response of reinforced concrete rectangular liquid containing

RESPONSE OF REINFORCED CONCRETE RECTANGULAR LIQUID CONTAINING STRUCUTRES UNDER CYCLIC LOADING

by

Reza Sadjadi Master of Applied Science,

Ryerson University, Toronto, Ontario, Canada, 2003

A dissertation

presented to Ryerson University

in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

in the Program of

Civil Engineering.

Toronto, Ontario, Canada, 2009

© Reza Sadjadi 2009

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ii

AUTHOR’S DECLARATION I hereby declare that I am the sole author of this thesis.

I authorize Ryerson University to lend this thesis to other institution or individuals for the

purpose of scholarly research.

I further authorize Ryerson University to reproduce this thesis by photocopying or by other

means, in total or in part, at the request of other institution or individuals for the purpose of

scholarly research.

iii

BORROWERS PAGE

Name

Address Date

iv

RESPONSE OF REINFORCED CONCRETE RECTANGULAR LIQUID CONTAINING STRUCUTRES UNDER CYCLIC LOADING

Reza Sadjadi, PhD, 2009 Department of Civil Engineering Ryerson University

ABSTRACT

The continual functioning of Liquid Containing Structures “LCS” is necessary for the well being

of a society especially during and after an earthquake. One of the main parameters used in the

seismic design of structures is the Response Modification Factor “R”. For LCS, there has not

been a justifiable guideline for determination of R factor and the empirical values have been

implemented in the design of such structures. While the seismic design criteria for the buildings

are mainly based on life safety and prevention of collapse, the concrete storage tanks should be

designed to meet the serviceability limits such as leakage. This study was aimed at evaluation of

the leakage behavior of ground supported open top rectangular RC tanks under the effect of

cyclic loading. Full-scale specimens representing a cantilever wall were designed and built to

simulate the leakage through the most critical region of the tank wall. A steel water pressure

chamber was installed at the wall foundation connection to simulate the effect of the water

pressure for the induced cracks at the critical location of the tank wall. Cyclic loading was

applied on the top of the wall while the critical region of the wall was subjected to pressurized

water. Different variables such as the thickness of the wall and the configuration of the shear key

were examined. The effect of retrofitting the damaged specimen with Glass Fiber Reinforced

Polymers (GFRP) sheets on the leakage behavior under cyclic loading was also investigated.

v

Analytical studies using a Finite Element software were followed to assess the viability of the

application of the FE analysis to the investigation of the cyclic behavior of the RC rectangular

tanks. FE analysis results correlated well with the experimental observations. A study on the

effect of R factor on the design loads was performed. Two major components of the R factor,

namely, the ductility factor, and the overstrength factors were discussed and determined for the

case of rectangular reinforced concrete LCS considering the leakage as the limit state. This study

is limited to rectangular tanks in which the wall dimensions promote a one-way behavior.

vi

ACKNOWLEDEMENTS I would like to express my thanks to a number of people without whom this project would not

have been possible.

I wish to express my deep gratitude to my supervisor, Dr. Reza Kianoush for his inspiring

guidance and support throughout this study.

Thanks are extended to Mr. Nidal Jaalouk for his great assistance and helpful comments in the

structural lab.

The assistance of some colleagues is also greatly appreciated. I am grateful to Mr. Arash

Akhavan Khalegi, Mr. Hamidreza Akhlagh Nejat, Mr. Mehdi Moslemi, and Mr. Nima Ziaolhagh

for their help during the experimental work.

The supports from the “Natural Science and Engineering Council of Canada” in the forms of

PGSD2 Scholarship is greatly appreciated.

Material supports from Cement Association of Canada, Innocon Ready Mix Concrete, C&T

reinforcing Steel are greatly appreciated

Finally, I would like to thank my family for their support and my wife for her continuous

encouragement, and support during the accomplishment of this work

vii

TABLE OF CONTENTS

AUTHOR’S DECLARATION ................................................................................................... ii

BORROWERS PAGE ............................................................................................................... iii

ABSTRACT ............................................................................................................................... iv

ACKNOWLEDEMENTS .......................................................................................................... vi

TABLE OF CONTENTS .......................................................................................................... vii

List of Tables ............................................................................................................................. xi

List of Figures ........................................................................................................................... xii

CHAPTER-1 ................................................................................................................................... 1

INTRODUCTION ...................................................................................................................... 1

1.1 General .............................................................................................................................. 1

1.2 Performance criteria of RC tanks ...................................................................................... 4

1.3 Objectives ......................................................................................................................... 5

1.4 Organization ...................................................................................................................... 6

CHAPTER-2 ................................................................................................................................... 8

BEHAVIOUR AND DESIGN ASPECTS OF LIQUID CONTAINING STRUCUTRES (LCS)

..................................................................................................................................................... 8

2.1 General .............................................................................................................................. 8

2.2 Damage to Liquid Storage Tanks ..................................................................................... 9

2.3 Previous studies on the seismic behavior of LCS ........................................................... 11

2.3.1 The Housner’s Model .............................................................................................. 14

2.4 Classification of ground supported tanks ........................................................................ 15

2.5 ACI guidelines for design of LCS .................................................................................. 15

2.5.1 Design loads ............................................................................................................. 19

2.5.1.1 Hydrostatic forces ............................................................................................. 19

2.5.1.2 Hydrodynamic forces ........................................................................................ 19

2.6 Dynamic force distribution above base for Rectangular tanks ....................................... 21

2.7 Effects of the tank dimensions on earthquake induced force properties ........................ 27

CHAPTER-3 ................................................................................................................................. 32

NONLINEAR RESPONSE OF RC STRUCTURES ............................................................... 32

viii

3.1 General ............................................................................................................................ 32

3.2 Inelastic response of a structure during an earthquake ................................................... 32

3.2.1 Effect of design parameters on the behavior ............................................................ 32

3.3 Response Modification Factor ........................................................................................ 42

3.3.1 Evolution of “Response Modification Factor” ........................................................ 42

3.3.2 Design Coefficients .................................................................................................. 49

3.4 Components of “Response Modification Factor” ........................................................... 52

3.4.1 Overview .................................................................................................................. 52

3.4.2 Strength Factor (Uang 1991) ................................................................................... 54

3.4.3 Ductility Factor ........................................................................................................ 58

3.5 Response modification factor for LCS ........................................................................... 73

3.6 Parametric study on the effect of R on the design loads ................................................. 75

3.6.1 Design load combinations ........................................................................................ 80

CHAPTER-4 ................................................................................................................................. 84

EXPERIMENTAL PROGRAM ............................................................................................... 84

4.1 General ............................................................................................................................ 84

4.2 Linear analysis of the behavior of the tank ..................................................................... 85

4.2.1 Long side loaded ...................................................................................................... 86

4.2.2 Short side loaded ...................................................................................................... 89

4.4 Preparation of the specimen ............................................................................................ 94

4.4.1 Installation of the strain gauges ............................................................................... 95

4.4.2 Construction of the Reinforcement Cage ................................................................. 96

4.4.3 Construction of the Formwork ................................................................................. 96

4.4.4 Casting of concrete ................................................................................................ 100

4.4.5 Preparation of the specimen for the experimental test ........................................... 106

4.4.6 Installation of the water pressure chamber ............................................................ 108

4.4.7 The floor brake ....................................................................................................... 115

4.5 General specifications of the specimens ....................................................................... 117

4.6 Details of design and construction for the specimens ................................................... 119

4.6.1 Specimen-1 ............................................................................................................ 119

4.6.2 Specimens-2, 3, and 4 ............................................................................................ 123

ix

4.6.3 Specimen-5 ............................................................................................................ 124

4.7 The Experimental Program ........................................................................................... 129

4.8 Key parameters in the experiments ............................................................................... 132

CHAPTER-5 ............................................................................................................................... 136

RESULTS OF THE EXPERIMENTAL PROGRAM ............................................................ 136

5.1 General .......................................................................................................................... 136

5.2 Specimen-1 ................................................................................................................... 136

5.2.1 The pretest .............................................................................................................. 137

5.2.2 The leakage test ...................................................................................................... 140

5.2.3 Test of the retrofitted Specimen-1 for leakage ...................................................... 150

5.3 Specimen-2 ................................................................................................................... 153

5.3.1 The leakage test ...................................................................................................... 154

5.4 Specimen-3 ................................................................................................................... 162

5.4.2 The leakage test ...................................................................................................... 163

5.4.3 Test of the retrofitted Specimen-3 for leakage ...................................................... 169

5.5 Specimen-4 ................................................................................................................... 173

5.5.2 The pretest .............................................................................................................. 174

5.5.3 The leakage test ...................................................................................................... 179

5.5.4 Test of Specimen-4 to failure ................................................................................. 184

5.6 Specimen-5 ................................................................................................................... 187

5.6.2 The pretest .............................................................................................................. 187

5.6.3 The leakage test ...................................................................................................... 191

5.6.4 Test of Specimen-5 to failure ................................................................................. 196

CHAPTER-6 ............................................................................................................................... 200

DISCUSSION OF THE EXPERIMENTAL RESULTS ........................................................ 200

6.1 General .......................................................................................................................... 200

6.2 Summary of the observations in the experimental program ......................................... 200

6.3 Discussion of the observations in the experimental program ....................................... 203

6.4 Determination of R factor ............................................................................................. 205

6.4.1 Overstrength factor ................................................................................................ 205

6.4.2 Ductility factor ....................................................................................................... 207

x

6.4.3 Determination of R factor ...................................................................................... 210

CHAPTER-7 ............................................................................................................................... 214

ANALYTICAL STUDY ........................................................................................................ 214

7.1 General .......................................................................................................................... 214

7.2 Material model for concrete in ABAQUS .................................................................... 215

7.2.1 Smeared cracking ................................................................................................... 215

7.2.2 Brittle cracking ....................................................................................................... 215

7.2.3 Concrete damaged plasticity (CDP)....................................................................... 216

7.2.3.1 Behavior of concrete in tension ...................................................................... 219

7.2.3.2 Behavior of concrete in compression .............................................................. 221

7.3 Reinforcement ............................................................................................................... 221

7.4 Validation ...................................................................................................................... 224

7.4.1 Reinforcement ........................................................................................................ 226

7.4.2 Concrete ................................................................................................................. 229

7.5 Application of the analytical study in the current research ........................................... 236

7.5.1 Response under cyclic loading............................................................................... 242

7.5.1 Pushover analysis ................................................................................................... 246

CHAPTER-8 ............................................................................................................................... 250

SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS.......................................... 250

8.1 Summary ....................................................................................................................... 250

8.2 Conclusions ................................................................................................................... 251

8.3 Recommendations for future work ............................................................................... 253

REFERENCES ....................................................................................................................... 255

APPENDIX A ............................................................................................................................. 268

DESIGN PARAMETERS ...................................................................................................... 268

APPENDIX B ............................................................................................................................. 280

ABAQUS INPUT FILE .......................................................................................................... 280

xi

List of Tables

Table 2-1 Response modification factors (ACI 350-3-06) ........................................................... 23

Table 3-1 Overstrength components for different structural systems ........................................... 58

Table 3-2 Coefficients to compute Rd proposed by Lai and Biggs ............................................... 63

Table 3-3 Parameters for the relationship proposed by Riddell, Hidalgo and Cruz ..................... 65

Table 3-4 Parameters for the relationship proposed by Nassar and Krawinkler .......................... 67

Table 3-5 Predominant period of ground motion for a set of earthquake records ........................ 70

Table 3-6 Parameters used for the definition of reduction factors ................................................ 74

Table 4-1 Mechanical properties for Grade B7 steel .................................................................. 107

Table 4-2 Properties of the tank used in the current research ..................................................... 118

Table 4-3 Design load components for the specimens ............................................................... 119

Table 4-4 Summary of the experimental test conducted in this study ........................................ 130

Table 4-5 GFRP properties (TYFO SEH-51A Composite Laminate – R.J. Watson, Inc.) ........ 132

Table 4-6 Key parameters used in the construction and design of the specimens ...................... 133

Table 4-7 Design load values and the reinforcement volume for the specimens ....................... 133

Table 4-8 Theoretical predictions for the specimens .................................................................. 135

Table 6-1 Design force components ........................................................................................... 212

Table 7-1 Material model information for steel .......................................................................... 227

Table 7-2 Stress-strain relationships for steel rebar .................................................................... 229

Table 7-3 Concrete behaviour in compression ........................................................................... 230

Table 7-4 Concrete “Tension Stiffening” and “Tension Damage” ............................................. 232

Table A-1 Design parameters for Specimen-1 ............................................................................ 268

xii

List of Figures

Figure 2.1 Illustration of the parameters in the Housner’s model 15

Figure 2.2 Non-flexible wall to base connections 16

Figure 2.3 Distribution of the hydrostatic pressure on a rectangular wall 20

Figure 2.4 Impulsive and convective mass factors vs. L/HL ratio 21

Figure 2.5 Impulsive and convective height factors vs. L/HL ratio 22

Figure 2.6 Components of the flexural moment at the base of the tank for impulsive and

convective pressures vs. L/HL ratio 24

Figure 2.7 Distribution of the impulsive pressure on a wall of a rectangular LCS 26

Figure 2.8 Distribution of the convective pressure on a wall of a rectangular LCS 27

Figure 2.9 Comparison of the period of vibration vs. HL/L ratio 29

Figure 2.10 Comparison of the ratio of seismic loads vs. HL/L ratio 30

Figure 2.11 Comparison of the ratio of seismic to static loads vs. HL/L ratio 31

Figure 3.1 Elastoplastic idealization of the inelastic response 36

Figure 3.2 Elasto-plastic system and its corresponding linear system 38

Figure 3.3 Ductility demand for elastoplastic system due to El Centro 39

Figure 3.4 Constant ductility response spectrum for elasto-plastic 41

Figure 3.5 Yield strength reduction factor of elastoplastic systems as a function of natural

vibration period Tn for μ = 1, 1.5, 2, 4, and 8 during El Centro ground motion (ξ=5%). 42

Figure 3.6 Use of R factor to reduce elastic spectral demands 46

Figure 3.7 Maximum considered earthquake response spectrum 49

Figure 3.8 Illustration of the inelastic and elastic response curves 51

Figure 3.9 Force-displacement relationship for an RC system 54

Figure 3.10 Increase in the moment capacity due to compression reinforcement 56

Figure 3.11 Linear and constant ductility nonlinear response spectra 61

Figure 3.12 Strength reduction factors proposed by Newmark and Hall 62

Figure 3.13 Strength reduction factors proposed by Lai and Biggs 63

Figure 3.14 Strength reduction factors proposed by Riddell, Hidalgo and Cruz 65

Figure 3.15 Strength reduction factors proposed by Arias and Hidalgo 66

Figure 3.16 Strength reduction factors proposed by Nassar and Krawinkler 67

xiii

Figure 3.17 Strength reduction factors proposed by Vidic, Fajfar and Fischinger 69

Figure 3.18 Strength reduction factors proposed by Borzi-Elnashai 74

Figure 3.19 Comparison of the value of the maximum seismic flexural moment for wall heights

of 3, 6, and 9 m. 77

Figure 3.20 Comparison of the proposed curve with ME / HL3 values for wall heights of 3 and 5

m. 78

Figure 3.21 Comparison of the proposed curve with ME / HL3 values for wall heights of 6, 8, and

10 m 80

Figure 4.1 The 3-D view of the tank in SAP2000 85

Figure 4.2 The coordinate system of the tank in SAP2000 87

Figure 4.3 Variation of the total flexural moment on the large side of the tank 87

Figure 4.4 Variation of the horizontal flexural moment on the large side of the tank with respect

to distance from bottom and corner of the tank wall 88

Figure 4.5 Variation of the vertical flexural moment on the large side of the tank with respect to

distance from bottom and corner of the tank wall 89

Figure 4.6 Variation of the total flexural moment on the short side of the tank 90

Figure 4.7 Variation of the horizontal flexural moment on the short side of the tank with respect

to distance from bottom and corner of the tank wall 91

Figure 4.8 Variation of the vertical flexural moment on the short side of the tank with respect to

distance from bottom and corner of the tank wall 93

Figure 4.9 The schematics of the test setup for the leakage test 95

Figure 4.10 Formwork for the foundation 97

Figure 4.11 The bracket supporting the actuator on the strong wall 98

Figure 4.12 Installation of the PVC pipe in the foundation 99

Figure 4.13 Dimensions of the hooks 100

Figure 4.14 Four pieces of lumber connected to the corners of the formwork 100

Figure 4.15 Configuration of the shear key in the experimental program (not to scale) 102

Figure 4.16 Preparation of the conventional shear key before casting of concrete 103

Figure 4.17 Erection of the water stop inside the wall reinforcement 103

Figure 4.18 Construction of the inverted shear key 104

Figure 4.19 The inverted shear key before placement of the wall’s formwork 104

xiv

Figure 4.20 The specimen before placement of the wall’s formwork 105

Figure 4.21 Casting of the concrete for the wall 106

Figure 4.22 The specimens after casting of the wall’s concrete 106

Figure 4.23 The connection of the actuator to the beam on the front side of the wall 108

Figure 4.24 The steel water chamber used for simulation of the water pressure 109

Figure 4.25 The gum-rubber sheet before installation on the wall 110

Figure 4.26 The installation of the gum-rubber sheet on the wall 110

Figure 4.27 Concrete anchors (12 ×178 mm) 111

Figure 4.28 Location of the concrete anchors 112

Figure 4.29 Installation of the plates inside the chamber 112

Figure 4.30 Steel rebars between plates and bottom angle 113

Figure 4.31 Top part of the water pressure chamber 114

Figure 4.32 Filling the water chamber with water 115

Figure 4.33 Complete setup including the water pressure chamber 116

Figure 4.34 The steel “floor brake” 116

Figure 4.35 The “floor brake” at the back of the foundation during the test 117

Figure 4.36 View of Specimen-1 118

Figure 4.37 Specimen-1 using conventional shear key configuration 121

Figure 4.38 Steel strain gauges for Specimen-1 123

Figure 4.39 Specimen-2 using inverted shear key configuration 125



Figure 4.42 Steel strain gauges for Specimens-2 through 5 127


Figure 4.44 Construction of the conventional shear key using a rubber water-stop 128

Figure 4.45 Specimen-5 before casting of concrete for the wall 129

Figure 4.46 Schematics diagram of the pretest 131

Figure 4.47 The moment curvature curve for Specimen-4 135

Figure 5.1 Specimen-1 with wall thickness of 400 mm and conventional shear key configuration

with water-stop 136

Figure 5.2 The applied lateral load on top of the wall 138

xv

Figure 5.3 The displacement of the top of the wall 138

Figure 5.4 First visible crack at the wall-foundation connection 139

Figure 5.5 Strain values for the wall rebars at the connection of the wall-foundation 139

Figure 5.6 Leakage of the water through the joint region at the initial stage of the test 140

Figure 5.7 The first leaking crack above the joint at the west side face of the wall 141

Figure 5.8 The first leaking crack above the joint at the east side face of the wall 141

Figure 5.9 Leakage at the east face of the wall at 260 kN 142

Figure 5.10 Cracking of the side faces of the wall after the test 143

Figure 5.11 The condition of the back face of the wall immediately after the test 143

Figure 5.12 Cracks at the front face of the wall 480 mm above the foundation 144

Figure 5.13 Cracks at the front face of the wall after conclusion of the test 144

Figure 5.14 Cracks at the front face of the wall after conclusion of the test 145

Figure 5.15 Extensive crack at the connection of the wall and foundation 145

Figure 5.16 The applied lateral load on top of the wall 146

Figure 5.17 Top displacement of the wall 146


Figure 5.19 Strain values for the top layer of the foundation reinforcement 100 mm in front of

the front face of the wall 148

Figure 5.20 Strain values for the top layer of the foundation reinforcement 100 mm behind the

back face of the wall 148

Figure 5.21 Horizontal displacement of the foundation 150

Figure 5.22 Variation of the hydraulic jack force during the test 150

Figure 5.23 Overlaps for the one layer FRP before application of the resin 151

Figure 5.24 The FRP on the foundation and the joint inside the chamber 152

Figure 5.25 The FRP inside the chamber 152

Figure 5.26 The specimen during the first stage of the test 153

Figure 5.27 Leakage of the specimen through the existing cracks 153

Figure 5.28 The joint before installation of the wall form 154

Figure 5.29 Leakage at the back face of the wall at the initial stage of loading (40 kN) 155

Figure 5.30 Applied lateral load on top of the wall 155

Figure 5.31 Lateral displacement of the top of the wall 156

xvi

Figure 5.32 Lateral load-displacement relationship for the top of the wall 156

Figure 5.33 Strain value of the steel at the connection of the wall-foundation 157

Figure 5.34 Leaking cracks at the west side face of the wall at 210 kN 158

Figure 5.35 Leaking cracks at the east side face of the wall at 280 kN 158

Figure 5.36 Leakage condition at 320 kN 159

Figure 5.37 Brittle failure of the wall at the end of the test 160

Figure 5.38 Displacement of the top of the wall during the entire test 160

Figure 5.39 Strain of the steel at the wall-foundation interface at the front side of the wall 161

Figure 5.40 Strain of the steel at wall-foundation interface at the back side of the wall 161


Figure 5.42 Specimen-3 having a conventional shear key configuration with water-stop 162

Figure 5.43 Specimen-3 having a conventional shear key configuration with water-stop

immediately after casting of the concrete for foundation 163


Figure 5.45 Leakage from the crack at west side face, about 430 mm 165

Figure 5.46 Leakage from the crack at west side face 166

Figure 5.47 Leakage from the cracks at west side faces, at load level of 200 kN 166

Figure 5.48 Leakage condition at the back face of the wall at load level of 210 kN 167


Figure 5.50 Strain values for the wall rebars at the wall-foundation interface 168


Figure 5.52 The retrofitted wall using two layers of GFRP 170

Figure 5.53 Leakage condition at the west side face of the wall at 30 kN 170

Figure 5.54 Leakage condition at the west face of the wall at 210 kN 171

Figure 5.55 Leakage condition at the back face of the wall at load level of 250 kN 171

Figure 5.56 The leaking crack at the back of the wall 172

Figure 5.57 Leakage at the back face through previously formed cracks 172

Figure 5.58 The retrofitted specimen after the test 173

Figure 5.59 The conventional shear key using a plastic water-stop for Specimen-4 174

Figure 5.60 Specimen-4 with wall thickness of 300 mm 174


xvii

Figure 5.62 Strain value of the steel at the connection of the wall-foundation region 176

Figure 5.63 Condition of the cracks at the east face of the wall 177

Figure 5.64 Condition of the cracks at the west face of the wall 178



Figure 5.67 West side face of the wall 181

Figure 5.68 Leaking cracks at the load level of 190 181

Figure 5.69 Back side of the wall at load level of 230 kN 182

Figure 5.70 Back side of the wall at load level of 250 kN 182



Figure 5.73 Loading for top of the wall 185


Figure 5.75 Extensive cracking of the front face of the wall after the test 186

Figure 5.76 Damage at the back face of the wall after the test 187

Figure 5.77 Specimen-5 with wall thickness of 400 mm 188

Figure 5.78 The conventional shear key using a rubber water-stop 188

Figure 5.79 The continuous crack at the front face at load level of 110 kN 189

Figure 5.80 The continuous crack at the front and west side faces of the wall 189

Figure 5.81 The continuous crack at the back and west side face of the wall 190



Figure 5.84 Leakage of the west side face at the beginning of the test 192

Figure 5.85 Leakage at the back side of the wall at the beginning of the test 193

Figure 5.86 Leakage at the east side face of the wall at the beginning of the test 193

Figure 5.87 Leakage of the wall at 210 kN load 194



Figure 5.90 Loading for top of the wall 197


Figure 5.92 Extensive cracking of the front face of the wall at 220 kN 198

xviii

Figure 5.93 Separation of the corner of the wall at connection to foundation 199

Figure 5.94 Extensive damage to the base of the wall at the end of the test 199

Figure 6.1 Relationships between the ductility component (Rd) and the displacement ductility

ratio (μ) at short period 210

Figure 7.1 Uniaxial behaviour of plain concrete (Hibbitt et al., 2007) 217

Figure 7.2 Effect of assignment of values of wt and wc on the behaviour of concrete 219

Figure 7.3 The postfailure behaviour of concrete in tension (Hibbitt et al., 2007) 220

Figure 7.4 The postfailure behaviour of concrete in compression (Hibbitt et al., 2007) 222

Figure 7.5 Comparison of isotropic and kinematic hardening 223

Figure 7.6 Meshing scheme for the column (approximate seed size of 75 mm) 226

Figure 7.7 Comparison of the effect of modelling the steel behaviour 228

Figure 7.8 Comparison of the experimental and analytical results 233

Figure 7.9 The column with coarser mesh (approximate seed size of 150 mm) 234


Figure 7.11 The proposed tri-linear material behaviour for steel 236


Figure 7.13 Modelling of Specimen-4 and the floor 237

Figure 7.14 Modelling the reinforcement for the wall and foundation 238

Figure 7.15 Special springs for anchoring the specimen to the floor 238

Figure 7.16 Boundary conditions defined for the model 240


Figure 7.18 Steel strain value at the front side of the wall at the wall-foundation interface 241

Figure 7.19 Steel strain value at the back side of the wall at the wall-foundation interface 241


Figure 7.21 Top displacement of the wall during the ABAQUS analysis 244

Figure 7.22 Steel strain value at the front side of the wall at the wall-foundation interface 245

Figure 7.23 Steel strain value at the back side of the wall at the wall-foundation interface 246


Figure 7.25 Comparison of the response of the wall and the pushover analysis 247

Figure 7.26 Analytical results for steel strain value at the interface of the wall-foundation 248

1

CHAPTER-1

INTRODUCTION

1.1 General

Liquid containing structures (LCS) are used for storage of water, petroleum products,

oxygen, nitrogen, high-pressure gas, LNG (Liquefied Natural Gas), LPG (Liquefied

Petroleum Gas), and so on. There are many types of such storage tanks depending on the

structure, construction material, content, volume, storage condition, etc. They include

cylindrical, spherical, and rectangular tanks. Since the pressure resistance of a spherical

tank is the highest compared to other shapes, it is ideal for storage of high-pressure gas

such as LPG. Most of the above mentioned tanks, particularly the cylindrical ones, have

been made of steel; however, concrete tanks are becoming more popular due to improved

service life and durability and also due to numerous observed failures of the steel tanks

during past earthquakes. Reinforced Concrete LCS have been used in environmental

engineering structures such as water reservoirs and sewage treatment tanks. While

cylindrical shapes may be structurally suitable for tank construction, rectangular tanks are

often preferred for water treatment process related purposes. Water tanks provide for the

storage of drinking water, fire suppression, agricultural farming and livestock, food

preparation and many other applications. The continual functioning of LCS is necessary

for the well being of a society. These structures may provide services necessary for the

emergency response of a community after an earthquake. The stored water might be used

for rescue work in case of fire or for drinking or sanitary purposes by surviving people,

where water shortage could increase the worry of epidemics. Also some of these LCS

might contain liquids such as oil or petrol or even hazardous materials. In the confusion

following an earthquake, due to the failure of the electricity system and the resulting loss

of lighting, the rescuing staff may not be able to immediately discover the leakage of

hazardous material. Leakage of such materials if accompanied by a fire might cause

damages many times greater than the earthquake itself.

2

Reinforced Concrete (RC) Liquid containing structures (LCS) are designed based on

serviceability criteria such as leakage. These structures are designed not only to have

functionality during the normal life cycle, but also to withstand the earthquake loading

without any extensive cracking that causes leakage. In RC tanks, cracking which leads to

leakage can be regarded as a possible mode of failure. Therefore a thorough

understanding of the cracking phenomenon in concrete member, especially during the

earthquake loading, is of main importance. Generally, the water tanks are under the

effects of the hydrostatic and gravity loads. During earthquake load reversals, the

situation becomes very complex as there are different components induced by the ground

movements. Therefore, the water tank should be designed considering all such effects.

An important aspect of the design of any structure for the earthquake loading is the

response modification factor (R factor), which is used for calculation of the design forces.

In zones of relatively frequent seismic activity, intense earthquakes are rare events. Most

structures might not experience a design earthquake and, therefore, design to resist such

events without damage would be economically impractical. In regions of strong ground

shaking, it may be impractical to design tanks for forces obtained from elastic (no

damage) response analysis without considering the nonlinear behavior and the ductility of

the structural system. R factors are assumed to represent the ratio of the forces that would

develop under a specified ground motion if the structure behaves entirely elastically to

the ones prescribed as design forces at the strength state assumed equal to a significant

yield level. Therefore, it is possible to design an RC structure for forces smaller than the

elastic forces and safely survive the ground motion excitation. When subjected to strong

shaking, tanks therefore are expected to respond in a non-linear fashion and may

experience some damage in terms of cracking. In LCS, the R factor needs to reflect

serviceability limits including leakage, which makes the determination of the appropriate

R value more complex. Little attention has been drawn into the behavior of RC

rectangular tanks. Also, no code existed in North America until 2001 to address the

seismic design of concrete tanks. The current R values included in the ACI Code (ACI

350.3/350.3R 2006) are empirical and without justification, and therefore, questionable

by designers.

3

The leakage failure mechanism under earthquake loading is different from that under

static loading. No cyclic load test to relate reinforcement and concrete stress/strain levels

with leakage limit state has yet been reported. It has been experimentally observed that in

an RC specimen under monotonic flexure, the compression zone significantly enhanced

the water-tightness of the specimen. It is possible that the R factor specified in the

relevant codes have not incorporated the necessary factors regarding the leakage behavior

of RC members. The reason might be that neither the leakage phenomenon under cyclic

load reversals is fully understood, nor the effect of factors such as compression zone,

ductility of the member, and over-strength has been accurately incorporated in the value

of R factor. Therefore, experimental tests are needed to examine the leakage limit state

due to reinforcement and concrete stress/strain levels, and to account for the stiffness

degradation and softening under cycling loading. This study is aimed at evaluation of the

leakage behavior of ground supported open top rectangular concrete tanks under the

effect of cyclic loading. Analytical studies have shown that the connection portion of a

wall-foundation at the middle of the larger side of a rectangular tank, in which the wall

behaves like a cantilever member, is the most critical region with respect to leakage. In

the current study, experimental tests are conducted on several cantilever wall

specimens that are representatives of full scale base slab-wall connection portion of

rectangular tanks. The walls are designed and built using the provisions of ACI

350.3/350.3R (2006). Cyclic loading is applied on the top of the wall to simulate the

earthquake loading while the critical region of the wall is subjected to water pressure to

investigate the leakage behavior. Assuming that a specified crack width that would

initiate leakage can be considered to be a function of the steel reinforcement stress/ strain,

then, one of the most important objectives of the experimental program would be the

determination of the stress/strain in the steel at the onset of leakage. The effect of

retrofitting of the cracked wall with Glass Fiber Reinforcing Polymer (GFRP) sheet with

respect to leakage under cyclic loads is also investigated.

The results and findings of the above experimental program are followed by analytical

simulation of the experimental tests using a finite element (FE) software, to facilitate the

generalization of the results to a broader range and more accurate determination of the R

factor for the RC rectangular LCS.

4

1.2 Performance criteria of RC tanks

In RC tanks, leakage can be regarded as one of the most critical modes of failure.

Leakage of tanks might also contribute to further damage of the tank by washing the

earth on which the tank is supported. The uneven settlement of the foundation can cause

the leaking cracks to expand. The structural design criteria of environmental engineering

structures for earthquake are quite different from those of general building structures.

