. A similar trend can be achieved for studied walls at a drift
ratio of 1.0% which the largest (21 %) and least (0.0%) increase rate of equivalent
damping obtained corresponds to boundary area ratios ranging from 0.45 to 0.60 and
0.15 to 0.30, respectively. It indicates that larger boundary area ratios have a more
significant effect on the equivalent damping.
5.3.3.4 Effect of Aspect Ratios
The effect of aspect ratios, H w/ Lw on wall structural performances is investigated for
four cases by changing the aspect ratio from 0.5 to 2.0 as listed in Table 5.24. The
tested RC wall, Specimen LWI with an aspect ratio of 1.125, is selected as a reference
wall in order to compare the analytical results with experimental ones in terms of
seismic performance. Different wall aspect ratios, H w/ Lw are achieved by varying
the wall height, H wand keeping the wall length, Lw same as that of the reference
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Cha/2.ter Five
wall. The reinforcement details of the walls studied are kept the same as that of the
reference wall.
5.3.3.4.1 Effect of aspect ratio on wall strength
Fig. 5.40 demonstrates the effect of wall aspect ratios on the backbone curves of
load-displacement loops and the wall strength capacity is observed to decrease
significantly with the augment of wall aspect ratios. As shown in the figure, the
percentage decrease of the analytical maximum strength is observed to be 47.7%,
63.3% and 70.7% with the increase of aspect ratios from 0.50 to 1.125, 1.625 and 2.0,
respectively. As shown in Fig. 5.40, a drift capacity of around 1.0% can be achieved for
walls studied other than for the wall with an aspect ratio of 0.5 for which a drift ratio of
0.5% is achieved.
5.3.3.4.2 Effect of aspect ratios on secant stiffness
For all RC walls with four different aspect ratios, the variation of the experimental and
analytical secant stiffness with the wall drift ratios is shown in Fig. 5.41. In general, the
experimental secant stiffness agrees well with the analytical ones with the observations
that wall secant stiffness degrades significantly with the increase of the wall drift ratios.
With the augment of wall drift ratios, the rate of decrease of the secant stiffness for RC
walls with lower aspect ratios is observed to be more significant than that of RC walls
with higher aspect ratios. For RC walls with an aspect ratio of 0.5, the secant stiffness
significantly reduces from 584.4 kN/mm to 136 kN/mm (around 300% degradation)
with the increase of wall drift ratios from 0.1 % to 0.5%. While for RC walls with
aspect ratios of 1.125, 1.625 and 2.0, this decrease rate is obtained to be 166%, 177%
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Chapter Five
and 180%, respectively.
5.3.3.4.3 Effect of aspect ratio on energy dissipation
Fig. 5.42 displays the variation of energy dissipation capacity with the drift ratios for
RC walls under four aspect ratios. In general, the energy dissipation capacity increases
with the added aspect ratios of RC walls studied at same drift ratios. However, up to a
drift ratio up to 0.5%, the increase of energy dissipated for all RC walls with aspect
ratios varying from 0.5 to 2.0 is observed to be insignificant, while for wall drift ratios
larger than 0.5% this increase of energy dissipated becomes obvious. With the augment
of aspect ratios from 1.125 to 2.0, the increase rate of dissipated energy is obtained to
be 11.6% corresponds to a wall drift ratio of 0.5% while this increase rate is 36.1 % for
RC walls at a drift ratio of 1.0%.
5.3.3.4.4 Effect of aspect ratio on equivalent damping
The effect of aspect ratios on equivalent damping of walls studied is shown in Fig. 5.43.
In general, the equivalent damping increases with higher aspect ratios of RC walls at
the same drift ratios. However, with an increase of wall drift ratios, the variation of
equivalent damping is observed to be different for all RC walls studied. At low drifts of
up to a ratio of 0.25% (prior to reinforcement yielding), analytical equivalent damping
for RC walls with four different aspect ratios, decreases significantly from maximum
(8.95%) to minimum equivalent damping (2.96%) with the drift ratios varying from
0.1 % to 0.25%. Whereas after that, the equivalent damping is observed to increase
significantly till the final stage for all of the RC walls studied. At an aspect ratio of
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Cha/2..ter Five
1.125, the percentage increase of equivalent damping is obtained to be 377%, while for
RC walls with an aspect ratio of 2.0, this increase rate is 402% with the augment of
drift ratios from 0.33% to 1.0%.
5.3.3.5 Effect of Construction Joints
The effect of construction joints on structural performances is investigated for both RC
structural walls with and without axial loads, and analytical results of these RC walls
with construction joints at the wall base are compared to those of RC walls without
construction joints. The poorly detailed construction joints are simulated by nonlinear
interface models as mentioned above and are incorporated to the base of the finite
element wall model. The tested specimen, Specimen LW1 with an aspect ratio of 1.125,
is selected as the reference wall. The overall dimensions and reinforcement details of
the studied walls with construction joints are kept same as the reference one.
5.3.3.5.1 Effect of construction joints on wall strength
The effect of construction joints on backbone curves of load-displacement relationships
of RC walls studied is presented in Fig. 5.44. It is shown from Fig. 5.44 that the
strength capacity is approximately the same, but strength degradation increases slightly
for specimens with construction joints at the base, although its ductility level remains
almost unchanged.
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Chapter Five
5.3.3.5.2 Effect of construction joints on secant stiffness
Fig. 5.45 shows the effect of construction joints on secant stiffness of RC structural
walls studied. The existence of construction joints has a minor effect on the secant
stiffness of RC walls studied since similar values of secant stiffness are observed for
walls with or without construction joints.
5.3.3.5.3 Effect of construction joints on energy dissipation
Fig. 5.46 demonstrates the effect of construction joints on energy dissipation of RC
walls studied. As indicated in Fig. 5.46, almost similar wall strain energy is dissipated
by walls with and without construction joints at a low drift ratio. However, with the
increase of wall drift ratios, it is observed that RC walls without construction joints
dissipate more strain energy than those with construction joints. This can be due to the
fact that with the presence of construction joints at the wall base, sliding deformation
along the construction joints may occur and thus the degree of the pinching of
load-displacement loops is increased which reduces the energy dissipation capacity of
RC walls.
5.3.3.5.4 Effect of construction joints on equivalent damping
The effect of construction joints on equivalent damping of walls studied is presented in
Fig. 5.47. In general similar analytical equivalent damping ratios are observed for RC
walls with or without construction joints. This indicates that the effect of construction
joints at the wall base on the wall equivalent damping is negligible.
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Chal2.ter Five
5.4 Conclusions
This study develops a nonlinear finite element procedure to explore the structural
performance of RC structural walls under cyclic loadings. Several conclusions derived
from this study can be made as follows:
1. After calibration against experimental results, the developed nonlinear finite
element procedure incorporating realistic material constitutive relations can
satisfactorily predict both global and local responses of tested RC structural
walls under cyclic loadings.
2. The investigations of the effect of four axial load ratios from 0.00 to 0.15 on
the structural performance of RC walls studied found that the presence of axial
loads significantly increases the wall strength but the rate of increase becomes
less when the RC walls subjected to a higher level of axial loads (from 0.10 to
0.15 etc). Moreover the contribution of axial load to the wall strength for RC
walls with a lower aspect ratio is found to be more significant than those with
higher aspect ratios. The secant stiffness of walls at the same drift ratio
increases with the added axial load, however, this effect reduces with the
increase of top drift. The presence of axial loads plays a beneficial effect on the
wall energy dissipation capacity at a high drift ratio, while at a low drift ratio
this effect is rather negligible. The equivalent damping of RC walls studied
decreases with the added axial load when the test progressed. At a low drift
ratio, a lower equivalent damping is obtained for RC walls under axial load
ratio of 0.05, whereas at a high drift ratio the equivalent damping is observed to
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Chapter Five
be higher than those of walls without axial load.
3. With the added longitudinal reinforcement content in boundary elements from
0.7% to 4.2%, the analytical maximum strength increases almost linearly and
the percentage increase of the maximum strength is observed to be 140%
around. The contribution of longitudinal reinforcement to the wall secant
stiffness is more effective for walls at higher drift ratios as the rate of increase
of secant stiffness increases more rapidly after the attainment of a wall drift
ratio of 0.33% for all four RC walls studied. At a high drift ratio (from 0.5% to
1.0%), the increase of longitudinal reinforcement in boundary elements have a
significant effect on the wall energy dissipation capacity as the rate of increase
of energy dissipation for RC walls at a drift ratio of 1.0% is obtained to be 51 %
With the added longitudinal reinforcement contents from 0.7% to 4.2%. While
at a low drift ratio (up to 0.33%) this effect is rather negligible as for wall drifts
of up to a ratio of 0.33%, the energy dissipation almost remains to be same. In
general, the equivalent damping for RC walls studied decreases with the added
longitudinal reinforcements.
4. The wall strength increases significantly with the increase of boundary column
area ratios, Ac / At from 0.15 to 0.60. Moreover, the trend of strength
increases is observed to be more significant as column area ratios tend to be
higher. With the change of area ratios from 0.15 to 0.30, the analytical
maximum strength increases relatively to be 0.11, while the increasing values
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Cha12.ter Five
are obtained to be 0.21 and 0.63 corresponding to the area ratios ranging from
0.30 to 0.45 and 0.45 to 0.60, respectively. The secant stiffness of RC walls
with same drift ratios increases with the added column area ratios from 0.15 to
0.60 as it increases by about 58% for RC walls at a drift ratio of 0.1 %.
However, the rate of increase of secant stiffness with the augment of column
area ratios, decreases more rapidly after a wall drift ratio of 0.10% for all four
studied RC walls. In general, the energy dissipation capacity and equivalent
damping increases with the added area ratios of boundary columns. However,
at a high area ratio, the increase of area ratios of boundary columns have a
significant effect on the wall energy dissipation capacity and equivalent
damping, while at low area ratios (from 0.15 to 0.30), this effect is rather
negligible.
5. The wall strength capacity is observed to decrease significantly with the
augment of wall aspect ratios as the percentage decrease of the analytical
maximum strength is observed to be 47.7%, 63.30/0 and 70.7% with the
increase of aspect ratios from 0.50 to 1.125, 1.625 and 2.0, respectively. With
the augment of wall drift ratios, the rate of decrease of the secant stiffness for
RC walls with lower aspect ratios is observed to be more significant than that
of RC walls with higher aspect ratios. In general, the energy dissipation
capacity and equivalent damping increases with higher aspect ratios of RC
walls studied at same drift ratios. However, up to a drift ratio up to 0.50/0, the
increase of energy dissipated for all RC walls with aspect ratios varying from
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Chapter Five
0.5 to 2.0 is observed to be insignificant, while for wall drift ratios larger than
0.5% this increase of energy dissipated becomes obvious.
6. The existence of construction joints has a minor effect on the secant stiffness of
RC walls studied since similar values of strength and secant stiffness are
observed for walls with or without construction joints. Almost similar wall
energy is dissipated by walls with or without construction joints at a low drift
ratio. However, with the increase of wall drift ratios, walls without construction
joints dissipate more strain energy than those with construction joints. Similar
analytical equivalent damping ratios are obtained for walls without or with
construction joints.
