Seismic performance of rc structural squat walls with ... · Reasonable strut-and-tie models for RC structural walls with and without axial loads are then developed to aid in better

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

ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library

drd

Rectangle

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.

- 1 -


drd

Rectangle

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.

- ii -


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.

- iii -


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 .

- IV -

I

II

IV

x

XI

1

1

3

5

7

7

8

10

10

11

11


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

- v-


CHAPTER THREE. SEISMIC PERFORMANCE OF MEDIUM-RISE


REINFORCEMENT ..

Abstract .


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 .


3.3.2.2 Local Response ..

3.3.3 Experimental Results of Specimen MW3 .


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. .

- vi -

72

72

73

75

75

7676

76

77

79

80

81

81

81

83

84

84

85

86

86

87

88

88

89

90

91


3.4.5 Stiffness Characteristics .

3.4.6 Energy Dissipation ..

3.5 Extrapolation of Experimental Results .

3.6 Conclusions .

CHAPTER FOUR. STIFFNESS CHARACTERISTICS OF


REINFORCEMENT .

Abstract .


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

- vii -

92

93

94

96

124

124

125

127

127

128

129

130

130

131

132

133

135

135

135

139

139

140

141

142

143


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 .


5.2 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 .

- V1II -

144144

146

147

171

171

172

175

175

180

181

183

184

186

187

189

190

192

193

194

196

197

198

198


'T

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

- ix -

II


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% .

Page

41

42

98

99

149

149

149

150

151

151

152

152

153

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 .

- x-

216

216

216

217

217


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 distribution in the horizontal web bar (R bar) ofSpecimen LW2 .

- Xl -

Page

43

43

44

44

45

46

46

46

47

47

47

48

48

49

49

49

50

50


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 .


Strain distribution in outermost longitudinal bars 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 ..


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 .


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 .

- xii-

50

51

51

51

52

52

52

53

53

54

54

55

55

55

56

56

56

57

57

57

58

58


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 #T 14 in horizontal bars ofSpecimen LW2 .





Stress-strain relationship for steel reinforcements .

Details of Specimen MW1 .

Experimental set-up .

Applied loading history .

LVDTs support arrangements in specimen walls .

- Xlll -

58

59

60

61

61

62

63

63

63

63

64

64

65

65

66

66

67

67

100

100

101

101

102


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 .


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 .


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 .

- xiv-

103

103

104

104

104

105

105

105

106

106

107

107

108

108

108

109

109

109

110

110

111

111


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 ..



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 .

- xv-

111

112

112

112

113

114

115

115

116

117

117

117

118

118

118

119

119

155

155

156

157

158

159

.-


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

- xvi -


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

- XVll -

--


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 .

- xviii-

244

245

245



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.

- 1 -


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

-2-

II

~


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

- 3 -


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

-4-

...

--


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

- 5 -


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.

- 6 -

.,.-

--


Chapter Two

CHAPTER TWO

SEISMIC PERFORMANCE OF

LOW-RISE STRUCTURAL WALLS WITH


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

- 7 -


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

- 8 -

...

-


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.

- 9-


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

- 10-

..,.

~


- 11 -

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


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

- 12-

~


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 .

- 13 -


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

- 14-

(2.8)

~


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

- 15 -


..

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

- 16-

.,..

~


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

- 17-


~

Chapter Two

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

- 18-

~

---


Chapter Two

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

- 19-


.J

Chapter Two

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)

- 20-

-.,

--


Chapter Two

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).

- 21 -


..

Chapter Two

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.


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

- 22-

---

----


Chapter Two

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.


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.

- 23 -


Chapter Two


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%.


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).

- 24-

~'."'f!".'I')~

--


Chapter Two


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).


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

- 25 -


Chapter Two

that there were dry construction joints at the base of wall LW4.


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.


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

- 26-

..

--


Chapter Two

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.


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.


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

- 27-


,JI

Chapter Two

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.


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

- 28 -

...

~


Chapter Two

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

- 29-


Chapter Two

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

- 30-

..

--


Chapter Two

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

- 31 -


drd

Rectangle

Chapter Two

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

- 32-

...