General building design criteria is based on life safety and collapse prevention,

consequently considering structural strength limit states. An environmental engineering

structure, however, requires serviceability limit states such as leakage, deflection, and

durability limit, due to the nature of its use. Of these serviceability limits, the leakage

limit state may govern. Currently, Canadian design standards do not directly address the

structural design of environmental concrete structures. A variety of other standards such

as New Zealand Standard (NZS-3106 1986), and British Standard 8007 (BS-8007 1987)

are available. However, the standard of American Concrete Institute (ACI 350 2006) is

one of the most comprehensive and preferred standards and will be referred to as the

Code throughout this study.

One of the leakage controls is through crack control. The size of crack for a liquid

containing structure was limited to approximately 0.25 mm (0.01 in) for normal

environmental exposure in the previous edition of ACI Code (ACI 350 2001). This

criterion was based on the performance of actual LCS under service loads and to a greater

extent a result of the research work done by Gergely and Lutz (1968) mainly based on

monotonic tension and/or flexural test. In the current Code (ACI 350 2006), the

maximum permissible reinforcement stress at service loads which is dependent on

reinforcement spacing is specified to limit the crack width. Numerous studies have been

conducted to evaluate the crack width. Available formulas are usually based on

simplifications and were subsequently incorporated into national codes. Some of the code

equations are based upon classical models, such as the bond-slip mechanism used in the

CEB/FIP (1990) approach, and others, such as the ACI approach, are based on empirical

equations obtained by regression analysis of test data. It has been shown that the use of

various code equations to estimate the crack width in the same member can result in

5

widely different values (Beeby 1979). While there is no agreement on the most

appropriate model to predict width and spacing of flexural cracks, there is an agreement

on the major factors affecting crack width. The average strain in the reinforcement

relative to that in the adjacent concrete has been stated as the most influential parameters

for crack width (Borosnyói and Balázs 2005). In structures where the control of crack

width and prevention of leakage are of paramount importance, the critical design

parameter can be considered to be the steel stress/strain.

In addition to the mentioned problems stated above, the crack width might not be a good

variable for prediction of the leakage because it does not consider several factors such as

the effect of compression zone and the condition of the RC member under cyclic loading.

It is important to note that in this research, because of the methodology used for

simulation of the water pressure, measuring the width of the cracks during the cyclic

testing of the specimens is almost impossible. As was mentioned above, the tensile strain

in the steel is the most influential parameter related to the crack width. Also the level of

the strain in the steel at both faces of the wall seems to be a better parameter for capturing

the behavior of the RC member during cyclic loading which is not possible by crack

width measurement (after the crack closes during cyclic loading, there is no parameter to

be measured). In this research, an attempt is made to correlate the leakage phenomenon to

the condition of the reinforcement strains at front and back sides of the wall. This has

another advantage as the level of the forces in the RC member can be easily correlated to

the level of the strain in the reinforcement using well-established theories of RC

members.

1.3 Objectives The focus of this study is on the understanding of the leakage failure mechanism, and on

clarifying the deciding factors with respect to leakage of the rectangular RC water tank

walls under cyclic loading. The experimental observation can lead to information

regarding the appropriate design values for earthquake loading which is reflected in the

“R factor”.

The objectives of this research program are to:

6

1. Understand the leakage failure mechanism of rectangular RC water tank walls

under cyclic loading.

2. Investigate the critical leakage failure zones of a cantilever wall.

3. Understand the deciding factors involved in the leakage failure mechanism of

rectangular RC water tank wall.

4. Investigate the effect of shear key configuration on the leakage under cyclic

loading

5. Investigate the effect of retrofitting of the cracked section with GFRP sheets with

respect to leakage.

6. Simulate the behavior of the RC wall under cyclic loading using finite element

techniques.

7. Assess the current values of R factors prescribed in the codes and recommend an

acceptable range of values of R for rectangular tanks.

1.4 Organization

This thesis consists of eight chapters.

Chapter-2 is an introduction into the behavior and design of LCS for seismic loading. It

also presents a literature review on the previous studies on the seismic behavior of LCS

including the mathematical models proposed for description of the behavior of such

tanks. The design procedure outlined in the current Code is also presented. Relevant

topics in dynamics of structures such as the response spectrum method for the elastic and

inelastic response of a system are presented in Chapter-3. A historical review on the

evolution of the R factor is presented followed by details of its constituents including the

overstrength factor, ductility factor, and also previous research dedicated to determination

of these constituents. A parametric study on the effect of R on the key design parameters

is also presented at the end of Chapter-3.

Chapter-4 discusses the details of the set up used for the experimental test in the

structural laboratory. The details of design and construction of the specimens will follow

accompanied by several photos to better present the experimental program part of the

current research. Chapter-5 is dedicated to reporting of the experimental observation of

different tests, such as pretest, leakage test, leakage test of the retrofitted specimen, and

7

test of the specimen up to failure for the five specimens in this study. The details of the

experimental observation are accompanied by several photos of different stages of each

test. Chapter-6 presents the summary of the observations made in the experimental tests

and also proposes a method to determine the R factor. Chapter-7 reports the analytical

studies of the current research. This Chapter presents the material models for concrete

and reinforcement available in the ABAQUS software (Hibbitt et al. 2007) which is used

in the current study. Then the validation of the ABAQUS software by comparing the

analytical and experiential results of the test on cyclic behavior of an RC cantilever

column available in literature is presented. The Chapter then concludes by simulation and

investigation of part of the experimental tests in the current research. Chapter-8 presents

the conclusion of the study along with some recommendations for future work in the field

of seismic design of RC LCS.

8

CHAPTER-2

BEHAVIOUR AND DESIGN ASPECTS OF LIQUID CONTAINING STRUCUTRES (LCS)

2.1 General

The safe performance of liquid containing structures (LCS) during and after earthquakes

is more important than the economic values of the tank and their contents, as the

consequence of failure of these structures might be more disastrous than their destruction

due to earthquake itself. It is important that utility facilities remain operational following

an earthquake to meet the emergency state requirements such as firefighting water and to

meet the public demands as a source of water supply for the survivors. On the other hand,

the containment of hazardous materials is important to meet the public safety after an

earthquake. For these reasons, serviceability limits becomes prime design consideration

in most of these structures. There has been a great deal of research carried out on the

seismic behavior of LCS. These researches include observing and identifying the causes

of damage and failure for LCS, proposing mechanical and mathematical models for the

induced forces in these structures during earthquakes, and experimental and analytical

studying of the seismic behavior of LCS.

The behavior of concrete tanks for LCS during earthquakes is quite different from that of

steel tanks. The most common damages in a steel tank can be considered to be the

buckling of the wall near its base and the failure of the anchorage system. Earthquake

loading results in an overturning moment on the tank. Steel tanks are flexible structures

and have inadequate resistance against lateral movement and overturning moments. As

water sloshes and moves to one side of a tank, the tank wall lifts off the ground. The tank

wall then impacts the ground when the tank rocks in the opposite direction, causing

severe compression loading and resulting in wall buckling near the base (also termed

elephant's foot buckling). However, in concrete tanks, large hydrodynamic forces could

develop in the highly stressed regions of the tank which could lead to severe cracking and

leakage which may eventually lead to failure of tank. Therefore, the leakage of concrete

9

tanks can be considered to be the most important design consideration for earthquake

loading effects. Accordingly, these structures are designed not only to have functionality

during the normal life cycle, but also to withstand the earthquake loading without any

extensive cracking which leads to leakage.

2.2 Damage to Liquid Storage Tanks There have been several catastrophic failures of storage tanks such as the failure of the

large steel molasses tank in Boston Massachusetts USA on January 14th, 1919. The

explosion of the large tank (around 90 feet in diameter and 50 feet in height) which was

filled to the top, killed 21 and injured 150 people. The disaster was caused by poor design

and construction, with a wall too thin to bear repeated loads from the contents. The

increased internal pressure due to production of carbon dioxide as a result of fermentation

occurring inside the tank is thought to have contributed to the disaster. The tank had only

been filled to capacity eight times since it was built a few years before, putting the walls

under an intermittent cyclical load.

Numerous reports have been published regarding the damage to tanks due to

earthquakes: Long Beach (USA), 1933 (Lund 2002); Alaska (USA), 1964 (Hanson

1977); Niigata (Japan), 1964 (Watanabe 1966); Coalinga (USA), 1983 (Manos and

Clough 1985); and Kocaeli (Turkey), 1999 (Lund 2000) among others.

During and after the Turkey Earthquake of August 17, 1999, the air release of 200 metric

tons of hazardous anhydrous ammonia; the leakage of 6500 metric tons of toxic

acrylonitrile (ACN) from ruptured tanks; the spill of 50 metric tons of diesel fuel from a

broken fuel loading arm, liquid petroleum gas leakages, and oil spills at oil refinery, and

the enormous fires were examples of catastrophes that can occur as a result of storage

tank failure. For instance, the environmental damage due to the ACN release included the

death of all animals on the grounds of the plant and all vegetation within 200 meters of

the tanks (Steinberg and Cruz 2004).

The fire that occurred as a result of the1964 Niigata Earthquake burned for more than 14

days consuming around 122 million liters of oil. Investigation of the cause of this fire led

to the conclusion that friction and impact between the roof and sidewall of the storage

10

tank led to sparking. The seal material between the roof and the sidewall was metallic,

and it was the seal that led to sparking when it scraped against the side wall. These sparks

ignited the petroleum vapors contained inside the tank, leading to a major fire.

Numerous damages to unanchored steel tanks have also been reported in Peek (1986).

The Northridge earthquake caused extensive damage to some major lifeline facilities in

the Los Angeles area. In the San Fernando Valley area, the earthquake caused damage to

five steel water storage tanks. Much of the observed damage was attributed to uplift of

the tanks during the earthquake. All of the damaged tanks suffered some form of

buckling at the tank walls. The roofs of several tanks collapsed owing to impact of the

sloshing water waves. The earthquake caused damage to many liquid storage tanks in the

power generating plant. Two 113,500 steel tanks shifted 100 mm from their original

positions, even though the tanks had been anchored to a reinforced concrete ring

foundation by 50.8 mm diameter steel anchor bolts. Two fire-fighting water storage tanks

were also damaged, losing their water contents. One of the fire-fighting water tanks was a

bolted steel tank unanchored at the base. The other was a welded steel tank with a very

strong anchorage system. The bolted tank failed in "elephant foot" buckling of the tank

wall around its base, whereas the welded tank anchorage system prevented the damage to

the welded tank wall (Lund 1996).

During past earthquakes concrete LCS have also suffered damages. In the San Fernando

earthquake of 1971, an underground water reservoir was subjected to an estimated

inertial force of 0.4g and suffered severe damages in terms of the collapse of the wall

(Jennings 1971). During the south-central Illinois earthquake of November 9, 1968, a

reinforced concrete ground-level tank opened along pre-existing hairline cracks, sending

a jet of water 50 feet (15.2 m) into the adjacent parking area. The cylinder, which had

walls 1 foot (305 mm) thick and 12 feet (3.6 m) high and an outside diameter of 52 feet

(15.8 m), was bound with five 3/4-inch (20 mm) bands that probably prevented the

complete failure (Gordon et al. 1970).

There are also other types of reported damage including failure of the supporting system,

failure of connection between the tank and piping or accessory system, and foundation

failure due to liquefaction of the supporting soil. Based on the observation of damages to

LCS during past earthquakes it is evident that these structures, in addition to the routine

11

hydrostatic loading, are subject to large hydrodynamic forces during seismic events. As a

result, high stresses can cause buckling failure in the steel tanks and could cause leaking

crack failure in concrete tanks. The importance of preventing such damages has attracted

many researchers to study and provide a better understanding of the seismic behavior of

LCS. These studies might lead to improvement of the seismic behavior of LCS for the

future events

2.3 Previous studies on the seismic behavior of LCS The dynamic response of liquid-filled tanks has been studied both experimentally and

theoretically by many researchers. Initial analytical studies involved the hydrodynamics

of liquids in rigid tanks resting on rigid foundations. Jacobsen (1949) calculated the

effective hydrodynamic masses and mass moments for the fluid inside a cylindrical tank

when the base of the tank experienced a horizontal translation. Jacobsen and Ayre (1951)

studied the effect of ground motions on cylindrical storage tanks. In their study, four

tanks, from 150 mm to 1200 mm in diameter, were subjected simultaneously to transient,

horizontal "ground motions" of simplified type. The data included samples of the wave

envelopes, photographic studies of the wave formation, maximum wave heights and the

locations of these maxima, and the fluid damping coefficients. Housner (1963) proposed

a widely used analytical model for circular and rectangular rigid tanks in which the

hydrodynamic pressures were separated into impulsive and convective components. Fluid

was assumed incompressible and the walls were assumed to be rigid. It was shown that a

part of the liquid moves in long-period sloshing motion, while the rest moves rigidly with

the tank wall. The latter part of the liquid known as the impulsive liquid experiences the

same acceleration as the ground, and contributes predominantly to the base shear and

overturning moment. Housner's theory has then served as a guideline for most seismic

designs of liquid storage tanks. However, failures of liquid storage tanks during past

earthquakes suggested that Housner's theory may not be conservative. The first finite

element method for evaluating the seismic behavior of flexible tanks was proposed by

Edward (1969). The hydrodynamic effects of fluid were taken into account as an added

mass matrix in the equation of motion of the coupled fluid-tank system (Ma et al. 1982).

Arya et al. (1971) studied the dynamic characteristics of liquid containers fixed at the

12

base and free at the top. Virtual mass due to the liquid was considered but sloshing effect

was ignored. The first experimental tests of the seismic response of a large scale thin-

shell liquid storage tanks were conducted by Clough and Clough (1977), and Clough and

Niwa (1979). The observations indicated that the pressures were much larger than those

obtained from Housner's equations. This difference in pressures was attributed to the

flexibility of the tank wall. Haroun (1984) presented a detailed analytical method for

rectangular tanks where the hydrodynamic pressures were calculated using classical

potential flow approach assuming a rigid wall boundary condition. Veletsos and Yang

(1974) analyzed the earthquake response of a circular liquid storage tank, assuming the

tank as a cantilever beam and considering a deformed shape of the tank system. The fluid

tank system was treated as a single degree of freedom system in terms of the lateral

displacement of the tank at the free surface level. The fluid inertial effect was considered

by means of an added mass concept in which an appropriate part of fluid mass is added to

the structural mass. The study was limited only to the impulsive component. Yang (1976)

found that for circular tanks with realistic flexibility, the impulsive forces are

considerably higher than those in a rigid wall. Clough (1977) tested different specimens

of broad cylindrical tanks on a shaking table at the University of California, Berkeley.

The results showed that the circumferential and axial wave modes of the shell theory

were strongly excited by seismic loads. Niwa (1978) conducted experimental tests on a

scaled model of a ground-supported, thin-shell, cylindrical liquid storage tank to assess

the applicability of current seismic design practice to such liquid storage tanks. Hunt and

Priestley (1978) studied the dynamic behavior of inviscid fluid contained in horizontally

accelerated cylindrical and rectangular tanks and developed mathematical equations

describing the fluid motion. Comparisons of predicted and measured free-surface

displacements of a model cylindrical water tank subjected to both sinusoidal and seismic

accelerations on a shaking table indicated close agreement between theory and

experiment. The solutions for seismic accelerations in one horizontal direction were then

generalized to include acceleration components in all three coordinate directions.

Balendra (1979) presented a finite element analysis of an annular cylindrical tank with an

axisymmetric elastic dome.. The fluid inside the tank was considered as inviscid and

incompressible and sloshing of the fluid was neglected. Clough et al. (1979) conducted

13

experimental studies on the earthquake response behavior of ground-supported, thin-

shell, cylindrical liquid storage tanks using the shaking table at University of California at

Berkeley. Several models fabricated from sheet aluminum, were subjected to simulated

earthquake accelerations with intensities up to 0.5 g. Shih and Babcock (1980) conducted

experimental evaluation of tank buckling under the effects of ground motion and

confirmed the important role of the fundamental mode in tank failures. Haroun and

Housner (1981) showed that the flexibility of the tank wall may cause the impulsive

liquid to experience accelerations that are several times greater than the peak ground

acceleration. They also conducted vibration tests on full-scaled liquid storage tanks

(Housner and Haroun 1979). Manos and Clough (1982) compared the static and dynamic

lateral load responses of a ground-supported cylindrical liquid storage tank model. A

rigid and a flexible foundation were studied. It was found that the rotational uplift

mechanism was accentuated by the static excitation as compared with that produced by

the dynamic input, particularly for the soft foundation material. Veletsos (1984)

considered the effect of the wall flexibility on the magnitude and distribution of the

hydrodynamic pressures and associated tank forces in circular tanks. It was assumed that

the tank-fluid system behaved like a single degree of freedom system and the base shear

and moment were evaluated for several prescribed modes of vibration. The changes in the

impulsive response in tanks supported on flexible foundations through rigid base mats

were studied by Veletsos and Tang (1990). They concluded that the base translation and

rocking resulted in longer impulsive periods and larger effective damping. Park et al.

(1992) studied the dynamic behavior of concrete rectangular tanks considering both

impulsive and convective components and using boundary element modeling for the fluid

motion and finite element modeling for the solid walls. The time-history analysis method

was used to obtain the dynamic response of liquid storage tanks subjected to earthquakes.

Kim et al. (1996) studied the dynamic behavior of a 3-D flexible rectangular fluid filled

isotropic container using Rayleigh-Ritz method ignoring the effects of sloshing and

considering only the walls perpendicular to the direction of the ground motion. Koh et al.

(1998) presented a coupled boundary element method-finite element method to analyze

three-dimensional rectangular storage tanks subjected to horizontal ground excitation.

The tank was modeled using the finite element method and the fluid domain using the

14

indirect boundary element method. Chen and Kianoush (2005) developed a procedure

called the sequential method for computing hydrodynamic pressures in two-dimensional

rectangular tanks in which the effect of flexibility of the tank wall was taken into account.

In spite of numerous researches regarding the seismic behavior of LCS, the Housner’s

model has been adopted with some modifications in most of the current codes and

standards such as ACI 350.3-06 (2006).

2.3.1 The Housner’s Model For seismic design of environmental structures, a hydrodynamic pressure formula is

developed by G. W. Housner (Housner 1963). Housner specified the hydrodynamic

pressure induced by the ground motion as an impulsive part and a convective part. In his

model, the fluid was assumed incompressible and the walls were assumed to be rigid. The

tank walls were considered to be of constant thickness and connected to its base so that

no sliding or uplift may happen. When a tank containing fluid of weight WL depicted in

Figure 2.1 is accelerated in a horizontal direction; a certain portion of the fluid acts as if it

were a solid mass of weight Wi in rigid contact with the walls. Assuming that the tank

moves as a rigid body with the bottom and walls undergoing the same acceleration, the

mass then exerts a maximum horizontal force directly proportional to the maximum

acceleration of the bottom of the tank. Housner defined this force as an impulsive force,

Pi. Thus, the impulsive force is the one associated with the inertia forces produced by the

fluid due to impulsive movement of the tank wall. The acceleration also induces

oscillations of the fluid, contributing additional dynamic pressures on the walls, in which

a certain portion of the fluid, of weight Wc, responds as a solid oscillating mass flexibly

connected to the walls, thus producing another horizontal force. Housner defined this

type of horizontal force as a convective force, Pc. In the model, it is assumed that Wc is

flexibly connected to the tank walls that produce a period of vibration corresponding to

the period of fluid sloshing. The sloshing pressures on the tank walls result from the fluid

motion associated with the wave oscillation. The period of oscillation of sloshing

depends upon the ratio of fluid depth to tank diameter and can be as long as several

seconds.

15

Figure 2.1 Illustration of the parameters in the Housner’s model

2.4 Classification of ground supported tanks

Ground supported tanks including on grade and below grade structures can be classified

according to their configuration (rectangular or circular) and the wall base joint type

(fixed, hinged, or flexible base). Rectangular tanks, which are the subject of the current

research, are categorized as fixed base and hinged base (ACI 350.3-06). Figures 2.2 (a)

and (b) show the base configurations for two types of fixed base and two types of hinged

base rectangular tanks, respectively. The base water-stop is not shown in the figures. The

most general of the liquid-containing system examined is upright rectangular or circular

tanks that are supported through a rigid foundation. This study is limited to the response

of above ground RC rectangular liquid-containing tanks that are anchored at the base and

filled with homogeneous liquid that is assumed incompressible and inviscid. The tank

walls are considered to be of constant thickness. The study is limited to the tanks in

which the wall length is sufficiently larger than its height to promote a one-way action of

the wall.

2.5 ACI guidelines for design of LCS

A properly designed tank must be able to withstand the applied loads without cracks that

would permit leakage. The goal of providing a safe RC tank that will not leak is achieved

16

by providing the proper amount and distribution of reinforcement, the proper spacing and

detailing of construction joints, and the use of quality concrete placed using proper

construction practices. Currently, Canadian design standards do not directly address the

structural design of environmental concrete structures. A variety of other standards such

as Eurocode-8 (Eurocode-8 1998), New Zealand Standard 3106-1986 (NZS 3106, 1986),

British Standard 8007 (BS 8007 1987) is available. However, the standard of American

Concrete Institute (ACI 350-06) is one of the most comprehensive and preferred

standards.

Figure 2.2 Non-flexible wall to base connections

(a) Fixed (b) Hinged

The American Concrete Institute formed Committee 350 in 1964 through the necessity of

providing guidance for the design of environmental engineering structures. General

guidance for the design of concrete sanitary engineering structures was given by ACI

17

350-71, which was consistent with ACI 318-71. In ACI 350-71, working stress method

was introduced as the primary design method. The seismic load consideration was added

in ACI 350-77 revision by recognizing impulsive and convective pressure of

hydrodynamic pressures due to an earthquake. Modified ultimate strength design was

introduced in ACI 350-85 report to reduce the stress of the reinforcement to the working

stress design level by introducing sanitary durability coefficient.

The first North American Code for design of environmental engineering concrete

structures was published by ACI in 2001 (ACI 350-01 and ACI 350.3/350.3R-01). In that

code, the variables of seismic base shear formula such as the R factor were given based

on different boundary conditions and tank types. The range of the R value for

environmental engineering structures such as LCS was different from the R range for

general building structure. In that Code, two design methods (working stress method and

modified ultimate strength method) were suggested, and have been calibrated to produce

similar but not necessarily identical designs. The basic design trend of ACI 350 standards

was to limit the reinforcing steel stresses under normal working stress level. This was

done explicitly through working stress design procedures in ACI 350-71 report. In the

modified ultimate strength design, this is done implicitly by controlling the crack width.

Gergely and Lutz (1968) developed a crack formula based on a statistical evaluation of

experimental crack width data, which has been used to limit the reinforcement stress

level. In their proposed equation, rather than calculating the crack width directly, the z

factor was introduced to implicitly control the reinforcement stress. ACI 350-01 did not

limit crack width, but limited the value of the quantity z below certain limits based on the

environmental exposure condition. In previous codes, provisions were given for

distributions of reinforcement that were based on empirical equations using a calculated

maximum crack width of 0.254 mm (0.01 in.) for normal environmental exposure. The

current ACI Code (ACI 350-06) guidelines limit the maximum stress in the steel at

service level to the values based on the environmental exposure condition.

In ACI 350-06, for normal environmental exposure areas, the calculated maximum stress

fsmax in reinforcement closest to a surface in tension at service loads is limited by Eq.2-1

(per MPa) and shall not exceed a maximum of o.6fy.

18

2-1

where s and db (in mm) are the spacing and diameter of the wall reinforcement,

respectively. β is defined as the ratio of distances to the neutral axis from the extreme

tension fiber and from the centroid of the main reinforcement.

ACI 350-06 is based on ultimate strength design method. In environmental concrete

structure design, the ultimate strength design of ACI is modified to account for the

serviceability limit states by including durability factors in the design load combinations.

The intent of the environmental durability factor is to reduce the effective stress in non-

pre-stressed reinforcement to a level under service load conditions, such that stress levels

are considered to be in an acceptable range for control of cracking. This durability factor

does not apply to the load combinations that include earthquake loading.

The current ACI 350.3-06 Code incorporates an equivalent dynamic model based on

Housner’s model, for calculating the resultant seismic forces acting on the ground-based

fluid container with rigid walls. In this model, the equivalent mass of the impulsive

component of the stored liquid Wi represents the resultant effect of the impulsive seismic

pressure on the tank walls. The equivalent mass of the convective component of the

stored liquid, Wc represents the resultant of the sloshing fluid pressures. In the model, it is

assumed that Wi acts rigidly with the tank walls at an equivalent height hi above the tank

bottom that corresponds to the location of the resultant impulsive force Pi. The impulsive

components are generated by the seismic accelerations of the tank walls so that the force

Pi is evenly divided into a pressure force on the wall accelerating into the fluid, and a

suction force on the wall accelerating away from the fluid. During an earthquake, the

force Pi changes direction several times per seconds, corresponding to the change in

direction of the base acceleration. Wc is the equivalent mass of the oscillating fluid that

produces the convective pressures on the tank walls with resultant force Pc, which acts at

an equivalent height of hc above the tank bottom and has a period of vibration

corresponding to the period of fluid sloshing.

2 2 4 0, m a x

2 2( ) 4 ( 2 )2 5 5 0

f s ds bβ

=

+ +

19

The forces Pi and Pc exert overturning moments at the base of the tank wall. The forces Pi

and Pc act independently and simultaneously on the tank. The force Pi (and its associated

pressures) primarily acts to stress the tank wall, whereas Pc acts primarily to uplift the

tank wall. The vertical vibrations of the ground are also transmitted to fluid, thus

producing pressures that act on the tank walls.

2.5.1 Design loads According to ACI Code the walls shall be designed for static and dynamic forces as

described below.

2.5.1.1 Hydrostatic forces Fluids exert pressure perpendicular to any contacting surface. When fluid is at rest, the

pressure acts with equal magnitude in all directions. The liquid pressure (P) at a given

depth depends only on the density of the liquid (ρ) and the distance below the surface of

the liquid (h).

P ghρ= 2-2

Based on Eq. 2-2, a column of water of one meter can exert a pressure equal to 4.9 kPa

on the surfaces located at the bottom of the water column. The distribution of the

hydrostatic pressure on a wall of a rectangular liquid containing structure is shown in

Figure 2.3 and has the following pattern:

( - )p H yhy L Lγ= 2-3

Where Phy is the pressure exerted on a unit width of the wall at the height y. γL and HL

denote the specific weight and height of the contained liquid, respectively.

2.5.1.2 Hydrodynamic forces The hydrodynamic forces include the following:

• Impulsive and convective pressures

• Inertia forces of the walls

• Inertia forces of the roof (ignored in this study)

• Effect of vertical acceleration

• Dynamic earth pressure from soil (ignored in this study)

20

Figure 2.3 Distribution of the hydrostatic pressure on a rectangular wall

The impulsive and the convective masses of liquid are lumped at equivalent heights. For

calculation of the equivalent weights of the impulsive and convective components and the

height above the base where they act, ACI (ACI 350-3R-06) proposes the following

equations for rectangular tanks:

2-4

2-5

For tanks with L/HL < 1.333

2-6

For tanks with L/HL ≥ 1.333

2-7

tanh[0.866( / )]

0.866( / )

W L Hi LW L HL L

=

0.264( / ) tanh[3.16( / )]Wc L H H LL LWL

=

0.5 0.09375( / )hi L HLHL

= −

0.375hi

HL=

21

For all tanks

2-8

where L is the length of the tank parallel to the direction of the earthquake. The variation

of impulsive and convective weight factors and height factors with respect to L/HL ratio

are shown in Figures 2.4 and 2.5, respectively. In Figure 2.4, Wi/W, and Wc/W denote the

ratios of the impulsive and convective component weights, respectively, with respects to

the liquid weight. In Figure 2.5, the quantities hi/h and hc/h denote the ratios of the

impulsive and convective equivalent heights respectively, with respect to the liquid

height.

0.000.100.200.300.400.500.600.700.800.901.00

0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5

L/HL

Wi/ W

& W

c/W

Impulsive

Convective

Figure 2.4 Impulsive and convective mass factors vs. L/HL ratio

2.6 Dynamic force distribution above base for Rectangular tanks

According to the Code (ACI 350-3-06), walls perpendicular to the earthquake force and

in the leading half of the tank shall be loaded perpendicular to their plane by the wall’s

own inertia force P′w, one-half the impulsive force, Pi, and one-half the convective force,

cosh[3.16( / )] 11

3.16( / )sinh[3.16( / )]

h H Lc LH H L H LL L L

−= −

22

Pc. Walls perpendicular to the earthquake force and in the trailing half of the tank shall be

loaded perpendicular to their plane by the wall’s own inertia force P′w, one-half the

impulsive force, Pi, one-half the convective force, Pc, and the dynamic earth and ground-

water pressure against the buried portion of the wall. However, from a logical point of

view, it is not clear how the water which is pulled away from the wall at the trailing half a

tank can exert such a significant suction force on the wall.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5

L/HL

hi/ h

& h

c/ h

ImpulsiveConvective

Figure 2.5 Impulsive and convective height factors vs. L/HL ratio

The dynamic impulsive and convective design force components above the base are

determined as:

2-9

2-10

Where Ci and Cc are the seismic response coefficients; I is the importance factor ranging

between 1 to 1.5; Wi and Wc are the impulsive and convective weight components

described above; and Ri and Rc are the response modification factors corresponding to the

impulsive and convective components, respectively, as reported in Table 2-1. Figure 2.6

illustrates the components of the vertical flexural moment at the base of the tank for

WiP C Ii i Ri= ×

WcP C Ic c Rc= ×

23

impulsive [(hi/h) × (Wi/W)] and convective [(hc/h) × (Wc/W)] pressures vs. L/HL ratio.

The vertical bending moment components exerted by the above mentioned components

on the entire tank cross section above the base of the wall are:

2-11

2-12

Table 2-1 Response modification factors (ACI 350-3-06)

In order to calculate the seismic design forces, it is necessary to obtain the required

parameters of the dynamic model of the system as follows.

ikm

ω = 2-13

ωi is the circular frequency of the impulsive mode of vibration, and m is the total mass

per unit width of a rectangular wall calculated as the sum of wall mass (mw) and the

impulsive component mass (mi). For a wall of uniform thickness (tw) and height of Hw:

310=

twm Hw w cρ 2-14

Type of structure

Ri

Rc On or above

grade Buried*

(a) Anchored, flexible-base tanks 3.25 3.25 1.0

(b) Fixed or hinged-base tanks 2.0 3.0 1.0

(c) Unanchored, contained, or uncontained tanks 1.5 2.0 1.0

(d) Pedestal-mounted tanks 2.0 ---- 1.0

* Buried tank is defined as a tank whose maximum water surface at rest is at or below

ground level. For partially buried tanks, the Ri value may be linearly interpolated

between that shown for tanks on grade, and for buried tanks

M P hi i i= ×

M P hc c c= ×

( ) ( )2

ii L L

L

W Lm HW

ρ= × × ×

24

2-15

2-16

36 ( )

4 10= ×

×c wE tk

h 2-17

2 2= =ii

mTk

π πω

2-18

Figure 2.6 Components of the flexural moment at the base of the tank for impulsive and

convective pressures vs. L/HL ratio

Ti is the fundamental period of oscillation of the tank which includes the impulsive

component of the contents. The circular frequency of oscillation (ωc) of the first

convective mode of sloshing and is calculated as:

c Lλω = 2-19

3.16 tanh[3.16( )]HLgL

λ = 2-20

Tc is the natural period of the first (convective) mode of sloshing calculated as

( )w w i i

w i

h m h mhm m

+=

+

25

2c

c

T πω

= 2-21

Ci is a period dependent spectral amplification factor for the horizontal motion of the

impulsive component calculated as:

For ≤ =T T C Si s i DS 2-22

1For > = ≤SDT T C Si s i DSTi

2-23

1S DTs S D S= 2-24

SDS= the design spectral response acceleration at short periods

23 aS S FDS s= 2-25

SD1 = the design spectral response acceleration at 1 s period

21 13 VS S FD = 2-26

SS and S1 are the “Mapped Spectral Response Accelerations” at short periods and 1-sec,

respectively, and shall be obtained from the Seismic Ground Motion maps 22-1 through

22-14 of ASCE 7-05, Section 22. Fa and Fv are the site coefficients and shall be obtained

from Tables 11.4-1 and 11.4-2, respectively, of ASCE 7-05, in conjunction with Table

20.3-1, “Site Classification” of ASCE 7-05.