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Cha12.ter Five
Table 5.3 - Comparisons of finite element predictions to test results
Tested Predicted Tested/Predicted
Specimen Fp,f(kN) ~ f(mm) Fp,p (kN) ~ (mm) F IF ~ I~p, p,p p,f p,p p,t p,p
(1) (2) (3) (4) (5) (6) (7)
Unit 1.0 483.2 17.75 504.6 18.50 0.96 0.96
S-Fl 393.3 20.19 355.0 20.00 1.11 1.01
32
0.058
16
0.030
8
0.025
Table 5.2 - Material properties and reinforcement ratio for
Unit 1.0 and Specimen S-Fl
- 216-
Table 5.1 - Coefficients for detennination of the fracture energy
dmax (mm)
Gfo(N· mml mm 2)
Properties Unit 1.0 Specimen S-Fl
Cross section shape Barbell Barbell
Concrete compressive strength (N/mm2) 25.0 35.0
Concrete fracture energy (Nmm/mm2) 0.051 0.072
Concrete compressive fracture (Nmm/mm2) 7.65 10.8
Yield strength (N/mm2) Boundary element 492.0 385.0
Vertical web 298.0 492.0
Horizontal web 298.0 492.0
Reinforcement ratio, p (%) at Boundary (Long.) 3.00 1.30
Vertical web 0.16 0.26
Horizontal web 0.16 0.24
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Chapter Five
Table 5.4 - Concrete material properties used for finite element analysis
~ LW1 LW2 LW3 LW4 LW5 MW1 MW2 MW3Properties
Ie· (N / mm 2) 40.2 41.6 34.8 39.8 35.6 41.2 39.6 40.3
~ (N/mm 2) 3.36 3.42 3.13 3.34 3.16 3.40 3.33 3.36
E c (N/mm 2) 34185 34578 32581 34072 32829 34467 34015 34214
Gf (N·mm/mm 2) 0.083 0.086 0.071 0.082 0.073 0.085 0.081 0.083
Gc (N· mm/mm 2) 12.4 12.8 10.7 12.3 10.9 12.7 12.2 12.4
Table 5.5 - Parameters investigated
No. Name Description Range Investigated
1 N/(fc'Ag ) Axial load ratio 0.00,0.05,0.10,0.15
2 Ph (%)Longitudinal reinforcement content
0.70, 1.40, 2.80, 4.20in boundary element
3 Ac / AtRatio of the sectional area of columns
0.15,0.30, 0.45, 0.60to the total sectional area
4 hw/lw Aspect ratio 0.5, 1.125, 1.625, 2.0
5 CJ Construction joints Cold Joints, No joints
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Cha12.ter Five
- 218 -
"for''''nn.ult
=--= • $~rn
Gfcr = 5.136 h I'
cnn.ult • J t
Strain
Secant unload/reload
1V(IT~.NtM
~~~~~Tl~'
Tt4ntP
Tension softening
{'r(J'~l1f
Hordijk
Ll/CwpPJ
-~-pP TU(IIP)
II
II
II
II
I Gc/hI
Ca.)
Lv1n_~f"
vr;
Unload/reload
Fig. 5.1 - Concrete in compression and tension
Parabolic compression model I fC
~T T!L ~v
'h_ . -;r~VT
(b)
Fig. 5.2 - Shear friction hypothesis [D1]
Compressionsoftening
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Fig. 5.3 - Coulomb friction criterion
I JI ,
I I
150 2200 150-
l IFig. 5.5 - The overall dimensions andreinforcement layout of Unit 1.0 [M1]
Chapter Five
Fig. 5.4 - Aggregate interlock relation [WI]
fI I I HI
2(+)I 11 1i i i-
t ,\, t jj \ 1 ~ \ 1
"T
,I i \ \ ~\\ II, t \\ \\ ';,
i ~- T '\1, '1, \ .\, t \
1 II ,\... I, -~~~1\ 1 .. 'I"~, \,,1 1 1_-+:--\'1'1 \ \ \ i \ I ; U
I" iI,,'" iii, 1
\ \:\ \, \ ,,l} __
\\ ,t ~/
I i I!
I :j
I i I
i
!
I I
,
Fig. 5.7 - Finite element idealization anddeformed shapes of Unit 1.0
at a drift ratio of 1.0%
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Chae.ter Five
Fig. 5.6 - The overall dimensions and
reinforcement layout ofSpecimen S-FI [W2]
~)
Fig. 5.8 - Finite element idealization anddeformed shapes of Specimen S-F 1 at a
drift ratio of 1.0%
Qon~o
.. 1>-
17M]12
20H8~ '7, I?1Q 1 'SO 1 <in I ',n 1 250 I W 181,2Q
7110
600
500
400
300
~ 200
"--' 100"'0~
.3 0
~ -100I-<ll)
~ -200~
-300
-400
-500
-600
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
Drift ratio (%)
Fig. 5.9 - Experimental and analytical hysteretic responses of Unit 1.0
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Chapter Five
40302010
.~~~':~:~'~':'~'-'~'~~ - - - - - ~ - - - --I I II I I
-----~----~-----~-----I
I -- Experimental-----./.
I ••••• - • AnalyticalI I I
_____ L ~ L _
I I II I I
o-10-20-30
II I I
----'-----r----T----I I II I I
----~-----~----~----
I I II I I
----'-----r----T----I I II I I
----: -----~~ ~ .~ .--.~·:7 .~ ~: :~ -~ ,
500
400
300
200
~ 100
""0~ 0.9-;; -100l-<(l)
~ -200...J
-300
-400
-500
-40
Displacement (mm)
Fig. 5.10 - Experimental and analytical hysteretic responses of Specimen S-F1
(a) Specimens LWI & LW2 (b) Specimen LW3 (c) Specimens LW4 & LW5
(d) Specimen MW1 (e) Specimen MW2 (f) Specimen MW3
Fig. 5.11 - Finite element idealization for all specimens tested
- 221 -
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Chall.ter Five
10 15 20 25 30
10 15 20 25
~"""I'"1'T"~'~-- ~ -- ~ -- ~ --
• 1 1 1 I
--1---1---1---1---1 I 1 11 "
- -,- - -Experimental -
1 ••• ··.·Analytical--r--, I I
1 1
I I 1
==~ -=~ ==i= ~ ~ -=i= =-K_P"I£VI.'/:./'j - - ~ - - ~ -~ Experimental __
- - - , - - r . ""'.Analytical1 _!... -- I __
I "I 1
-20 -15 -10 -5
--:-n-~ -~ :(-) -:----:- D -:----1- -1--
1 1--1- , , -,-
__ 1_ Specimen LW2 _1__I I 1
__1__ ..J __ L
1 1
--~nlP(-)--~: DC=> 1--__ 1i --i--- - ~Specimen LW4- - ~ - -
--: --: --_L~j --
700
600
500
400
300
200
100
Displacement (mm)
(b)
-100
·200
·300
-400
-500
-600
-700 I-25 -20 -15 ·10 -5
~~~
..9""§
<l)
~..J
10 15 20 25
I 1 1 1- - -1- - -1- - -1- - -1- - -
I 1 1 I
- - -:- - --Experimental'-
1 ·······Analytical- - -1- - -1- - -1- - -1- - -
1 1 I 1
, 1
--~111C:(.l __ :__
1 1
1 _ -1--
- -i D __:__ -hl/i"~'" 41C .C
- - ~Specimen L~1 1
- :-- :- -:~,-100
500
300
200
400
-400
-500
-25 -20 -15 -10 -5
-200
-100
-300
Displacement (mm)
(a)
~~~
..9""§
<l)
~..J
700 500
600 --:-n'-:-- I 1 1- - 1- - -I - - I - -
400500 __ 1- _1__ ;······;l..:.·..:.··.:]".:.. _.1 __
==!: D =t:1 1 1
300400 1---17"-+--
~1 1 I
~300 1- -.' - - -t - - 200
200 __ :_ Spe~imen ~W3.:__ 1 I I-1---1--'--
~ __ 1__ .J __ - _1__ ·'1 1 1 ~100
~100
1 1 ·r -~ 1- - -I - -1- -~
..9 0 B~
.1' I 1 1
~-100 .:-t.-"':" +- - -1- - ..., - - -+ - - -100~ 1 I 1 1 1 ~<l) -200 <l)
~--,--,--,-...,--,--
~1 1 1 1 I -200..J -300 - - "I - - 'I - - 1- - -, - - I - - ..J
-400__ J __ L __ I__ .J __ .1 __
-3001 1 _ Experimental
-500 ---+--+--, 1 .. ····Analytical -400
-600 ---r--r---,-,---t---700
I 1 I 1 1 -500
-25 -20 -15 -10 -5 0 5 10 15 20 25 30 -25
Displacement (mm) Displacement (mm)
(c) (d)
10 15 20 25 30 35
..=:r .. ~ ... :. 1 1 1- -I - -1- - 1- - 'I - 1- -I - -
1 I 1 1 1 1- -I - -1- - 1- - T - I - -I - -
1 1 1 1 1 1- -I - -1- - 1- - T - , - -, - -
-"':1 - -:- .-Experimental._1 1 ·-··· .. Analytical
- ..., - -1- -, - , - , - ..., - -
1 1 1 1 1 1
~-~.~:+:I 1 1 1 1 _ 1
- , - .., - -I - - 1- - r T
1 1 1 I I 1
-~-l!';1;-, -~-~--~- D-~-~-
I 1 I
--l-- -"--+-1 1 1
-+-- - ... -+-1 Specimen MW1 I 1_.L L_l._
1 1 I 1•.... 1.·····(.·...I I j 'f b ';#;
200
500
300
600
400
100
-500
-600
-35 -30 ·25 -20 -15 -10 -5
-100
-200
-300
-400
~~~
..9~t~..J
10 15 20 25 30
• ~ 1 1 1.·4'.c;·':" r- - - r- - -I - - -r - -
1 1 1 1 1- - , - - , - - 1- - -I - - , - -
I 1 1 1 I- - "I - - 'I - - 1- - -I - - I" - -__ J __ L __ L __I __ J __
1 1 - Experimental---+--+-- .
1 1 ·······Analytical- - .., - - , - - 1- - -I - - , - -
1 1 1 I 1
-20 -15 -10 -5
n-~-~~(·)-:--
__ 1_ _ L_
==i: D =t =~- ~"H"/""
700 I600
500
400
300
200
100
-100
-200
-300
-400
-500
-600
-700
-25
~~~
..9~t~..J
Displacement (mm) Displacement (mm)
(e) (0
- 222-
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Chapter Five
I J.• ··I I I I I I"'::'~"':':f-- -1- -1- -1- -! - -+ - +-
~ I I I I I I I I: -I--I--I--I--I-~--+-+-
I I I I I I I I- I- - I- -1- -1- -1- ~ - -+ - +-_
I I I_ I- _I- _1 __ -Experirrenlal _
I I I ··· .. ··Arnlylical- I- - I- -1- -1- -1- -' _ -+ _ -l- _
I I I I I I I I
600I I I
500 -1- -I -,-I I I
400 -1- -I -,-I 1 I
300 -1- -I -,-I I I
200 -I- -I -,-Specimen MW3 I
100 - 1- -I - -I - -, -
-100
·200
·300
-600 '---'----'----'_-'-----'----'-_'----'----'----J._-'-----'---'-_'-----'--_
-35 -30 -25 -;m -15 -10 -5 0 5 10 15 20 25 10 15 40 45
-500
-400
10 1 5 20 25 30 35
.•.•.•.•• ~... I I I I- - 1- - r- - r- - i - i - 4 -
I I I I I IR?'4,Wlf-A':":':'-i- -1- - r - r - i - T - 4 -
I I I I I I- - 1- - r - r - i - i - 4 -
__ :__ ~ _ - Experimental _
I I ·· .. · .. Analytical- - 1- - r - r - i - T - 4 -
I I I I I I
=tl=l![1~.) ~--:-, ,- -I-
I I- r- - ,- -I-
I I I- r- -" ,- -I-
I Specimen MW2 I I
- ~ -: : :., ··r·~I~.:
400
300
500
200
100
-400
-200
-100
·600 '--~~~~~-~~-.l--~-~~~-~~-----'
-35 -30 -25 ·20 ·15 -10 -5
·300
·500
Displacement (mm) Displacement (mm)
(g) (h)
Fig. 5.12 - Experimental and analytical hysteretic responses of all specimens tested
I.." /: •.• 0
......~~
----. .. ,).....".. )-
Specimen LW1
!J[_.-
•.~.. _... ········o
200 400 600 800 1000 1200 1400 1600 1800 2000
Wall width marked from the left (mm)
....... DIANA 0.1% drift
-- DIANA 0.25% drift....... DIANA 0.5% drift
-.-.- DIANA 1.0% drift
Test 0.1% drift
Test a.25% drift
Test 0.5% drift
Test 1.0% drift
0.01
0.008
0.006
0.002
-0.002
c.~ 0.004Ci5
. 0Specimen LW1.~.,
1 .