~


Chapter Two

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

- 33 -


Chapter Two

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-

...

~


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

- 35 -


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-

--

__-.AI


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-


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

- 38-

...

~----~ ~


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-


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-

...

~


drd

Rectangle

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


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-

~


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 -----------------------------------------------------


drd

Rectangle

-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-

.. ~


drd

Rectangle

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


--.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 ~


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 -


]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-

~


drd

Rectangle

drd

Rectangle

drd

Rectangle

drd

Rectangle

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


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-


-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


- 50-

.

................., .--


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 -


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


-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


(a)At a drift ratio of 0.17%


drd

Rectangle

-wChapter Two


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


/ /////-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


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


- 52-

~-------~------- ~


.{).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


T14 T15T13

.. ,....

- 53 -


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 %



...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)


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


- 54-

I!il'

~-------....

M······,::>·<~·::I~;:

~




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


600


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)



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


. -" ~""'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%


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)


- 56-

1-..-.. -----.-..------------- ~


drd

Rectangle

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


-0.002

600


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





-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



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-

~


drd

Rectangle

drd

Rectangle

drd

Rectangle

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..


~..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- - .

........'.'.""', .--


drd

Rectangle

drd

Rectangle

drd

Rectangle

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


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-


drd

Rectangle

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


-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'' "'; .

~


drd

Rectangle


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


294kN

B


600

-600


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


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


- 66-

III:

1:1

11:1

II'!

1:-_-------- ~


drd

Rectangle

0.00195

Gauge #T14

#T bnr

Strain

~l3 __ Tl~

Tl4 TIS

0.00155

Specimen LW5

P(-)=541.2 kN

F


D

A

0.00115

Chapter Two

- 67-

~... '\Y 160.3' \':11' ;-'.-."

1850

- , ~::- .~

I "~I--- ~f- /

r M~~ 160.3 ~ 380. /

1294 kN

c


600

400 -. -

200ZCfl)C,)~

eB 5~fl)

..dr./)

-400

-600



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



[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.

- 68-


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.

- 69-


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

- 70-

jJI


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

- 71 -


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

- 72-


Chapter Three


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.

- 73 -


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

- 74-


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.

- 75-


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:

- 76-

~


- 77-

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


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

- 78-

~ ---'-


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

- 79-


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

- 80-


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


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

- 81 -


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

- 82-


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.


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%.

- 83 -


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.


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

- 84-


drd

Rectangle

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%.


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

- 85 -


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.


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

- 86-


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.


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).

- 87-


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.

- 88 -


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

- 89-


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

- 90-

I

I


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,

- 91 -


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

- 92-

I


drd

Rectangle

drd

Rectangle

drd

Rectangle

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

- 93 -


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.

- 94-

1 ----------


drd

Rectangle

drd

Rectangle

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

- 95 -


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

- 96-


drd

Rectangle

drd

Rectangle

drd

Rectangle

drd

Rectangle

drd

Rectangle

drd

Rectangle

drd

Rectangle

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.

- 97-


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>


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

- 99-


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

- 100-


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


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

- 102 -

!(


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


-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%


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

.. -. -


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~


R8

T16

R6 R7

T14 T15

R5

- 'j .f • - -, • = . , - .. • • = ''")i;1 ; "i

T13

,..

RB

T16

Gauge location

Gauge location


- 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


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-


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


Chapter Three

."".

1000 1200 1400 1600 1800 2000


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


-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

~


-

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%

'---. ....... :'>(.--


""' 'IX: ............ " ,.[~...,

(c)At a drift ratio of 1.33 %

Fig. 3.16 - Crack patterns of Specimen MW3

- 109-


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 -


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




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


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-


- 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


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.


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


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 -


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


drd

Rectangle

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 -


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



- 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


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


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


[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-


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,


[T2] Thomas N. Salonikios, Andreas J. Kappos, Ioannis A. Tegos, and Georgios



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 -


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


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



Ratio of moment of resistance at overstrength to moment

- 122 -


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 -


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 -


Chapter Four


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 -


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-


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

- 127 -


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

- 128-

(4.2)


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.

- 129-


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.

- 130-


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.