Cc which is a period dependent spectral amplification factor for the horizontal motion of

the convective component is calculated as:

1.51.6 1For 1.5≤ = ≤SDT C Sc c DST Ts c

2-27

2

2.41.6For > =SDST Cc cT Ts c

2-28

The distributions of the impulsive and convective pressures on a wall of a rectangular

26

LCS are shown in Figures 2.7 and 2.8, respectively.

Figure 2.7 Distribution of the impulsive pressure on a wall of a rectangular LCS

In shallow tanks most of the water sloshes (convective loading); whereas in tall tanks,

most of the water contributes to the impulsive load. The analysis approach takes into

account the period of the tank contents and the energy included in the response spectrum

at that period. As a result, per unit weight, sloshing water contributes much less lateral

loading on the tank wall than does water causing impulsive loading. The sloshing water

"runs up" the tank walls. If the tank has a roof, the sloshing water can damage the roof

structure if adequate freeboard is not provided (Ballantyne 2002).

To account for the effect of the vertical acceleration on the walls, the hydrostatic load

from the tank content is multiplied by the vertical spectral acceleration. In the absence of

a site-specific response spectrum, a value not less than 2/3 for the ratio of the vertical to

horizontal acceleration is recommended.

And lastly, the seismic induced load in the RC concrete wall due to its weight is

calculated as:

'' WwP C Iw i Ri

ε= × 2-29

P′w is the lateral inertia force of one accelerating wall, perpendicular to the direction of

the earthquake, and W′w is the equivalent weight of one wall perpendicular to the

27

direction of the earthquake. The effective mass coefficient, ε, is dependent on the ratio of

length of the tank to the height of the liquid and has a value of less or equal to unity.

Figure 2.8 Distribution of the convective pressure on a wall of a rectangular LCS

As the resultant forces (impulsive force and convective force) due to earthquake may not

simultaneously occur during and after earthquake, for the total horizontal base shear the

“Square Root Sum of Squares” approach is used.

2 2 2( )= + + +i w c vV P P P P 2-30

where V= total horizontal base shear, Pi = total lateral impulsive force, Pw = lateral inertia

force of the accelerating wall, Pc = total lateral convective force, and Pv = effect of

vertical acceleration.

The above procedure shows that several load components are influenced by the

magnitude of Ri, and therefore, the base shear and base moment are not linearly

proportional to the Ri value. This fact makes the determination of the Ri value a rather

complex process.

2.7 Effects of the tank dimensions on earthquake induced force properties

A study was conducted to study the effects of different variables on the design values of

rectangular RC tank walls. The height and thickness values of the walls varied from 3 to

28

10 m and from 300 mm to 1000 mm (equal to 10% of the height), respectively.

The length of the tank (in the direction of earthquake) was varied to determine the effect

of the ratio of liquid height to length of the tank (HL/L) on the response. The (HL/L) ratios

varied from 0.05 (a shallow tank) to 1 (a tall tank). Figure 2.9 shows the code prescribed

value of the period of the impulsive component (Ti) and convective component (Tc), vs.

HL/L for wall heights (Hw) ranging from 3 to 10 m. It was assumed that the height of the

wall is 500 mm above the free surface of the liquid. Ss, S1, Fv, Fa, I, Ri, and Rc were taken

as 150%, 60%, 0.8, 0.8, 1, 2, and 1, respectively. The values of 150% and 60% represent

a high seismic zone (e.g. California).

The impulsive period ranges from 0.04 sec for tanks with wall height of 3 m to around

0.18 sec for tanks with wall height of 10 m. The ratio of HL/L does not have much

influence on the period of vibration of a tank with a certain wall height. The period of

convective component is sensitive to HL/L ratio for shallow tanks. For shallow tanks with

HL/L equal to 0.05, the period of convective component varies from 20 sec for tanks with

wall height of 3 m to 40 sec for tanks with wall height of 10 m. For tall tanks, the period

of convective component is similar and below 5 sec for the range of the wall height

between 3 m to 10 m.

To investigate the seismic load effect on the RC rectangular tank wall, a study was also

conducted. Figures 2.10(a) and (b) illustrate the ratio of seismic flexural moment divided

by square of the wall height (ME/(HW)2), and the ratios of seismic shear force divided by

the wall height (VE/HW), respectively, with respect to HL/L. The figures indicate that the

value of (ME/(HW)2), (VE/HW) are almost constant for a wide range of HL/L, but increase

due to the increase in the height of the wall.

29

Period of vibration (Ti)

0

0.04

0.08

0.12

0.16

0.2

0 0.2 0.4 0.6 0.8 1 1.2

HL/L

Perio

d (s

ec)

35810

Period of vibration (Tc)

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8 1 1.2

HL/L

Perio

d (s

ec)

3

5

8

10

Figure 2.9 Comparison of the period of vibration vs. HL/L ratio

(a) impulsive component (Ti) (b) convective component (Tc)

Figures 2.11(a) and (b) show the ratio of seismic to static flexural moment and shear

force maximum values at the base of the wall, respectively, with respect to different

HL/L. The earthquake effect is more critical for shorter water tank walls (3 to 5 m) than

the higher water tank walls (6 to 10 m). The ME /MS and VE/VS ratio become smaller as

the height increases.

(a)

(b)

Hw

Hw

30

Flexural moment

020406080

100120140

0 0.2 0.4 0.6 0.8 1

HL/L

ME/(

H W)2 3

5810

Shear force

0

5

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1HL/L

V E/H

w 35810

Figure 2.10 Comparison of the ratio of seismic loads vs. HL/L ratio

(a) flexural moment (b) shear force

(b)

(a)

Hw

Hw

31

Flexural moment

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 0.2 0.4 0.6 0.8 1 1.2

HL/L

ME/M

S 35810

Shear force

0.00

0.20

0.40

0.60

0.80

0 0.2 0.4 0.6 0.8 1 1.2

HL/L

V E/V

S

35810

Figure 2.11 Comparison of the ratio of seismic to static loads vs. HL/L ratio

(a) flexural moment (b) shear force

(a)

(b)

Hw

Hw

32

CHAPTER-3

NONLINEAR RESPONSE OF RC STRUCTURES

3.1 General Building codes require design procedures to provide a low probability of failure or

collapse under any probable occurrence of the loading such as earthquake, and to provide

sufficient stiffness such that the serviceability of the structure can be maintained

following routine loading. Most buildings may not experience a design earthquake and,

therefore, design to resist such events without damage would be economically

impractical. In regions of strong ground shaking, it is sometimes impractical to design a

building for forces obtained from elastic (no damage) response analysis. Elastic forces

can be so large that they are reduced by a Response Modification Factor “R” to obtain the

design forces. Therefore, it is possible to design an RC structure for forces smaller than

the elastic forces and safely survive the ground motion excitation. This is mainly because

the RC concrete structures possess an inherent ductility that enables them to deform

inelastically without significant damage. Another important point is that the actual

response of the structure is generally better than stipulated by the design force level

mainly because of the effects of over-strength and redundancy. In this Chapter a brief

introduction into the dynamics of structures including the response spectrum method is

presented. A historical background on the evolution of the “R” factor is discussed

followed by the introduction of the ductility and overstrength components of “R” factor.

A summary of the extensive research which has been conducted with regards to the

mentioned components is also described. The design method based on the ACI 350 code

including the appropriate design load combinations incorporating the R factor and a

parametric study is presented.

3.2 Inelastic response of a structure during an earthquake 3.2.1 Effect of design parameters on the behavior (Chopra 2001, Tedesco et al. 1999)

33

For many engineering applications the maximum absolute values of the relative

displacement, relative velocity, and absolute acceleration of the system during a ground

motion are of main interest. These quantities are referred to as spectral displacement (Sd),

spectral velocity (Sv), and spectral acceleration (Sa), respectively. Plots of spectral

displacement (Sd), spectral velocity (Sv), and spectral acceleration (Sa) versus the un-

damped natural period of vibration (or frequency), for different values of damping factors

(ζ) are called earthquake response spectra. A set of such curves for different values of

damping factor is called the response spectra.

The response of a SDOF system to an earthquake is dependent on its natural period of

vibration and damping properties. Such spectra are very useful as they enable the

designers to assess the response of a structure with known natural period of vibration and

damping to a specific earthquake. Considering the practical range of the damping

parameters and using some simplifications the following approximations can be derived

between the spectral quantities:

1d vS S

ω≅ 3-1

a vS Sω≅ 3-2

For the engineering applications the pseudo-spectral velocity (Spv) and pseudo-spectral

acceleration (Spa) are employed as discussed in the following.

pv dS Sω= 3-3

2pa dS Sω= 3-4

Spv and Spa are called the pseudo-spectral velocity and pseudo-spectral acceleration,

respectively. The equivalent static force of the system (base shear) is the product of the

displacement, u (t), and lateral stiffness of the system (k). The definition of the pseudo-

spectral acceleration (Spa) results in the expression of the maximum equivalent static

force in terms of the product of the mass and the pseudo-spectral acceleration.

2max ( )f t kS m S mSs d n d paω= = = 3-5

34

Eq. 3-5 can be expressed in terms of the product of base shear coefficient, S pa

g, and the

weight of the structural system (W) as shown by Eq. 3-6.

SW pamS S Wpa pag g= = 3-6

It is very important to note that the maximum base shear is equal to product of mass and

pseudo-spectral acceleration (Spa) (and not the peak acceleration); therefore, for this

purpose the pseudo-acceleration response spectrum is sufficient for calculation of the

equivalent force (base shear).

The response of a structure to a past earthquake will be different than its response to

future earthquakes. However, certain similarities exist among earthquake ground motions

recorded under similar conditions. On the basis of these similarities, response spectra

from earthquakes with common characteristics have been averaged to create design

spectra. A design spectrum is a specification of the seismic design force or displacement

of a structure having a specified period of vibration and damping ratio. The design

spectrum consists of a set of smooth curves with one curve for each level of damping. A

design spectrum differs from a response spectrum. The response spectrum is a plot of the

peak response of all possible SDF system and is a description of a particular ground

motion. The smooth design spectrum is a specification of the level of seismic design

force, as a function of natural vibration period and damping ratio.

Past experience has indicated that many structures subjected to severe earthquake ground

shaking can undergo deformations beyond their elastic limit. Indeed, many structures

designed in accordance with the equivalent lateral force procedures specified in most

seismic design codes are expected to exhibit inelastic behavior during an extreme seismic

event. The equivalent static lateral force specified in the seismic design codes does not

represent the maximum dynamic forces expected to be exerted on a structure during a

major earthquake. These design seismic forces are equivalent to the actual earthquake

forces in the context that the structure designed to resist such forces (elastically) should

be able to:

1- Resist minor earthquakes without damage (neither structural nor nonstructural)

35

2- Resist moderate earthquakes without structural damage, but possibly experience

some nonstructural damage.

3- Resist a major earthquake without collapse, but possibly with some structural

damage as well as nonstructural damage.

Therefore, the inelastic response of structures during severe ground shaking represents a

critical aspect of earthquake engineering. The inelastic resistance-displacement

relationship exhibited by most ductile structures can be idealized by the dashed curve

shown in Figure 3.1. However, for convenience and to simplify the response calculations,

the actual resistance-displacement relationship is often idealized as a linear elastic-

perfectly plastic (or elastoplastic) representation as shown by the solid line in the same

figure. The elastoplastic approximation to the actual curvilinear resistance-displacement

curve is drawn so that the areas under both curves are the same at the value of the

maximum permissible displacement xm. With this type of force-displacement relation, the

maximum restoring force exhibited by the system is FSy (corresponding to the yield

strength).

The governing equation of motion for a SDOF elastoplastic system excited by ground

acceleration ( )gx t&& is given by:

( ) ( )mx cx F x mx ts g+ + = −&& & && 3-7

( )2 ( )

F xsx x x x tgmωζ+ + = −&& & && 3-8

where ζ is the damping ratio of the system and Fs(x) is the elastoplastic restoring force

characterized by the resistance-displacement relationship shown in Figure 3.1. The time

domain solution of Eq. 3-8 for any specified set of parameters and prescribed earthquake

excitation can be obtained using the step-by-step integration procedures. However, for

design purposes, the maximum system response is of primary interest, for which inelastic

response spectra and design spectra can be employed.

36

Figure 3.1 Elastoplastic idealization of the inelastic response

Studies conducted by Housner and Blume resulted in development of inelastic response

spectra. From observation of response spectra, it was concluded that:

(1) In the long-period region, the elastic and inelastic systems exhibit the same total

displacement,

(2)- In the intermediate period range, the elastic and inelastic systems absorb the same

total energy; and

(3) In the small-period region, the elastic and inelastic systems have the same restoring

force. In terms of the system parameters, these observations may be summarized as:

1- Long period range

x xep el≅ 3-9

FSFsy μ≅ 3-10

2- Intermediate period range

2 1x xep el

μμ

≅−

3-11

37

12 1

F Fsy Sμ≅

− 3-12

3- Small period range

x xep elμ≅ 3-13

F Fsy S≅ 3-14

where xel = elastic displacement

xep = elastoplastic displacement

FS= elastic restoring force

FSy= elastoplastic or yield restoring force

m = ductility factor (will be described in this section)

The response of an elastic SDF system is different than the response of corresponding

elastoplastic system under same earthquake excitation. Both systems have the same mass

and damping properties and also the same stiffness before yielding, and therefore, the

same natural period of vibration before yielding. FE and Fy are the peak forces induced by

the earthquake in the linear elastic system and the elastoplastic system, respectively.

While the displacement of the linear elastic system is limited to uE, the elastoplastic

system yields at the displacement uy, and deforms without any increase in the force value

up to the maximum displacement at um. It can be assumed that FE is the design strength of

a building intended to remain elastic during the earthquake and Fy is the design strength

of the building intended to yield and response nonlinearly and deform up to um. Two very

important definitions are needed to be introduced with reference to Figure 3.2. The first

definition is the “ductility factor”, denoted by m, which is the ratio of the ultimate

deformation to the deformation corresponding to the yield deformation. The other

definition is the “yield strength reduction factor”, symbolized as Ry, and is assumed to

represent the ratio of the force that would develop under a specified ground motion if the

structure behaves elastically to the load prescribed as design forces at the strength state

equal to a yield level, Fy.

38

umuy

μ = 3-15

FERy Fy= 3-16

From Eqs. 3-15 and 3-16 and with reference to Figure 3.2, the ratio of the maximum

displacement of an elasto-plastic system to that of the corresponding linear system is

obtained as a function of the ductility ratio and the yield strength reduction factor.

umu RE y

μ= 3-17

Figure 3.2 Elasto-plastic system and its corresponding linear system

Ductility demand is a requirement on the design of the system in the sense that its

ductility capacity should exceed the ductility demand. For a given excitation, the ductility

demand and the relationship between um and uE depends on the natural period of vibration

and on the yield strength reduction factor. Figure 3.3 shows the ductility demand for

elastoplastic system due to El Centro ground motion for ξ = 5% and Ry = 1, 2, 4, and 8.

For very long period systems in the displacement-sensitive region of the spectrum, the

deformation um of an elastoplastic system is independent of Ry and is essentially equal to

39

the peak deformation of the corresponding linear system. This implies that for a given μ,

the design yield strength for elasto-plastic system approaches 1/μ times the strength

required for the system to remain elastic. For systems with natural period in the velocity-

sensitive region of the spectrum, um may be larger or smaller than uE; both are affected

irregularly by variations in Ry. For systems with natural period in the acceleration-

sensitive region of the spectrum, um is larger than uE, and um increases with increasing Ry

and decreasing natural period of vibration. Therefore, the ductility demand can be much

larger than Ry.

Figure 3.3 Ductility demand for elastoplastic system due to El Centro

ground motion for ξ=5% and Ry =1, 2, 4, and 8.

The ductility demand on very-short-period systems may be very large even if their

strength is only slightly below that required for the system to remain elastic. Thus,

extremely-short-period systems should be designed for yield strength the same as the

elastic strength FE; otherwise, the inelastic deformation and ductility demand may be

excessive.

40

One of the most important considerations in the design of the yielding structures is the

determination of the yield strength so that the deformation and ductility demands are not

excessive. Before discussing the mentioned concept, it is important to introduce some

definitions regarding the properties of the elasto-plastic system. Assume that uy is the

deformation of the elasto-plastic system corresponding to its yielding. A plot of uy against

Tn for a fixed value of ductility factor is called the yield-deformation response spectrum.

Similar to the definition of the pseudo-velocity and pseudo-acceleration spectra for the

linear systems, these parameters for the corresponding elasto-plastic system become:

V uy yω= × 3-18

2A uy yω= × 3-19

The yield strength of the elasto-plastic system can then be expressed in terms of the

pseudo-acceleration value and the weight of the system:

Ayf kD mA Wy y y g= = = 3-20

Figure 3.4shows the constant ductility response spectrum for elasto-plastic systems and

El Centro ground motion (ξ=5%), for ductility ratio values of μ = 1, 1.5, 2, 4, and 8.

It is possible to determine the design level strength for an SDF system to be designed for

an allowable ductility m corresponding to a given excitation. Corresponding to the

allowable ductility and the known value of natural period and damping factor, the values

of Ay /g is read from the spectrum of Figure 3.4. Eq. 3-20 gives the yield strength

necessary to limit the ductility demand to the allowable ductility. The peak deformation is

.u um yμ= and f yuy k

= 3-21

The ductility factor and the peak deformation represent design requirements associated

with the design force Fy, and the designer should design and detail the structure to posses

the required deformation capacity. Figure 3.4 shows that the design yield strength is

reduced with increasing values of the ductility factor. Even small amounts of inelastic

41

deformation, corresponding to m = 1.5, produce a significant reduction in the design

force. Additional reductions are achieved with increasing values of m but at a slower rate.

Figure 3.4 Constant ductility response spectrum for elasto-plastic

systems and El Centro ground motion (ξ=5%).

To study these reductions quantitatively, Figure 3.5 shows the yield strength reduction

factor of elastoplastic systems as a function of Tn for four values of m equal to 1, 1.5, 2, 4,

and 8. From Figure 3.4, for each value of Tn the m = 1 curve gives FE /W and the curve

for another m gives the corresponding Fy/W. The yield strength reduction factor is then

calculated using Eq. 3-16. For example, consider systems with Tn = 0.5 sec in Figure 3.4;

FE = 0.919W and Fy = 0.179W for m = 4; and the corresponding Ry = 5.13. Such

computations for m = 1, 1.5, 2, 4, and 8 give Ry = 1, 2.26, 2.70, 5.13, and 8.33,

respectively. Repeating such computations for a range of Tn leads to Figure 3.5. The

practical implication of these results is that a structure may be designed for earthquake

resistance by making it strong, by making it ductile, or by designing it for economic

combinations of both properties. Consider again an SDF system with Tn = 0.5 sec and ξ=

5% to be designed for the El Centro ground motion. If this system is designed for a

strength FE = 0.919W or larger, it will remain within the linearly elastic range during this

42

excitation; therefore, it need not be ductile. On the other hand, if it can develop a ductility

factor of 8, it need be designed for only 12% of the strength required for elastic behavior.

Alternatively, it may be designed for strength equal to 37% of FE and a ductility capacity

of 2; or strength equal to 19.5% of FE and a ductility capacity of 4. For some types of

materials and structural members, ductility is difficult to achieve, and economy dictates

designing for large lateral forces; for others, providing ductility is much easier than

providing lateral strength and the design practice reflects this. The strength reduction

permitted for a specified allowable ductility varies with Tn. Ry varies from 1, implying no

reduction, at the short-period end of the spectrum; to Ry = m at the long-period end of the

spectrum. In between, Ry determined for a single ground motion varies in an irregular

manner. However, smooth curves can be developed for design purposes. The yield

strength reduction factor for a specified ductility factor also depends on the damping ratio

(ζ), but this dependence is not strong.

Figure 3.5 Yield strength reduction factor of elastoplastic systems as a function of natural

vibration period Tn for μ = 1, 1.5, 2, 4, and 8 during El Centro ground motion (ξ=5%).

3.3 Response Modification Factor

3.3.1 Evolution of “Response Modification Factor” (Hamburger 2002)

43

Building code provisions for earthquake-resistant design do not intend that structures be

capable of resisting seismic design loading within the elastic or near-elastic range of

response. They intend that structures resist large earthquake loading without life-

threatening damage and, in particular, without collapse; that is, some level of damage is

permitted. Most buildings will never experience a design earthquake and, therefore,

design to resist such events without damage would be economically impractical.

Early building code provisions for seismic resistance focused on prohibiting certain types

of construction observed to behave poorly in past earthquakes. In the early 20th century,

building codes around the world began to introduce requirements that structures intended

to resist earthquakes be provided with sufficient strength to resist a specified lateral force.

These requirements are retained in most building codes today as a basic design method

and are frequently termed the equivalent lateral force (ELF) technique.

Following the powerful earthquake of December 1908 in the Messina Strait, Italy, a

committee of practicing engineers and engineering professors studied buildings that

performed well, and by estimating the ground shaking and forces to which they were

subjected, inferred seismic coefficients that would have been adequate for their design.

They found that the first story of buildings should be designed for a horizontal force

equal to one-twelfth of the weight above and the second and third stories be designed for

one-eighth of the building weight above. This appears to have been the first formal

recommendation to provide earthquake resistance by providing lateral strength equal to a

fraction of supported weight of the structure. The first edition of the Uniform Building

Code (PCBO, 1927) is believed to be the first modern code containing seismic

provisions. This building code required design of structures for the simultaneous

application of a lateral force at each roof and floor level equal to 10% of the structure's

weight tributary of that floor. In the 1937 edition of the UBC the concept of

differentiating seismic risk by means of a zonation map was introduced. This first map

divided the continental United States into three seismic zones and the required lateral

strength for a structure was modified based on these seismic zones. Based on

observations that tall structures seemed to perform better in earthquakes than low-rise

construction, a base shear formula was introduced into the 1946 edition of the UBC to

include the effect of decrease in the base shear with increasing number of stories. Short

44

structures were designed for the most severe lateral forces, equivalent to 10% of the

structure's weight, while the design forces for taller structures could be reduced in

proportion to the number of stories. In 1952, the first recommendations for relating

design lateral forces to structural period, based on spectral response concept were

introduced. These recommendations were incorporated by “Structural Engineers

Association of California” (SEAOC) into the first edition of the Recommended Lateral

Force Requirements and Commentary (SEAOC 1999), commonly known as the Blue

Book, and were adopted in the 1958 edition of the UBC. During 1958 to 1970, the Blue

Book recognized that the actual forces imposed on structures by strong earthquakes were

significantly larger than those that had historically been used for design purposes. Based

on the observation of actual structure behavior it was rationalized that structures designed

for a fraction of the real imposed loading could survive earthquake shaking with damage

but not collapse, as long as they were provided with continuous and tough lateral-force-

resisting systems. Recognizing that actual earthquake forces resulting from design

earthquake shaking were potentially larger than the design strength of the structure,

SEAOC recognized that it was inevitable that structures designed in this manner would

be damaged. SEAOC proposed that structures designed in accordance with the Blue

Book recommendations would provide the following performance capabilities:

- Resist minor earthquake shaking without damage

- Resist moderate earthquake shaking without structural damage but possibly with some

damage to nonstructural features.

- Resist major levels of earthquake shaking with both structural and nonstructural

damage, but without endangerment of the lives of occupants.

- Resist the most intense levels of ground shaking ever likely to affect the building

without collapse.

After the February 9, 1971, earthquake near Sylmar, California (magnitude 6.6) it was

observed that code complying buildings were not capable of meeting the performance

goals suggested in the Blue Book. It was evident that major revisions to the codes were

necessary. As a result, SEAOC formed the Applied Technology Council (ATC) which

published its first report “ATC-3-06”in 1978 (ATC 1978). The report introduced

response spectrum analysis methods as the preferred procedure for design and

45

reformatted the equivalent lateral force procedure to clarify its use as a simplification of

the more exact technique. ATC-3-06 differed from previous seismic codes and contained

several new concepts that included the use of R factors. The intent of ATC-3-06 was to

develop R factors that could be used to reduce expected ground motions presented in the

form of elastic response spectra to lower design levels by bringing modern structural

dynamics into the design process. Figure 3.6 illustrates the use of R factors to reduce

elastic spectral demands to design force levels. The base shear equation for structures for

which the period of vibration was not calculated took the following form (ATC 1995a):

3-22

In this expression, V is the seismic base shear force; Aa is the effective peak acceleration

of the design ground motion (expressed as a fraction of g), R is the response modification

factor, and W the total reactive weight. The factor of 2.5 is a “dynamic amplification

factor” that represents the tendency for a building to amplify accelerations applied at the

base. For structures for which the fundamental building period was calculated, the base

shear equation in ATC-3-06 was given as:

1.20.67A SvV W

RT= 3-23

In this expression, Av is the effective peak velocity related acceleration, S is a soil profile

coefficient, and T is the fundamental period of the building. The soil profile coefficient is

used to account for soil properties that could amplify the bedrock motion; ranging from

1.0 to 1.5. The base shear of Eq. 3-23 provides an upper limit on the base shear calculated

using Eq. 3-24.

2.5AaV WR

=

46

Figure 3.6 Use of R factor to reduce elastic spectral demands

to the design force level (ATC 1995)

The values of R selected in ATC-3-06 represented the consensus opinion of the experts

involved in its development. The first step in assigning R values was to set a maximum

value of R for the structure types considered to provide the best seismic performance,

with the highest reserve strength or ductility. Bertero (1986) and ATC (1995b) have

identified major shortcomings in the values and formulation of the response modification

factors used in seismic codes in the United States. These shortcomings include the

following:

1. A single value of R for all buildings of a given framing type, irrespective of building

height, plan geometry, and framing layout, cannot be justified.

2. The response modification factor is intended to account for the ductility of the framing

system. Recognizing that a constant ductility ratio cannot be used to uniformly reduce

elastic spectral demands to design (inelastic) spectral demands (measured typically as

base shear), R must be period-dependent. This dependence is recognized in the Eurocode

and the Mexican Code.

3. The reserve strength (strength in excess of the design strength) of buildings designed in

different seismic regions will likely vary substantially. Assuming that reserve strength is

47

a key component of R, R should be dependent on either the seismic zone or some ratio of

gravity loads to seismic loads.

ATC-3.06 report was not adopted as the basis for building codes until the seismic

provisions of the 1988 edition of UBC were rewritten by SEAOC. In the mid-1980s, the

Building Seismic Safety Council “BSSC” was formed with a mandate to convert the

ATC-3.06 recommendations into a set of nationally applicable seismic provisions

applicable by building codes nationwide. The BSSC provisions were first published in

1985 as the “National Earthquake Hazards Reduction Program” “NEHRP Provisions”

(BSSC, 1985). The NEHRP Provisions assume significant amounts of nonlinear behavior

to occur under design level events. The NEHRP provisions design for desired limiting

levels of nonlinear behavior for a single design earthquake intensity level, termed

maximum considered earthquake (MCE). In most regions of the United States, the MCE

is defined as that intensity of ground shaking having a 2% probability of exceedance in

50 years. The MCE is thought to represent the most severe level of shaking ever likely to

be experienced by a structure, considering very limited possibility of occurrence of more

severe motions. The maps used in the NEHRP Provisions through 1994 provided the

effective peak acceleration coefficient and the effective peak velocity-related acceleration

coefficient values to use for design. By 1997, significant additional earthquake data had

been obtained that made the mentioned maps out of date. For the 1997 Provisions, a joint

effort involving the BSSC, the Federal Emergency Management Agency (FEMA), and

the U.S. Geological Survey (USGS) was conducted to develop both new maps for use in

design and new design procedures reflecting the significant advances made in the

previous years. The BSSC’s role was to develop new ground motion maps for use in

design and design procedures based on new USGS seismic hazard maps. A detailed

description of the development of the maps is contained in the USGS Open-File Report

96-532, National Seismic-Hazard Maps: Documentation, June 1996, by Frankel, et al.

(1996).

The recent NEHRP Provisions incorporate a series of national seismic hazard maps for

the United States and territories. One set of the map presents contours of MCE, 5%

damped, elastic spectral response acceleration at a period of 0.2 sec, termed SS. The other

set presents contours of MCE, 5% damped, elastic spectral response acceleration at a

48

period of 1.0 sec, termed S1. In both cases, the spectral response acceleration values

should be adjusted to account for the sites with different subsurface conditions. To

facilitate this process, a site is categorized into one of six site class groups. Once a site

has been categorized within a site class, a series of coefficients are provided that are used

to adjust the mapped values of spectral response acceleration for site response effects.

These coefficients were developed based on observed site response characteristics in

ground motion recordings from past earthquakes. These two coefficients are the Fa

coefficient which is used to account for site response effects on short period ground-

shaking intensity; and the Fv coefficient, which is used to account for site response effects

on longer period motions.

The values of these coefficients are determined as a function of site class, and mapped

MCE ground-shaking acceleration values. Site-adjusted values of the MCE spectral

response acceleration parameters at 0.2 and 1 sec, respectively, are found from the

following equations.

S F SMS a S= 3-24

1 1S F SM v= 3-25

The two site-adjusted spectral response acceleration parameters, SMS and SM1 permit a

5% damped, maximum considered earthquake ground-shaking response spectrum to be

constructed for the site. This spectrum is constructed as indicated in Figure 3.7 and

consists of a constant response acceleration range, between periods of T0 and TS, a

constant response velocity range for periods in excess of TS, and a short period range that

ramps between an estimated zero period acceleration given by SMS /2.5 and SMS. T0 is the

period at which the maximum spectral acceleration is reached, and TS is the period at

which the spectral acceleration starts to decrease from its maximum value.

Regardless of whether site-specific spectra or spectra based on mapped values are used,

the actual design values are taken as two thirds of the MCE values. The resulting design

parameters are labeled, respectively, SDS and SD1 and the design spectrum is identical to

the MCE spectrum, except that the ordinates are scaled down to two thirds of the MCE

49

values as the design procedures are believed to provide a margin against collapse of at

least 150%.

3.3.2 Design Coefficients The Static Lateral Force Method, although does not produce estimates of nonlinear

response quantities; it is a valuable analysis and design tool for the design professional

because it is simple to use and does not require the designer to have an understanding of

structural dynamics. It can be used to develop preliminary sizes of components and

elements of a structural framing system for later evaluation by more rigorous methods.

The Static Lateral Force Method accounts for nonlinear response in a system by the use

of a response modification factor (termed R in the NEHRP Provisions). This factor, first

introduced in ATC-3-06 (ATC 1978), reduces the base shear force (Ve) calculated by

elastic analysis using a 5% damped acceleration response spectrum (Sa,5) for the purpose

of calculating a design base shear (Vb):

,5S WV aeVb R R= = 3-26

Figure 3.7 Maximum considered earthquake response spectrum

50

Under the NEHRP Provisions, the required seismic design forces are determined by

elastic methods of analysis, based on the elastic dynamic response of structures to design

ground motion. However, due to inelastic behavior of the system when responding to the

design ground motions, it is recognized that linear response analysis may not provide an

accurate representation of the actual earthquake demands. Therefore, when linear analysis

methods are employed, a series of design coefficients are used to adjust the computed

elastic response values to suitable design values that consider probable inelastic response.