. _L··~·..;,.-..;.~:.::·i···o· •
.••.... DIANA 0.1% drift
-- DIANA 0.25% drift....... DIANA 0.5% drift
-.-.- DIANA 1.0% drift
Test 0.1% drift
Test 0.25% drift
Test 0.5% drift
Test 1.0% drift
.".~:;<'
."
0.01
0.006
0.008
0.002
·.~:;:iOr40~ 600 800 1000 1200 1400 1600 1800 2000
-0.002 -.,:. . Wall width marked from the left (mm)
c.~ 0.004Ci5
(a) Section 1-1 of Specimen LW 1 (b) Section 2-2 of Specimen LWI
- --- DIANA 0.1% drift £1'0.02 --- DIANA 0.25% drift....... DIANA 0.5% drift
-·---DIANA 1.0% drift0.015 Test 0.1 % drift /
Test 0.25% drift • SpeCiitle.n ~W2Test 0.5% drift \
,f.0.01 • Test 1.0% drift \c
\.§,I
C;; • \
0.005 ..Wall width marked from the left (mm)
200 400 600 800 1000 1200 1400 160018002000
-0.005
0.02 -- -- DIANA 0.1% drift
--DIANA 0.25% drift £1-'0.015 ... _._. DIANA 0.5% drift
-·-·-DIANA 1.0% drift
)l: Test 0.1% drift Specimen LW20.01c Test 0.25% drift.~
Test 0.5% driftU5 it.1
0.005 it. Test 1.0% drift
c_-~_·.:::::~-.----:--.o 0•••••• '0'o.------~-~
1000 1200 14001600 1800 2000
Wall width marked from the left (mm)..
- _I-0.005
(c) Section 1-1 of Specimen LW2 (d) Section 2-2 of Specimen LW2
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Chae.ter Five
0.02 ---- DIANA 0.1% drift
D[0.02., -- - - DIANA 0.1% drift
JD[--DIANA 0.25% drift
--DIANA 0.25% drift ....... DIANA 0.5% drift....... DIANA 0.5% drift
0.015 -·_·-DIANA 1.0% drift0.015 -I -·---DIANA 1.0% drift
::K Test 0.1 % drift ::K Test 0.1% drift
Test 0.25% drift • Test 0.25% driftTest 0.5% drift Specimen MW1
0.01 0 Test 0.5% drift Specimen MW1 .001 ] ~ Test 1.0% drift "~ " c
~ Test 1.0% drift ;' ~ ~
J0.005
- . - - -'-._._.-._-.
~.~
/
loA.~ en / ~
;' 0.005 ;'
/
.r> ......k:::~.-::~··~~····· ······0··0.·(;)/
_.. ~ ,~""';~"':""',"'~'";~i 0. P2"O~>400 600 800 1000 1200 1400 1600 1800 2000 " ":£20 400 600 800 1000 1200 1400 1600 1800 2000~/
-0.005 ~ Wall width marked from the left (mm)-D.005 Wall width marked from the left (mm)
(e) Section 1-1 of Specimen MW 1 (f) Section 2-2 of Specimen MWI
Fig. 5.13 - Verification of longitudinal strain distribution along wall length
~'1
R8
I
R7
"~~ ",.. -I,
R6R5
o Test 0.50% drift
oA. Test 1.0% drift
+ Test 0.25% drift
)( Test 0.10% drift
R8
Gauge Location
Specimen LW1 II'~-'-R Bar
....... DIANA 0.1 % drift"='IIt&,,.,~ _
--- DIANA 0.25% drift,
.-- ... - DIANA 0.50% drift -...... I.-t.. _. - DIANA 1.0% drift .
0002~ I0.0018 P<~
~~~:: n0.0012
.£ 0.001~en 0.0008
0.0006
~:~~: t tL~-~~~·L~0
-0.0002R5 R6 R7
(a) R bar in Specimen LWI
0.002 --I I
0.0018 f--Rp~ Specimen LW1 lI-'-T Bar0.0016 f-- .....-
TL~ TI" ........ "13 1'1"
1'''' Tl~ 0 d 'ft '....... 1'L<4 TL:!l0.0014 f-- /' ._. __ '-'.'''''' DIANA 0.1 Yo n ,
0.0012 ;' ;' • k---- DIANA 0.25% drift --.:......,---'=1r-r----t---------t-
.£ 0.001 I' •..••. DIANA 0.50% drift .
~ /,.' /,// .... ·_·-DIANA 1.0% drift "'-'''- '
U5 0.0008 '/', H' ~1/// ":l( Test 0.10% drift "_~,,'. \
0.0006 />:.J./'---/ + Test 0.25% drift ',", ".\. I /0.0004/./:::,:; 0 Test 0.50% drift '., . \
0.0002 .7 ~ Test 1.0% drift --"'" '.,·t....:;-.~~-.:-:::-:::-=.~.o . , , "., - T "
-0.0002T13 T14 T15 T16 T13
Gauge LocationT14 T15 T16
(b) T bar in Specimen LW1
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Chapter Five
R8R7R6R5
)I( Test 0.10% drift
Test 1.0% drift
• Test 0.25% drift
o Test 0.50% drift
..•.... DIANA 0.50% drift
-·_·-DIANA 1.0% drift
R8
Gauge Location
'.
---4---+--------+---- Specime n LW2 --+------+--ll'~-)R Bar -
. .. - -. - DIANA 0.1% drift -----+--- -1>5.'.7... _
.. DIANA 0.25% drift
0.002
~'n0.0018
0.0016
: ~.,,07 ••0.0014
0.0012
c 0.001.~ / ,U5 0.0008
0.0006
0.0004/// "
'0,
0.0002~,
0 ....... __ ..............
-0.0002R5 R6 R7
(c) R bar in Specimen LW2
I
"-._-_.- DIANA 0.1% drift
. ". DIANA 0.25% dri« '.....
"-"-
...... DIANA 0.50% drift
~·_·-DIANA 1.0% drift t
)I( Test 0.10% drift ~'.
• Test 0.25% drift '''.....
,/ .0 Test 0.50% drift
./ • Test 1.0% drift,/
T13 T14 T15 T16 T13
Gauge LocationT14 T15 T16
(d) T bar in Specimen LW2
Specimen MW1 II'~-) -R Bar
--·····DIANAO.1%drift -
........... DIANA 0.25% drift -----+--- "" •••7 •• ~
..•.... DIANA 0.50% drift
-·_·-DIANA 1.0% drift
• Test 1.0% drift ~.....
R8R7R6
~_.-
R5
Test 0.50% drift
• Test 0.25% drift
)I( Test 0.10% drift
R8
Gauge Location
_::...~--
0.002
~n0.0018
0.0016
== ~~.. ~0.0014
0.0012
c: 0.001
~ 0.0008
0.0006
0.0004
0.0002
0 ~':-':-::'=-:-:-=-"-'-'-"~ ..-_ ..~ .-
-0.0002R5 R6 R7
(e) R bar in Specimen MWI
- 225-
.......-------------------------------------------_...
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Cha/l.ter Five
'.
T16T15T14
~n. ns
I I~-)I
T16 T13
Gauge Locati on
(f) T bar in Specimen MW 1
--.------.7"... ---.. .... Test 1.0% drift
T14 T15T13
o~~:~: ~pn~----!.-----!--- Specimen MW1 ---+-
0.0016 '-- I I I T Bar
0.0014 '-- TlJ TIO1I4Tl':> -----...-----1;.....__,---- ······DIANAO.1%drift 1-
0.0012 l-- " "'. /------- DIANA 0.25% dnft -C L---J~ ~t-----="'~---
.~ 0.001 . . ' , , ...... DIANA 0.50% dnft "f ~ ", [t) 0.0008 I . I- . - . - DIANA 1.0% drift /
0.0006 f " .;K Test 0.10% drift• Test 0.25% drift
0.0004 -r-----t-----;:::---"''------+---+-----.-. 0 Test 0.50% drift
0.0002 . ..'
,0000: ' . . .. .. '" I .. 0 ... 0 ... o. of I'::-.-·~'·::~::~ ..l l'C;:'. "T
Fig. 5.14 - Verification of horizontal strain distribution in selected horizontal bars
- 226-
Fig. 5.15 - Representation of the secant stiffness, energy dissipation capacityand equivalent damping factors
~m ~
CI h =_1_. Ah
eq 21l Fm~m
Ah = Area(ABCDEF)D
K _Fm
Aeq ---
~m F
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
Chapter Five
o. - - .• 0.•••••••••••• £-.0 ..
(a) RC walls with an
aspect ratio of 1.125
252015
Displacement (mm)
Drift ratios
0.5% 1.00%
..........
Specimen LW 1 tested
- ••••• - •• c" "~"~'~."" •
10
----*- Specimen LW2 tested
. -.•. _. Axial load ratio: 0.00
.. 'II
5
0.25% .;xXj!--- oX'
.xRxx
800
1000
-5
..'
-10-15-20
Axial load ratio=~j~ Ag
! ! P(-)
n-25
Drift ratios
1.00% 0.50%
xX~x
?' -800XX 0.25%
-1000
. ..•... Axial load ratio: 0.05
----+- Axial load ratio: 0.10
.. ,x", Axial load ratio: 0.15
. 1 d . N800
Axial oa ratIO - f' A
~ 0.25 0.5% 1.00%( g
n- "'t:l600
.xxDrift ratiosro x-
.Q .X
~x
~ro400 .................:l
.-', •. -- .. iI· _.. -. _.......•
'.. '
(b) RC walls with anDisplacetrent (rrnn)aspect ratio of 1.625
-35 -30 -25 -20 -15 -10 5 10 15 20 25 30 35.. ' .'•...... .......~- .... "~. ...•. -- Axial load ratio: 0.00
-..- Specimen MWI testedA'---oo
." "•... Axialload ratio: 0.05
x --+- Axial load ratio: 0.10Drift ratios x·x -600
XX .. ·x-· - Axial load ratio: 0.15
1.00% 0.25%
-800
Fig. 5.16 - Effect of axial load on backbone curves of load-displacement loops
- 227-
............-_--------------------------~
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Chal2.ter Five
0.150.100.05
~ RC walls with an aspect ratio ,,'"of 1.125 ,,,"',,""
--- RC waDs with an aspect ratio" - - - - - - - - - - - -_~:-~;-~"181 N
- ---;~:~~wtth a~ aspectratio~_~of1.125 (Expernnenta~ ,?,,"',,'" ~
..~:- - - - - - - - -I - - - - - - - - - - - - - - - - - - - -
III1
I 1- - - - -1- - - - - - - - - - - - - - - - - - - - -I - - - - - - - - - - - - - - - - - - - -
I 1
1
1
IIo ...... .., ..,
0.00
5
4
~~
"'-""1 3~1:;
~6 ~
I I"~~~ 21:;
~~.