- 131 -


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

- 132 -


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:

- 133 -


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

- 134-


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

- 135 -


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

- 136 -

(4.8)


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)

- 137-

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.


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

- 138-


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

- 139 -

(4.16)

(4.17)

(4.18)


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

- 140 -


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.

- 141 -

I'ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library

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),

- 142-


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.

- 143 -


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

- 144-


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.

- 145-


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.

- 146-


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

- 147 -


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.

- 148 -


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

- 149-


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)

- 150-


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

- 151 -


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

- 152 -


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

- 153 -


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

- 154-


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]

\

~

\

\

- 155 -


Concrete compression strut

Chal!.ter Four

- 156 -

Lcs

a

fix As

CD

Fig. 4.3 - Compression strut of wall panel


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

- 157 -


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

- 158 -


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

- 159 -


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)


(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


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 -

•


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

- 161 -


Cha12.ter Four

REFERENCES

[A 1] ACI Committee 318, "Building Code Requirements for Structural Concrete


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 -


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 -


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

- 164-

Chal!.ter Four


n

s

~Yf

~YV

~s

~sh

r

Length of the cantilever wall

Yield moment

Axial loads

Modular ratio


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


Shear distortion

Yield curvature

The error value

- 165 -

Chapter Four


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

- 166-


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

- 167 -

I,


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

- 168-


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

- 169 -


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

- 170-


Chapter Five

CHAPTER FIVE

FINITE ELEMENT PARAMETRIC STUDY OF THE

BEHAVIOR OF STRUCTURAL WALLS WITH


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

- 171 -


Cha12.ter Five


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

- 172 -


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

- 173 -


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.

- 174-


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

- 175 -


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

- 176-


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

- 177 -


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).

- 178 -


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

- 179-


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

- 180-


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

- 181 -


- 182 -

Chae.ter Five

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


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

- 183 -


l

Cha12.ter Five

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

- 184-


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

- 185 -


Chal2.ter Five

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.

- 186-


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

- 187 -


v

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.

- 188 -


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

- 189-


Cha12.ter Five

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

- 190-


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

- 191 -


- 192-

same as those in the tests.

chapters are selected as reference walls. The overall dimension and reinforcement

(5.7)

Cha/2.ter Five

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!

Iii


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

- 193 -


Chal2.ter Five

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

- 194-


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

- 195 -


Cha12.ter Five

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

- 196-


--.,'...", ..,....."".,.,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,

- 197 -


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

- 198 -


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.

- 199-


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

- 200-


Chapter Five

ratios.

5.3.3.2.3 Effect of longitudinal reinforcement content on energy dissipation


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

- 201 -

---------------------j


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

- 202-


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.

- 203-


Cha12.ter Five

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

- 204-


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

- 205 -


Cha12.ter Five

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

- 206-


Chapter Five

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

- 207-


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%

- 208-


Chapter Five

and 180%, respectively.

5.3.3.4.3 Effect of aspect ratio on energy dissipation


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

- 209-


Ijlli

ill!III

II

II,iiiIi

liilil,11

II'!""""",I"

1'1

(

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.

- 210-


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.

- 211 -


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

- 212-


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

- 213 -


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

- 214-


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.

- 215 -


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


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

- 217 -


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


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%

- 219-


drd

Stamp

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

- 220-


drd

Stamp

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 -


drd

Stamp

drd

Stamp

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


(e) (0

- 222-


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


(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


....... 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

- 223-


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

- 224-


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-

.......-------------------------------------------_...


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


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-

............-_--------------------------~


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-


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!


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-


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


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-


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.


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-


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


Fig. 5.27 - Contribution of longitudinal reinforcement content to secant stiffness

- 235 -

...._---------------------------------_...........


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


Fig. 5.29 - Contribution of longitudinal reinforcement content to the energy dissipation

- 236-


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


Fig. 5.31 - Contribution of longitudinal reinforcement content to equivalent damping

- 237 -

"


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 -


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


Fig. 5.35 - Contribution of boundary columns to secant stiffness

- 239-

..,


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-


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


Fig. 5.39 - Contribution of boundary columns to equivalent damping

- 241 -

..