The proportioning of structural elements is typically based on the distribution of internal

forces computed based on linear elastic response spectrum analyses using response

spectra that are reduced from the anticipated design ground motions. As a result, under

the severe levels of ground shaking, the internal forces and deformations produced in

most structures will exceed the point at which elements of the structures start to behave in

an inelastic manner. Reduction factors prescribed in seismic codes are intended to

account for energy dissipation capacity as well as for overstrength. The improvement of

reduction factors has been identified as a way to improve the reliability of present

earthquake-resistant design provisions. One of the main factors influencing the R factor is

the nonlinear behavior in the structure which takes into account the energy dissipation

capacity of the structure. The inelastic action results in a significant amount of energy

dissipation, also known as hysteretic damping. In a properly designed structure, with

increased loading which causes the formation of additional plastic hinges, the capacity

increases until a maximum is reached. The extra capacity obtained by this continued

inelastic action provides the reserve strength necessary for the structure to resist the

extreme motions of the actual seismic forces. Some structures have far more energy

dissipation capacity than do others. The extent of energy dissipation capacity available is

largely dependent on the amount of stiffness and strength degradation the structure

undergoes as it experiences repeated cycles of inelastic deformation.

Figure 3.8 contains an elastic design response spectrum, an elastic response line, and an

inelastic response curve for a structure, all plotted in terms of lateral inertial force (base

shear) vs. lateral roof displacements. The dashed diagonal line represents the elastic

response of the structure. It is a straight line because a structure responding in an elastic

manner will have constant stiffness. The intersection of this diagonal line with the design

51

response spectrum indicates the maximum total lateral base shear, VE, and top

displacement, ΔE, the structure would develop if it responded to the design ground

motion in an elastic manner. The third plot in the figure represents the inelastic response

characteristics of the same structure, referred to as a pushover curve. The pushover curve

has an initial elastic region having the same stiffness as the elastic response line. The

point Vy, Δy on the pushover curve represents the end of the elastic behavior. Beyond Vy,

and Δy, the curve is represented by a series of segments, with sequentially reduced

stiffness, representing the effects of inelastic softening of the structure. The lateral base

shear force, VU at the peak of the pushover curve, represents the maximum lateral force

that the structure is capable of developing at full yield. The response modification

coefficient, R, is used in the provisions to set the minimum acceptable strength, at which

the structure will develop its first significant yielding, Vy. The following relationships

govern the mentioned parameters, where Rd, and Ro will be described in Section-3.4. Sa,5

denotes the base shear coefficient assuming an elastic response.

VE =WSa,5 3-27

VU=WSa,5/Rd 3-28

VY=WSa,5/Rd Ro 3-29

Figure 3.8 Illustration of the inelastic and elastic response curves

52

3.4 Components of “Response Modification Factor”

3.4.1 Overview In the mid-1980s, data from an experimental research program at the University of

California at Berkeley were used to develop an improved understanding of the seismic

response of code-compliant steel braced frame buildings and to propose a draft

formulation for the response modification factor (Uang and Bertero 1986), (Whittaker et

al. 1987). Base shear-roof displacement relationships were established by plotting the

roof displacement at the time corresponding to the maximum base shear force for each

earthquake simulation and each model. Using the experimental data, the Berkeley

researchers described R as the product of three factors that accounted for reserve strength,

ductility, and added viscous damping:

R = Ro Rd Rξ 3-30

In this equation, Ro is a strength factor; Rd is a ductility factor and Rξ is a damping factor.

Using data from the most severe earthquake simulation test, the strength factor was

calculated to be equal to the maximum base shear force (Vu) divided by the design base

shear force (Vb) at the strength level. The ductility factor was calculated as the base shear

for elastic response (Sa,5) divided by the maximum base shear force (Vu). The damping

factor was set equal to 1.0. Further studies ATC-34 (ATC 1995a) supported a new

formulation for R in which R was expressed as the product of three factors: R = (Ro

Rd)Rr, where Ro , Rd , and Rr were a period-dependent strength factor; a period-dependent

ductility factor; and a redundancy factor, respectively. The redundancy factor, developed

as part of Project ATC-34 (ATC 1995b), is proposed to quantify the improved reliability

of seismic framing systems that use multiple lines of vertical seismic framing in each

principal direction of a building. Any evaluation of the key components of R must

address the fact that the components are not independent of each other.

Damping is often used to characterize energy dissipation in a building frame, irrespective

of whether the energy is dissipated by hysteretic behavior or by viscous damping.

Damping in a building responding in the elastic range is generally termed equivalent

viscous damping and is generally assigned a value equal to five percent of critical value.

A viscous damping factor could be used to reduce displacements in a yielding frame but

53

may not proportionally reduce force demands. If seismic design practice shifts to

displacement-based procedures rather than force-based procedures, it may be appropriate

to include a damping component in the formulation for R factor.

A typical force-displacement relationship for an RC member is shown in Figure 3.9. Line

OE denotes the linear response of the RC member if it is stiff enough to remain linear

elastic during the design earthquake loading. This stiff RC member is designed based on

the forces obtained from linear response spectrum. This elastic response force is denoted

as FE. The result is an RC member with unnecessary excessive size of sections rendering

it impractical and uneconomical. Considering the low probability of occurrence of the

severe motion of the design earthquake and also nonlinear behavior of the RC section, it

is possible to design the section based on design forces that are reduced from FE by R

factor. This design level force is denoted by FY in Figure 3.9. The result is a member

which can behave in or near a linear elastic response as expected during the routine

service load and serve its target purpose in a nonlinear fashion after strongest probable

earthquake with a very low possibility of failure or collapse. This behavior is illustrated

by the solid curve passing point A. The extent, to which the curve passed point A,

depends on several parameters such as ductility, overstrength, and redundancy. A

structure can display additional resistance if it is redundant and if yielding takes place in

a sequence rather than all at once. In a redundant RC system or a system where capacity

design philosophy governs the behavior, different members will yield sequentially as

shown by points B and C until the ultimate capacity of the system FU is reached at point

D. This increase in the strength of the system is due to the over-strength. ΔY denotes the

displacement corresponding to the first yielding and ΔU denotes the maximum

displacement of the system before failure. The ability of the system to deform beyond ΔY

is called “ductility” and the ratio between ΔU and ΔY is called “displacement ductility

ratio” denoted by μ as was described by Eq. 3-15. As a result of ductility, the structure

has a capacity to dissipate hysteretic energy. Because of this energy dissipation capacity,

the elastic design force can be reduced to a yield strength level by the factor Rd.

FERd FU= 3-31

54

The reserve strength that exists between the actual structural yield level and first

significant yield level is defined in terms of the overstrength factor Ro:

FURo Fy= 3-32

Appendix C of the 1996 Recommended Lateral Force Requirements and Commentary

(widely known as the SEAOC Blue Book) proposed splitting the response modification

factor into two components, Ro and Rd, representing reserve strength and global ductility,

respectively. The response modification factor is set equal to the product of Ro and Rd

divided by an importance factor that is set equal to 1.0 for nonessential buildings.

Values for the strength and ductility factors are provided in the Blue Book. For each

framing system, the values were selected such that the product of Ro and Rd was no less

than the value assigned to R in the 1994 edition of NEHRP Provisions.

Figure 3.9 Force-displacement relationship for an RC system

3.4.2 Strength Factor (Uang 1991) The importance of overstrength in the survival of buildings during severe earthquake

shaking has long been recognized (Bertero 1986); (Blume 1977); (Rojahn 1988). Its

importance has been confirmed by the shaking table testing of multistory reinforced

55

concrete (Bertero et al. 1984); (Shahrooz and Moehl 1987) and steel structures (Uang and

Bertero 1986); (Whittaker et al. 1989). Its role is even more significant for structures with

short periods because ductility is ineffective in reducing the required elastic strength in

this period range. The maximum lateral strength of a building will generally exceed its

design lateral strength because components are designed with capacities substantially

greater than the design actions; material strengths generally exceed specified nominal

strengths; and drift and detailing requirements often require the use of stronger

components than that required for strength alone. Material overstrength results from the

fact that the design values used to proportion the elements of a structure are specified to

be conservative lower bound estimates of the actual strengths of the structural materials

and their effective strengths in the as-constructed structure. All structural materials have

considerable variation in the strengths that can be obtained in given samples of the

material from a specific grade. A statistical study (Ellingwood et al., 1980) showed that

the overstrength to account for the difference between actual static yield strength and

nominal static yield strength for structural steel can be taken as 1.05. The design

requirements typically base proportioning requirements on minimum specified values

that are further reduced through material reduction (φ) factors. The actual expected

strength of the as-constructed structure is significantly higher than this design value and

should be calculated using the mean strength of the material, based on statistical data, by

removal of the φ factor from the design equation, and by providing an allowance for

strain hardening, where significant yielding is expected to occur. The overstrength to

account for the increase in yield stress as a result of the strain rate effect during an

earthquake excitation was proposed as 1.1 (Ellingwood et al. 1980).

For a given structure, the ratio of Ro will vary as a function of seismic zone and building

height (or fundamental period). For members that are proportioned to resist significant

gravity loads, a substantial percentage of the overall capacity may be available since

actual loads are probably at levels far below the design value at the time of the

earthquake. The seismic overstrength factor will be higher if the building is located in

low seismic zones because gravity and wind loads are more likely to govern the design.

The study of the variation of overstrength factor using inelastic static analyses for a

typical interior frame of an office building located in a region of high seismic risk (Assaf

56

1989) showed that the low-rise buildings usually have a higher overstrength. The value of

overstrength factor will be even higher for building structures with less than four stories,

where the design is governed by gravity loads; and with more than 12 stories where the

design is governed by stiffness (or story drift). Differences in regional construction

practices will also affect the value of the strength factor, but in less predictable ways.

In some instances, geometry or other detail code provisions will dictate larger member

sizes and hence greater capacities than those solely based upon conformity to strength

requirement provisions. Code-mandated limits on interstory drift may require the use of

member sizes in flexible (long-period) framing systems that are greater than those

required for strength alone.

It is important to note that some RC members have reinforcement placed at the top and

bottom of the section, although they are designed based on design force considering only

one layer of tensile reinforcement (such as the RC walls in a tank). In reality these

member behaves as a section with doubly reinforced section. Figure 3.10 shows the

increase in the moment capacity of a RC section due to addition of compression

reinforcement. For normal ratios of tension reinforcement (ρ<0.015) the increase in

moment capacity is less than 5% (MacGregor and Bartlett 2000).

Figure 3.10 Increase in the moment capacity due to compression reinforcement

(MacGregor and Bartlett 2000)

57

The proposed revisions to the NBCC (2005) included an explicit overstrength-related

force modification factor, Ro, to account for over-strength. To account for the various

components contributing to the overstrength-related force modification factor, Ro, the

following formulation was suggested (Mitchell et al. 2003).

o size yield sh mechR R R R R Rφ= × × × × 3-33

Rsize is the overstrength arising from restricted choices for sizes of members and elements

and rounding of sizes and dimensions. RΦ is a factor accounting for the difference

between nominal and factored resistances, equal to 1/Φ, where Φ is the material strength

reduction factor. The factor RΦ is included in Eq. 3-33 because it is appropriate to use

nominal resistances when designing for an extremely rare event such as earthquake

effects corresponding to a return period of 2500 years.

Ryield is the ratio of “actual” yield strength to minimum specified yield strength; Rsh is the

overstrength due to the development of strain hardening; and Rmech is the overstrength

arising from mobilizing the full capacity of the structure such that a collapse mechanism

is formed. The factor Ryield accounts for the fact that the minimum specified material

strength typically underestimates the actual strength. Rsh accounts for the ability of strain

hardening to develop in the material at the anticipated level of deformation of the

structure. Therefore, it varies with the type of material and the extent of inelastic action

that can develop in the structural system. Hence, more ductile structures, designed with

higher Rd values, have larger Rsh values. Rmech accounts for the additional resistance that

can be developed before a collapse mechanism forms in the structure. A structure can

display this additional resistance only if it is redundant and if yielding takes place in a

sequence rather than all at once.

The values of the overstrength-related force modification factors, Ro, for concrete

structural systems are given in Table 3-1. The component Rsize accounts for the fact that

designers choose reinforcing bars that are available and hence often provide an excess of

steel. In addition, bar spacings are usually rounded down and member sizes are rounded

up. To account for these factors Rsize has been assumed to be 1.05.

58

For many members the strength is governed by yielding of the reinforcement. The

resistance reduction factor for reinforcing bars, Φs, is 0.85, and hence RΦ is 1.18.

Although the actual average reinforcing bar yield is somewhat above the specified value

(Mirza and MacGregor 1979), a conservative value of 1.05 was assumed for Ryield, since

this effect seems to be less pronounced for the larger bar sizes. The component Rsh

accounts for the development of strains well into strain hardening, resulting in stresses

above the yield stress. This effect is significant for ductile members that have excellent

confinement of the concrete and are able to prevent the premature buckling of the

longitudinal bars. Rsh is taken as 1.25 only for the ductile cases and 1.10 for the

moderately ductile cases.

Table 3-1 Overstrength components for different structural systems

Type of structural system Rsize RΦ Ryield Rsh Rmech Ro

Ductile MRF 1.05 1.18 1.05 1.25 1.05 1.71

Moderately ductile MRF 1.05 1.18 1.05 1.10 1.00 1.43

MRF with conventional

construction

1.05 1.18 1.05 1.00 1.00 1.30

Ductile coupled walls 1.05 1.18 1.05 1.25 1.05 1.71

Ductile partially coupled walls 1.05 1.18 1.05 1.25 1.05 1.71

Ductile shear walls 1.05 1.18 1.05 1.25 1.00 1.63

Moderately ductile shear walls 1.05 1.18 1.05 1.10 1.00 1.43

Shear walls with conventional

construction

1.05 1.18 1.05 1.00 1.00 1.30

3.4.3 Ductility Factor (Miranda 1993a, Miranda and Bertero 1994)

The ductility factor is a measure of the global nonlinear response of a framing system and

not the components of that system. Assuming that a multistory building can be modeled

as a single-degree-of-freedom (SDOF) system, and that estimates of global displacement

ductility are available, relations between the ductility factor and the displacement

ductility can be developed. These relations for single-SDOF systems have been the

subject of much research in recent years. Although overstrength factors may be estimated

59

by analytical methods, it is more reliable to establish ductility capacities of structural

systems by experimental means.

The first attempt to study the characteristics of an ensemble of “linear elastic response

spectra” (LERS) of recorded ground motions was made by Housner (1959), who

computed the average LERS of eight ground motions recorded during four earthquakes.

Newmark and Hall (1969) proposed a design response spectrum which consisted of a

trapezoidal spectrum based on acceleration, velocity, and displacement controlled regions

defined as the product of the corresponding maximum ground-motion parameters and

amplification factors. The vertical component of ground motion was first considered by

Mohraz et al. (1972), who studied the response of linear elastic SDOF systems subjected

to 14 vertical motions and 28 horizontal motions. The first statistical study to explicitly

consider the effect of soil conditions in LERS was conducted by Hayashi et al. (1971). In

their study, they averaged the LERS of 61 accelerograms recorded in 38 Japanese

earthquakes. They concluded that linear spectral shapes are site dependent. After the

1971 San Fernando earthquake; based on the investigation of 104 horizontal ground

motions recorded on four different group of soils; Seed et al. (1976) concluded that there

are clear differences in linear spectral shapes for different soil and geological conditions,

and recommended the consideration of these effects in selecting earthquake resistant

design criteria. Similar results and conclusions were presented by Mohraz (1976), who

studied horizontal ground motions as well as the vertical component of 54 earthquake

records whose majority was recorded during the San Fernando earthquake.

The influence of soil conditions on reduction factors was first studied by Elghadamsi and

Mohraz (1987), who computed constant yield displacement “inelastic response spectrum”

(IRS) of SDOF systems with an elastic-perfectly plastic hysteretic behavior. Their study

was based on a set of records which did not include very soft soil sites. Krawinkler and

Nassar (1990) studied average IRS of bilinear and stiffness degrading SDOF systems

subjected to 33 horizontal ground motions recorded during the 1987 Whittier Narrows

earthquake. They concluded that reduction factors are independent of epicentral distance

and are only slightly modified by the type of hysteretic model.

The level of inelastic deformation experienced by the system under a given ground

motion is typically given by the displacement ductility ratio, μ. The ductility dependent

60

factor of the response modification factor, denoted by Rd, is defined as the ratio of the

elastic strength demand to the inelastic strength demand.

( 1)

( )

FyRd Fy i

μ

μ μ

==

= 3-34

where Fy (μ= 1) is the lateral yielding strength required to avoid yielding in the system

under a given ground motion and Fy (μ= μi) is the lateral yielding strength required to

maintain the displacement ductility ratio demand, μ, less than or equal to a pre-

determined target ductility ratio, μi, when subjected to the same ground motion.

In general, for structures responding inelastically during earthquake ground motions,

inelastic deformations increase as the lateral yielding strength of the structure decreases.

For a given ground motion and a maximum tolerable displacement ductility demand, μi,

the problem is to compute the minimum lateral strength capacity Fy (μ= μi) that has to be

supplied to the structure in order to avoid excessive ductility ratio demands larger than μi.

As shown in Figure 3.11, lateral strengths Fy (μ= 1) and Fy (μ= μi), if normalized by the

weight of the system, correspond to ordinates of a linear elastic response spectrum and a

constant displacement ductility ratio nonlinear response spectrum, respectively. For

design purposes, Rd corresponds to the maximum reduction in strength that is consistent

with limiting the displacement ductility ratio demand to the pre-determined target

ductility μi in a structure that will have a lateral strength equal to the design lateral

strength. For a given ground motion, computation of Fy (μ= μi) involves iteration, for

each period and for each target ductility, of the lateral strength Fy until the computed

ductility demand (μ) is, within a certain tolerance, close to the target ductility (μi).

An Rd spectrum can be constructed by plotting the strength reduction factors computed

using Eq. 3-34 of a family of SDOF systems (with different periods of vibrations)

undergoing different levels of inelastic deformation, μi, when subjected to a certain

ground acceleration. The magnitude of Rd factors is primarily a function of the maximum

tolerable displacement ductility demand, the period of the system, and the soil conditions

at the site. Other factors that may affect the magnitude of the Rd factor, but to a much

lesser degree, are the type of hysteretic behavior and damping of the structure, as well as

the distance to the epicenter of the earthquake. In the following section, some of the

61

previous studies that have addressed the Rd factors are reviewed and proposed

expressions to estimate Rd are presented.

Figure 3.11 Linear and constant ductility nonlinear response spectra

(Miranda and Bertero 1994)

Newmark and Hall (1969): Based on elastic and inelastic response spectra of the NS

component of the El Centro, California earthquake of May 18, 1940, as well as on

previous studies of the response on simple systems to pulse-type excitations and two

other recorded ground motions, the following observations were made:

1) in the low-frequency and medium frequency spectral regions, an elastic and an

inelastic system have approximately the same maximum displacement;

2) in the extremely high-frequency region, an elastic and an inelastic system have

the same force; and

3) in the moderately high-frequency region, the principle of conservation of energies

can be used by which the monotonic load-deformation diagram of the elastic

system up to the maximum deformation is the same as that of an elastic-perfectly

plastic system subjected to the same excitation.

These observations resulted in the recommendation of a procedure to construct inelastic

spectra from the elastic spectra. The Rd factors proposed by Newmark and Hall are

plotted in Figure 3.12.

62

Lai and Biggs (1980): Design inelastic response spectra were proposed based on mean

inelastic spectra computed four sets of five artificial accelerograms of 10, 20, 30 and 40 s

length in a statistical study to investigate the influence of strong motion duration,

ductility and damping on the inelastic acceleration and displacement response spectra.

Four ductility ratios corresponding to μ =2, 3, 4, and 5 were considered. The study was

limited to elasto-plastic systems. They presented a set of coefficients, referred to as

inelastic displacement and acceleration response ratios, at several control periods

between 0.1 and 4 s which can be used to estimate inelastic displacement and

acceleration spectra from the elastic. The Rd factors are given by the following equation:

(log )R Td α β= + 3-35

where coefficients α and β depend on the displacement ductility ratio and the spectral

period range as shown in Table 3-2. The Rd factors proposed by Lai and Biggs are plotted

in Figure 3.13.

Newmark & Hall

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12

Period (sec)

R d

μ=2μ=3μ=4μ=5μ=6

Figure 3.12 Strength reduction factors proposed by Newmark and Hall

Riddel and Newmark (1979): An improved set of factors was based on a statistical

analysis of inelastic response spectra for elasto-plastic systems with 2, 5, and 10 percent

damping, and for bilinear and stiffness degrading systems with 5 percent damping and for

ductility ratio values from 1 to 10. The study concluded that peak responses of elasto-

plastic, bilinear and stiffness degrading systems are very similar, and that the use of an

elastic-plastic spectrum for inelastic analysis is generally conservative. The study, the

63

first to consider a statistical analysis of inelastic spectra of recorded ground motions,

considered ten earthquake ground motions recorded on rock and alluvium sites. The

proposed inelastic spectra were computed using factors which depend on displacement

ductility, the spectral region, and also on the damping ratio.

Table 3-2 Coefficients to compute Rd proposed by Lai and Biggs

Period range Coefficient μ=2 μ=3 μ=4 μ=5

0.1≤T<0.5sec α 1.6791 2.2296 2.6587 3.1107

β 0.3291 0.7296 1.0587 1.4307

0.5≤T<0.7sec α 2.0332 2.7722 3.3700 3.8336

β 1.5055 2.5320 3.4217 3.8323

0.7≤T<4.0sec α 1.8409 2.4823 2.9853 3.4180

β 0.2642 0.6605 0.9380 1.1493

Lai & Biggs

00.5

11.5

22.5

33.5

44.5

0 1 2 3 4 5

Period (sec)

Rd

μ=2μ=3μ=4μ=5

Figure 3.13 Strength reduction factors proposed by Lai and Biggs

Elghadamsi and Mohraz (1987): The procedure proposed by Elghadamsi and Mohraz,

referred to as the yield displacement spectrum (YDS) used an elastic-plastic SDF system

with different yield displacements to compute the spectra for several damping ratios and

base excitations. Using an elastic-plastic model, inelastic design spectra were computed

for two ensembles; 50 horizontal components representative of alluvium, and 26

horizontal components representative of rock. The results indicated that for a given

64

ductility the factors increase slightly as damping increases. Since the elastic spectral

ordinates decrease significantly with an increase in damping, the inelastic spectral

ordinates decrease as damping increases. Therefore, it can be concluded that the effect of

damping on the inelastic spectrum stems primarily from its effect on the elastic spectrum.

According to Elghadamsi and Mohraz, the Rd factors for rock are in general higher than

those for alluvium, particularly for lower periods and for higher ductilities. This

phenomenon can be explained by indicating that elastic systems with high and

intermediate periods amplify the ground motion from alluvium more than from rock.

However, when a system undergoes large ductilities, its period increases, and

consequently, it would amplify the motion from alluvium more than it would from rock.

As ductility increases, the difference between alluvium and rock sites becomes

significant. This study concluded that Rd factors are not significantly influenced by soil

conditions and that their effects stem primarily from their effects on elastic response

spectra. For given ductility and period of vibration, one may reduce the elastic forces

more for a structure on rock than for a structure on alluvium.

Riddell, Hidalgo and Cruz (1989): This study was based on inelastic spectra computed

for four sets of earthquake records computed for SDOF systems with an elasto-plastic

hysteretic behavior and with 5% damping. Simplified strength reduction factors were

proposed based on approximate mean strength reduction factors. The strength reduction

factors proposed in this study consisted of two linear segments given by Eqs. 3-36 and 3-

37.

* 1*0 1 *RT T R Td T

−≤ ≤ ⇒ = + 3-36

* * T T R Rd≤ ⇒ = 3-37

The value of T* was proposed to vary between 0.1 and 0.4 seconds for ductility ratios

between 2 and 10. The value of R was proposed to be equal to μ for 2<μ<5 and smaller

than μ for 5<μ<10 as shown in Table 3-3. Strength reduction factors computed with the

above equations are shown in Figure 3.14.

65

Table 3-3 Parameters for the relationship proposed by Riddell, Hidalgo and Cruz

Parameter μ=2 μ=3 μ=4 μ=5 μ=6 μ=7 μ=8

R* 2.0 3.0 4.0 5.0 5.6 6.2 6.8

T* 0.1 0.2 0.3 0.4 0.4 0.4 0.4

Riddell, Hidalgo and Cruz

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12

Period (sec)

Rd

μ=2μ=3μ=4μ=5μ=6μ=7μ=8

Figure 3.14 Strength reduction factors proposed by Riddell, Hidalgo and Cruz

Hidalgo and Arias (1990): Based on the approximate mean strength reduction factors

computed by Riddell, Hidalgo and Cruz, this study proposed an expression to compute Rd

factors consisting of a nonlinear curve that is applicable in the whole period range of

interest. The proposed expression is given by Eq. 3-38.

1

1

TRd TkTo μ

= ++

−

3-38

The factor k To was reported to vary for different groups of ground motions. For the draft

of the Chilean code the study recommended a value of k =0.1. Rd factors computed with

the above equation assuming To == 0.2 are shown in Figure 3.15.

66

Arias and Hidalgo

0123456789

0 2 4 6 8 10 12

Period (sec)

Rd

μ=2μ=3μ=4μ=5μ=6μ=7μ=8

Figure 3.15 Strength reduction factors proposed by Arias and Hidalgo

Nassar and Krawinkler (1991): This study considered the response of SDOF nonlinear

systems when subjected to 15 ground motions recorded in the Western United States with

magnitude between 5.7 and 7.7. The records used were obtained from alluvium and rock

sites, however, the influence of site conditions was not explicitly considered. The

sensitivity of mean strength reduction factors to the epicentral distance as well as

structural system parameters such as natural period, yield level, strain-hardening ratio and

the type of inelastic material behavior (i. e. bilinear versus stiffness degrading) were

examined. The study concluded that epicentral distance and stiffness degradation have a

negligible influence on strength reduction factors. Based on mean strength reduction

factors, Eqs. 3-39 and 3-40 were proposed to estimate strength reduction factors:

1

[ ( 1) 1]cR cd μ= − + 3-39

( , )1

aT bc T a TTα = +

+ 3-40

where a is the post-yield stiffness as a percentage of the initial stiffness of the system,

and the parameters a and b are given in Table 3-4. Strength reduction factors proposed by

Nassar and Krawinkler are shown in Figure 3.16.

67

Table 3-4 Parameters for the relationship proposed by Nassar and Krawinkler

α a b

0.00 1.00 0.42

0.02 1.00 0.37

0.10 0.80 0.29

Nassar and Krawinkler

0123456789

10

0 2 4 6 8 10 12

Period (sec)

Rd

μ=2μ=3μ=4μ=5μ=6μ=7μ=8

Figure 3.16 Strength reduction factors proposed by Nassar and Krawinkler

Vidic, Fajfar and Fischinger (1992): Based on mean strength reduction factors

computed from the standard records of California and of the 1979 Montenegro,

Yugoslavia earthquake (20 records altogether) simplified expressions were proposed to

estimate strength reduction factors. In order to study the influence of input motion, in

addition to the groups of records from California and Montenegro, three other groups of

records were used. The proposed equations can be used for the calculation of the reduced

strength and the maximum displacement of the system in combination with any smooth

elastic spectrum with the exception of spectra recorded on very soft soil deposits. The

study considered SDOF systems with bilinear and stiffness degrading hysteretic behavior

with 10% hardening of slope after yielding, and two types of viscous damping. The study

concluded that the reduction factors increase with decreasing damping, and as a

conservative approximation, the reduction factor for 5% damping can be used instead of a

reduction factor for a lower damping percentage. The simplified expressions proposed in

the study consist of two linear segments. In the first segment which corresponds to the

68

short-period region, Rd increases linearly with increasing period from Rd = 1 to a value

that it is equal or nearly equal to the ductility factor. In the second segment, the strength

reduction factor maintains a constant value. The details of the proposed relations depend

on the hysteretic behavior and damping of the system. The following expressions were

proposed for calculation of Rd in different period ranges and assumed ductility:

( 1) 10 10

C TRT T R cd Tμ≤ ⇒ = − + 3-41

( 1) 10 1CRT T R cd μ≥ ⇒ = − + 3-42

0 2 1cTT c Tμ= 3-43

21

VevTAea

ϕπϕ

= 3-44

The coefficients C1, C2, CR and CT depend on the hysteretic behavior and damping and

are reported elsewhere (Vidic, Fajfar and Fischinger 1992). Strength reduction factors

computed using system with a Q-model hysteretic behavior, 5% mass-proportional

damping, and mean amplification factors eaϕ and evϕ of the 20 ground motions

considered in the study (i.e., eaϕ =2.5, and evϕ =2.0) are shown in Figure 3.17.

Miranda (1993a): In order to study the influence of site conditions on strength reduction

factors, a group of 124 ground motions recorded on a wide range of soil conditions

during various earthquakes was considered. Based on the local site conditions at the

recording station, ground motions were classified into three groups: ground motions

recorded on (1) rock (2) alluvium; and (3) very soft soil deposits. Rd factors were

computed for bilinear SDOF systems undergoing displacement ductility ratios between 2

and 6. In addition to the influence of soil conditions, the investigation considered the

influence of magnitude and epicentral distance on strength reduction factors. The study

concluded that while soil conditions may influence significantly the reduction factors

69

(particularly for soft soil sites), magnitude and epicentral distance have a negligible effect

on mean strength reduction factors.

Vidic, Fajfar and Fischinger

0123456789

0 2 4 6 8 10 12

Period (sec)

Rd

μ=2μ=3μ=4μ=5μ=6μ=7μ=8

Figure 3.17 Strength reduction factors proposed by Vidic, Fajfar and Fischinger

Miranda & Garcia (2002): Clough and Johnston (1966) investigated the effects of

stiffness degradation on SDOF systems subjected to four earthquake ground motions.

They concluded that ductility demands in the degrading systems were not significantly

different from those observed in ordinary elasto-plastic structures, except for structures

having a period of vibration less than 0.5 s, and recommended further investigations on

the earthquake resistance of degrading stiffness structures in the short period range. Other

early studies reached similar conclusions. The study by Riddell and Newmark (1979)

concluded that ordinates of the average spectra did not vary significantly when various

nonlinear models were used, and that the use of the elastoplastic idealization could

provide a conservative estimate of the average response to a number of earthquake

motions. Some recent studies have provided quantitative measures of the effect of

stiffness degradation on the seismic response of SDOF systems. For example, Nassar and

Krawinkler (1991) have shown that strength reduction factors of stiffness-degrading

structures are smaller than those of non-degrading structures for periods of vibration

smaller than about 0.4 s. For longer periods, the strength reductions factors of stiffness-

degrading structures are, on average, up to 20% larger than those of non-degrading

structures, which means that in this spectral region lateral strength demands of non-

70

degrading structures are larger than those with stiffness degradation. In the short period

range, the larger value of the ductility ratio increases the difference between the strength

reduction factors of stiffness-degrading and non-degrading structures. However, no clear

trend with change in the level of ductility ratio could be observed in the medium- and

long-period range. Recent studies have concluded that seismic demands on structures

built on very soft soil can be significantly different from those of structures on rock or on

firm sites. In particular, these studies have concluded that seismic demands of structures

built on soft soil are strongly dependent on the ratio of the period of vibration of the

structure to the predominant period of the ground motion (Tg). By idealizing the spectrum

as a bilinear curve, Tg is defined as the period at which the two strait lines intersect. This

period is approximately equal to the period at the intersection of constant-velocity and

constant-acceleration regions of the earthquake response spectrum. Values of

predominant period of the ground motion for a couple of earthquake records are listed in

Table 3-5.

Table 3-5 Predominant period of ground motion for a set of earthquake records

Earthquake Station Component Predominant period (sec)

Imperial Valley (1940)

El Centro

N90E 0.55

Washington (1949)

Olympia

S86W 0.60

Kern County (1952)

Taft

S69E 0.44

San Fernando (1971) Pacoima Dam

S16E 0.40

Loma Prieta (1989)

Corralitos

S00E 0.44

The effect of stiffness degradation on the lateral strength demands of inelastic SDOF

systems subjected to soft soil records was investigated by Miranda and Garcia (2002).