"--
Axial load ratio, N /(Ie' Ag
)
Fig. 5.17 - Contribution of axial load ratio to the wall strength
- 228-
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Chapter Five
320
280
240
S 200
~
2 160rf.Jrf.J(])
~ 120~
rf.J
§ 80Q(])
en40
,-- - - - - ,-- - - - - 1- - - - - 1- - - - - 1- - - - -1- - - - - - - - I - - - - I - - - - I - - - - I - - - - I - - - - l
1 I I I I 1 r q I I
: Axialload ratio = ---.!/-: : \: ---)I(- - - LWI tested :
:c - - - -:- -1- -:- -C~~:-;-t ----:-t---~- ~_. ---Axial had ratio: 0.00 :
i---JI I[ - --:-- -- -,-It--- ~X -- =~: ::::tio: 0.05 1
L_ -- --_1- ---_I{_ - ---~ \J _. ··-e-· - Axial had ratio: 0.10 .
l .-_L __ --1--- - -1- - - - _']-- :; -- - - juk: "El __ .~xiaIIo~r~~:"0.15_:: (a) RC walls with an: :" ',~:::::1 1 r I' I I 1 1 1
~ __ aspect ratio of1.125 __:__ &'~:__ ~ ~,-~~ ~ ~ ~ ~ ~1 I I I 1 I' . I' 1 I 1 1 I
: : : : :.0: i ): .'1 : : : : :I 1 1 1 j' I" I', 1 I 1 1
~ ----~ ----:- ----:- --.~O-:- - .~~::.' \~ ..~ _~ Q.~ __ ~ ~ ~ ~
1 1 1 1 I): A, 1 1 1 11 I 1 1 I' I'" I 1 1 11 I 1 I 1 _ 1 1 I~ - - - - ,-- - ~.~ ~~~ =-,~ - - - - ~ - - - 1 :- ,- - - I .- - ~ - - - - I - - - - ~
1 - I 1 I
I 1 1 1o I 1 1 1
-1.2 -1 -0.8 -0.6 -0.4 -0,2 0 0.2 0.4 0.6 0.8 1.2
Drift ratio (%)
1.20.80.60.40.2
- - - - 1- - - - - 1- - - - - 1- - - - -1- - - - - 1- - - - -I
I I 1 1 1 1I 1
: ---+- Axialload ratio: 0.00 :
-1-:-_ .0 -.- o. MW1 tested :• 1 1
: ------ Axialload ratio: 0.05 :1 1
I 1
I - 0 -e··· Axial load ratio: 0.10 II
-I
-e- Axial load ratio: 0.15 :1
1 1
1 1
1 1
1 I- - - - - - - - - - - - - - - -1- - - --I
I 1I 1
I 1
1 1
I'· 1 1
1 _ 1 ~ 1 1 1 1----~. ----1-----1-----1-----1
1 10
1 I 1 I
I I 1 1 1I __ .1 1 1
I 1I 1I 11 1
o-0.2-0.4-1 -0.8 -0.6
o-1.2
90
30
60
150
120
Drift ratio (0/0)
Fig. 5.18 - Effect of axial load on secant stiffness of walls studied
- 229-
I
I
III&... I!
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300 ,-,------------,--------------,------------,
0.150.05 0.10
Axial load ratio, N /( fc.· Ag
)
--+-- RC walls with an aspect ratio of 1.125 (Drift :ratio: 0.1%) :
----- RC walls with an as pect ratio of 1.625 (Drift I
ratio: 0.1%) : ~1~~- - -.- - . RC walls with an as pect ratio of 1.125 (Drift :
ratio: 0.33%) :- - -)K- - - RC walls with an aspect ratio of 1.625 (Drift :
ratio: 0.33%) ,.~ RC walls with an as pect ratio of 1.125 (Drift - - - - - - - -:- - - -
ratio: 1.00,10) 186%--e--- RC walls with an aspect ratio of 1.625 (Drift
ratio: 1.00,10)
- - - - - - - - - - - - - - - - - - - - :- - - - -~ 130.o/.ik: :: .....
,~/ .'6;
___________________~ 0 :.~ c:;;;~:~ ~ii;_~ _: 99~,,:::,·" :,~1J~~" :
'64% :- -.- 60% - - - - - - - - - - - - - - - -:- - - - - - - - - - - - - - - - - - - -
57% :,,
I
o F :
0.00
50
250
~ 200'-"
Cha12.ter Five
~
6
~
~ 150~
6
~I~.
~- 100'--
Fig. 5.19 - Contribution of axial load ratio to the wall secant stiffness
- 230-
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Chapter Five
16 --,--------r------r-------,-------,-------,--------,
~1
1- - -1- - - - - - - -
1
(a) RC walls with anaspect ratio of 1.125
1
I1
I1
I• 1 I
r,~' - - - - - -t - - - - - - - -1- - - - - - - -
I II 1.e·
-+- LW1 tested
.. -.- -- Axial load ratio: 0.00
--.- LW2 tested
•• -E)'" Axial load ratio: 0.05
---Q-- Axial load ratio: 0.1 0
••• El· •• Axial load ratio: 0.15
II
IG
:11
1
II I I
- ... - - - - - - - ...j. - - - -.:... - - -1- - - _I I' II I 1I II II 1I I:I tI :1I : I
I .' 1- I" - - - - -.- - I" - -1
II 1
: : .0I I '1 I.'I ,1
1 1 ,- - - - - - - -t - - - - - - - -I - - - - - - ~
I 1I 11 I1
III
8
4
O+--~--+---=----+----+------+-----+-------1
12
o 0.2 0.4 0.6 0.8 1.2
Drift ratio (%)
16 -,-----------,-----,-----------,----~-----,--------,
" I~ Axial load ratio: 0.10
--.- MW1 tested
...•-. - Axial load ratio: 0.00
.. -+- - - Axial load ratio: 0.15
IIo
:1III1
I I. I- - -E>- - - Axial load ratio: 0.05 - ~ - - - - - - - - ~ - - " - - - - I - - - - - - - -
1 I ,~I I·I I,'1 ,1 I
I • I II ,. 1 I
I I II I I. I I
-------I--------r-------I----~--- -----1--------
: : : ,/ :I I I 1I I I II I I : II I' I
I I :1. II 1 : I II I • I I I
- - - - - - - 1 - - - - - - - - r - - - -.- - .'1 - - - - - - - - r - - - - - - - I - - - - - - - -
1 I.' I 1 I
: I ,0,": (b) RC walls with anI I .••••• I
I I , • I aspect ratio of 1.625I .'.1. I
I I1 I
o +----.JI~====-:..,_----__rl----_Ir----_,-- -,-- -1
12
8's~;>., 8OJ)lo-;(I)~(I)
~.~
tlr.rJ
4
o 0.2 0.4 0.6 0.8 1.2
Drift ratio (%)
Fig. 5.20 - Effect of axial load on energy dissipation of walls studied
- 231 -
III
I
.... ............11
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Chal2.ter Five
16 -1r
1.00% ----+- Drift ratio: 0.10% (Aspect ratio: 1.125)31~Q.'--· ....... Drift ratio: 0.10% (Aspect ratio: 1.625)
--.- Drift ratio: 0.17% (Aspect ratio: 1.125)12
•• - , •• 0 Drift ratio: 0.17% (Aspect ratio: 1.625)56% ----+- Drift ratio: 0.25% (Aspect ratio: 1.125)
S _. -)1(" - Drift ratio: 0.25% (Aspect ratio: 1.625)
~ ~ Drift ratio: 0.33% (Aspect ratio: 1.125)g
8.. ·e··· Drift ratio: 0.33% (Aspect ratio: 1.625)
>-.. -----G-- Drift ratio: 0.50% (Aspect ratio: 1.125)OJ) 0.67%~ 39% .... ···-.- .. ~ .
• 0 • E)••• Drift ratio: 0.50% (Aspect ratio: 1.625)0)
~ ............0) .-' -.- Drift ratio: 0.67% (Aspect ratio: 1.125)~ .. ,
..........;
...•--- Drift ratio: 0.67% (Aspect ratio: 1.625)~ 37%ri5 4 --- Drift ratio: 1.00% (Aspect ratio: 1.125)13% .. -•... Drift ratio: 1.00% (Aspect ratio: 1.625)
0.50%$..... ___ .. _~.-_o--
---
---
---
0.150.10
Axial load ratio, N /( fe' Ag )
0.05
21%
o-,:::: :::: ::::: :::::::: ::::: :::::::: :::f :::: ::::: ::: ::::: ::::: ::: ::::: :... --i-'" -- '.. ", -- -- ... "g - _ri-: .,
0.00
Fig. 5.21 - Contribution of axial load ratio to the energy dissipation
20 .,----,----,------,.--------,-----,------,
II---1-------1
1
1
II1
~LWI tested
- -.- - Axial load ratio: 0.00-...-LW2 tested• - -E)o 0 0 Axial load ratio: 0.05
---+- Axial load ratio: 0.10- - .E]••• Axial load ratio: 0.15
.' 1
I. 1
- - - -: -; £ - - -;'" - -:-
'I
1 ,0'1
I•
1
1
: ~ !(a) RC walls with an : I,; ,/, " 1
aspect ratio 0 f 1.125 -- - - - - -: - - - - - - - -:- - -" - .. - - - ~ - - - - - - -1 1
1 1
I 1
II1
1
1 1-------1----------1
1
4
8
II II
____' 1
l T-:~ ~Iv ......-- 1 ,
\ : :'".. I / ~I
.. I • I •
" • - :.:-,;'- .- ;II'- ~.~ .~ -
~.:" ..,~.:
•.• ' 0" - 1
o I [3' I' I1 II
16
12
-..~OJ)
.6
S.gE0)
c;>.;0~
o 0.2 0.4 0.6 0.8 1.2
Drift ratio (%)
- 232-
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Chapter Five
1.20.8
••• E) ••• Axial load ratio: 0.05
...•... Axial load ratio: 0.00
....... Axialload ratio: 0.15
0.6
Drift ratio (%)
0.40.2
1
I
•1
1 I , 1
- - - - - (b) RC walls with an - - - - -: - - - - - - - -:- - - - ~.:.- - - ~ - - - - - - -
aspect ratio of 1.625 : : ~.' ~1 I' I
1 1 1 I,' 1- - - - - - - .j... - - - - - - - --I - - - - - - - -I - - - - - - 1- - - ;- ::.. - - - .j... - - - - - - -
1 I 1 1 11 I I 1
1 I I. II 1 1 ' I
I 1 I. 1-------t ------- ~ -------.,-;~ +1 I 1 '
1 II ,'I
I • 1
---a-- 1 , • 1
~ - - - - - - - ~ -.:~ - -:-,- - -:- ---.- MWI tested1 I, 11 "I 11 ,,' I I
- - - - - -. ~ ~. -'~ ~~ :-. ~ ~ - - - - - - -:-I • ' 1 1 -.- Axial10ad ratio: 0.10
G. 1 L',,' .-0 1 1
"0'~"~ 1 1
~•••••• 1 1••..•• f·· ••••• 1 I
24
20
~16
~bl)
.S
S 12~
"'0
"E(])
~ 8:>
'S0'"~
4
0
0
Fig. 5.22 - Effect of axial load on equivalent damping of walls studied
0.150.10.05
Axial load ratio, N /( fe' Ag )
::;:: :::::: .. :-::.:- ~ ~ .._-,-:.:" ::::;:::: .. :: .