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-


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-


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-


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-

•


Chal2.ter Five

REFERENCE

[AI] ACI Committee 318, "Building Code Requirements for Structural Concrete



[C1] Comite Euro-International du Beton, "CEB-FIP Model Code 1990,"

Thomas Telford Services Ltd. London, 1993,437 pp.

[D1] DIANA - "Finite Element Analysis User's Manual," Release 8.1, Second

Edition, TNO Building and Construction Research, Delft, Netherlands,

2003.

[D2] Dan Palermo and Frank 1. Vecchio, "Simulation of Cyclically Loaded

Concrete Structures Based on the Finite Element Method", ASCE J. of

Structural Engrg., Vol.133, No.5, 2007, pp. 728-738.

[E1] Elmorsi M, Kianoush MR, and Tso WK, "Nonlinear Analysis of Cyclically

Loaded Reinforced Concrete Structures," ACI Structural Journal, V. 95,

No.6, Nov.-Dec. 1998, pp. 725-739.

[F1] Foster, S. J., and Marti, P., "Cracked membrane model: FE implementation."

J. Struct. Eng., Volume 129, Issue 9, pp. 1155-1163,2003.

[K1] Khatri, D., and Anderson, J. C., "Analysis of Reinforced Concrete Shear

Wall Components Using the ADINA Nonlinear Concrete Model,"

Computers & Structures, Vol. 56, No. 2/3, 1995, pp. 485-504.

[K2] Kwak, H. G., and Filippou, F., "Nonlinear FE Analysis of RIC Structures,"

Computers & Structures, Vol. 65, No.1, 1997, pp. 1-16.

[K3] Kwak, H. G., and Kim, D. Y., "Material Nonlinear Analysis of RC Shear

Walls Subjected to Cyclic Loadings," Engineering Structures, Vol. 26, 2004,

pp. 1423-1436.

- 246-


•

Chapter Five

[K4] Kabeyasawa, T., Shiohara, H., and Otani, S., "U.S.-Japan Cooperative

Research on RIC Full-Scale Building Test, Part 5: Discussion on Dynamic

Response System," 8th World Conference on Earthquake Eng., Vol.6, San

Francisco, U.S.A., 1984.

[K5] Kim, T. W., "Performance Assessment of Reinforced Concrete Structural

Walls for Seismic Loads," PhD dissertation, Civil and Environmental

Engineering, University of Illinois at Urbana-Champaign, 2004.

[K6] Kim, T. H., Lee, K. M., Chung, Y. S., and Shin, H. M., "Seismic Damage

Assessment of Reinforced Concrete Bridge Columns," Engineering

Structures, Vol. 27, 2005, pp. 576-592.

[L1] Linde, P., "Numerical Modeling and Capacity Design of

Earthquake-Resistant Reinforced Concrete Walls," Institute of Structural

Engineering, Swiss Federal Institute of Technology (ETH), Zurich, Report

No. 200, 1993.

[M1] Mestyanek J. M., "The Earthquake of Resistance of Reinforced Concrete

Structural Walls of Limited Ductility," Master thesis, University of


[M2] Mattock, A.H., and Hawkins, N.W., "Shear Transfer in Reinforced

Concrete," PCI Journal, V.17, No.2, Mar.-Apr. 1972, pp.55-75.

[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.

- 247-


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


No.3, May-June, 1995.

[P5] Pilakoutas, K., and Elnashai, A., "Cyclic Behavior of Reinforced Concrete


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.

- 248-


..

Chapter Five

[T1] Thomas N. Salonikios, Andreas J. Kappos, loannis A. Tegos, and Georgios


Walls: Design Basis and Test Results," ACI Structural Journal, V.96, No.4,


[T2] Thomas N. Salonikios, Andreas J. Kappos, loannis A. Tegos, and Georgios



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.

- 249-


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


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

- 250-


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

- 251 -


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

- 252-


«

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

- 253 -


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.

- 254-


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

- 255 -


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

- 256-


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.

- 257 -


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

- 258 -


«

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:

- 259-


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

- 260-


+

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.

- 261 -


Documents

Seismic performance of rc structural squat walls with ... · Reasonable strut-and-tie models for RC structural walls with and without axial loads are then developed to aid in better