The modified-Clough model (Mahin and Bertero 1976) was used to represent structures

that exhibit significant stiffness degradation when subjected to reverse cyclic loading and

the elastic-perfectly-plastic model was used to represent non-degrading structures. A total

of 69,600 strength-reduction factors corresponding to 50 periods of vibration, six levels

of inelastic deformation, and two types of hysteretic behavior were computed. Mean

71

strength-reduction factors corresponding to each hysteretic behavior (i.e., modified-

Clough model and elastic-perfectly plastic model) were then computed for each period

and each displacement ductility ratio.

It was shown that the estimation of the predominant period of the ground motion was

very important in order to adequately assess seismic demands on both linear elastic and

nonlinear structures built on soft soils. The strength reduction factors of ground motions

recorded on soft soil are characterized by very distinct peaks, where the strength

reduction factors are significantly larger than the displacement ductility ratio. These

peaks coincide with the predominant period of vibration of the ground motion. For

systems with periods of vibration that are more than two times the predominant period,

the strength reduction factor is on average, approximately equal to the displacement

ductility ratio. For systems with periods of vibration smaller than about two-thirds of the

predominant period of the ground motion, the strength reduction factor is smaller than the

displacement ductility ratio. The opposite is true for systems with periods of vibration

between about two thirds and two times the predominant period of the ground motion.

For systems with periods of vibration that are approximately equal to the predominant

period of vibration the reduction in lateral strength demand due to inelastic behavior can

be very large. This indicates that, for these structures, inelastic behavior is efficient in

reducing lateral strength demands.

The strength-reduction factors vary significantly with the ratio of the period of vibration,

T, to the predominant period of the ground motion, Tg, for both stiffness degrading and

non-degrading systems. With the exception of systems with periods of vibration that are

two or more times the predominant period of the ground motion, strength reduction

factors computed with ground motions recorded on soft soils are significantly different

than those computed from ground motions recorded on rock or firm soil sites.

For structures whose period of vibration is close to or larger than the predominant period

of vibration of the ground motion, lateral strength demands for stiffness degrading

systems are, on average, smaller than those of non-degrading system. Thus, in this

spectral region stiffness degradation is, on average, beneficial to the structure by reducing

lateral strength demands. Structures with stiffness degradation, and with periods of

vibration shorter than the predominant period of vibration of the ground motion, can

72

experience lateral strength demands larger than those of non-degrading structures in the

same period range. The shape of inelastic response spectra differs significantly from the

shape of elastic response spectra. This difference depends on the level of inelastic

deformation, the local site conditions, and the period of vibration. Thus, direct scaling by

using a period-independent factor of elastic spectra to obtain inelastic strength demands is

neither rational nor conservative.

Borzi-Elnashai (2000): The accelerograms of the dataset were recorded during 43

earthquakes of magnitudes between 5.5 and 7.9, at a distance from the nearest point on

the fault of up to 260 km. The data was well-distributed with respect to magnitude,

distance and site classification. This study confirmed that, in a force-based method, the

hysteretic behavior does not significantly change the level of force for which the structure

has to be designed, in order to reach a fixed level of displacement ductility. Inelastic

spectra were derived using two models: an elastic perfectly plastic (EPP) representation

and another more complex system which has a yield point, a maximum force point and a

post-ultimate branch which may represent hardening as well as softening. Four different

levels of post-yield stiffness (E2) corresponding to 0, 10%, -20%, and -30% were used in

the study. Because of the two parameters definition of the EPP model, level of force-

resistance and stiffness, the influence of strong-motion, distance and soil condition on

inelastic response spectra may be better visualized. Displacement ductility ratios of 2, 3,

4 and 6 were considered. The inelastic spectra were defined between 0.05 s and 3 s. The

results of parametric investigation indicated that the parameter with the strongest

influence on inelastic spectra is the slope of the post yield branch. The reduction

coefficient was equal to 1 at a zero period and increased linearly up to a period T1, which

was defined as the period at which the behavior factor reaches the value R1. A second

linear branch was assumed between T1 and T2. The value of the reduction coefficient

corresponding to T2 was denoted as R2. For periods longer than T2 the behavior factor

maintained a constant value equal to R2. R factor was slightly dependent on the period in

the long period range and almost corresponded to the ductility value. On the other hand in

the short period range, the R factor was dependent on both ductility and period. A

moderate, though not negligible, influence from the hysteretic behavior was observed

73

throughout the period range. It was observed that the influence of input motion

parameters on inelastic spectra was similar to that for the elastic spectra. Moreover, the

hysteretic models assumed only mildly influence on the inelastic acceleration spectra,

therefore, the level of force imposed on structures was not heavily influenced by their

global hysteretic behavior. The relationships proposed by Borzi-Elnashai are presented

by Eqs. 3-45 through 3-47.

( 1) 1 T<T1 11

TR RT

= − + 3-45

1( ) T <T<T1 2 1 1 22 1

T TR R R R

T T

−= + −

− 3-46

T>T2 2R R= 3-47

The values R1, R2, T1 and T2 that allow the definition of approximate spectra for all

relevant ductility levels and hysteretic behavior can be obtained using Eqs. 3-48 through

3-51.

0.251T = 3-48

0.163 0.602T μ= + 3-49

1 1 1R a bμ= + 3-50

2 2 2R a bμ= + 3-51

The values of a1, a2, b1, and b2 correspond to the different hysteretic behavior patterns.

These values are reported in Table 3-6. Figure 3.18 illustrates the strength reduction

factors proposed by Borzi-Elnashai for the five levels of ductility ranging from 2 to 6.

3.5 Response modification factor for LCS

It is reasonable that, similar to the method employed for R factors in buildings, the R

factors for LCS be considered as a product of several components including strength,

74

ductility, damping, and redundancy factors. For LCS such as water tanks, the redundancy

component seems irrelevant because as soon as the leakage starts at the most critical part,

the structure can be assumed to have lost its functionality.

Table 3-6 Parameters used for the definition of reduction factors

Hysteresis behavior a1 a2 b1 b2

EPP 0.69 1.01 0.90 0.24

E2=0 0.55 1.33 1.37 0

E2=10% 0.32 0.96 1.69 0.51

E2=-20% 0.38 1.24 1.67 0

E2=-30% 0.29 1.21 1.83 0

Borzi & Elnashai

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5

Period (sec)

Rd

μ=2μ=3μ=4μ=5μ=6

Figure 3.18 Strength reduction factors proposed by Borzi-Elnashai

According to ACI 350.03-06, damping ratios of 5% and 0.5% are assumed for the

impulsive and convective components of LCS, respectively. As was mentioned in

Section-3.4.1, in a force-based procedure similar to the current practice for computing the

design forces, the damping factor can be assumed as unity and its effect on the response

modification factor is through its effect on the ductility factor component. In this study it

is assumed that the value of R is the product of only two components, namely the strength

related reduction factor (Ro), and the ductility related reduction factor (Rd). Assessment

of the above factors is not possible before experimental tests are conducted on the

75

leakage behavior of the tanks under cyclic loading. As was shown in Figure 2.9 (a), the

impulsive component of earthquake has a very short period of vibration ranging from

0.04 s for wall height of 3 m to 0.18 s for wall height of 10 m. The relationships between

μ and Rd, as discussed previously, showed the variation of ductility related reduction

factor in this period range, based on the assumed level of ductility. Before experimental

tests on the leakage behavior of the LCS, no accurate estimate of the safe level of

ductility for LCS can be made. Moreover, the leakage tests can provide the stress or

strain level of steel corresponding to the leakage failure of LCS. The stress or strain level

of steel at start of leakage is necessary for determination of the components of Ro factor

such as the strain hardening factor. If the leakage starts before yielding of the steel, then

no strain hardening factor is relevant. On the other hand if the leakage starts after the

steel has gone well into the nonlinear range, appropriate estimate of the strain hardening

factor becomes important. The effect of the compression reinforcement on the

enhancement of the flexural strength of the wall is negligible due to close proximity of

the compression reinforcement layer to the neutral axis of the wall.

3.6 Parametric study on the effect of R on the design loads

A parametric study is conducted to investigate the effects of different variables on the

design values of rectangular RC tank walls. The height and thickness values of the walls

vary from 3 to 10 m and from 300 mm to 1000 mm (equal to 10% of the height),

respectively. The length of the tank (in the direction of earthquake) varies to determine

the effect of the ratio of liquid height to length of the tank (HL/L) on the design

parameters. The ratios vary from 0.05 (a shallow tank) to 1 (a tall tank). It is assumed that

the height of the wall is 500 mm above the level of the liquid (i.e. constant freeboard of

500 mm). Ss, S1, Fv, Fa, I, and Rc are taken as 150%, 60%, 0.8, 0.8, 1, and 1, respectively,

as discussed in Section-2.8.

A parametric study is conducted on the effect of variation of Ri on the seismic induced

vertical flexural moment ME at the bottom of the larger wall of the tank. As was

mentioned earlier, the middle vertical strip of the large side of the tank can be regarded as

a cantilever section under the effect of static and dynamic loads. Considering the R factor

for the convective component as unity (which is the case in the ACI Code), the variation

76

of Ri influences the impulsive components as well as the inertia force of the wall and the

vertical acceleration effect. Therefore the relationship is not linear. The value of the

vertical seismic flexural moment at the base of the wall is calculated with respect to

different values of Ri ranging form 1 to 10, and different ratios of HL/L ranging from 0.05

to1. The calculated values of ME are scattered among walls of different heights. Figure

3.19 compares the charts corresponding to wall heights of 3, 6, and 9 m. As shown in the

figure, for a certain height of the wall, the effect of variation of Ri is more pronounced for

smaller value of Ri; while for values of Ri more than 5, the effect of variation of Ri on ME

becomes almost constant. For simplicity in the current parametric study, R denotes the

impulsive component response modification factor for the remainder of the current

Chapter as the convective component response modification factor is equal to unity.

To further facilitate the study on the effect of variation of R on ME, the values of ME are

normalized by HL3. As a result, the values of ME / HL

3 become more appropriate for

comparison purposes among walls of different heights. Attempt is made to propose

mathematical formulations for the curve which can be regarded as the relationship

between ME and R. Because of the difference between the values of the seismic induced

force for different wall heights, two relationships were proposed based on the height of

the wall.

For wall heights of 3, 4, and 5 m, Eq. 3-52 is proposed.

3-52

The curve representing to the above equation is shown as a thick black curve in Figure

3.20 for each wall height. For wall heights of 6 through 10 m, Eq. 3-52 overestimates the

seismic load; therefore, Eq. 3-53 is proposed.

3-53

The curve representing to the above equation is shown as a thick black curve in Figure

3.21 for each corresponding wall height. As can be observed, the curves can be

conservatively regarded as the relationship between ME and R.

2.73/ 0.7 1.5M HE L R= +

1.93/ 0.45 1.5M HE L R= +

77

Figure 3.19 Comparison of the value of the maximum seismic flexural moment for wall

heights of 3, 6, and 9 m.

78

Figure 3.20 Comparison of the proposed curve with ME / HL

3 values for wall heights of 3

and 5 m.

Based on Eqs. 3-52 and 3-53, and corresponding figures, it is observed that by increasing

the value of R in the equations, the value of the seismic design force decreases as

expected. However, the rate of decrease is not linear and varies for different range of R.

As can be seen in Figures 3.20 and 3.21, increasing the value of R above five has

negligible effect on the decrease of the seismic design force. The rate of decrease is more

79

significant for lower values of R. For instance, for tanks with liquid height ranging

between 3 to 5 m, increasing the value of R in Eq. 3-52 from one to two, reduces the

ratio of ME / HL3 from 3.4 to 1.65, while increasing the value of R from two to three,

reduces the ratio of ME / HL3 from 1.65 to 1.22. This signifies the importance of assigning

an accurate value for R factor when its range is close to unity.

80

Figure 3.21 Comparison of the proposed curve with ME / HL

3 values for wall heights of 6,

8, and 10 m

Eqs. 3-52 and 3-53 can also be used to estimate a conservative value for the design

seismic load based on the height of the tank wall by using the appropriate value of R. For

instance the seismic design flexural moment values for a tank with wall and liquid

heights of 5 m and 4.5 m, respectively, assuming the value of the R factor of two (based

on ACI 350.03-6), for a shallow (HL/L=0.05) and a tall (HL/L=1) tank are 138 kN.m and

112 kN.m, respectively. Using Eq. 3-52, results in the value of seismic design flexural

moment for a tank with wall and liquid heights of 5 m and 4.5 m, respectively, equal to

150 kN.m. This value is independent of the HL/L ratio and provides a conservative

estimate for the calculated values of 138 and 112 kN.m. Eqs. 3-52 and 3-53 can be used

to provide a conservative estimate of the value of the vertical flexural moment at the

base. This will eliminate the time consuming calculations of different components of

seismic design components.

3.6.1 Design load combinations Based on ACI 350-06 Code, liquid-containing structures subjected to earthquake-induced

forces can be designed in accordance with the strength design method or the alternate

design method. Design codes usually specify a variety of load combinations which

include factored loads for each load type in order to ensure the safety of the structure

81

under different probable loading scenarios because of uncertainty and variability in

different loading effects. The factor assigned to each load is influenced by the degree of

accuracy to which the load effect usually can be calculated and the variation that might

be expected in the load during the lifetime of the structure. In assigning factors to

combinations of loading, some consideration is given to the probability of simultaneous

occurrence. Structural members should be designed to have strengths at all sections at

least equal to the required strengths calculated for the factored loads and forces in such

combinations. In ACI 350-06, two load combinations should be considered for seismic

design of liquid containing structures when only hydrostatic and hydrodynamics loads are

applied on the structure. These combinations are U1 = [1.4 × (D + F)], and U2 = [(1.2 × F)

+ (1 × E)], where D and E correspond to the dead load and earthquake load effects,

respectively. F denotes the loads due to the liquid pressure which can be found by the

following equation in terms of the flexural moment at the bottom of the wall using the

specific weight of the liquid γL.

3-54

For the design of environmental engineering structures, the required strength shall be

multiplied by the environmental durability factor (Sd) when durability, liquid-tightness, or

similar serviceability are of main concern. This durability factor shall not be used for

designs using service loads and permissible service load stresses.

3-55

where

where fs is the permissible tensile stress in reinforcement under service loads calculated

using Eq. 2-1 for the normal environmental condition. In environmental engineering

concrete structures, durability and long-term service life are of main importance. The

resulting stresses in non-prestressed reinforcement using normal building code load

factors are higher than would be desirable in environmental engineering concrete

factored loadunfactored load

γ =

3 / 6M HStatic L Lγ= ×

1.0yd

s

fS

fφγ

= ≥

82

structures. The intent of the environmental durability factor is to reduce the effective

stress in non-pre-stressed reinforcement under service load conditions, such that stress

levels are considered to be in an acceptable range for control of cracking. When a liquid-

containing structure is designed in accordance with the strength design method, the

environmental durability factor (Sd) need not be applied to load combinations that include

earthquake loads. The current edition of the ACI Code (ACI 350-06) proposes Eq. 3-55

for the calculation of environmental durability factor (Sd), where the maximum

permissible stress in the steel is calculated based on Eq. 2-1 for normal environmental

exposure areas. Therefore, Sd is a function of the spacing between steel rebars.

Using simplifications as allowed by the Code, the following relationships can be used to

determine the value of Sd based on the thickness of the wall (h).

20.138 ( ) 25 for h < 400 mm25

= +dsS 3-56

3-57

where s (in mm) is the bar spacing. Considering the specific weight of water, the design

load combinations with respect to the mentioned load combinations incorporating, Eqs.

3-54 and 3-55 become:

Combination-1: 31.4 / 6S Hd L Lγ× × ×

or 2 3L

s0.355 ( ) 25 H for h < 400 mm25

+ × 3-58

2 3L

s0.315 ( ) 25 H for h 400 mm25

+ × ≥

For wall heights of 3, 4, and 5 m:

20.156 ( ) 25 for h 400 mm25

= + ≥dsS

83

Combination-2: 2.7 3 3[(0.7 ) ] [1.2 / 6]1.5 H HL L LRγ+ × + × × 3-59

or

For wall heights of 6 through 10 m:

Combination-2: 3-60

or

Eqs. 3-59 and 3-60 are important for the purpose of the current study as they can relate

the design forces to the value of the R factor.

2.7 3(2.65 )1.5 HLR+ ×

1.9 3 3[(0.45 ) ] [1.2 / 6]1.5 H HL L LRγ+ × + × ×

1.9 3(2.40 )1.5 H LR+ ×

84

CHAPTER-4

EXPERIMENTAL PROGRAM

4.1 General

Analytical studies show that the wall base connection in the middle of the larger side of a

rectangular tank is the most critical region with respect to leakage. The magnitude of

vertical flexural moment is the highest in this region and the contained liquid also has the

highest pressure near the base of the tank, leading to possible leakage after cracking.

Therefore, any experimental research on the leakage behavior of RC rectangular tanks

should focus on this critical region. If the length of the tank wall is relatively large

compared to the wall height, its behavior approaches that of a one-way slab and the

vertical middle strip at the middle of the tank wall behaves similar to a cantilever

member.

For the experimental work in the current study, full scale wall-base slab specimens were

built to represent a cantilever wall. The concrete for the wall and for the foundation were

cast separately by providing a shear key. Two different shear key configurations were

used in the experiments which will be discussed later in this Chapter. A steel water

pressure chamber was installed at the connection of the wall-foundation to simulate the

liquid-tank interaction and to enable the detection of possible leakage within the critical

regions of the tank wall. The attachment of water pressure chamber to the tank wall was

rather challenging since the chamber had to be installed in a way to act independent of

the tank wall while the tank wall was subjected to cyclic loading. The connection of the

chamber and wall had to be watertight in such a way that no pressurized water could

escape from the interface. In the following, the linear analysis of a rectangular tank for

determination of the distribution of the flexural moments on the tank walls will be

discussed. Afterwards, the steps taken for preparation of the specimens in the

experimental program are discussed. The detailed information used for the design of each

specimen along with description of the specimens and theoretical prediction of the key

behavior parameters are also presented.

85

4.2 Linear analysis of the behavior of the tank

An RC rectangular water tank was analyzed with properties such as L (larger side of the

tank wall) = 20 m, B (shorter side of the tank wall) = 10 m, tw (thickness of the wall) =

300 mm, HW (height of the wall) = 4 m, HL (height of the water) = 3.5 m, γL (specific

weight of water) = 9.8 KN/m3, and γC (specific weight of concrete) = 23.6 kN/m3. Using

0.25×0.25 m shell elements for the walls (no sensitivity study was conducted), the design

forces according to ACI code were incorporated in an analysis using SAP2000 (CSI

2004) using the Joint Pattern option to apply the mentioned distributions with respect to

the height of the tank on the walls. The seismic design parameters were selected to

correspond to a high seismicity zone in California. The R factors for the impulsive and

convective components (Ri and Rc) were taken as unity, and therefore, the response was

assumed to be elastic and no reduction in the design forces due to nonlinear behavior was

considered. Figure 4.1 illustrates the 3-D view of the tank model in SAP2000.

Figure 4.1 The 3-D view of the tank in SAP2000

86

In all of the analyses the hydrostatic load is applied on all four sides of the tank, where

for the seismic loads, first the large side walls were only loaded, and then the shorter side

walls were only loaded. The total load in each analysis consisted of the sum of the

hydrostatic load and the seismic induced loads. For simplification it is assumed that the

impulsive and convective weights of the liquid at one direction are equal to those in the

perpendicular direction although they may differ slightly due to different dimensions (B

and L) of the plan of the tank.

Assuming the z axis to be in the direction of the height of the tank wall, the applied

seismic load (per kN/m2) on the tank walls based on the ACI 350.3 Code provisions are:

1 0.8 0.42 28.3The wall inertia force = = 9.6 1

× × ×

55.5[4 3.5 6 1.3125 (6 3.5 12 1.3125) ( )]2 3.5The impulsive pressure = 13.87 -3.423.5

× − × − × − × ×=

z

z

19.7 [4 3.5 6 1.79 (6 3.5 12 1.79) ( )]2 3.5The convective pressure = 2.61 0.11 23.5

z

z× − × − × − × ×

= +

Effect of vertical acceleration = 0.16 9.8 (3.5 - ) 5.49 1.57 × × = −z z

4.2.1 Long side loaded For the first part of the analysis, the mentioned seismic forces are only applied on both

walls at the larger side of the tank. Figure 4.2 shows the coordinate system used for this

analysis. Figures 4.3(a) and (b) show the distribution of the total (seismic plus

hydrostatic) vertical and total horizontal flexural moments on the wall at the larger side of

the tank, respectively. Figures 4.4(a) and (b) show the variation of the magnitude of the

seismic and the total horizontal flexural moment on the large side of the tank,

respectively. Figures 4.5(a) and (b) show the variation of the seismic and the total vertical

flexural moment on the large side of the tank, respectively. It is shown that the magnitude

of the horizontal flexural moment is highest at the top corners of the tank walls and has a

minimal value at the bottom of the wall at the corners. The magnitude of the horizontal

flexural moment in the middle portion of the wall increases as the height coordinate

changes from the top to the bottom of the wall. However, the maximum magnitude of the

87

horizontal flexural moment at the bottom of the middle of the wall is about 50% of that at

the top corners of the wall.

Figure 4.2 The coordinate system of the tank in SAP2000

Figure 4.3 Variation of the total flexural moment on the large side of the tank

(a) Vertical moment (b) horizontal moment

(a)

(b)

88

Seismic horizontal moment (M11)

0

5

10

15

20

0 5 10

Distance from corner (m)

Mom

ent (

kN.m

/m)

z=0z=1z=2z=3z=4

Total horizontal moment (M11)

-100

102030405060

0 5 10Distance from corner (m)

Mom

ent (

kN.m

/m)

z=0z=1z=2z=3z=4

Figure 4.4 Variation of the horizontal flexural moment on the large side of the tank with

respect to distance from bottom and corner of the tank wall

(a) seismic induced moment (b) total moment

The maximum vertical flexural moment occurs at the middle of the wall and at the

connection of the wall and foundation. The maximum vertical moment diminishes as the

coordinates move from the middle towards the corners and also from bottom to the top of

the wall.

(a)

(b)

89

Seismic vertical moment (M22)

0102030405060

0 5 10


Mom

ent (

kN.m

/m)

z=0

z=1

z=2

z=3z=4

Total vertical moment (M22)

-50

0

50

100

150

0 5 10


Mom

ent (

kN.m

/m)

z=0z=1

z=2z=3

z=4

Figure 4.5 Variation of the vertical flexural moment on the large side of the tank with



4.2.2 Short side loaded For the next part of the analysis, the mentioned seismic forces are only applied on both

walls at the shorter side of the tank while the hydrodynamic load is applied on all four

walls. Figures 4.6(a) and (b) show the distribution of the total vertical and horizontal

flexural moments on the wall at the shorter side of the tank, respectively. Figures 4.7(a)

(a)

(b)

90

and (b) show the variation of the magnitude of the seismic and the total horizontal

flexural moment on the short side of the tank, respectively.

Figures 4.8(a) and (b) show the variation of the seismic induced and the total vertical

flexural moment on the large side of the tank, respectively. The variation of the

horizontal and vertical flexural moment on the short side wall follows the same trend as

that of the long side wall; however, due to the smaller surface area of the wall in the short

direction, the magnitudes of the flexural moments is smaller compared to those on the

larger side.

Figure 4.6 Variation of the total flexural moment on the short side of the tank

(a) Vertical moment (b) horizontal moment

(a)

(b)

91


-5

0

5

10

15

20

0 2 4 6


Mom

ent (

kN.m

/m)

z=0

z=1

z=2z=3

z=4

Total horizontal moment (M11)

-100

102030405060

0 2 4 6Distance from corner (m)

Mom

ent (

kN.m

/m)

z=0z=1z=2z=3z=4

Figure 4.7 Variation of the horizontal flexural moment on the short side of the tank with



(b)

(a)

92

From the result of the analyses, it can be concluded that the maximum vertical moment

occurs at the middle of the wall at the larger side of the tank and at the connection of the

wall and foundation. The maximum vertical moment diminishes as the coordinates move

from the middle towards the corners and also from bottom to the top of the wall. Because

of the relative dimensions of the larger side of the tank, the middle strip at the middle of

the lager side wall can be assumed to have cantilever behavior. With regards to leakage,

the connection of the wall foundation at this region is also critical because the water

pressure is the highest at the bottom of the wall leading to a more critical condition for

leakage. It is believed that any safe design procedure should consider the potential

leakage failure of the tank in this region. The maximum values of vertical seismic and

total flexural moment at the base were obtained by SAP2000 analysis. The values at the

middle of the larger side were 48.3 kN.m/m and 115.5 kN.m/m respectively, while at the

middle of the shorter side they were 35.4 kN.m/m and 82 kN.m/m respectively.

Assuming the response modification factors equal to unity and following the procedures

outlined in ACI 350-3-06, the maximum values of vertical seismic and total flexural

moment for the large side (based on the cantilever wall approximation) were calculated

as 50.5 kN.m/m and 120.6 kN.m/m, respectively, which correlated well with the

SAP2000 analyses results for the larger side walls.

The maximum values of horizontal seismic and total flexural moment, obtained by the

analysis, at the top corner of the longer side were 9.7 kN.m/m and 23.1 kN.m/m

respectively. The corresponding values for the shorter side were 7.1 kN.m/m and 16.4

kN.m/m, respectively. The distribution of the horizontal moments shows that the

maximum value corresponds to the top edge of the wall at the corners of the tank.

Knowing that the pressure of the liquid is minimal at the top level of the liquid height;

and frequently a freeboard is provided, this region was not considered to be as critical

with respect to leakage.

4.3 Test setup

To investigate the leakage behavior of a rectangular tank, similar to the one just

mentioned, it is not practical to build a full scale complete tank for experimental tests. To

93

focus on the leakage behavior of the most critical part, the middle portion of the wall at

the larger side of the tank was selected and full scale wall-foundation specimens were

built to represent this region.

Seismic vertical moment (M22)

0

10

20

30

40

0 2 4 6


Mom

ent (

kN.m

/m)

z=0

z=1

z=2

z=3z=4


-5

0

5

10

15

20

0 2 4 6


Mom

ent (

kN.m

/m)

z=0

z=1

z=2z=3

z=4

Figure 4.8 Variation of the vertical flexural moment on the short side of the tank with



(a)

(b)

94

Figure 4.9 illustrates the schematics of the test setup in the current experimental research.

An actuator was used to apply cyclic loading on top of the wall to simulate the

earthquake induced cyclic reversal loads. A steel water pressure chamber was installed at

the connection of the wall-base slab to simulate the water pressure as desired. As will be

described later in the current Chapter, a hydraulic jack was used for installation of the

water pressure chamber which is also shown in Figure 4.9. Provided that the loading

condition is able to induce the desired pattern of stresses for the specimen, then it would

be possible to investigate the leakage due to a certain pressure of water through the

induced cracks in the critical region. Because of availability of one actuator in the lab, the

load was applied at a certain height where it could produce the same maximum shear

force and flexural moment at the base of the wall, simultaneously.

For calculation of this height, the value of flexural moment corresponding to the sum of

the unfactored seismic and static flexural moments was divided by the value of shear

force corresponding to the sum of the unfactored seismic and static shear forces.

Because of the cantilever type behavior of the wall, the portion of the wall above this

height was not necessary and was not considered in the experimental program.

Observation of the force applied by the hydraulic jack on the pressure chamber, which

was schematically depicted in Figure 4.9, showed that this force was reduced as the

actuator pushed the wall and was increased as the actuator pulled the wall. Effect of this

force was ignored in the calculation of the point of application of actuator load to avoid

significant complexity.

4.4 Preparation of the specimen

A total of five specimens were constructed for the experimental tests in the current study.

The specimens corresponded to the wall in a rectangular tank with height of 4 m. Two

different thickness values of 300 mm and 400 mm were considered for the specimens.

The test specimens were designed based on the provisions of ACI 350 Code. The details

of the tests conducted on each specimen are presented in Chapter-5. In the following the

steps taken for construction and preparation of the specimens are discussed.

95

Figure 4.9 The schematics of the test setup for the leakage test

4.4.1 Installation of the strain gauges For measuring the level of the strain in the reinforcing steel at a desired location, a steel

strain gauge was installed in that location. A thorough procedure was utilized to install

the strain gauges at appropriate steel bars regions. The ribs of the deformed bars were

first removed using a power grinder until the surface became smooth. A coat of “A

Water-Based Acid Surface Cleaner” M-Prep conditioner containing phosphoric acid was

applied on the surface followed by application of “A Water-Based Alkaline Surface

Cleaner” M-Prep neutralizer containing ammonia water to clean the desired location from

harmful material. Then the strain gauge was attached from the top surface, with the gauge

circuit facing up, to a small piece of scotch tape where Catalyst-C drops were applied to

its bottom surface. Then a layer of M-bond adhesive was applied to the surface of the

steel bar and the scotch tape containing the gauge was attached to the steel bar while care

was taken to align the gauge properly with respect to the longitudinal axis of the rebar.

After few seconds the scotch tape was removed and the strain gauge was waterproofed

with three subsequent coats of “M-Coat-A” air-drying polyurethane coating, allowing 20

minutes air-drying time between applications. Hot wax was then brushed onto the strain

gauge for protection during casting and was followed by the application of self-adhesive

96

aluminum foil. It should be noted that one should not expect a fool-proof strain reading

from a strain gauge. Any misalignment from the horizontal axis of the rebar due to errors

in installation of the strain gauge or even movement and misalignment of the rebar during

casting of the concrete might reduce the accuracy of the results.

4.4.2 Construction of the Reinforcement Cage The foundation dimensions were 600 mm for height, 1800 mm for length (perpendicular

to the wall) and 1500 mm for width (parallel to the wall) (except 1900 mm for the fifth

specimen). For all specimens, the foundations were designed based on the forces obtained

by a linear elastic analysis performed in SAP2000. The volumes of rebars were then

increased to strengthen the foundations against major possible cracking in order to

eliminate or minimize any leakage from cracks that may form within the foundation. As a

result, the foundation had 2 orthogonal layers of 10 No. 25 bars with spacing of 150 mm

in the main direction (perpendicular to the wall) and 10 No. 25 bars with spacing of 185

mm in the other direction (parallel to the wall). These bars were placed both at the top

and the bottom of the section. To facilitate the placement of the top layer, both ends of

the hooks of the bars at the sides of top layer were welded to the corresponding bars from

the bottom layer. These 2 top bars served as a sawhorse for the top layer. It is important

to note that for the top reinforcement layers, the main rebar layer (perpendicular to the

wall) was placed under the rebar layer oriented at the perpendicular direction (parallel to

the wall) layer. This was done to facilitate the installation of the 15 mm rubber water-stop

for the shear key and to avoid its interference with the top layer of reinforcement as will

be discussed later in this Chapter. After construction of the foundation reinforcement

cage, the wall vertical bars (No. 20) were installed, followed by installation of the

horizontal bars (No. 20). In the case of the wall with 300 mm thickness, the end hooks for

the horizontal bars were rotated upward to accommodate for inadequate space for the end

hooks in the wall formwork.