22%
1.00%1.4%
!~.%
0.67%
..~~%
35%0.50%
----+- Drift ratio: 0.10% (Aspect ratio: 1.125)...•... Drift ratio: 0.1 0% (Aspect ratio: 1.625)
-.- Drift ratio: 0.17% (Aspect ratio: 1.125)_..•... Drift ratio: 0.17% (Aspect ratio: 1.625)
~ Drift ratio: 0.25% (Aspect ratio: 1.125)•• ')1('" Drift ratio: 0.25% (Aspect ratio: 1.625)
---{}- Drift ratio: 0.33% (Aspect ratio: 1.125).. ·e··· Drift ratio: 0.33% (Aspect ratio: 1.625)
--e- Drift ratio: 0.50% (Aspect ratio: 1.125)_. 'E!••• Drift ratio: 0.50% (Aspect ratio: 1.625)
---- Drift ratio: 0.67% (Aspect ratio: 1.125)...•-.. Drift ratio: 0.67% (Aspect ratio: 1.625)
------- Drift ratio: 1.00% (Aspect ratio: 1.125)_..•... Drift ratio: 1.00% (Aspect ratio: 1.625)
24
20
.--....C 16bl)~
'0..8 12~
"'0
~(])
~:> 8'30'"~
4
0
0
Fig. 5.23 - Contribution of axial load ratio to the equivalent damping
- 233 -
II:
IIIIL. ~:J.
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
Chap..ter Five
800 '1r---r------.------.........-
0.25%
25
1.00%
20
Drift ratios
15
Displacement (mm)
0.5%
10
.o .....• O·····-e--·-----------o
5
Longitudinal reinforcement content inbOlmdary elements
-----*- 0.7%
··· .. ··1.4%
--- 1.4% (tested)
--·e--· 2.8%
-...-4.2%
0.25%
400
600
-ROO
-400
-600
0'
-5
0'
~""g..9S~ro....:l
-10-15-20
PbJ!(-)
_.~......•.... _..-.. -
o·0-· - - - - - -······0 -. -. _-00" -'
Longitudinal reinforcementcontent in boundary elements
-25
Fig. 5.24 - Effect of longitudinal reinforcement content on backbone curves ofload-displacement loops
4.2
140%
:0.40
3.52.8
1
III
1 1
---------+------------~--
I 1
1
1
1
1
2.11.4
0,48
34%
~ RC walls with an aspect ratio of1.125
-..- RC walls with an aspect ratio of1.125 (Experimental)
III1
1 1- - - - - - - - - - - -1- - - - - - - - - - - - -
1
1
- - - - - 84% ~~ - - - - - - - - ~ - - - - - - - - - - --1
1
1
1 I
_ < '\ 0.36 - - - - - t----- -------i------------1 1
1 I
1 1 I
------------~-----------~------------T------------r------------
: 28%: : :I I 11 1 I
1 1 I
o .... '0.7
150
120
~~
§ 90~6.........
---§~6 60I
§
S30
Longitudinal reinforcement content in boundary elements (%)
Fig. 5.25 - Contribution of longitudinal reinforcement content to the wall strength
- 234-
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Chapter Five
1.20.80.60.4
: Longitudinal reinforcement
: content in bOlUldary elements
0.2
- - - - I - - - - - 1- - - - - T - - - - --, - - - - - 1- - - - - l
I I I I I II 1 1 1 1 I
t
t
t
1
I- - -,- --j - - - - - 1- - - - - T - - - - ""1 - - - - - r- - - - - l
", : : .. ·x·,· 0.70% :01 I I
',: : ---+- 1.40% :I'. t 1
~_oJ:- __ :__ ···&···1.40% (tested) .~
: G,: -2.80% :'I I
';. _. -E)- - - 4.20% :I" I
I' I I I I1- - ~- - ... - - - - ~ - - - __ 1 .J
'. I I I I, I t I I·t. I I I
: '0,: : :~ - I II I I
...J---..::.e----- JI t
t
t
IIII
---x "'I •'J
I'
:'~-,I X'I',
- - - - ...J - - - - ..: I:"" ~ :6..-1 I " 'I I x,I II II II II I
o-0.2-0.6 -0.4-0.8-1
,- - - -"1 - - - - -1- - - - -., - - - - "1 - - - - -1- - --
: : : :P(-): :: ¢=:: ?
~--- D --1-----i--:: Ph :: ,"I 1 I 'I I I,'1 I 101 I I I I I,I I I I I ,1f- - - - -, - - - - -1- - - - - +- - - - -, - - ,0_ 1I I I I I 0' II I I I I 'I I I I I,I I I I 1I t I I "1I til I
I I I I 0' IL- __ - - --" - - - _ -1 .... _ ... __
I I I I,'
I I I L'I I I ,'1I I I <3 I
I I I.' I1 I ,'I'
I I. IL G.:. _I IIIIIII
o-1.2
60
30
90
150
120
Drift ratio (%)
Fig. 5.26 - Effect of longitudinal reinforcement content on secant stiffness
4,2
144%
3.52.82.11.4
~Driftratio:O,10%
.. '.',. Drift ratio: 0.33%
-G- Drift ratio: 1.00%
I II II I
- - - - - - - - - ,_ - - - - - - - - - - - - - - - - - - _1- 1 _I I I
I II
------:------ 0.43 1- ~.-.-.-i13Yo
:A':--"I .,' I
- - - - - -1- - - - -.,,;- \- 0.36 . - - - - - -84% .... - I
I 'I, :.to I
- - - - - - - - - ,- - - - - - - - - - - - - - - - - - ,'~I- - - - - - - - - -,- - - - - - - --
: : 0.41 . 82% :I I. 1 I
- - - - - - - - - ~ - - - - - - - - -1- :- '- - - - - - -1- - - - - - - - - -I - - - - - - - - -
I - I I 52%: ....(\0.43: :I I I
- - - - - - - - -,- - - :-.~ - - -1- - - - - - - - - -1- - - - --
I ,,' I --=----~-
:6~O.~ ~. :_ 33% : _. : 22% .,...-.:;_~---\~-0.23 - :
I Io I
160
140
120
~
~:::::-- 100
~o......
80.........
~c
I60
~-:~
40
20
0
0.7
Longitudinal reinforcement content in boundary elements (%)
Fig. 5.27 - Contribution of longitudinal reinforcement content to secant stiffness
- 235 -
...._---------------------------------_...........
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Chal!.ter Five
16 -,-,----,--------,----,------,-----,-------,
...' I
I
••• t)••• 2.80°,10
~4.20%
--+-0.70%
...•... 1.40%
--.- 1.4% (tested)
III
_____ -.J II ----I-
I II II II I h
II
IIII
Longitudinal reinforcerrent content in :
- - - - -~~~~~_e~~~~ - - - - - - - ,- - - - - - - -i- --;7----T - - - - - - - -I I II I II I QI 1 .,' I
I I " II II II I.-----A~"C---------------I
4
o I 1:1'. - I 50
8
12
8s~
>->~~
=~='al-<[/)
o 0.2 0.4 0.6
Drift ratio (%)
0.8 1.2
Fig. 5.28 - Effect of longitudinal reinforcement content on energy dissipation
16 -r,-------r------~------..,--------____,__-----____,
32%
-+- Drift ratio: 0.10%III
- - .)t(_. - Drift ratio: 0.25% :
---EI--- Drift ratio: 0.33% :. . I I I
...•... Drift ratIo: 0.50% : ~ ~ __ .-::-. :-.~.",.. _.-'.'..::.::"
~Driftratio:0.67% I I ····1.... 51%. . : -*....... :.. . t.... Drift ratIo: 1.00% ...... of· .. ••• --.... I 32% I
_.... $...............: : :
: 14% : : :- - - - - - - - - - - - '1 - - - - - - - - - - - - -1- - - - - - - - - - - - - I - - - - - - - - - - - - '1 - - - - - - - - - - - - -
I I I II 1 I II I I II I I II I I I
I I 116% I
I 7.6% I.~.~ .-.~. ~.~~.~.~ .~~.~t -. -.-.~~. ~.- .~~. ~.~ ·T·~· ~.~ .-. ~.~.~~. ~.~.~ :f'~~~; ~.~.~~. ~.:.-: c.7·i .-. c.o.~ ~.,.7- --~~~: 000 / I : :
• /0 I II I II I
12
Ss~
>-> 8b1)l-<~
=~='a.l:lr:rJ
4
4.23.52.82.11.4
o I ~0.7 "' ......................•...........I ••••••• .. ··t
Longitudinal reinforcement content in boundary elements (%)
Fig. 5.29 - Contribution of longitudinal reinforcement content to the energy dissipation
- 236-
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Chapter Five
20 ,..------,-------,---------r----r-------.---------,
---+---------I1
o1
I
-1- - .. ~I
-+-0.70%
- -.- - 1.40%
--A-- 1.4% (tested)
···e··· 2.80%
~4.20%
I AI'_ I _ L _
1#'1
; I
I••I. __ ~'
II
Longitudinal reinforcerrent content in
boundary elements-------~-------~-------~------
I II II II 1I 1I I
___ l J __I 1I 1I 1
1
I1
I
IIIIII
- - - - - - - -I-I II II II II II Io -+------.;------;.-1 --;I ---;.- ---i- ---1
4
8
16
12
o 0.2 0.4 0.6 0.8 1.2
Drift ratio (%)
Fig. 5.30 - Effect of longitudinal reinforcement content on equivalent damping
4.2
44%
3.52.8
-+- Drift ratio: 0.10% .-.~.. - Drift ratio: 0.25%
-----.- Drift ratio: 0.67% . -.•. -- Drift ratio: 1.00%
2.11.4
IIII
I I
- - - - - - - - - - - - ~ - - - - - - - - - - - - ~ - - - - - ~ Drift ratio: 0.33% . -.EJ- • - Drift ratio: 0.50%
- -- ,-.-.~ .. ?~~. :I I
- - - - - - - - - - - - ~ - - - - - - - - - _....: ~'~~.~.--:.:-. -:-_~.- - - - - - -:- - - - - - - - - - - - - ~ - - - - - - - - - - --
I I ·····- •.••• 126% I
__________ ~ _I!~~ j L~ ~. ~.~ ~ ~. ~.~ .~ ~. ~L~ ~.~ .~ ~ __ ~3~~~~I I I 1I I I
-.- ••••• I 21% I I 35% I
~-dt~._._••..:..• .:..:-':,,-=,',:,,:-.:..-=.. ..:...':":-":""':.:'''':''''':':-....:..:4-L~•.:.:••:.:.-:....:••:..:.:....:••:....:..~••:..:•.:.....:..:.''':':..:''~'.:.:'~.-:::::.:::::••~.======-~~=:=:::~I:;;:;::;;::::==.- 34%I I I ~r~·_=_--=·_..:'=--·-=-·_=':_·=__.~.--"._- - - - - - - - - - - - - - - - - - - - - - - - - 'I - - - - - - - - - - - - -I 23% I
I I II I II I II I II I I
- ------------ --------------------------1------------.~ _- ;... : :
: :" -- - ~-. _ .. -.. -- _. ~ - -_. - _ .. -.I I I I
24
20
--~ 16Ol)
.50..S 12~
""0
~0)
-;;.> 8'3c-~
4
0
0.7
Longitudinal reinforcement content in boundary elements (%)
Fig. 5.31 - Contribution of longitudinal reinforcement content to equivalent damping
- 237 -
"
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Chal2.ter Five
252015
Displacement (nun)
0.5% 1.00%
10
--+---- 0.15
···e··· 0.30
---*- 0.30 (tested)
••• )1(••• 0.45
---0.60
Area ratio, Ac / At
5-5-10
0.50% Drift ratios -40010.25%
-15
n(-l 500
Z400 J
0.25%C Drift ratios"0~
..9]2
D====r==fr- Ac
~
.....:l
At
Ac Section area ofcolwrns
At Total section area
-20
1.00%
-25
-500
Fig. 5.32 - Effect of boundary columns on backbone curves of load-displacement loops
15 .,.-----------,---------------,-------------,
0.6
14.2%
0.63
0.450.3
-+- Analytical
---&- Experimental
I1
1
1
1
II1
1
1
- - - - - - - - - - - - - -1- - - - - - - - - - -
I1
1
1
1
I 1
I I1 I
1 4.8% 1
- - - - - - - - - - - - - - - - - - ~ - - - - - - - - - - - - - - - - _1- _
III1
1.7% 1
1
I1
1
- 1
1
I
o ~ v:e:-:t:= =: 0.7% :I
0.15
§ 5~
~ 10§
~o.........-..