4.4.3 Construction of the Formwork The formwork for each specimen consisted of 2 separate components; the formwork for

the foundation and the formwork for the wall. The formwork for the foundation consisted

of flat pieces of plywood joined together to form cross-sectional area of 2 m ×2 m for the

97

bottom of the foundation, and a rectangular box made of 3/8" plywood which were

assembled using 50×50 mm studs on four corners. The plywood for the rectangular box

was additionally reinforced with “SPF” lumbers as horizontal bracing at the bottom and

the middle of its elevation. This would decrease the effects of deflection of sides during

casting of concrete. To further reinforce the foundation formwork, six 3/8" threaded rods

(three at the bottom and three at the middle of the box elevation) were passed through

opposite “SPF” lumbers to restrain the relative movement of the two bracing lumbers at

parallel sides as can be seen in Figure 4.10. It is recommended that all necessary

precautions be taken into consideration to strengthen the formwork, as its failure during

the casting of the concrete would be very undesirable.

Figure 4.10 Formwork for the foundation

The formwork for the wall was a rectangular box made of 12 mm plywood pieces which

were assembled using 50×50 mm studs on four corners. Horizontally and vertically

installed lumbers were used to reinforce the formwork against deflection. Threaded rods

were passed through these lumbers to additionally reduce the effect of the deflection of

the formwork as the deflection would cause problems during installation of the water-

pressure chamber and also would decrease the accuracy of the results in terms of the

thickness of the wall. Four holes were drilled at appropriate places in the formwork using

38 mm wood drill bit and four small 25 mm PVC pipes were installed at the top part of

the wall formwork to provide the holes necessary for connection of the loading system to

the specimen’s wall. The inner surface of the wall formwork was coated with

98

polyurethane to prevent the water from absorption by the wood. Also a hole was drilled

in the back side of the wall for strain gauges wires to pass through the back of the wall.

These wires were connected to the data acquisition system after casting the concrete for

the wall.

The strong floor and the strong wall in the Structural Laboratory at Ryerson University

have a grid of holes placed orthogonally at distance (center to center) 600 mm from each

other. Because of the sizes of the specimen and the water pressure chamber and also the

limited space, it was only possible to anchor the specimen to the floor with two threaded

rods in the foundation in the front side of the wall and two threaded rods in the

foundation at the back side of the wall. It is important to mention that because the

actuator was supposed to be installed in the middle of the large steel bracket shown in

Figure 4.11, and this bracket was installed on the strong wall with supports that were 1.8

m apart (on the grid of the hole on the wall with center to center of 600 mm as shown in

Figure 4.11) it was not possible to use an odd number (e.g. 3) for the number of the rods

for anchoring the foundation at each side of the wall. Also using four rods at each side of

the wall for anchoring the foundation would require a larger specimen which would

exceed the limit of the actuator’s applicable loads at the desired level of performance of

the specimen.

Figure 4.11 The bracket supporting the actuator on the strong wall

99

During the placement of the foundation reinforcement, the location of the foundation

holes (for anchoring rods) were marked and four PVC pipes with inner diameter of 75

mm were installed at appropriate places and were fixed to the reinforcement cage as can

be seen in Figure 4.12. Because of the relative sizes of the threaded rod and the hole,

there was a small tolerance for the error in placement of the PVC pipe and any

carelessness would result in very undesirable consequences. Before casting the concrete

for foundation, the location of these four pipes were double-checked to ensure that the

holes through the foundation would coincide with those in the laboratory floor. After

installation of the pipes, four No.15 inverted U shape hooks were placed near the four

corners of the specimen to provide the hooks for lifting and moving of the specimen by

the crane. The dimensions for theses hooks are illustrated in Figure 4.13. It is important

that the location of these hooks in front of the wall does not interfere with the location of

the water-pressure chamber. The foundation formwork was further reinforced with four

short pieces of lumber, which diagonally connected the corners of the foundation’s

formwork as shown in Figure 4.14 to restrain the corners against opening due to fresh

concrete pressure.

Figure 4.12 Installation of the PVC pipe in the foundation

100

Figure 4.13 Dimensions of the hooks

Figure 4.14 Four pieces of lumber connected to the corners of the formwork

4.4.4 Casting of concrete All specimens were cast using concrete provided by a local ready-mix supplier.

Compressive strength of 30 MPa, maximum aggregate size of 20 mm, and no air

101

entrainment and no superplastisizer were specified. To follow the practice which is used

in the industry, the concrete for the foundation and the wall, were poured separately by

creating a construction joint where a water-stop was installed in the shear key region

(referred to as conventional shear key) to inhibit the leakage through the shear key.

Water-stops are commonly used at the cold joint in concrete structures such as water

tanks to prevent the seepage of the fluid through the joint (Malik 2006). The size of the

conventional shear key used in this experimental program was 50 mm deep and 100 mm

wide. The configuration of the conventional shear key used in the current experimental

program is shown in Figure 4.15(a). As was mentioned earlier, the rebar layers at the top

of the foundation were arranged in a way to reduce the conflict between the 150 mm

water-stop and the top layer of the foundation.

Another configuration referred to as the inverted shear key without a water-stop, shown

in Figure 4.15(b), was also constructed only for the second specimen and its leakage

performance was evaluated. The size of the inverted shear key used in this experimental

program was 50 mm deep and 100 mm wide. Before casting of the concrete for

foundation, the shear key region was prepared. For the conventional shear key the

dimensions of the key were cut out of the foundation formwork from both sides with

respect to the centerline of the wall as shown in Figure 4.16. Two long pieces of 50×50

mm studs were used to hold the water-stop and to maintain the shape of the key during

casting of concrete for foundation. The ready-mix concrete was delivered to the structural

laboratory by truck mixers, and discharged inside the formwork using a 0.5 m3 bucket

and the crane in the laboratory. Care was taken during vibrating the concrete in order to

prevent the movement of the PVC pipes.

Before the concrete reached the desired elevation under the wall, the rubber water-stop

was installed using two long pieces of 50×50 mm studs holding the water-stop inside. It

is important to mention that the 50×50 mm studs were slowly removed 6 hours after

casting the concrete to provide the shape of the shear key. In one case the studs were left

for 24 hours and the removal became very difficult. Before placing the wall formwork,

the rubber water-stop was pierced at its top part and at several locations. Short pieces of

tie wire were used as shown in Figure 4.17, to erect the water-stop in its place during

casting of the wall’s concrete.

102

Figure 4.15 Configuration of the shear key in the experimental program (not to scale)

(a) Conventional shear key (b) Inverted shear key

For the specimen with inverted shear key, immediately after casting of the concrete, two

long pieces of 50×50 mm studs covered with a special oil, were placed on the foundation

between the two layers of the wall rebars. Two small pieces of lumber were used between

these studs at both ends to adjust and maintain the distance between the studs as shown in

Figure 4.18 (a). Afterward, the space was filled and troweled with concrete as is shown in

Shear key

Shear key Water-stop

(a)

(b)

103

Figure 4.16 Preparation of the conventional shear key before casting of concrete

Figure 4.17 Erection of the water stop inside the wall reinforcement

Figure 4.18 (b). For elimination of any disturbance for this shear key, the studs were

removed 48 hours after casting of the concrete. Figure 4.19 shows the shear key after

casting of the concrete for foundation. Figure 4.20 shows the specimen after casting of

the concrete for foundation and before placing of the formwork for the wall.

104

Approximately one week after placement of the foundation concrete, the concrete for the

wall was cast using ready mix concrete by means of the 0.5 m3 bucket and the crane as

shown in Figure 4.21. A rubber hammer was used to compact the concrete inside the

formwork as the use of vibrator was not possible due to limited accessibility. Figure 4.22

illustrates the specimens after casting of the concrete for the wall along with the measures

taken to reinforce the formwork including the studs and the threaded rods.

Figure 4.18 Construction of the inverted shear key

(a) before pouring the foundation concrete (b) after pouring the foundation concrete

Figure 4.19 The inverted shear key before placement of the wall’s formwork

105

It is important to mention that the first batch of the concrete that is discharged from the

truck mixer is generally segregated. This segregation will produce a honey-combed

concrete which is not suitable for a liquid containing tank. This situation is critical due to

the fact that this concrete will be placed at the bottom of the wall which is the most

critical location with respect to leakage. As a possible solution to this problem, the first

batch of the concrete was discarded, and the next batch discharged from the truck mixer

was used.

Figure 4.20 The specimen before placement of the wall’s formwork

Similar to the foundation, the wall was water cured for seven days after casting of the

concrete. The compressive strength of the concrete for each specimen was determined by

testing standard cylinders (300 × 150 mm). For each concrete sample, six cylinders were

cast. The cylinders were kept near the specimens and were cured similar to the

specimens. The cylinders were tested after 28 days for compressive strength. The actual

tests on the specimens were conducted few days after their corresponding 28-day strength

tests.

106

Figure 4.21 Casting of the concrete for the wall

Figure 4.22 The specimens after casting of the wall’s concrete

4.4.5 Preparation of the specimen for the experimental test

After each specimen was transferred to the designated location for testing, four high

strength (Grade B7) threaded rods were used to anchor the specimen to the strong floor.

107

The extra length of the PVC pipe above the foundation was cut. 20 mm thick plates were

used as washers between the nuts and the top of foundation, and between the nuts and the

basement ceiling for each threaded rod. The distance between the specimen wall and the

strong wall was measured to be equal at both ends before the nuts were tightened using a

500 lb-ft torque wrench.

The next step was the attachment of two steel beams to the top of the wall at the front and

back side. These two beams, which were strengthened with stiffeners and a plate on the

flange, had four holes with diameter of 30 mm on the flange. The patterns of these holes

on both beams were identical with the ones provided on top of the specimen’s wall. Four

high strength (Grade B7) threaded rods were used with high strength washers and nuts to

attach these two beams through the RC concrete wall.

It is important to mention that these threaded rods were initially of normal strength

material with diameter of 12 mm. However, due to failure of one of the washers during a

test, and also due to their elongation, they were replaced by high strength (Grade B7)

rods with diameter of 19 mm. The beam on the front side of the wall had a 25 mm thick

square shape plate welded to its middle for connection of the actuator to the beam on the

front side of the wall. This square shape plate had four 25 mm holes near its corners for

attachment to the actuator using four high strength (Grade B7) threaded rods with

diameter of 19 mm as can be seen in Figure 4.23. Table 4-1 shows the mechanical

properties of the Grade B7 steel used in the experimental test setup.

Table 4-1 Mechanical properties for Grade B7 steel

Grade

Diameter (mm) Tensile strength (MPa) Yield strength (MPa)

B7 63.5 and under 862 724

63.5 to 101.6 793 655

101.6 to 117.8 689 517

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Figure 4.23 The connection of the actuator to the beam on the front side of the wall

4.4.6 Installation of the water pressure chamber The steel water chamber shown in Figure 4.24 had to be installed at the connection of the

wall-foundation. The reason for using the chamber was to simulate the water pressure for

the induced cracks at the bottom portion of the front face of the wall. The pressure of

water was expected to have an influence on the leakage through the induced cracks.

However, there were some complexities in the installation of the steel water pressure

chamber to the connection of the wall-foundation. The connection of the steel chamber to

the concrete wall should be water-tight to prevent the pressurized water to escape during

the test. It should be mentioned that the water inside the chamber was under a certain

pressure to simulate the pressure exerted by the column of the water contained in the tank

(i.e. 4 m head). Also, the steel chamber should not be directly attached to the concrete

wall, as it can affect the stiffness of the concrete wall. Several options were considered

and examined, and finally, a 25 mm thick gum rubber sheet was used as a medium

between the steel chamber and the concrete wall. The gum rubber thickness would allow

the edges of the steel chamber to penetrate enough inside the rubber to prevent the

pressurized water to escape from the interface. The maximum penetration length was

considered to be 8-10 mm to allow for the back and forth movement of the wall during

cyclic loading.

The size of the openings of the bottom and front side of the chamber were cut through the

rubber sheet as can be seen in Figure 4.25. Because of the imperfections on the surface of

109

the concrete and also for a water-tight attachment of the rubber and the concrete, bathtub

silicon glue was applied between the gum-rubber sheet and the concrete. It is important to

note that the silicon glue was very flowable in the beginning and would flow away under

the rubber sheet. Therefore, the silicon glue was left to gain strength for 15 minutes and

then, the rubber sheet was attached as shown in Figure 4.26. After attaching the rubber to

the wall, several straight lumbers were used to hold the rubber during the time that the

silicon glue was drying out. It was very important that the final surface of the rubber

sheet becomes smooth; otherwise, the risk of leakage of pressurized water from the small

gaps between the rubber and the chamber would exist. Three days after installation of the

rubber sheet, the steel chamber was placed in its position on the foundation and in contact

with the rubber. This time was considered enough for the silicon glue to dry out and gain

strength.

As was mentioned, the steel chamber edges had to penetrate into the gum rubber between

8-10 mm to eliminate the escape of pressurized water. The force necessary for the

penetration (approximately 40 kN) was provided by a hydraulic jack and a large steel

beam as was shown in Figure 4.9. In case of leakage that may occur between the chamber

edges and the gum rubber sheet during the test, the chamber could be pushed further by

manually increasing of the hydraulic jack’s force.

Figure 4.24 The steel water chamber used for simulation of the water pressure

110

Figure 4.25 The gum-rubber sheet before installation on the wall

The vertical movement of the chamber should be restrained while some small movement

in the horizontal direction was allowed (for penetration into the rubber sheet). Therefore,

two thick plates were used to vertically restrain the bottom angle at the inner perimeter of

the steel chamber. The bottom angle was highlighted in Figure 4.24. Each plate had three

holes through which three concrete anchors (12 ×178 mm), shown in Figure 4.27, were

used to fix each plate to the foundation.

Figure 4.26 The installation of the gum-rubber sheet on the wall

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First, the chamber was forced to penetrate into the gum-rubber sheet up to the appropriate

depth of penetration (i.e. 8 mm) by using the hydraulic jack and the steel beam. Then, the

steel plates were placed on the mentioned bottom angle with appropriate small distance to

the edge of the angel. This small distance which was around 10 mm was used to allow for

back and forth movement of the chamber. The locations of all six holes were marked and

the steel chamber was removed and the holes were drilled. It is important to mention that

the size of the drill bit used for drilling the foundation concrete should be identical to the

size of the concrete anchors (12 mm), and the drill should be kept completely vertical

through the drilling. During hammering on the concrete anchors; two nuts were placed on

top of the anchors, with the top nut only covering the tip of the anchor. This was done to

prevent damage to the top part of the anchor as a result of hammering. In the event of

damage to any of concrete anchors, the anchor should be cut, followed by drilling a new

hole in the plate and in the foundation.

Figure 4.27 Concrete anchors (12 ×178 mm)

During the concrete anchor installation, the depth of penetration of the anchors into the

foundation was around 50 mm. After the necessary depth was drilled, the top of the hole

was intentionally chamfered with the tip of the drill held inclined. The concrete anchors

were then placed inside the holes; epoxy glue was poured on the chamfered surface of the

foundation around the anchors to prevent the rotation of the anchors during tightening of

the nuts. The glue failed due to extensive force applied for tightening of the nuts. Several

options were tested and finally the FRP resin showed to be the only option. The liquid

resin can penetrate inside the hole covering the anchor and restrain it against rotation.

Figure 4.28 shows the plates installed in their place to ensure the accuracy of the

procedure. In the case of any error, the hole on the plate could be widened or drilled at a

new location. After conclusion of this part of the setup, the plates were removed and the

chamber was placed back at its position. The hydraulic jack was used again to push the

chamber to its approximate final position. Before installation of the steel plates the entire

112

connection between the chamber and rubber, both inside and outside of the chamber, was

sealed with silicon glue and left undisturbed for 48 hours.

Figure 4.28 Location of the concrete anchors

Following the drying of the silicon glue, the installation of the plates started. As can be

seen in Figure 4.29, thick steel plates were used as support under the two mentioned

plates to act as a support to force the plates to drive down the chamber’s bottom angle as

the concrete anchors were tightened. To facilitate the back and forth movement of the

chamber, round steel rebars were put between the plates and the bottom angle as shown

in Figure 4.30.

Figure 4.29 Installation of the plates inside the chamber

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Figure 4.30 Steel rebars between plates and bottom angle

After the plates were fixed in their position and the chamber was forced down on the gum

rubber sheet, the chamber was gradually filled with water. The step filling of the chamber

could facilitate the detection of the leaking region in case any was observed. The water

chamber was almost filled up to its top as can be seen in Figure 4.31. The chamber had

24 high strength bolts installed on its top part, where the top plate would connect to it

through 24 coinciding holes.

It is important to mention that some of the bolts did not go through the corresponding

holes due to the deflection of the chamber as the result of the force from the hydraulic

jack on the chamber. The problem was fixed by gradual bending of the problematic bolts

using very slight hitting by a hammer. Following the application of the silicon glue to the

top surface of the chamber, the top plate was placed after one hour to eliminate the flow

of fresh silicon glue. The next day, the top plate was tightened to the chamber using the

24 nuts that were covered by fresh silicon glue for elimination of leakage through the

bolts and holes.

114

Figure 4.31 Top part of the water pressure chamber

The chamber had a slender steel tube welded to the top plate as the inlet for water. After

the top plate was placed, the chamber became sealed and water could not move into the

chamber because of the presence of trapped air. The problem was resolved by using a tee

and a shut off valve as shown in Figure 4.32. The city water was connected by a hose to

one end of the shut off valve while the other end of the shut off valve was connected to a

tee by a tube. The tee would connect the city water to the chamber and also to another

short plastic tube. First, the short plastic tube was bent to prevent the water or air to

escape, and then the tap water was opened. A few seconds later the tap was closed and

the short plastic tube was released from bending. As a result the trapped air was able to

escape through the short tube as it was replaced by water inside the chamber. This

procedure was continued until water replaced all the trapped air inside the chamber. After

this, the gauge shown in Figure 4.32 was attached to the short plastic tube to measure the

pressure of the water inside the chamber. As the city water tap was opened, the water

pressure inside the chamber increased until it reached the desired level and then the tap

was closed. It is important to mention that during the experimental tests, due to leakage

of water through the induced cracks, the water pressure decreased. However, opening of

the tap would again increase the pressure to the desired level. The complete setup before

the leakage test including the installed water pressure chamber is shown in Figure 4.33.

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Figure 4.32 Filling the water chamber with water

4.4.7 The floor brake It was initially assumed that tightening of the thread rods that attached the specimen to

the floor was adequate to restrain the horizontal back and forth movement of the

specimen. However, during the first test, horizontal movement of the specimen was

observed. Therefore, it was decided to construct a device called the “floor brake” to

restrain the horizontal movement of the foundation. The “floor brake” which is shown in

Figure 4.34 consisted of a steel box section where two steel cylinders were passed

through two holes through the box and were welded to it. The steel cylinders were

inserted into the holes in the laboratory floor. The gap between the floor brake and the

foundation was shimmed with steel plates as shown in Figure 4.35. Measurement of the

displacement of the foundation during the experimental tests showed that this device was

very successful for its intended purpose.

116

Figure 4.33 Complete setup including the water pressure chamber

Figure 4.34 The steel “floor brake”

117

Figure 4.35 The “floor brake” at the back of the foundation during the test

4.5 General specifications of the specimens

A total of five full-scale wall-foundation specimens, similar to the one shown in Figure

4.36, were constructed for the experimental program. The experimental program was

aimed to provide information about cyclic behavior of RC walls with respect to leakage

due to pressurized water. Different tests were conducted on the specimens including the

initial test where the specimen was subjected to loading until the first crack was

observed. Other part of the experimental tests included the investigation of the leakage

behavior of the specimens while subjected to lateral incremental cyclic loading. The

leakage performance of the specimens was assessed with respect to the configuration of

the shear key. The effect of retrofitting of the damaged specimens using GFRP sheet was

also investigated. After the leakage test, some of the specimens were loaded up to failure.

Wall thickness values of 400 mm for Specimens-1 and 5; and 300 mm for Specimens-2,

3, and 4 were considered. All specimens were designed using the provisions of ACI 350-

06. The spacing between the No.20 horizontal rebars for the wall was 300 mm for all

specimens as specified in ACI 350-06 for shrinkage and temperature reinforcement

requirements. The spacing of longitudinal rebars for the wall was calculated base on the

design load according to ACI 350-06. The properties of the rectangular tank and other

relevant design information used in this research are presented in Table 4-2.

118

Figure 4.36 View of Specimen-1

Table 4-2 Properties of the tank used in the current research

B (Width of a rectangular tank, perpendicular to the direction

of the ground motion) 40.0 m L (Length of a rectangular tank, parallel to the direction of the

ground motion) 20.0 m tw (Average wall thickness) 300 or 400 mm HL (Designed depth of stored liquid) 3.5 m HW (Wall height) 4.00 m γL (Density of the liquid) 9.8 KN/m3 f'c (28 days compressive strength of concrete) 30 MPa Ss (mapped maximum considered earthquake 5% damped

spectral response acceleration parameter at short periods,

expressed as a fraction of the acceleration due to gravity

1.5

S1 (mapped maximum considered earthquake 5% damped

spectral response acceleration parameter at a period of 1 s,

expressed as a fraction of the acceleration due to gravity)

0.6

Fa (short-period site coefficient (at 0.2 s period)) 0.8 Fv (long-period site coefficient (at 1.0 s period)) 0.8 I (Importance factor) 1 Ri (Response modification factor for impulsive component) 2 Rc (Response modification factor for convective component) 1

119

4.6 Details of design and construction for the specimens In the following section the design information along with construction details for each

1500 mm wide specimen is presented. It is important to note that all design values

correspond to a strip of 1000 mm width of the tank wall and should be multiplied by 1.5

for comparison to the corresponding values of the actual specimens. Two wall

thicknesses of 300 mm and 400 mm were considered for the specimens. The design

information for the specimens including the value and point of effect of each load

component is shown in Table 4-3.

Table 4-3 Design load components for the specimens

Force component Magnitude

(kN)

Distance to the base from

the centroid of force

(m)

Flexural moment at

base of wall

(kN.m)

Hydrostatic pressure 60.1 1.17 70.1

Impulsive force 27.7 1.31 36.4

Convective force 9.9 1.79 17.7

Wall inertia effect 6.4 (4.8)* 2.00 12.8 (9.6)*

Vertical ground

motion effect

9.6 1.17 11.2

* Values corresponding to the 300 mm thick wall

4.6.1 Specimen-1 For Specimen-1, the horizontal seismic induced shear force on the connection of the wall

foundation is:

( ) ( ) ( )2 2 2(27.7 6.4) 9.6 9.9 36.8 kN/mseismicV = + + + =

The seismic induced moment on the connection of the wall foundation is:

( ) ( ) ( )2 2 2(27.7 1.31 6.4 2.0) 9.6 1.17 9.9 1.79 53.4 kN.m/mseismicM = × + × + × + × =

The horizontal static shear force on the connection of the wall foundation is:

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60 kN/mstaticV =

The static moment on the connection of the wall foundation is:

1.17 60 70.2 kN.m/mstaticM = × =

For the design of Specimen-1, it was assumed that earthquake induced load in the pull-

direction only involves the flexural moment related to the concrete wall inertia of 12.8

kN.m, and the load in the push direction includes all the seismic related components

equal to 53.42 kN.m. A static flexural moment of 70.2 kN.m was considered in both

directions. In the design of Specimen-1, the effect of pulling load of wall inertia during

the earthquake was ignored due to its relative small value and the loading scheme

considered only pushing the specimen and pulling it back to the initial neutral position.

The design of the wall reinforcement for Specimen-1 is presented in Appendix-A. For

Specimen-1, eight No.20 bars with spacing of 200 mm at the front side and the seven

No.20 bars with spacing of 225 mm at the back side of the wall were provided as shown

in Figure 4.37. The reinforcement ratio calculated based on the total area of

reinforcement and the gross concrete section area for Specimen-1 is 0.75%. Several strain

gauges were installed in different regions of the specimen. Figure 4.38 illustrates the

schematics of the locations of the strain gauges for Specimen-1. These locations include

the point on the wall longitudinal bars at the interface of the wall-foundation at the front

face of the wall denoted by SWF-1, SWF-2, and SWF-3 in Figure 4.38(a); and at the

back face of the wall denoted by SWB-1, and SWB-2 in Figure 4.38(b). Figure 4.38(c)

illustrates the strain gauge on top layer bar of the foundation at a location 100 mm in

front of the front face of the wall denotes by SFT-1, and the strain gauges located 100

mm behind the back face of the wall as SFT-2. Figure 4.38(d) illustrates the strain gauges

on the steel installed on the bars in bottom layer of the foundation, namely SFB-1, and

SFB-2.

121

Figure 4.37 Specimen-1 using conventional shear key configuration

(a)

122

(b)

(c)

123

Figure 4.38 Steel strain gauges for Specimen-1 located at

(a) front side of the wall (b) back side of the wall

(c) top of the foundation (d) bottom of the foundation

Specimen-1 had a conventional shear key at the joint as was shown in Figure 4.37 after

casting of the concrete for foundation. The foundation concrete had a 28-day f’c of 36

MPa and the wall concrete, which was poured separately seven days after casting of the

concrete for foundation, had a 28-day f’c of 31 MPa.

4.6.2 Specimens-2, 3, and 4 Specimens-2, 3, and 4 represented cantilever walls with thickness of 300 mm and height

of 4 m, filled with water up to 3.5 m height. The design information for this specimen is

identical to that of the first specimen shown in Table-4.2, except that because of the

smaller thickness of the wall (300 mm) as compared to 400 mm thickness of the wall for

Specimen-1, the wall inertia changed to 4.8 kN/m and the earthquake shear force and

flexural moment changed to 35.3 kN/m, and 50.5 kN.m/m, respectively. For the design of

the longitudinal rebars for these specimens it was assumed that the loads in the push and

pull directions have the same magnitude. Therefore, 10 No. 20 rebars were used at the

(d)

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front face as well as the back face spaced uniformly at 155 mm and shown in Figures

4.39, 4.40, and 4.41 for Specimens-2, 3, and 4, respectively. The reinforcement ratio

calculated based on the total area of reinforcement and the gross concrete section area for

these specimens is 1.3%.

Specimen-2 had inverted shear key configuration as described in Section-4.3.4 and was

shown in Figures 4.18 and 4.19 (see also Figure 4.39). Specimens-3 and 4 had

conventional shear key configuration as shown in Figures 4.40 and 4.41, respectively,

before casting the concrete for foundation. The strain gauge locations for these specimens

were similar to Specimen-1, except that the foundation reinforcement did not have any

strain gauges and the strain gauges were installed on the wall vertical bars as shown

schematically in Figure 4.42. This was mainly due to the low level of the strains in the

foundation bars which were observed during the tests for Specimen-1 as will be discussed

in Chapter-5.

An important note to mention is that the point of application of the actuator load was

lowered down about 100 mm for Specimens-4 and 5. This is due to the fact that the

plywood sheets for the wall formwork had a width of 1200 mm. For the wall of 1400 mm

another piece of plywood was used at the top of the 1200 mm plywood. However, due to

different curvatures of the pieces of plywood, after casting of the concrete for the wall,

the wall was not finished smoothly at the place where the loading beam was installed. For

this reason the loading beam was attached at a height of about 1150 mm above the

foundation to provide a smooth surface behind the loading steel beam.

The concrete for Specimens-2 and 3 were poured at the same time. The concrete for the

foundation had a 28-day f’c of 31 MPa and the concrete for the wall had a 28-day f’c of

29.5 MPa. For Specimens-4, the concrete for foundation had a 28-day f’c of 30 MPa and

the concrete for the wall had a 28-day f’c of 25 MPa.

4.6.3 Specimen-5 Specimen-5 had a wall thickness of 400 mm representing a cantilever wall with same

thickness and height of 4 m filled with water up to 3.5 m height. The design information

for this specimen is similar to Specimen-1, except that for Specimen-5 loading was

considered to be similar in pull and push directions (as was the case for Specimens-2, 3,

and 4). Because of the limitation of the actuator load, the flexural moment resistance of

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the wall was decreased by using six No. 20 bars with spacing of 280 mm at both faces of

the wall. This spacing was determined based on the corresponding design load

combinations with the load factors equal to unity. The environmental durability factor

was also assumed as unity. The reinforcement ratio calculated based on the total area of

reinforcement and the gross concrete section area for Specimen-5 is 0.6%. Due to

decreased flexural resistance of the wall, it was expected that the behavior up to failure

can be better captured. The 28-day maximum compressive of the concrete for the

foundation and the wall, which were cast separately, were 30 MPa, and 25 MPa,

respectively. Figures 4.43 and 4.44 illustrate Specimen-5 including the set-up for the

conventional shear key using a rubber water-stop, before and after casting of the concrete

for foundation, respectively. Specimen-5 was the only specimen with 1900 mm width of

the foundation. This was due to the availability of the certain size of foundation rebars in

the structural laboratory. Therefore, the foundation dimension was 200 mm longer on

each side of the length of the wall as can be observed in Figure 4.45 showing the 1500

mm formwork for the wall placed on the finished foundation.

.

Figure 4.39 Specimen-2 using inverted shear key configuration

126



127

Figure 4.42 Steel strain gauges for Specimens-2 through 5 located at


(b)

(a)

128


Figure 4.44 Construction of the conventional shear key using a rubber water-stop

129

Figure 4.45 Specimen-5 before casting of concrete for the wall

4.7 The Experimental Program

The tests included pretest, leakage test, leakage test of the retrofitted specimen, and test

of the specimen up to failure of the wall for selected specimens as outlined in Table 4-4.

The parameter in the last column denotes the diameter of the threaded rods used to

anchor the specimen to the structural laboratory strong floor. During the experimental

program, the position of the specimen in the structural laboratory was in a way that the

front face of the wall, where the actuator was connected to, was towards the north

direction as indicated in Figure 4.46.

The pretest was aimed at providing information about initial stage of behavior of a RC

cantilever wall. This stage of loading includes initial cracking of the specimen which is

expected to occur at and near the connection of the wall-base slab. Following the

conclusion of the pretest, the leakage test was conducted. The water pressure chamber

was installed at the connection of the wall-foundation as described in Section-4.2.6. As

was mentioned, the force needed to push the water pressure chamber into the gum rubber

is provided by applying about 40 kN using the hydraulic jack. This force is observed to

130

fluctuate significantly during the test. The point of application of the jacking load is about

500 mm above the top of foundation. In all the experimental results, the values of the

actuator force reported are independent of this hydraulic jacking force.

Table 4-4 Summary of the experimental test conducted in this study

Specimen Name of

the test

Description Remarks Threaded rod

diameter to anchor the

slab to the strong floor

1 SP-1 Pretest 31 mm

SL-1 Leakage test 31 mm

SR-1 Retrofitted specimen

leakage test

One layer of

GFRP

31 mm

2 SL -2 Leakage test 31 mm

3 SL -3-1 Leakage test 31 mm

SL -3-2 Leakage test 44 mm

SR -3 Retrofitted specimen

leakage test

Two layers of

GFRP

44 mm



SF-4 Test to failure 44 mm



SF-5 Test to failure 44 mm

The leakage test is aimed at providing information regarding the leakage behavior of the

RC specimen under the effect of quasi-static reversed cyclic loading. For this

experimental work the test started by applying incremental forced controlled cyclic

loading with three cycles of push and pull. Several tests on cyclic performance of RC

members have been reported (Clyde et al. 2000, Ricles and Paboojian 1993) in which,

after the specimen has experienced the first yielding of the longitudinal reinforcement,

the testing is carried out using displacement-control. In the experimental tests for the

131

current study, the forced controlled cyclic test is continued until the last part of the test.

This is mainly due to the presumed level of leakage which was thought to occur near the

yielding of vertical reinforcement due to expected extensive cracking.