§~o
I
Boundary area ratio, Ac / At
Fig. 5.33 - Contribution of boundary columns to the wall strength
- 238 -
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Chapter Five
1.20.80.60.40.2
1
1
I1
II 1
____ ..l --.J __ '
1 II I1 I
1 I1 I1 1
1 1
----T----~-----r----T----~-----I
I 1 1 1 1 II I I 1 1 II I I 1 1 1I I I 1 1 1I I I
: Area ratio, Ac / At : :-~----~-----~----~----~-----I
1 I I I I 1
1 1 1
1 1 • - -)K-' - 0.15 1
1 1 1
1 I I
: : -+-0.30 :+-----l-- -----1
, 1 I I
~I : ···.···0.45 :"1 1 1
1. : -0.60 :I' 1 1
---l-----I------ ... ----~-----I1 1 1 1 1
1 I 1 1 I
I I 1 1 I1 1 1 I
I I 1 I1 I 1 I1 1 1 I
____ .L I 1
1 I 1
I 1
1
1
1
1
1
o-0.2-0.4-0.6-0.8-1
Drift ratio (%)
I - - - - T - - - - -I - - - - - I - - - - I - - - - -I - - - -
: : I ~ P(_) : ::: ~::liD 1 I1 1 1 I
~----f -~----~-1 I I 11 I I I1 1 I 1
1 I I 1
~ ~ ~_Ac ~ :1 I ~~- 1 1
: : :At
: : 1
1 1 1 1 1
I 1
I A 1leiI-- - - ... - - - - -, - - - - - ~ - - - - _ - -
: At: Total section area :1 1 1
1 1 1
1 1 1
1 I -
1 1 1 I
L -1 I L
1 I 1
1 I
1
1
III
o-1.2
90
30
60
150
120
Fig. 5.34 - Effect of boundary columns on secant stiffness
64%
17%
-2.33 - - - - - -
1
1
I----- ... ------------
I1
_. -a.-" Drift ratio: 0.33%
~Driftratio:1.00%
-+- Drift ratio: 0.10%
1
1
1
1 29%---------------~-------------
1
1
1
1
- - - - - - - - - - - - - - - -I - - - - - - - 1.82I1
II
1
1
1
1 1
----------------l---------------- ... -------------- -I1
II
_________ -1 _
1
1
1
II---------1----1
1
70
60
50
~
::<0 40
"""~::<0 30I
~-:
'- 20
10 7.1% ----
1 0
0.3 0.45 0.6
Boundary area ratio, Ac / At
Fig. 5.35 - Contribution of boundary columns to secant stiffness
- 239-
..,
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Chal2..ter Five
16 Ti------r-------,----,----..,---------,------_
I-T--------
I
II
--T--------II
-9,. I
.,." .~IIII1- .-'- - .... '-
I,',,'1
IIIIIII
I I,_____ II 1- - - --
I II II II I
IIII,---III;III
A c / At
---+- 0.15
.. ·····0.30
-.- 0.30 (tested)
-- -e- - - 0.45
-G--0.60
Area ratio,
8
4
IIIII
- - - I II ------T--------
I II II II II II II I
01 ~- ::
12
i~>-.
t~v~
"§[/)
o 0.2 0.4 0.6 0.8 1.2
Drift ratio (%)
Fig. 5.36 - Effect of boundary columns on energy dissipation
16 ,1--------------,--------------,---------------,
0.60
50%
89%
0.450.30
--.-- Drift ratio: 0.10%
--·x··· Drift ratio: 0.25%
~Drift ratio: 0.33%
---.- -- Drift ratio: 0.50%
--e-- Drift ratio: 0.67%
8
IIIIII
I I
- -: - - - - - - - - - - - - - - - - - - - - - ~ - - - - - - - - - - -;....,- ---:.... ':"- - - --
---1:1--- Drift ratio: 1.00% : 20%:._ .. -····
---_._--_.------_._ -~~% _ -.•...... _ - -..:I I
____ I I---T------------- I: ---- ;------------------97%
I II I
1 ' ', 39%'
4 ~ 16% ~
..................................~.~r~~~.~ ~.~.- .~~. ~.-.... ~. ~.~-. ~-~~~~ ~.~: c.c.~~ ••~.,,< - - - -• :I I
0o.;~ ..·..·.. ····· ---- ·········C···· t
12
i~~v~v~
"§~
Boundary area ratio, Ac / AI
Fig. 5.37 - Contribution of boundary columns to the energy dissipation
- 240-
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Chapter Five
25 ,-----------,-------,--------,--------,-----,---------,
II------T-------III
___ 0.60
. -•. ·0.30
-.-0.30 (tested)
···e··· 0.45
Area ratio,
II
I I
------~-------+-------
I
II
II.
_'-I'
."
I1
1
1
I
I "--------I-~.O----... 't-
,1-
IIIII
I I- - - - - - - -I - - - - - - - -1-
1-------1-------1
1
IIII
5
, ' 1
~......... ~ -"1 ~ - - -:::.. - -1- - - - - - - - r - - - - - - - i - - - - - --
I' ,rt' I
'II--+-- 0.151 1I , 1
1
,1 1.J 1 _
I 1
1 1
1 I1 1
1 I1 I
o +-----r---------,I------,-I------,-----------,-------i
15
20
10
o 0.2 0.4 0.6 0.8 1.2
Drift ratio (%)
Fig. 5.38 - Effect of boundary columns on equivalent damping
24
20
---~ 16bl)
.S0..S 12C'::l
""0
E~
~8;>
'30'"~
4
-+-Driftratio:0.10% -.. )1:.-. Drift ratio: 0.25% :
~Driftratio:0.33% -··El·-· Drift ratio: 0.50% :1
.~.~.=~~~~~:.O:6~O~.:::~::~.~~.~~~:1~~~.~.~~.~.~.~-.~~:~J~.~.C"__r-_~00/1 5.7% °1 38%
10 1 1- - - - - - - - - - - - - - - - - - - - - r - - - - - - - - - - - - - - - - - - - - -1- - - - -
1 25'X'1
9%:--=.-=-=:..=-'-_-::_-=-=-=-=-=-=--- - - - - - - - r-- - - - - - - - - - - - - ,-: ~ --~'.: .-:''':- ~t~·~ ------ -- ---------
I -' II I
····- ••• _-_ ••••••••• 1._ •• -· II1
- - - - - - - - - r-- - - - - - - - - - - - - - - - - - - - - -1- - - - - - - - - - - - - - - - - - - -
I1
I 1I 1
- - - - - - - - - - - - - - - - - - I _ :-•.,._.,: :.: ,:,. : _
-------------------------- •. - ;t- .. -. 1
1 1
1 1
1 1
1 1
0.600.450.30
0+------------+----------------1-----------------1
0.15
Boundary area ratio, Ac / At
Fig. 5.39 - Contribution of boundary columns to equivalent damping
- 241 -
..
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Chal!.ter Five
800
Aspect ratio = HwlLw ~
1.2
Drift ratio (%)
0.80.60.4
--4-- Aspect ratio: 0.500
--..... Aspect ratio: 1.125
---- Aspect ratio: 1.125 (tested)
---~-. - Aspect ratio: 1.625
-.- Aspect ratio: 2.000
0.2
.~~
-600
""g 600.Q]£C\l
.....:l
-0.4-0.6-0.8
~._~
-1
lII:I I
-1.2
-800
Fig. 5.40 - Effect of aspect ratio on backbone curves of load-displacement loops
600
500
i 400---g
rf.Jrf.J 300(l)
~~rf.J
~ 200~C)(l)
rn100
1.20.80.60.40.2
- - - - 1- - - - , - - - -I - - - - r - - - I - - - - I;(
',: •• ')K' •• Aspect ratio: 0.500 :: I I• I I
- - ~ -:- -+- Aspect ratio: 1.125 :, I I
:': ...~... Aspect ratio: 1.125 :___ ~:_ (tested) ":
.~ : --- Aspect ratio: 1.625 :~I I
'Jf ••• E)••• Aspect ratio: 2.000 :- - - -I~ "I
I $K II ' II II , II x, I I I I
- - - - 1- - - -, ,I - - - - - - - - I" - - - -I - - - - I
I I', I I II I I I II I'X I I II I I I II I I I I- - - - 1- - - - I" - - - - - - - - ;- - - - -I - - - - I
--J I I I I
I I I II I I
I
o
r---'----r---T---~----r---
I I I I I II I
: Aspect ratio = HwlLw :I I I I I ~~---~----~---T--------r-~-
I lIC-) I I
~---- D¢::::l r,HW -:---I I IiI I'I I (
~ - - - -; I : il -: - - - ~,~ - - -I I I Lw I I 'K.. II I I I.' II I I I • II I I I 1)1( II" - - - I - - - - 1- - - - T - - - -I -: ,- - - I - - -I I I I J' II I I I ,'I I
: : : :;(: :L J L 1___ _ __ L_~_
I I I I I ."