Figure 4.46 Schematics diagram of the pretest

Also due to the expected low level of ductility of the RC wall, the failure was expected to

happen at low levels of deformation, and in most cases, the loading did not continue far

into the inelastic range. For investigation of the effect of retrofit of the cracked specimen

with respect to leakage, the cracked section was retrofitted with GFRP sheet. This implies

that there was no intention for strengthening the response of the wall against the applied

loads. After conclusion of the leakage test of the specimen and removal of the water

pressure chamber, the gum rubber sheet is detached from the wall. The surface of the

wall and foundation within and near the area where the chamber is to be attached is

carefully scrubbed by a steel brush attached to a drill. This is done to clean the surface

from any dirt and remainders of the silicon glue of the specimen and to create scratch on

the concrete surface for a better bonding development between GFRP layer and the

concrete surface. Then the specimen is vacuum cleaned to ensure a clean concrete wall

surface for the application of the GFRP sheet. After attaching the GFRP sheet on the

wall, the installation steps for the water pressure chamber were repeated. The properties

N

132

of the GFRP used in this test are presented in Table 4-5. Since no attempt was made to

strengthen the wall with respect to its load carrying capacity, no actual test on the

properties of the GFRP material was performed. Two elements of epoxy (resin and

hardener) are mixed up in an appropriate proportion and are blended until turning into a

single smooth liquid. Then the epoxy is applied with a soft painting brush on the surface

of concrete, while the GFRP sheet is installed on top of the epoxy simultaneously.

For those specimens that were not retrofitted, the forced controlled cyclic loading was

applied until failure of the specimen was observed.

Table 4-5 GFRP properties (TYFO SEH-51A Composite Laminate – R.J. Watson, Inc.)

Property ASTM

Method

Typical Test Value Design Value

Ultimate tensile strength in

primary fiber direction (MPa)

D 3039 575 460

Elongation at break (%) D 3039 2.2 1.76

Tensile modulus (GPa) D 3039 26.1 20.9

Nominal laminate thickness (mm)

D 1777

1.18

1.18

4.8 Key parameters in the experiments

A summary of the key parameters used in the current experimental program is presented

in Tables 4-6 and 4-7. These parameters include, the thickness of the wall, the shear key

configuration, the value of the design flexural moment at the base of the wall calculated

using the ACI 350-06 load combinations, the volume of longitudinal rebars at the front

and back side of the wall, and the maximum compressive strength of the concrete

cylinder sample measured 28 days after casting.

Theoretical predictions for cracking load at the base of the wall, yielding of the steel at

the front side of the wall at the interface of wall-foundation for the 1500 mm wide

specimens were obtained to provide some insight regarding the behavior of the

133

specimens. Reinforced concrete sectional analysis program, Response-2000 (Bentz and

Collins 2000) was also used to obtain the moment-curvature curves for the specimens.

The moment curvature curve for Specimen-4 is shown in Figure 4.47.

Table 4-6 Key parameters used in the construction and design of the specimens

Table 4-7 Design load values and the reinforcement volume for the specimens

Specimen Design load

combination-1

(kN.m)

Design load

combination-2

(kN.m)

Reinforcement

at front side of

wall

Reinforcement

at back side of

wall

1 167 138 8 No.20 7 No.20

2 137 135 10 No.20 10 No.20

3 137 135 10 No.20 10 No.20

4 137 135 10 No.20 10 No.20

5 70* 123* 6 No.20 6 No.20

*Amounts of reinforcement are based on design load combinations with values of environmental durability factor and load factors of unity The height of the point of application of the actuator load with respect to the top of

foundation is also presented in Table 4-8. The values of depth of the neutral axis (N.A.)

to the compression face of the wall corresponding to the yield load were also calculated

and presented in Table 4-8. The load values in Table 4-8 are calculated for 1000 mm and

Specimen Wall thickness

(mm)

Shear key

configuration

28-day f′c of concrete (wall)

(MPa)

1 400 Conventional 31.0

2 300 Inverted 29.5

3 300 Conventional 29.5

4 300 Conventional 25

5 400 Conventional 25

134

1500 mm wide walls. The values corresponding to 1000 mm wide walls facilitate the

comparison to the design load values in Table 4-7 which are based on 1000 mm strip of

the wall width, while the values corresponding to 1500 mm wide walls facilitate the

comparison with the experimental observation of the corresponding loads

The theoretical value for yielding flexural moment for 1000 mm width of Specimen-1

was 206 kN.m. The governing load combination for this specimen was 167 kN.m as

shown in Table 4-7. It was investigated that the presence of reinforcement at the

compression side of the wall did not have a major effect on the increase in the flexural

moment capacity of the wall (for all specimens) mainly due to close proximity of

reinforcement to the neutral axis. However, the rounding down of the spacing from 213

mm to 200 mm was responsible for such difference. Calculating the flexural moment

corresponding to yielding of the specimen with the amount of the reinforcement

calculated based on the design load values (spacing of 213 mm) would result in 182

kN.m. Considering the reduction factor of 0.9 incorporated in the design procedure for

the reinforcement which was not applied to the theoretical predictions, the difference

between the design load and the theoretical predictions becomes negligible (around 1%).

For Specimens-2, 3, and 4 the theoretical yielding moment for 1000 mm width of the

specimen was 179 kN.m. The governing load combination for this specimen was 137

kN.m as shown in Table 4-7. The spacing value calculated based on the design load value

was 170 mm. However, due to a certain value for width of the wall, using 10 No.20

rebars would result in the spacing value of 155 mm which was considered acceptable.

Calculating the flexural moment corresponding to yielding of the specimen with the

amount of the reinforcement calculated based on the design load values (spacing of 170

mm) would result is 153 kN.m. Applying the same methodology described above for

Specimen-1, the difference between the design load and the theoretical predictions

becomes negligible (less than 1%).

135

Table 4-8 Theoretical predictions for the specimens

Moment-curvature

050

100150200250300350400

0 0.05 0.1 0.15 0.2 0.25 0.3

Curvature (rad/m)

Mom

ent (

kN.m

)

Figure 4.47 The moment curvature curve for Specimen-4

Specimen Cracking moment (kN.m)

Height of actuator load (m)

Yielding moment (kN.m)

Depth of N.A. (mm)

1 m width

1.5 m width

1 m width

1.5 m width

1 97 145 1.28 206 309 80

2 55 82 1.28 179 268 71

3 55 82 1.28 179 268 71

4 51 76 1.15 179 268 71

5 86 129 1.15 153 230 71

136

CHAPTER-5

RESULTS OF THE EXPERIMENTAL PROGRAM

5.1 General In this Chapter the results of the experimental tests conducted on the wall-base slab

specimens described in Chapter-4 are discussed. The results include the observation of

the cracking of the specimens during the loading of the top of the wall and due to the

magnitude of the loads. The observations also include the detection of leakage from the

sides of the wall, and in some cases from the back of the wall. The load-displacement

relationships for top of the wall for different tests are also presented as well as the strain

values for the installed strain gauges at each sequence of the displacement of top of the

wall.

5.2 Specimen-1 This specimen has a wall thickness of 400 mm with a conventional shear key

configuration as shown in Figure 5.1. The wall is reinforced with eight No.20@200 mm

and seven No.20@230 mm at the front and back faces, respectively. The locations of the

steel strain gauges were shown in Figure 4.37 in Chapter-4.

Figure 5.1 Specimen-1 with wall thickness of 400 mm and conventional shear key

configuration with water-stop

137

5.2.1 The pretest The pretest is conducted on Specimen-1 to establish the first visible cracking of the

specimen. The loading scheme is determined in such a way to apply the force on the wall

in push direction and reversing the load to pull back the wall to its initial neutral position

as shown in Figure 5.2. As will be discussed later, the specimen was not adequately

restrained against horizontal movement of the specimen on the floor; and due to

horizontal movement of foundation; the top of the wall did not return back to its initial

neutral position as shown in Figure 5.3.

The pre-test started with the application of force-controlled cyclic loading of pushing the

wall and pulling it back to its initial neutral position with three cycles at each increment

of 10 kN. The first visible crack was observed at the wall-foundation interface at a load

level of 110 kN (compared to theoretical value of 114 kN) and is shown in Figure 5.4.

The steel strain values in the wall reinforcement at the wall-foundation interface at the

front side of the wall which were measured by the strain gauges SWF-1, SWF-2, and

SWF-3 (shown in Figure 4.37) is shown in Figure 5.5(a) experiencing a maximum strain

of 580 με. The same information for the reinforcement at the back side of the wall

measured by the strain gauges SWB-1 and SWB-2 (shown in Figure 4.37) is shown in

Figure 5.5(b) having a maximum value of -105 με (in compression).

Measured but not shown, the foundation top layer steel strains at the top of the foundation

located 100 mm in front of the face of the wall corresponding to strain gauges SFT-1,

(shown in Figure 4.37) had a maximum strain of 35 με. The strain gauge located 100 mm

behind the back face of the wall corresponding to strain gauge SFT-2 (shown in Figure

4.37), had a maximum strain of -21 με. The strain gauges at the bottom of the foundation

SFB-1, SFB-2 (shown in Figure 4.37) fluctuated between strains of 5, and -5 με. The test

was kept at the maximum load for a short period of time for careful detection of the crack

as is observable by the last parts of the graphs in Figure 5.5. The steel strains at this stage

for the front and back sides of the wall are shown by constant value of the strains

measured as 580, and -105 με, respectively.

138

-20

0

20

40

60

80

100

120

No. of cycles

Late

ral f

orce

(kN

)

Figure 5.2 The applied lateral load on top of the wall

-101234567

No. of cycles

Top

disp

lace

men

t (m

m)

Figure 5.3 The displacement of the top of the wall

139

Figure 5.4 First visible crack at the wall-foundation connection

0100200300400500600700

No. of cycles

Stra

in (×

10-6)

-120-100

-80-60-40-20

020

No. of cycles

Stra

in (×

10-6

)

Figure 5.5 Strain values for the wall rebars at the connection of the wall-foundation


(a)

(b)

140

5.2.2 The leakage test After the installation of the water pressure chamber, the pressure of water inside the

chamber was maintained at 3.5 m head, and the water chamber was pushed forward by

the hydraulic jack. The leakage test started by application of a forced-controlled cyclic

loading with increments of 10 kN with three cycles in each increment with the same

loading scheme as described for the pretest. At the load level of 70 kN leakage of water at

the front side of the wall through the joint region at the wall foundation interface started

as is shown in Figure 5.6. With the increase in the magnitude of the load to 120 kN, new

leaking cracks started to propagate at both side faces of the wall. These cracks which

measured around 150 mm deep, were located 160 mm above the foundation as shown in

Figures 5.7 and 5.8 for the west and east side faces, respectively. It is expected that these

cracks have been formed along the width of the wall. However, due to presence of the

water pressure chamber, it was not possible to observe the entire length of these cracks.

Figure 5.6 Leakage of the water through the joint region at the initial stage of the test

141

Figure 5.7 The first leaking crack above the joint at the west side face of the wall

Figure 5.8 The first leaking crack above the joint at the east side face of the wall

As the load magnitude increased, the depth of the leakage at the mentioned cracks

increased until the load level of 220 kN was reached. At this stage new leaking cracks

were observed 480 mm above the foundation and at both side faces of the wall. Figure

5.9 shows the east side face of the wall at the load level of 260 kN; where the depth of the

leaking cracks 160 mm and 480 mm above the foundation, were 250 mm and 180 mm,

142

respectively. After the leakage depth became intensified at the load level of 340 kN,

passing through the mid-depth of the section, the number of load cycles was increased to

around 50 cycles, while the same load levels of 340, and afterward 350, and 360 kN were

maintained for each 50 cycles. This was done to investigate the possible effect of the

number of cycles on leakage at this stage. The test was continued up to a load level of

360 kN when it was stopped due to extensive cracking observed at both side faces of the

wall. As can be seen in Figure 5.10, the neutral axis depth has extensively increased with

respect to the tension face of the wall, indicating significant yielding of the front face

rebars in tension. The maximum strain gauge reading for the front side reinforcement was

about 15000 με. However, no leakage was observed at the back side of the wall as can be

seen in Figure 5.11 even with the repeated 50 cycles of loading near the capacity of the

member. Figures 5.12, 5.13, and 5.14 show the cracks at the front face of the wall shown

by arrows after conclusion of the test and removal of the water pressure chamber. Figure

5.15 shows the extensive crack at the connection of the wall and foundation. The applied

load and the displacement of the top of the wall are shown in Figures 5.16 and 5.17,

respectively.

Figure 5.9 Leakage at the east face of the wall at 260 kN

143

Figure 5.10 Cracking of the side faces of the wall after the test

(a) west side face (b) east side face

Figure 5.11 The condition of the back face of the wall immediately after the test

144

Figure 5.12 Cracks at the front face of the wall 480 mm above the foundation

Figure 5.13 Cracks at the front face of the wall after conclusion of the test

145

Figure 5.14 Cracks at the front face of the wall after conclusion of the test

Figure 5.15 Extensive crack at the connection of the wall and foundation

146

0

50

100

150

200

250

300

350

400

No. of cycles

Late

ral f

orce

(kN

)

Figure 5.16 The applied lateral load on top of the wall

-10

0

10

20

30

40

50

60

No. of cycles

Top

disp

lace

men

t (m

m)

Figure 5.17 Top displacement of the wall

Figures 5.18(a) and (b) illustrate the strain values for the wall rebars at the connection of

the wall-foundation at the front side (SWF-1, SWF-2, and SWF-3) and the back side

(SWB-1 and SWB-2) of the wall, respectively. The front steel strained significantly and

147

all the strain gauges failed during the test. The back side steel rebars were first in

compression. Near the end of the test, due to moving of the neutral axis towards the

compression face of the section, back side steel rebars started to experience tensile strain.

The steel strain at the top of the foundation 100 mm in front of the face of the wall

measured by strain gauges SFT-1 and shown in Figure 5.19 had a maximum strain of

about 750 με. The steel strain at the top of the foundation 100 mm behind the back face

of the wall measured by strain gauges SFT-2 and shown in Figure 5.20 had a maximum

strain of about 120 με. Because of the high level of damage to this specimen and the fact

that none of the foundation steel strain levels became critical, it was decided not to install

the strain gauges for the foundation reinforcement for the next specimens.

Front side steel strain

-5000

0

5000

10000

15000

20000

0 500 1000 1500 2000 2500 3000

Time (sec)

Stra

in

Back side of the wall

-500

0

500

1000

1500

2000

0 500 1000 1500 2000 2500 3000

Time (sec)

Stra

in (×

10-6)



(a)

(b)

-5000

50010001500200025003000

0 500 10

148

-200

0

200

400

600

800

0 500 1000 1500 2000 2500 3000

Time (sec)

Stra

in (×

10-6)

Figure 5.19 Strain values for the top layer of the foundation reinforcement 100 mm in

front of the front face of the wall

-140-120-100

-80-60-40-20

020

0 500 1000 1500 2000 2500 3000

Time (sec)

Stra

in (×

10-6)

Figure 5.20 Strain values for the top layer of the foundation reinforcement 100 mm

behind the back face of the wall

After the analysis of the data, several problems became evident. The movement of the

foundation introduced extensive complexity into the interpretation of the data in terms of

the absolute displacement of the wall. The horizontal displacement of the foundation

measured by an LVDT which was installed at the back side of the foundation is shown in

Figure 5.21. Also applying the hydraulic jack force to the water-pressure chamber caused

some difficulties in the interpretation of the data. The test started by applying 40 kN force

149

to the chamber to penetrate the chamber edge enough into the gum rubber to prevent

leakage. As shown in Figure 5.22, this force was fluctuating during the test due to back

and forth movement of the wall; increase and decrease in the pressure of the water; and

sometimes by manually increasing the force to prevent leakage from the interface of the

chamber and rubber. A drawback in the test setup was the fact that the hydraulic jack and

the actuator loads were collected by different data acquisition systems which could not be

correlated. It would be desirable to use a system which could have made it possible to

correlate these loads (with different point of application) in order to obtain the net load on

the wall with respect to the base.

The other problem was the way that the load was applied. The data showed that the

actuator load in the push direction has increased incrementally while the load in the pull

direction never came back to its initial position as shown in Figure 5.16. Another problem

in the test was due to using one data acquisition system to record the load and

displacement exerted by the actuator, and another independent data acquisition system to

measure only the steel strain values. Therefore, there was no way to correlate the data

regarding the strain values to the level of the load or displacement of the wall, as they

were recorded by two different data acquisition systems. This caused the steel strain

values for Specimen-1 to be presented with respect to time. For the next specimens an

LVDT was installed at the back of the wall and was connected to the data acquisition

system which was used to measure the steel strains. Therefore, it was possible to correlate

the load displacement of the top of the wall and the steel strain values for the next

specimens by correlating the top displacement values obtained by the two independent

data acquisition systems.

Although the test for this first specimen was regarded as a control test to check the testing

method and possible deficiencies, valuable results could still be obtained from the test.

150

Figure 5.21 Horizontal displacement of the foundation

Hydraulic jack force

0

20

40

60

80

100

120

0 500 1000 1500 2000 2500 3000Time (sec)

Forc

e (k

N)

Figure 5.22 Variation of the hydraulic jack force during the test

5.2.3 Test of the retrofitted Specimen-1 for leakage For the retrofitted Specimen-1 the application of one layer of GFRP to the cracked

regions of the wall was assessed. The entire front face of the wall up to a height of 700

mm was covered with GFRP sheet and this was extended to the sides of the wall to cover

the cracked area. The layer was also extended 400 mm on the foundation at the front of

the front face of the wall. Wherever the length of the GFRP sheet was not enough to

cover the area, another GFRP sheet was installed with 300 mm overlap as can be seen in

151

Figure 5.23. After installation of the gum-rubber sheet and the pressure chamber, the

interface of the water-chamber and the GFRP sheet was sealed with silicon glue as shown

in Figures 5.24 and 5.25 to eliminate the chance of leakage through the interface between

the gum rubber and the GFRP. This ensures that any possible leakage would pass through

the GFRP sheet installed on the specimen and through the previously formed crack at the

front face of the wall. Due to failure of the strain gauges during the previous test, no

information could be obtained regarding the strain condition of the wall reinforcement.

At the start of the test a 40 kN force was applied on the pressure chamber by the

hydraulic jack. A static load of 70 kN was applied using the actuator, however no leakage

was observed through the existing cracks on the side faces of the wall. Considering the

distances between the points of application of the load by the hydraulic jack and the

actuator as 400 mm and 1250 mm above the foundation, the total applied flexural

moment at the base of the wall was about 100 kN.m. For comparison, the flexural

moment at the base of the tank due to 3.5 m height of water was calculated as 70 kN.m.

The cyclic load with a pattern described in the pretest and leakage test, was followed

beyond the 70 kN load in increments of three cycles each. The wall started to leak at 117

kN. The leakage was on the sides of the wall at existing crack locations (at the wall base

and above). However, the leakage was beyond the point where the FRP was discontinued

as can be seen in Figures 5.26 and 5.27.

Figure 5.23 Overlaps for the one layer FRP before application of the resin

152

Figure 5.24 The FRP on the foundation and the joint inside the chamber

Considering the point of application of the actuator as 1250 mm above the foundation,

the 117 kN produces a 146 kN.m flexural moment at the base. Adding the flexural

moment caused by the hydraulic jack of approximately 15 kN.m, the total flexural

moment is approximated as 160 kN.m at the start of the leakage. It is important to note

that the design load based on the first design load combination (the load combination

without seismic loads) for this specimen as stated in Table 4-7 was 167 kN.m for a unit

width of the wall which is equivalent to 250 kN.m for this 1500 wide specimen. This

design load corresponded to a value of 1.7 for the environmental durability factor

incorporated in the load combination, based on the spacing provided for the longitudinal

reinforcement.

Figure 5.25 The FRP inside the chamber

153

Figure 5.26 The specimen during the first stage of the test

Figure 5.27 Leakage of the specimen through the existing cracks

5.3 Specimen-2

Specimen-2 had a wall thickness of 300 mm with inverted shear key configuration at the

joint as described in Section-4.3.4 and shown in Figure 5.28. The wall thickness was

chosen as 300 mm to investigate the effects of the thickness of the section and also

possible effect of the reduced spacing of the longitudinal reinforcement compared to a

154

thicker wall section as used for Specimen-1 (with thickness of 400 mm). The wall has ten

No.20@155 mm at the front and back sides of the wall. It was assumed that the loads in

the push and pull directions have the same value, considering the possibility of a tank

having water on sides of the wall. This situation may exist in a multi cell tanks

representing an interior wall. Because of the low level of the value of strains observed in

the foundation rebars for the damaged Specimen-1 (less than 40% of yielding strain), no

strain gauge was installed for the foundation steel bars. In the remainder of this Chapter,

SWF and SWB denote the strain gauges at the front and at the back sides of the wall,

respectively, at the wall-foundation interface.

Figure 5.28 The joint before installation of the wall form

5.3.1 The leakage test After installation of the water pressure chamber including application of the hydraulic

jack force, the test started by applying cyclic loading which consisted of three cycles of

forced controlled push and pull with similar values at each increment (10 kN increments).

During the cycles corresponding to the actuator load of 40 kN, leakage was observed at

the back side of the wall at the connection of the wall and foundation as shown in Figure

5.29. The side faces of the wall near the water chamber were also leaking at a slow rate

with a leaking crack length of 100 mm at this stage. Figures 5.30 and 5.31 illustrate the

applied lateral load and displacement at the top of the wall, respectively. The lateral load-

displacement relationship for the top of the wall is shown in Figure 5.32. Figures 5.33(a)

and (b) show the variation of strain values of rebars, with respect to the top displacement,

155

at the connection of the wall-foundation region at the front (SWF gauges) and back

(SWB gauges) of the wall, respectively.

Similar to the test of Specimen-1, four threaded rods with diameter of 31.7 mm were used

to fix the specimen to the lab’s floor. However, to restrain the horizontal movement of

the specimen during the test, a “floor brake” was used. The analysis of the data recorded

by an LVDT showed that the floor brake was successful for its intended purpose and the

horizontal movement of the foundation was almost equal to zero during all tests.

Figure 5.29 Leakage at the back face of the wall at the initial stage of loading (40 kN)

Actuator force

-60-40-20

0204060

Cycles

Forc

e (k

N)

Figure 5.30 Applied lateral load on top of the wall

156

Top displacement

-3.0-2.0-1.00.01.02.03.04.05.0

Cycles

Dis

plac

emen

t (m

m)

Figure 5.31 Lateral displacement of the top of the wall

Top force-displacement

-60

-40

-20

0

20

40

60

-4 -3 -2 -1 0 1 2 3 4

Top displacement (mm)

Forc

e (k

N)

Figure 5.32 Lateral load-displacement relationship for the top of the wall

-200

-100

0

100

200

300

-4 -2 0 2 4 6


Stra

in (×

10-6)

157

-100-50

050

100150200250

-4 -2 0 2 4 6


Stra

in (×

10-6)

Figure 5.33 Strain value of the steel at the connection of the wall-foundation


Due to leakage associated with the back face of the wall, the cyclic loading stopped at 40

kN. After this stage, monotonic load was applied with increments of 5 kN to further

investigate the behavior of the wall. The reason for application of monotonic loading at

this stage was to determine if the leakage was due to the effect of cyclic loading. As the

monotonic load was increasing, the crack at the wall-foundation joint started to penetrate

more into the depth of the wall followed by leaking from the side faces of the wall. As

the load reached 115 kN, a new leaking crack with a length of 70 mm formed 150 mm

above the joint at the west side face of the wall. Also the length of the joint crack was 150

mm reaching the mid-section of the wall when the strain in the steel was at a quarter of

the yield value (500 με). At 210 kN a new crack with a length of 100 mm and a new

crack with length of 50 mm were observed at 320 mm and 420 mm above the joint at the

west side face of the wall, respectively, as can be seen in Figure 5.34. At this stage the

length of the leaking crack at the west side face, 150 mm above the joint, was increased

to 120 mm.

158

Figure 5.34 Leaking cracks at the west side face of the wall at 210 kN

The pattern of the cracks at the east side face of the wall was similar to the west side face

and is shown in Figure 5.35 at the load level of 280 kN. At the load level of 320 kN the

side faces at the joint was leaking severely, followed by a slower rate of the leakage at the

upper cracks at both side faces as shown in Figures 5.36(a) and (b). The interesting point

was that the rate of leakage at the back of the wall was not as severe and was very

slightly higher than the rate at the initiation of the back face leakage at 40 kN cyclic load.

At the load level of 340 kN, due to a sudden increase in the applied load, the wall failed

at displacement of 92.9 mm as shown in Figure 5.37.

Figure 5.35 Leaking cracks at the east side face of the wall at 280 kN

159

Figure 5.36 Leakage condition at 320 kN

(a) west side face (b) east side face and back of the wall

The data regarding the top-displacement of the wall recorded by LVDT is shown in

Figure 5.38. The steel strains variation with respect to the top displacement are shown in

Figure 5.39 for front side of the wall (gauges SWF), and Figure 5.40 for back side of the

wall (gauges SWB) for the entire test including the initial cyclic loading stage. The

reinforcement at the front side of the wall was so strained that the strain gauges failed at

(a)

(b)

160

the top displacement of about 50 mm after recording a maximum strain of 6000 με. The

steel strain for the back side of the wall was first in compression and at the last stage of

the test experienced tensile strain in excess of 2000 με, indicating that the compression

zone became very small.

Figure 5.37 Brittle failure of the wall at the end of the test

-100

102030405060708090

100

Time

Top

disp

lace

men

t (m

m)

Figure 5.38 Displacement of the top of the wall during the entire test

161

-2000

0

2000

4000

6000

8000

0 20 40 60 80 100


Stra

in (×

10-6

)

Figure 5.39 Strain of the steel at the wall-foundation interface at the front side of the wall

As was mentioned, the test started by applying 40 kN force by the hydraulic jack to the

chamber. As can be seen in Figure 5.41, this force was fluctuating at the initial cyclic

portion of the test. In order to penetrate the edge of the water chamber into the gum

rubber, the force was manually increased whenever leakage of water through the

interface of chamber and rubber was observed.

-500

0

500

1000

1500

2000

2500

0 20 40 60 80 100


Stra

in (×

10-6

)

Figure 5.40 Strain of the steel at wall-foundation interface at the back side of the wall

162

0

10

20

30

40

50

0 200 400 600 800 1000 1200 1400

Time (sec)

Forc

e (k

N)


5.4 Specimen-3

Specimen-3 was identical to Specimen-2 except that instead of the inverted shear key at

the joint, a conventional shear key, shown in Figures 5.42 and 5.43, was used. The wall

had a thickness of 300 mm and was reinforced with ten No.20@155 mm at the front and

back sides. For this specimen the strain gauges were installed only for the wall rebars at

front and back sides and at the location corresponding to the wall-foundation interface.


163


immediately after casting of the concrete for foundation

5.4.2 The leakage test After installation of the water pressure chamber, the cyclic loading started with three

cycles of push and pull in increments of 5 kN. At load level of 50 kN, leakage of water at

corners at the connection of the wall-foundation at both side faces of the wall (near the

front face of the wall) was observed. After the cyclic load level reached about 80 kN,

rocking of the foundation became significant. This rocking was mainly due to the 31 mm

threaded rods which were used to tighten the specimen to the strong floor. The test was

stopped at this time to tighten the rods. At 90 kN the bolts which were used to connect

the steel beams through the top of the wall were observed to have elongated, and a gap

could be observed between the wall and the front side steel beams. At 105 kN a slightly

leaking crack was observed at the east side face of the wall and another crack at the west

side face of the wall, both around 430 mm above the foundation. The cyclic loading was

repeated at this level for an arbitrary number of 50 cycles. Due to rocking of the

foundation the test was again stopped for tightening of the rods. This showed that

stronger rods with higher cross-sectional area were necessary. Using the same threaded

rod, the test restarted from the zero load condition, and at 90 kN, a small crack was

observed at the west side face of the wall. At 115 kN, one of the ½" bolts connecting the

steel beams through the top of the wall failed. It was decided to replace these rods with

164

higher diameter (¾") rods with high strength properties. Figure 5.44 shows the load-

displacement relationship for the top of the wall before the failure of the threaded rod at

the top of the wall.

The test was repeated after the four threaded rods with diameter of 31.7 mm were

replaced by stronger threaded rods with a diameter of 44.4 mm to fix the foundation to

the floor, and the top of the wall threaded rods (normal strength material with diameter of

½") were replaced by high strength (Grade B7) rods with diameter of ¾". Before

application of the cyclic load using the actuator, during increasing the water pressure to

the predetermined level corresponding to 3.5 m head of water, small leakage was

observed at the connection of the wall-foundation and the side faces of the wall starting

from the front side of the wall. The leakage was through the cracks at the wall-foundation

connection which were formed during the previous test on the specimen.

-150

-100

-50

0

50

100

150

-20 -15 -10 -5 0 5 10


Late

ral l

oad

(kN

)


The cyclic loading started with three cycles of push and pull with increments of 10 kN.

At 80 kN, leakage from the crack at both side faces, about 430 mm above the base, was

observed and shown in Figure 5.45. For the west side face, the length of the leaking crack

was measured as 50 mm. These were cracks that formed during the previous tests. At 150

kN, water was pouring out from the wall-foundation joint at the sides (near the front face)

in the push direction of the cycle and stopped pouring when the load direction was

reversed. At 190 kN, the joint leakage became significant and also new leaking cracks

165

were observed 120 mm above the joint at the side faces as shown in Figure 5.46 for the

west side face. At 200 kN, the lengths of the leaking cracks at 120 mm, and 430 mm

above the foundation were around 150 mm (mid-depth of the wall) as shown in Figure

5.47 for the west side face. At the load of 210 kN, leaking was observed at the back side

of the wall at 430 mm above the foundation, and at a slower rate at 120 mm above the

foundation as shown in Figure 5.48. The test was continued for twelve more cycles (four

sets of three cycles) up to 240 kN when the test was stopped. The length of the leaking

crack 430 mm above the base reached around 250 mm at the side faces of the wall and

stopping of the test was mainly to prevent the complete failure of the specimen. It is

important to mention that the rocking of the foundation and opening of the steel beams on

top of the wall were minimized due to replacement of the previous rods with stronger

ones, as were visibly obvious.

Figure 5.45 Leakage from the crack at west side face, about 430 mm

above the base at load level of 80 kN

166

Figure 5.46 Leakage from the crack at west side face, about

120 mm above the base at load level of 190 kN

Figure 5.47 Leakage from the cracks at west side faces, at load level of 200 kN

167

Figure 5.48 Leakage condition at the back face of the wall at load level of 210 kN

The lateral load-displacement relationship for the top of the wall during this cyclic stage

is shown in Figure 5.49. Figures 5.50(a), and (b) show the variation of strain values of the

steel, with respect to the top of wall displacement, at the wall-foundation interface at the

front (gauges SWF) and back (gauges SWB) sides of the wall, respectively. The front

side strain gauges experienced the maximum tensile strain of 2247 με corresponding to

the wall top displacement of 18.7 mm in the push direction. The back side strain gauges

experienced the maximum tensile strain of 1547 με corresponding to the wall top

displacement of 13.6 mm in the pull direction. Like previous test, the test started by

applying 40 kN force to the chamber to penetrate the chamber edge enough into the gum

rubber to prevent leakage. As can be seen in Figure 5.51, this force was fluctuating

during the test due to back and forth movement of the wall, increase and decrease in the

water pressure, and sometimes by manually increasing the force to prevent leakage from

the interface of the chamber and rubber.

It was decided to investigate the effect of retrofitting of the specimen on the leakage

behavior of the damaged wall as discussed in the following section.

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5.4.3 Test of the retrofitted Specimen-3 for leakage For investigation of the effect of retrofitting the section with GFRP sheet, two layers of

GFRP were applied to the cracked regions of the specimen. The entire front face width of

the wall up to a height of 700 mm was covered with two layers of GFRP while the layers

were extended 400 mm on the foundation at the front of the front face of the wall as

shown in Figure 5.52. As can be seen, the GFRP layers were not extended to the side

faces of the wall; however, they covered the front face in such a way that any possible

leaking water had to pass through the GFRP layers. Before application of the cyclic load

using the actuator, during increasing the water pressure to the predetermined level of 3.5

m head, no leakage was observed at the connection of the wall-foundation. The cyclic

loading started with three cycles of push and pull in increments of 10 kN. At the initial

stage of the test, the crack at the west side face 120 mm above the joint became wet up to

the mid-depth of the wall at 30 kN as shown in Figure 5.53.