: : : : ~ - ._,~"I I I I
I ~.- .. -~I--.·tto,-----,' --.-1.2 -1 -0.8 -0.6 -0.4 -0.2
Drift ratio (%)
Fig. 5.41 - Effect of aspect ratio on secant stiffness
- 242-
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Chapter Five
16 -,--------,------,.-----r------,-------,----------.,
-1- - - - - --
1
1
1...' 1
1
- - ..: - - _I _, 1
III1
1
1
1
1
- - - - -I - - - - - -
1
1
IIIII
---&- Aspect ratio: 2.000
'.- .. - Aspect ratio: 1.125
••• E)- _. Aspect ratio: 1.625
--+- Aspect ratio: 0.500
-.-Aspect ratio: 1.125 (tested)
1
I
1
1
1
1
1
- - - - - - - 1- 1 _I1
I
II
4
8
I'I 1
1 1
,fi .' 1
1 • ' <•~ : _:~- I
I1
IIII1
o +---=---=--,....------,,.--------r-------,-------,-I--------1
12
o 0.2 0.4 0.6
Drift ratio (%)
0.8 1.2
Fig. 5.42 - Effect of aspect ratio on energy dissipation
1.2
A
0.8
'1, 1
1
1
1
1
----r------ -------IIII 1
----~ --~--~-------, 1
0.60.4
1 ".,',,'I
-~~-~-,~ --~------
1
II
I," • 1 I
~~~~-----~------~-------,'<OJ 1 1
;' # 1 I 1
'I I 1
1 I 1
~-----~-------~------~-------
1 1 1
I I 1I I 1I 1 1
_~ L L ~ _
1 I 1 1
1 I 1 1
1 I 1 I
1 1 1 I
0.2
~ Aspect ratio: 0.500
••••• Aspect ratio: 1.125
----.- Aspect ratio: 1.125 (tested)
., ·e,·, Aspect ratio: 1.625
-----.-- Aspect ratio: 2.000- - - - - - -1- - - - - - - .., - - - - - - - t- - -
1 I 1
1 I 1 0'I II 1
- - - - - - -1- - - - - - - -I - -I 1I 1
1
1 1
- - -1- - - - - - - -I1
1
1
1
28
24
r-.... 20
Cb.Ol::: 16'a8'""'0
"S 12aJ~;>'3 80"'~
4
0
0
Drift ratio (%)
Fig. 5.43 - Effect of aspect ratio on equivalent damping
- 243-
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Chal2.ter Five
,..... Ii
t··_·_·····_············_·~.cJ,.
25
-. -'.1.0%
2015
Displacement (mm)
0.50%
-.- -,,-- ..
105
No constructionjoints (Withaxialloads)
. -.•-. - With construction joints(With axial loads)
---?r- No constructionjoints(Without axial load)
.. .•. .. With construction joints(Without axial loads)
,...'
0.25%
Drift ratios
,~
~
800
400
600
-5
zc~~
""§~.....:l
-10
..-600
Drift ratios0.500/0. I 0.25%
... ,.,' ..... -
-15
Construction jointsat the wall base
ll<Ol
-20
1.0%
.- ".
-25
-800
Fig. 5.44 - Effect of construction joints on backbone curves of load-displacement loops
1.20.80.60.40.2o-0.2-0.4-0.6-0.8-1
200 r - - - I - - - -1- - - - T - - - -1- - - - I - - - - - - - r - - - I - - - -1- - - - T - - - -1- - - - 1
I I I I I I II I I I 1 I I Nt" . II I P(_) I I I ~ 0 cons ructtonJomts I: :LIF
:: I (with axial loads) :I I I I: I
~ ~ __: ~ _ _ _ : .. -., .. W~h co~structionjoints :: II :: - : (WIth axmlloads) 'I
I I II I ::: - No constructionjoints I
I I I (.th. I
: Construction joints I ' I WI out axmlloads) :I I I I
100 ~ __ , at the wall base I~_ _ _ •• -e· .. With construction joints ~: I I I (without axial loads) I
I I I I II I I I I I I II I I I I I I II I I I I I I II I I I I I I II I I I I I I
50 ~ - - - ~ - - - -:- - - - ~ - . - ~ - - - ~ - - - -:- - - - ~I I I I I II I ,,'1 1 ..... I I II I I ' • .1 I I
1 : 1 I'" I
I I I II I I
I I Io I I I I
-1.2
rfJrfJ(l)
~~
rfJ
E~u(l)
r:n
,-, 150
]g
Drift ratio (%)
Fig. 5.45 - Effect of construction joints on secant stiffness
- 244-
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Chapter Five
15 ,---------,-------,--------,----r------,-------_
~ No construction joints(Without axial loads)9
6
3
)-.._a--.. No constructionjoints / .+
(With axial loads) I I / ." I
. --.- -- With construction joints : - - - - - - - :- - -1..-:. ~ -:- -------(With axial loads) : : /:: :
: : // :
~I ----AI:'::---.~·~""I:- -------"" "."". With construction joints '
(Without axial loads) : ! .: : :I " l .. ' I
-------~-------~-------~- ---- ~-------~-------I I I,. 1 II I I I I1 I I II I .I' I I1 I "I I II I I II I I I I
-------~-------~-- ----------~-------~-------
I I I II I 1 II I II I II I I
o +--~~~~~::::~"~,.~·-~----~---~I ~I----~
12
o 0.2 0.4 0.6
Drift ratio (%)
0.8 1.2
Fig. 5.46 - Effect of construction joints on energy dissipation
1.20.8
,.'
0.6
Drift ratio (%)
0.40.2
II1
II I
- - - -' - - -1- - - - - - - --I - -.:..... I I .
III
24
20
16~eon 12~
'a~
"'08"E
(])
c;>'3 40'"~
0
0
Fig. 5.47 - Effect of construction joints on equivalent damping
- 245-
•
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Chal2.ter Five
REFERENCE
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•
Chapter Five
[K4] Kabeyasawa, T., Shiohara, H., and Otani, S., "U.S.-Japan Cooperative
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[M1] Mestyanek J. M., "The Earthquake of Resistance of Reinforced Concrete
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[M2] Mattock, A.H., and Hawkins, N.W., "Shear Transfer in Reinforced
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[M3] Mattock, A.H., "Shear Friction and High-Strength Concrete," ACI
Structural Journal, V.98, No.1, Jan.-Feb. 2001, pp.50-59.
[M4] Mikame, A. H. et al., "Parametric Analyses of RC shear walls by FEM,"
Structural Design, Analysis, and Testing, Proceedings of the sessions related
to design, analysis and testing at Structures Congress' 89, ASCE, May 1989,
pp.301-310.
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Chae.ter Five
[Nl] Nasir, N., "Nonlinear Finite Element Analysis and Parametric Investigation
of Low-rise Reinforced Concrete Shear Walls," Master thesis, Department
of Civil Engineering, University of Ottawa, July 1999.
[01] Orakcal, Kutay, Conte, Joel, P., and Wallace, John, W., "Nonlinear
Modeling of RC Structural Walls," 7th U.S. National Conference on
Earthquake Engineering, Boston, Massachusetts, 2002.
[PI] Park, Y. 1., and Hofmayer, C. H., "Finite Element Analysis for Seismic
Shear Wall, International Standard Problem," Department of Advanced
Technology, Brookhaven National Laboratory, April 1998.
[P2] Paulay, T., and Priestly, M. J. N., "Seismic Design of Reinforced Concrete
and Masonry Buildings," John Wiley & Sons, New York, 1992, 744 pp.
[P3] M.H. Phillips, "Horizontal Construction Joints in Cast-in-situ Concrete",
Master of Engineering Report, University of Canterbury, Christchurch, New
Zealand, 1972.
[P4] Pilakoutas, K., and Elnashai, A., "Cyclic Behavior of Reinforced Concrete
Cantilever Walls, Part I: Experimental Results," ACI Material Journal, V.92,
No.3, May-June, 1995.
[P5] Pilakoutas, K., and Elnashai, A., "Cyclic Behavior of Reinforced Concrete
Cantilever Walls, Part I: Experimental Results," ACI Material Journal, V.91,
No.2, May-June, 1995.
[S 1] Sittipunt, C., and Wood, S. L., "Finite Element Analysis of Reinforced
Concrete Shear Walls," PhD dissertation, Department of Civil Engineering,
University of Illinois at Urbana-Champaign, Urbana, Ill., 1993.
[S2] Stevens, N. J., Uzumeri, S. M., and Collins M. P., "Reinforced Concrete
Subjected to Reversed Cyclic Shear-experiments and Constitutive Model,"
ACI Structural Journal, Vol. 88, 1991, pp. 135-146.
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Chapter Five
[T1] Thomas N. Salonikios, Andreas J. Kappos, loannis A. Tegos, and Georgios
G. Penelis, "Cyclic Load Behavior of Low-Slenderness Reinforced Concrete
Walls: Design Basis and Test Results," ACI Structural Journal, V.96, No.4,
July-August 1999, pp. 649-660.
[T2] Thomas N. Salonikios, Andreas J. Kappos, loannis A. Tegos, and Georgios
G. Penelis, "Cyclic Load Behavior of Low-Slenderness Reinforced Concrete
Walls: Failure Modes, Strength and Deformation Analysis, and Design
Implications," ACI Structural Journal, V.97, No.1, Jan.-Feb. 2000, pp.
132-142.
[VI] Vulcano, A., and Berto, V. V., "Analytical Models for Predicting the Lateral
Response of RC Shear Walls: Evaluation of Their Reliability," Report No.
UCB/EERC-87/19, EERC, University of California, Berkeley, 1987.
[WI] Walraven, J. C., and Reinhardt, H. W., "Theory and Experiments on the
Mechanical Behavior of Cracks in Plain and Reinforced Concrete subjected
to Shear Loading," Heron 26, 1(a), pp. 5-68,1981.
[W2] Wu Hui, "Design of Reinforced Concrete Walls with Openings for Strength
and Ductility," Ph.D thesis, Nanyang Technological University, 2004, 486
pp.
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Ah
Ah,i
Ah,o
Ag
c
D
fe'
Fm
Fp,p
Fp,t
It
Ge
Gf
h
h eq
heq,i
heq,o
K eq
Kj,a
Chae.ter Five
NOTATIONS
Energy dissipation capacity
Energy dissipation at the ith level of parameters investigated
Energy dissipation at initial level of parameters investigated
Gross cross section area
Cohesion
Tangential stiffness matrix
Cylinder strength of concrete
Peak lateral load
Predicted peak loads
Tested peak loads
Concrete tensile strength
Compressive fracture energy
Tensile fracture energy
Crack band width
Equivalent damping factor
Equivalent damping at the ith level of parameters investigated
Equivalent damping at initial level of parameters investigated
Secant stiffness
Analytical secant stiffness at the ith level of parameters investigated
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Chapter Five
K O,a Analytical secant stiffness at initial level of parameters investigated
N Axial load
P Clamping force perpendicular to the sliding plane
~,max Maximum strength at the ith level of parameters investigated
PO,max Maximum strength at initial level of parameters investigated
Pb Flexure reinforcement ratio in boundary element
£::.ult Ultimate crack strain
t Force traction vector
tn Tensile traction
J1 Friction coefficient
V Shear force
t5 Separation between the two halves
If/ Dilatancy angle
t/J Friction angle
l1u Relative displacements vector
11 m Peak lateral displacement
11 p,p Predicted associated displacements
11 p,t Tested associated displacements
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Cha/2.ter Six
CHAPTER SIX
CONCLUSIONS AND RECOMMENDATIONS
6.1 Conclusions
An experimental program is carried out to explore the seismic performance of eight
squat RC structural walls with limited transverse reinforcement. The local and global
responses of the RC structural walls tested under cyclic loadings are described in detail.
The influence of several design parameters, such as axial loads, transverse
reinforcements in the wall boundary columns and the presence of construction joints at
the wall base, on the local and global responses of these RC structural walls are also
reported in this experimental program. Based on the observed experimental results,
reasonable strut-and-tie models for squat RC structural walls with and without axial
loads are developed to aid in better understanding the force transfer mechanism and
contribution of reinforcement in RC walls tested.
Next, an analytical procedure incorporating both shear and flexure deformations, is
presented to properly evaluate the initial stiffness for RC structural walls with low
aspect ratios. A comprehensive parametric study including a total of 180 combinations
is carried out and a simple expression is proposed to determine the wall initial stiffness
as a function of three factors: yield strength of the outermost longitudinal reinforcement,
applied axial compression and wall aspect ratios. The proposed stiffness formulae are
validated with experimental results and its effectiveness in stiffness predictions are
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«
Chapter Six
further found by comparison with previous stiffness formulas.
Finally, a nonlinear finite element analytical procedure is used in this study to
investigate the global responses of squat RC structural walls under cyclic loadings. By
employing this nonlinear analytical procedure, the global responses such as strength
capacity, stiffness characteristics, energy dissipation capacity and equivalent damping
of these squat RC structural walls under reversed seismic loadings are described in
detail and calibrated against the experimental ones. Furthermore, based on the verified
analytical procedure, an extensive parametric study is carried out to investigate the
influence of several design parameters such as axial loads, longitudinal reinforcements
in the wall boundary elements, aspect ratios, area ratio of boundary columns and the
presence of construction joints at the wall base, on the global responses of squat RC
structural walls.
Several conclusions, as listed below, can be drawn based on these experimental and
analytical investigations:
1. In general, all eight tested specimens with limited transverse reinforcement
behave in a flexural manner and are capable of developing their flexural
capacity prior to failure. Values of drift at initial cracking range from 0.1 % to
0.17% and 0.17% to 0.25% for specimens with aspect ratios of 1.125 and 1.625,
respectively.
2. Specimens with an aspect ratio of 1.125, specimens LWl, LW2, LW4 and LW5,
generally exhibit more ductile behavior than expected, even if they have
insufficient confinement reinforcement corresponding to 70% and 25% of the
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Cha12.ter Six
NZS 3101 and ACI 318 specified quantity of confining reinforcement,
respectively. The displacement ductility factors of the four specimens are more
than 3.0 and can generally experience an average story drift of at least 1%
without significant strength degradation. By contrast, Specimen LW3,
containing 30% and 10% of the NZS 3101 and ACI 318 specified quantity of
confining reinforcement respectively, shows quite critical seismic performance
with respect to the strength and deformation capacities achieved. The
displacement ductility of Specimen LW3 is observed to be less than 3.0, which
proves to exhibit only limited ductile behavior of the wall as stipulated in NZS
3101 code.
3. For specimens with an aspect ratio of 1.625, the content of transverse
reinforcements at the wall boundaries, a reduction to 30% and 10% that is
required by NZS 3101 and ACI-318 code corresponding to fully ductile walls,
might be considered as an effective measure for confining the concrete in the
compression zone in terms of the limited ductile performance of walls.
4. Moreover, for the specimens with more content of the transverse reinforcements
in wall boundaries, the flexural contribution of the total deformation increases
while the shear component of total displacements decreases. This indicates
clearly that seismic performance such as drift, ductility and energy dissipation
capacity can be enhanced by increasing the amount of the transverse
reinforcement at the boundary elements of a wall. However, it is also found that
the content of transverse reinforcement in wall boundary element and the
presence of construction joints at the wall base have negligible effects on the
stiffness characteristics of the RC walls tested.
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Chapter Six
5. The tests showed that the peak load attained for structural walls with
construction joints are almost the same as those without construction joints and
both walls can sustain the ultimate loads well into the post-elastic range.
However, the drop off in load for structural walls with construction joints, after
the peak had been attained, was larger than that for structural walls without
construction joints. This indicates that the stiffness degradation for structural
walls with joints, especially for structural walls with an aspect ratio of 1.125, is
large after the peak load has been attained. However, in terms of limited ductile
criteria, structural performances such as ductility, strength and energy
dissipation capacities for tested walls with construction joints at the base can
still be considered to be adequate. Therefore, the usual object that the
surrounding concrete acts as a monolithic part of the member can be achieved
in the design of squat RC walls with a construction joint.
6. From the testing, it also can be concluded that the level of axial compression
has a minor effect on the degradation rate of secant stiffness despite the fact that
the presence of axial compressions in specimens can lead to higher secant
stiffness in contrast with those without axial loads. For the wall specimens
subjected to axial compression, the amount of energy dissipated is larger than
that corresponding to specimens without axial loads due to the favorable effect
of the axial compression with regard to controlling the pinching of hysteresis
loops.
7. By decomposing the total lateral deformation into flexural and shear
components as well as sliding components, it can be demonstrated that the bulk
of the energy dissipation is due to flexure. The amount of energy dissipated due
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Chal!.ter Six
to shear components does not change much under the condition of axial
loadings on the specimen. With regards to the energy dissipation contributed by
sliding components, it is found to increase slightly due to the presence of
construction joints at the wall base, but still remained at a low level up to the
final stage of testing.
8. The developed strut-and-tie analytical models for specimens tested with and
without axial load respectively, accounting for different contributions of
horizontal and longitudinal web reinforcement, are found to be able to
accurately reflect the force transfer mechanisms of RC structural walls under
cyclic loadings. Moreover, the analytical tensile strains in horizontal web bars
of RC structural walls predicted by use of the developed strut-and-tie models
agree well with the tested data. This provides evidence that the assumed
strut-and-tie models are reasonable for the flow of forces and contribution of
web reinforcements in walls tested.
9. By comparison with the results from the current tested results, it is found that
the developed analytical approach, incorporating both the flexure and shear
deformations, can effectively evaluate the initial stiffness of RC low-rise
structural walls with limited transverse reinforcement.
10. Analysis of the parametric study, shows that three critical parameters: yield
strength of outermost longitudinal reinforcement, applied axial load, and wall
aspect ratios, influence the initial stiffness of low-rise structural walls most.
Based on this parametric study, a simple expression to evaluate the wall initial
stiffness accounting for both flexure and shear deformations is proposed and
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Chapter Six
validated with the tested data. The results obtained are in good agreement with
the experimental work.
11. By applying the concrete constitutive law as a total strain rotating crack model
and a reinforcing bar as the Von Mises plastic material considering the strain
hardening rule, a nonlinear finite element analysis procedure is used to
extensively explore structural performance of squat RC structural walls under
cyclic loadings. Based on regular Newton-Raphson iteration method, this
developed analytical procedure is found to be able to satisfactorily predict both
global and local responses of tested RC structural walls under cyclic loadings.
12. The investigations of the effect of four axial load ratios from 0.00 to 0.15 on the
structural performance of squat RC structural walls found that the presence of
axial loads significantly increases the wall strength but the rate of increase
becomes less when the RC walls subjected to a higher level of axial loads (from
0.10 to 0.15 etc). Moreover the contribution of axial loads to the wall strength
for RC walls with a lower aspect ratio is found to be more significant than those
with a higher aspect ratio. The secant stiffness of walls at the same drift ratio
increases with the added axial load, however, this effect reduces with the
increase of the top wall drift. The presence of axial loads on RC walls is found
to playa beneficial effect on the wall energy dissipation capacity at a high drift
ratio, while at a low drift ratio this effect is negligible. The equivalent damping
of RC walls studied decreases with the added axial load when the test
progressed. At a low drift ratio, a lower equivalent damping is obtained for RC
walls under axial load ratio of 0.05, whereas at a high drift ratio the equivalent
damping is observed to be higher than that of walls without axial load.
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Cha12.ter Six
13. With the addition of longitudinal reinforcement content in boundary elements
from 0.7% to 4.2%, the analytical maximum strength increases almost linearly.
The contribution of longitudinal reinforcements to the wall secant stiffness
increases more rapidly after the attainment of a wall drift ratio of 0.33%. The
increase of longitudinal reinforcement in boundary elements have a more
significant effect on the energy dissipation capacity for RC walls at a high drift
ratio (from 0.5% to 1.0%) than for those up to a drift ratio of 0.33% as its
energy dissipation almost remains the same. In general, the equivalent damping
for RC walls studied decreases with the addition of longitudinal reinforcements
in boundary elements.
14. The wall strength increases significantly with the augmentation of boundary
column area ratios, Ac / At from 0.15 to 0.60 and the rate of increases is
observed to be higher as column area ratios tend to be larger. The secant
stiffness of RC walls increases with the added column area ratios, however, the
rate of increase of secant stiffness with the augmentation of column area ratios,
decreases rapidly after a wall drift ratio of 0.10%. In general, the energy
dissipation capacity and equivalent damping increases with the addition of area
ratios of boundary columns and the rate of increase becomes more significant as
area ratios of boundary columns tend to be larger.
15. The wall strength capacity is observed to decrease significantly with the
augmentation of wall aspect ratios as the percentage decrease of the analytical
maximum strength is observed to be 47.7%, 63.3% and 70.7% with the increase
of aspect ratios from 0.50 to 1.125, 1.625 and 2.0, respectively. With the
augment of wall drift ratios, the rate of decrease of the secant stiffness for RC
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«
Chapter Six
walls with lower aspect ratios is observed to be more significant than those of
RC walls with higher aspect ratios. In general, the energy dissipation capacity
and equivalent damping increases with the addition of aspect ratios of RC walls
at same drift ratios. However, up to a drift ratio of 0.5%, the increase of energy
dissipated for all RC walls with aspect ratios varying from 0.5 to 2.0 is observed
to be negligible, while for wall drift ratios larger than 0.5% this increase of
energy dissipated becomes significant.
16. The structural interface model based on the shear friction theory is adopted for
RC structural walls with construction joints at the base. The analytical results
using this model correspond well with the experimental ones and this indicates
that this shear friction model is suitable for simulation of joint behaviors if a
suitable friction coefficient for the surface is chosen. The existence of
construction joints has a minor effect on the secant stiffness of walls studied
since similar values of strength and secant stiffness are observed for walls with
or without construction joints. Also, almost similar wall energy is dissipated by
walls with or without construction joints at a low drift ratio. However, with the
increase of wall drift ratios, walls without construction joints dissipate more
strain energy than walls with construction joints. Similar analytical equivalent
damping ratios are obtained for both squat RC walls with and without
construction joints.
6.2 Recommendations for Future Works
Several issues related to this study deserve further investigation. This includes:
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Cha12.ter Six
1. Experimental programs should be carried out to further investigate the effect of
boundary column areas on the seismic performance of squat RC structural walls
with limited transverse reinforcement.
2. The analytical initial stiffness predicted by the proposed nonlinear finite
element procedure is normally larger than that from testing. This may due to the
perfect bond assumed between the longitudinal reinforcement and associated
concrete. Accordingly in the future analytical works a reasonable structural
interface model between the reinforcement and concrete should be developed
and incorporated to the fmite element analytical model.
3. More experimental stiffness data for squat RC structural walls under cyclic
loadings are needed to calibrate the effectiveness of the proposed stiffness
formulas.
4. In current nonlinear finite element analysis of squat RC walls, the structural
interface model simulating the construction joints is based on the shear friction
analogy, and parameters such as normal and shear stiffness of the interface
affecting the models are mostly shown to be semi-empirical. Moreover, the
friction coefficient in the model is assumed to be constant throughout the
analysis, which does not represent the true behavior of the interface. In fact, the
true behavior of the interface including aggregate interlock of the surface along
the construction joints and dowel action of the reinforcement bars crossing this
surface will change significantly as the test progressed and has not been
explicitly represented in this finite element analysis due to its complexity.
Accordingly, extensive research should be carried out to investigate the true
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+
Chapter Six
behavior of the interface along the construction joints when the test progressed
and more precise finite element models involving the aggregate interlock of the
interface and dowel action of reinforcement bars crossing the interface should
be incorporated to the future work.
5. By employing the proposed finite element analytical procedure, the local
responses of squat RC structural walls under cyclic loadings should be explored
further.
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