At 130 kN load, the leakage through this crack became severe. At this load the crack at

430 mm above foundation at the west side face started to leak. No leakage was observed

at the east side and also the connection regions at both sides. At 190 kN, leakage was

observed at the back face of the wall through the crack 120 mm above the foundation. At

210 kN, the leaking crack at the west face connected to the leaking crack at the back side

forming a connected leaking crack at 120 mm above the foundation as shown in Figure

170

5.54. Figure 5.55 illustrates the top crack at the back of the wall, 430 mm above the

foundation which started to leak at 250 kN.

Figure 5.52 The retrofitted wall using two layers of GFRP

Figure 5.53 Leakage condition at the west side face of the wall at 30 kN

At 290 kN the joint at the west face started to leak severely. Because of the limited

capacity of the actuator, the test was continued with application of 40 cycles with

constant load of 250 kN. It was only at this stage of loading that the east face started to

171

leak at a slow rate, along the crack that initiated at the front face and 430 mm above the

foundation. Figures 5.56 and 5.57 show the condition of the leaking cracks at the back of

the wall, and Figure 5.58 shows the condition of the specimen after the conclusion of the

test and removal of the water pressure chamber.

Figure 5.54 Leakage condition at the west face of the wall at 210 kN

Figure 5.55 Leakage condition at the back face of the wall at load level of 250 kN

172

Figure 5.56 The leaking crack at the back of the wall

Figure 5.57 Leakage at the back face through previously formed cracks

173

Figure 5.58 The retrofitted specimen after the test

5.5 Specimen-4

Specimen-4 was identical to Specimen-3 where the wall had a thickness of 300 mm and

was reinforced with ten No.20@155 mm at the front and back sides. The specimen had a

conventional configuration for shear key where a 150 mm rubber water-stop was installed

in the shear key region as shown in Figure 5.59. The purpose of constructing and testing

this specimen, which was similar to Specimen-3, was to verify the findings of the leakage

test on Specimen-3. The other reason was to conduct a pretest and also to test the

specimen up to failure in order to assess the ultimate state behavior which was not

captured for Specimen-3 due to application of GFRP sheets. These tests can facilitate the

comparison of the analytical (using FE models) and experimental results with respect to

cyclic behavior of the specimen without the complex effects of the water pressure

chamber as will be discussed in Chapter-7. As can be seen in Figure 5.60, for this

specimen the strain gauges were installed only for the vertical bars at front and back sides

of the wall.

174

Figure 5.59 The conventional shear key using a plastic water-stop for Specimen-4

Figure 5.60 Specimen-4 with wall thickness of 300 mm

5.5.2 The pretest The pretest started by application of a monotonic force controlled load in the push

direction while data on the strain values was being collected. The load was increased in

increments of 10 kN and then paused for detection of the possible cracks. At the load of

70 kN, the first visible crack was observed at the front side of the wall at the connection

of the wall-foundation (compared to the theoretical value of 69 kN). The load was

increased and after it reached 90 kN, the test was paused for careful detection of possible

175

cracks. Except the mentioned crack, no other observable crack was formed. Top load-

displacement relationship for this monotonic loading stage is shown in Figure 5.61. The

strain values at the front face of the wall (gauges SWF) and at the back face of the wall

(gauges SWB) are shown in Figures 5.62(a) and (b), respectively. The front side strain

gauges experienced the maximum tensile strain of 915 με corresponding to the wall top

displacement of 6 mm in the push direction. The back side strain gauges experienced the

maximum compressive strain of 254 με corresponding to the wall top displacement of 6

mm in the push direction.

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Figure 5.62 Strain value of the steel at the connection of the wall-foundation region


After the initial monotonic loading stage, the cyclic loading stage of the pretest followed.

It is important to note that the water pressure chamber was not installed at the pretest.

This was done to reduce the complexities associated with loadings exerted by the

pressurized water on the wall and the hydraulic jack on the pressure chamber and

consequently on the wall. This would also facilitate the detection of the possible cracking

at the front face of the wall at the initial stages of loading.

The cyclic loading started by applying 3 cycles of push and pull with increments of 10

kN on top of the wall. At the load level of 90 kN, the crack at the connection of the wall-

foundation at the front face was visibly opening and closing, however, because of the

location of the crack it was not possible to measure the crack width. At the load level of

110 kN, the same observation was made at the back side of the wall. At the load level of

140 kN, the crack at the front face of the wall became visibly wider. At the load level of

170 kN, several cracks were observed. The test was paused for careful detection of the

possible cracks. A crack was observed at the distance 230 mm above the foundation at

the east side face of the wall which started from the back side of the wall. The width of

this crack was measured as 0.28 mm. This crack was named Crack-1 as shown in Figure

5.63. Another crack was observed at the west side face of the wall which started from the

front face and 320 mm above the base. The width of this crack was measured as 0.2 mm.

This crack was named Crack-2 as shown in Figure 5.64. At 180 kN, several other cracks

were observed. At the east side face, three cracks initiated from back side of the wall at

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177

120 mm, 410 mm, and 530 mm above the base. Three cracks were also observed at the

east face starting from the front face at 120 mm, 290 mm, and 410 mm above the base as

shown in Figure 5.63. At the west side face, cracks that initiated from the back side were

observed at 120 mm, 290 mm, 410 mm, and 500 mm above the base. The cracks that

initiated from the front side were at heights 120 mm, 320 mm, 420 mm, and 520 mm

from the base as shown in Figure 5.64.

Figure 5.63 Condition of the cracks at the east face of the wall

At the load 190 kN, the test was paused at the maximum push level, and afterward, at the

maximum pull level to check the continuous cracks at the front and back faces of the

wall, respectively. It was observed that several continuous cracks have formed at the

178

front and back faces connecting the mentioned cracks at the east and west side faces of

the wall. At the load level of 190 kN, the observed cracks became wider. The width of

Crack-1 and Crack-2 became 0.32 mm, and 0.3 mm, respectively. The lateral load-

displacement relationship for the top of the wall during this cyclic stage is shown in

Figure 5.65. Figures 5.66(a), and (b) show the variation of strain values of the steel, with

respect to the top displacement, at the wall-foundation interface at the front side (gauges

SWF) and back side (gauges SWB) of the wall, respectively. The front side strain gauges


displacement of 11 mm in the push direction. The back side strain gauges experienced the

maximum tensile strain of 1716 με corresponding to the wall top displacement of 12 mm

in the pull direction. It is important to note that the theoretical yielding load for

Specimen-4 is 233 kN.

Figure 5.64 Condition of the cracks at the west face of the wall

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5.5.3 The leakage test Although the wall was cracked and leakage was expected to happen at the side faces of

the wall during the leakage test, the main purpose of this research program was to

observe and establish the criteria for the leakage at the back side of the wall. After

conclusion of the pretest, the water pressure chamber was installed, the hydraulic jack

force was applied, and the appropriate water pressure was maintained. The cyclic loading

started with three cycles of forced controlled push and pull in increments of 10 kN. At 20

kN, leakage started from the connection of the wall-foundation from both side faces

through existing cracks. At 40 kN, leakage started at east side face from the crack located

120 mm above the joint; and at 60 kN, this leakage length was measured as 50 mm. At

120 kN, leakage at the connection of the wall-foundation reached the mid-depth of the

wall; while for the crack 120 mm above the joint, the leakage length was around 100 mm

as can be seen in Figure 5.67(a). At 140 kN, for the crack 120 mm above the joint, the

leakage length was measured as 145 mm. At 160 kN, the length of the leakage from the

wall-foundation connection remained at the mid-depth of the wall section (because of the

water-stop at the joint), and for the crack 120 mm above the joint the leakage length was

around 180 mm as shown in Figure 5.67(b). At 190 kN, the crack 320 mm above the joint

started to leak at the west side face, and the crack 410 mm above the joint at the east side

face started to leak as shown in Figures 5.68(a) and (b), respectively.

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At 210 kN, yielding of the front face steel was reported (compared to theoretical value of

233 kN excluding the effect of the hydraulic jack force). The depth of the leakage at the

crack 120 mm above the joint was 215 mm. At 230 kN, yielding of the back side steel

was observed and the back face of the wall became wet along the crack located 410 mm

above the foundation. At this level, the crack 120 mm above the joint at the back side of

the wall became very slightly wet as can be seen in Figure 5.69. The crack located 410

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181

mm above the foundation started to leak at 250 kN while a new leakage was observed

along a crack line 120 mm above the foundation as shown in Figure 5.70.

Figure 5.67 West side face of the wall at load level of

(a) 120 kN (b) 160 kN

Figure 5.68 Leaking cracks at the load level of 190

(a) west side face (b) east side face

182

Figure 5.69 Back side of the wall at load level of 230 kN

Figure 5.70 Back side of the wall at load level of 250 kN

The lateral load-displacement relationship for the top of the wall is shown in Figure 5.71.

Figures 5.72(a), and (b) show the variation of strain values of the steel, with respect to the

top displacement of the wall, at the connection of the wall-foundation region at the front

(gauges SWF) and back (gauges SWB) of the wall, respectively. The front side strain

gauges experienced the maximum tensile strain of 2295 με corresponding to the wall top

displacement of 17.9 mm in the push direction. The back side strain gauges experienced

the maximum tensile strain of 2002 με corresponding to the wall top displacement of 14

mm in the pull direction. The front side reinforcement experienced yielding at the top

183

displacement of 15 mm corresponding to a lateral load of 210 kN by actuator on top of

the wall.

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5.5.4 Test of Specimen-4 to failure The assessment of the behavior of the specimen was further continued without the water

chamber and up to the failure of the specimen. After removal of the water pressure

chamber and the gum rubber sheet off the front side of the wall, the test started from no

load condition by application of force controlled cyclic loading with increments of 10 kN

having three cycles of push and pull in each increment as shown in Figure 5.73(a).

Because of the different capacity of the loading actuator in the push and pull directions

(355 kN in push, and 240 kN in pull direction), after the load in pull direction reached

200 kN, the loading pivot point was shifted intentionally as can be observed from Figure

5.73(b). This caused more displacement in push direction and less displacement in pull

direction as the incremental loading level was increasing. The maximum load that was

sustained by the specimen was 338 kN followed by gradual softening of the behavior

until it failed at a displacement of 120 mm. Lateral load-displacement relationship for the

top of the wall is illustrated in Figure 5.74.

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Figure 5.73 Loading for top of the wall

(a) lateral load (b) lateral displacement

The extensive opening of the wall-foundation connection and extensive cracking at the

front face 190 mm above the foundation was observed as shown in Figure 5.75. At the

failure, spalling of the concrete at the back face of the wall along a surface within 100

mm above the foundation was observed as shown in Figure 5.76.

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Figure 5.75 Extensive cracking of the front face of the wall after the test

187

Figure 5.76 Damage at the back face of the wall after the test

5.6 Specimen-5 Specimen-5 had a wall thickness of 400 mm with a conventional shear key configuration

as shown in Figures 5.77 and 5.78. Specimen-5 was reinforced with six No.20@280 mm

at the front and back sides. As shown in Figure 5.77, for this specimen the strain gauges

were installed only for the wall bars at front and back sides. The reason for constructing

and testing of this specimen was to investigate the leakage behavior of a wall of 400 mm

depth for the effect of cyclic loading with equal loads in push and pull direction which

was not done for Specimen-1 (the only specimen with wall depth of 400 mm tested prior

to this specimen). Due to limitation in actuator load capacity as was observed during the

testing of Specimen-1, the reinforcement volume was decreased for Specimen-5 to better

capture the ultimate load stage. As was discussed, the reinforcement volume was

determined based on the reduced design loads by incorporating the design load factors of

unity.

5.6.2 The pretest The pretest started by application of cyclic loading consisted of applying 3 cycles of

equal push and pull in increments of 10 kN on top of the wall. At the load level of 110

kN (compared to theoretical value of 112 kN), cracking at the connection of the wall-

foundation at the front face and back face observed. Figure 5.79 shows the cracking at the

188

front face. At the load level of 150 kN in the push direction, a continuous crack was

observed at the front face of the wall 400 mm above the wall-foundation connection.

Figure 5.77 Specimen-5 with wall thickness of 400 mm

Figure 5.78 The conventional shear key using a rubber water-stop

This crack extended to both side faces of the wall as shown by the blue line in Figure

5.80. The crack lengths at the east and west side faces of the wall were similar and equal

to 200 mm. At the load level of 150 kN in the pull direction, a similar continuous crack

was observed at the back face of the wall 400 mm above the wall-foundation connection.

189

This crack extended to both side faces of the wall as shown by the red line in Figure 5.81.

The crack lengths at the side faces of the wall were similar and also equal to 200 mm.

The pretest was stopped at load level of 170 kN due to the rebars strains approaching

yielding value. Note that the theoretical value for yielding of the steel rebars was 203 kN.

Figure 5.79 The continuous crack at the front face at load level of 110 kN

Figure 5.80 The continuous crack at the front and west side faces of the wall

190

Figure 5.81 The continuous crack at the back and west side face of the wall

The lateral load-displacement relationship for the top of the wall during this cyclic stage

is shown in Figure 5.82. Figures 5.83(a), and (b) show the variation of strain values of the

steel rebars, with respect to the top displacement, at the connection of the wall-foundation

region at the front and back of the wall, respectively. The front side reinforcement

experienced a maximum strain of about 1950 με, while the back side reinforcement

experienced a maximum strain of about 1970 με.

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5.6.3 The leakage test After conclusion of the preliminary testing, the water pressure chamber was installed, the

hydraulic jack force was applied, and the appropriate water pressure was maintained.

However, as soon as the water pressure was increased to the equivalent pressure of 3.5 m

head of water, the leakage started through the side face of the wall along the existing

cracks at the base and located 400 mm above the base as shown in Figure 5.84, and also

the existing crack at the back face of the wall 400 mm above the wall-foundation

(b)

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192

connection started to get wet as shown in Figure 5.85. No other leakage was observed at

the back of the wall. The cyclic loading started with three cycles of forced controlled

equal push and pull with increments of 10 kN. At 60 kN, the length of the leakage 400

mm above the base at east side face was 200 mm, reaching the mid-depth of the section

as shown in Figure 5.86. At this stage the crack located 400 mm above the base at the

back side of the wall was leaking slightly, while no leakage was observed at the back face

at the base. At 150 kN, the lengths of the crack at both side faces of the wall were around

300 mm; while the length of the leaking crack at the joint in spite of severe leaking did

not exceed the mid-depth (because of the water-stop). At 210 kN, water was

intermittently pouring out of the crack located 400 mm above the base at the back of the

wall during the push and pull loading. Figures 5.87(a) and (b) show the leakage at the

back side and east side faces, and back side and west side faces, respectively. At 250 kN,

the leakage length through the joint was so significant that a considerable amount of

water was escaping the chamber and the required water pressure could not be maintained.

At this load level the leakage length at the side face of the wall reached the full-depth of

the section and the test was stopped at this point.

Figure 5.84 Leakage of the west side face at the beginning of the test

193

Figure 5.85 Leakage at the back side of the wall at the beginning of the test

Figure 5.86 Leakage at the east side face of the wall at the beginning of the test

194

Figure 5.87 Leakage of the wall at 210 kN load

(a) back side and east side face (b) back side and west side face

The lateral load-displacement relationship for the top of the wall is shown in Figure 5.88.

Figures 5.89(a), and (b) show the variation of strain values of the steel, with respect to the

top displacement, at the connection of the wall-foundation interface at the front and back

of the wall, respectively. The front side strain gauges experienced the maximum tensile

(a)

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195

strain of 4076 με corresponding to the wall top displacement of 12 mm in the push

direction. The back side strain gauges experienced the maximum tensile strain of 2498 με

corresponding to the wall top displacement of 14 mm in the pull direction.

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5.6.4 Test of Specimen-5 to failure The assessment of the behavior of the specimen was further continued without the water

chamber and up to the failure of the specimen. The test started from no load condition by

application of force controlled cyclic loading in increments of 10 kN having three cycles

of push and pull in each increment as shown in Figure 5.90(a). Figure 5.90(b) shows the

top displacement of the wall. The maximum load that was sustained by the specimen

before conclusion of the test was 240 kN in the push direction. Lateral load-displacement

relationship for the top of the wall is illustrated in Figure 5.91. At the load level of 190

kN the joint was opening significantly as shown in Figure 5.92. The wall started to slide

very slightly at this stage. The bottom of the wall had several cracks at this stage

especially near the base in a region approximately 150 mm above the joint. At the load

level of 230 kN the wall was sliding back and forth very significantly. Also four wedge

shape pieces of concrete from corners of the wall at the connection to the base became

separated from the wall as shown in Figure 5.93. At the load level of 240 kN, due to

extensive damage to the base of the wall shown in Figure 5.94, the test was concluded.

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Figure 5.92 Extensive cracking of the front face of the wall at 220 kN

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Figure 5.93 Separation of the corner of the wall at connection to foundation

Figure 5.94 Extensive damage to the base of the wall at the end of the test

200

CHAPTER-6

DISCUSSION OF THE EXPERIMENTAL RESULTS

6.1 General Five full-scale wall-foundation specimens were constructed and tested during the

experimental program in the current study. The height of the walls in all specimens

corresponded to a tank with a wall height of 4 m. Two different wall thicknesses of 300

mm and 400 mm were considered. Also leakage performance of two different

configurations for the shear key was evaluated. The tests were aimed at providing

information regarding the behavior of the RC rectangular tanks under cyclic loading

including and mainly with respect to their leakage behavior. In the following, the results

of the test are summarized and discussed. Based on the conclusions reached from the

experimental program, an attempt is made to determine the appropriate values for

response modification factor for RC rectangular tanks.

6.2 Summary of the observations in the experimental program

Specimen-1

Specimen-1 had a wall thickness of 400 mm and a conventional shear key incorporating a

water-stop in the joint region. The wall had eight No.20@200 mm and seven No.20@230

mm at the front and back faces of the wall, respectively. Throughout the entire tests for

Specimen-1, the loading scheme was in a way to push the specimen forward and then pull

it back to its neutral position. Therefore, the back face of the wall did not experience any

tensile stress/strain, and the front face of the wall did not experience any compressive

stress/strain.

The specimen (without the water chamber) was first subjected to a monotonic stage of

loading (test SP-1) when a first visible cracking was observed at the wall-foundation

interface at a lateral load of 110 kN compared to theoretical prediction of 114 kN. The

observation showed that the back face of the wall did not experience any visible cracking.

After the water chamber was installed, the leakage test (test SL-1) was conducted. During

this test, due to the loading scheme, the back side of the wall did not experience cracking.

201

At the last stages of the test, the leakage penetrated deep into the wall as was observed

from the side faces of the wall; and also the reinforcement strain at the front side of the

wall exceeded far beyond the yield level. After conclusion of the test, the observed crack

widths at the front face of the wall were rather excessive. However, no leakage at the

back side of the wall was observed. Following the leakage test, the specimen was

retrofitted with one layer of GFRP and tested for leakage (test SR-1). The specimen

sustained monotonic loading in excess of the static load without leakage, however, during

the cyclic loading stage of the test at the load level corresponding to 60% of the design

load (including the hydrostatic load), water showed to have passed through the GFRP

layer.

Specimen-2

Specimen-2 had a wall thickness of 300 mm and an inverted shear key at the joint region.

The wall had ten No.20@155 mm at the front and back sides of the wall. The cyclic

loading scheme included same magnitudes of load in the push and pull directions.

Therefore, the front and back faces of the wall experienced compressive and tensile

stress/strain. Initially a leakage test (test SL -2) using cyclic loading was conducted on

the specimen, and leakage at the back face of the wall was observed at the early stages of

cyclic loading. Afterwards, the specimen was under the application of the monotonic

loading up to failure where no apparent increase in the rate of the leakage, compared to

that during the cyclic loading stage, was observed.

Specimen-3


water-stop in the joint region. The wall had ten No.20@155 mm at the front and back

faces of the wall. The cyclic loading scheme included same magnitudes of load in the

push and pull directions. Therefore, the front and back faces of the wall experience

compressive and tensile stress/strain. The leakage test on the specimen was conducted

(test SL -3). It was observed that leakage occurred only after the reinforcement layers at

front side of the wall experienced yielding. It is necessary to mention that cracking was

observed at front and back faces of the wall at the initial stages of test (test SL -3) as

202

detected on the side and back faces of the wall. After the conclusion of the leakage test,

the specimen was retrofitted with two layers of GFRP and tested for leakage (test SR-3).

At the initial stage of the cyclic loading water showed to have passed through the GFRP

layers, and as the load magnitude increased, the leakage became more pronounced.

Specimen-4


water-stop in the joint region. The wall had ten No.20@155 mm at the front and back

sides of the wall. The cyclic loading scheme included same magnitudes of load in the

push and pull directions. Therefore, the front and back faces of the wall experience

compressive and tensile stress/strain. The specimen (without the water chamber) was first

subjected to a monotonic stage of loading (test SP-4) when a first visible cracking was

observed at the wall-foundation interface at the lateral load of 70 kN compared to

theoretical prediction of 66 kN. The pretest continued with the application of cyclic

loading with same magnitudes of load in the push and pull directions, and therefore

cracking was observed at the front as well as the back face of the wall. The pretest

continued until the strain in the reinforcement approached the yield value. After the water

chamber was installed, the leakage test (test SL-4) was conducted. It was observed that

leakage occurred only after the reinforcement layers at front and back sides of the wall

experienced yielding. After conclusion of the leakage test, the pressure chamber was

removed and the specimen was tested up to failure (test SF-4) under cyclic loading. The

specimen experienced softening after it reached the maximum lateral load at an

approximate displacement ductility ratio of four, and failed at an approximate

displacement ductility ratio of eight.

Specimen-5


water-stop in the joint region. The wall had six No.20@280 mm at the front and back

sides of the wall. The cyclic loading scheme included same magnitudes of load in the

push and pull directions. Therefore, it is likely that the front and back faces of the wall

experience compressive and tensile stress/strain.

203

The specimen (without the water chamber) was first subjected to a cyclic stage of loading

(test SP-5) when the first visible cracking was observed at the wall-foundation interface

at the lateral load of 110 kN compared to theoretical prediction of 112 kN. The pretest

continued until the reinforcement strain approached very close to the yield value. After

the water chamber was installed, the leakage test (test SL-5) was conducted. As soon as

the test started, slight leakage was observed at the back of the wall which increased due

to the increase in the magnitude of loading. After conclusion of the leakage test, the

pressure chamber was removed and the specimen was tested up to failure (test SF-5)

under cyclic loading. The specimen failed due to significant back and forth sliding of the

wall on the foundation and spalling and separation of the concrete particles from the

bottom of the wall.

6.3 Discussion of the observations in the experimental program

From the observation during the experimental program conducted on five wall-

foundation specimens the following observations were made.

1- Specimen-1 was subjected to a cyclic loading condition in which the front face

experienced severe cracking, while the back face did not experience any cracking.

This was due to the fact that the concrete at back face of the wall did not

experience significant tensile stress/strain. Although the specimen was

additionally subjected to more number of cycles at the near failure loading where

the depth of he compression block had decreased to about 10% of the thickness of

the wall, no leakage was observed at the back of the wall. It is postulated that the

compression block at the back side of the wall was able to prevent the leakage and

also cracking at both faces of the wall is necessary for leakage to occur.

2- For the specimen with inverted shear key configuration without a water-stop,

leakage at the back side of the wall was observed at the initial stage of cyclic

loading where the steel strain level was below 15% of yield value. The ensuing

monotonic loading stage did not show any apparent increase in the rate of leakage

indicating that the leakage was due to the effects of cyclic loading.

204

3- For Specimens-3, 4, and 5 the conventional shear key incorporated a water stop at

the joint region. The loading condition was in a way that both faces of the wall

experienced cracking during the cyclic loading stage. Although the base of the

wall was the first region that experienced cracking and the crack width showed to

be wider in this region, no leakage was observed at the back face of the wall at the

base. It is postulated that the water stop was effective in preventing the leakage at

the base of the wall.

4- For Specimens-3, 4, and 5, the level of steel strain was beyond the yield value

(except for Specimen-5 where the strain level was slightly below the yield value)

when leakage at the back face of the wall was observed. It is possible that the

reinforcement for Specimen-5 had experienced yielding but the strain gauges

showed a lower value. This can be due to the misalignment of strain gauge on the

rebar or the rebar in the wall. Therefore, it may be appropriate to assume that

leakage occurs soon after the yielding of the reinforcement. It is also possible that

the excessive spacing of the reinforcement was the reason for leakage close to

yielding. However, it is difficult to reach such general conclusion based on the

results of the test of only one specimen.

5- Retrofitting of the cracked wall with Glass Fiber Reinforcing Polymer (GFRP)

sheet with respect to leakage was effective only for monotonic loading situation

and also very low level of cyclic loading. The cyclic straining of the GFRP and its

matrix in tension and compression might be the cause for leakage of water past

the GFRP layer(s). Although the GFRP layer is effective in tension, its

performance (with respect to leakage) when subjected to cycles of

tension/compression is questionable. In the test of the retrofitted Specimen-1, the

cyclic load subjected the GFRP layer in repeated states of tension and neural

stress (no compression stress). The GFRP was able to prevent the leakage below

the load levels corresponding to about twice the hydrostatic force. However, the

loading scheme for Specimen-3 subjected the GFRP layers to repeated cycles of

205

tensile and compressive stress/strain, resulting in leakage at relatively low levels

of cyclic loading.

6- Under cyclic loading, the crack opening starts as the load is increased beyond the

tensile strength of the concrete. When the load direction is reversed and part of

the section containing the crack, experience compressive stress/strain, the crack

closes and the compression block is able to prevent the leakage. This crack

opening and closing would continue until the reinforcement is in the linear elastic

range and retrieves its initial condition when the load is removed. The reinforcing

bars at both sides would start to deform linearly until yielding during cyclic

loading. After the reinforcement experiences plastic deformation as a result of

yielding in tension, it requires more compressive force to retrieve its original

condition and to close the crack(s). If opening of the crack(s) at the side which is

not in contact with the liquid, happens before complete closing of the crack(s) at

the other side (in contact with the liquid), leakage may occur.

6.4 Determination of R factor Based on the observations in the experimental program of the current study for the

specimens with conventional shear key incorporating a water stop, leakage did not occur

before the yielding of the reinforcement. Although more experimental tests on specimens

with different variables are needed to verify the above statement, the methodology

presented in the following can be followed for determination of the R factor for RC

rectangular tanks.

It was concluded in Chapter-3, that the R factor can be assumed to be the product of

overstrength and ductility components. In the following, attempts will be made to

determine each of these two components.

6.4.1 Overstrength factor The overstrength factor was decomposed into several components according to Eq. 3-33

which is used in the current Chapter to determine the Ro factor.

206

o size yield sh mechR R R R R Rφ= × × × × 3-3

Rsize is the overstrength arising from restricted choices for sizes of members and elements

and rounding of sizes and dimensions. For water tanks, since the design procedure is

based on selection of a specific thickness for the wall and providing the necessary

spacing for the reinforcement, the reinforcement spacing is believed to play an important

role for the Rsize factor. In the current study the main reason for such overstrength was

observed to be the rounding down of the spacing which resulted the Rsize factor to be 1.13

for Specimen-1 (resulting from reducing the bars spacing from 213 mm to 200 mm) and

1.17 for Specimens-2, 3, and 4 (resulting from reducing the bars spacing from 170 mm to

155 mm). However, as the length of the wall is much larger in the actual water tank

compared to the 1500 mm length of the wall for the specimens in the current study, the

effect of rounding down of the spacing on the overstrength becomes less effective than

the above values. Therefore, a value of 1.05 referred to in Table-3.1 and adopted for all

types of RC structural systems in the corresponding code is considered reasonable for this

factor.

RΦ is a factor accounting for the difference between nominal and factored resistances,

equal to 1/Φ, where Φ is the material strength reduction factor. No reduction factor is

incorporated for material strength reduction factor in the design procedures for RC tanks,

however, due to incorporation of the reduction factor of 0.9 for flexural moment used in

the design procedures, it is reasonable to use a value of 1.1 (=1/0.9) for this factor. Also

considering the load factors incorporated in the design load combination which includes

the seismic load U2= [(1.2 × F) + (1 × E)], another overstrength factor should be

considered. It was pointed out in Chapter-3 in Eqs. 3.58 through 3.60 that the load

combination U2 can be conservatively taken as:

2.7 3(2.65 )2 1.5U HLRi

= + × for wall heights of 3, 4, and 5 m

1.9 3(2.40 )2 1.5U HLRi

= + × for wall heights 6 through 10 m

Ignoring the load factor of 1.2 from the liquid pressure component of the load

combination, the design load combination will be reduced to:

3-59

3-60

207

2.7 3(2.33 )2 1.5U HLRi

= + × for wall heights of 3, 4, and 5 6-1

1.9 3(2.08 )2 1.5U HLRi

= + × for wall heights 6 through 10 m 6-2

For practical ranges of R, the ratio of the design load combination U2 incorporating the

load factor as calculated by Eqs. 3-59 and 3-60 to the same (U2) without the load factor

calculated by Eqs. 6-1 and 6-2 can be conservatively taken as 1.06. Therefore a value of

1.17 (=1.1×1.06) is considered for RΦ.

Ryield is the ratio of “actual” yield strength to minimum specified yield strength. This

value was obtained in this study for a sample of three bars as 1.07. However, the value of

1.05 in Table-3.1 was adopted as it was based on a significantly larger sample size and

comprehensive statistical studies regarding the Ryield factor as reported in (Mitchell et al.

2003). For the current study regarding the RC water tank the value of Ryield factor will be

considered as 1.05.

Rsh is the overstrength due to the development of strain hardening; and is taken as unity

as the leakage is expected to happen not long after yielding, and therefore, no strain

hardening factors seems appropriate. Rmech is also taken as unity as the failure of an RC

tank is assumed to initiate at the first leakage. This value was also taken as unity in

Table-3.1 for all types of RC structural systems except for the “Ductile MRF” (which

was taken as 1.05).

According to the above procedure, the numerical value of the overstrength component is

determined as follows.

Ro = 1.05 × 1.17 × 1.05 × 1 × 1 = 1.29

6.4.2 Ductility factor For a given acceleration time history the ductility reduction factor is primarily influenced

by the period of vibration of the system and the level of inelastic deformation.

The normal trend in the ductility reduction factor variation with respect to the period of

the system shows an increase from a value of Rd =1 for period of vibration of near zero,

208

to a value approximately equal to the target displacement ductility ratio for periods of

vibration more than one second for different values of target displacement ductility after

which it remains almost constant. The impulsive component of earthquake in an LCS has

a very short period of vibration ranging from 0.04 s for wall height of 3 m to 0.18 s for

wall height of 10 m as discussed in Chapter-2. The relationships between Rd and μ, in the

mentioned period range are shown in Figure 6.1 based on the work of the corresponding

researchers.

Newmark & Hall

0

0.5

1

1.5

2

2.5

3

3.5

0 0.05 0.1 0.15 0.2 0.25

Period (sec)

R d

μ=2

μ=3

μ=4

μ=5

μ=6

Riddell, Hidalgo and Cruz

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.05 0.1 0.15 0.2 0.25

Period (sec)

Rd

μ=2μ=3μ=4μ=5μ=6μ=7μ=8

(a)

(b)

209

Arias and Hidalgo

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2 0.25

Period (sec)

Rd

μ=2μ=3μ=4μ=5μ=6μ=7μ=8

Nassar and Krawinkler

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.05 0.1 0.15 0.2 0.25

Period (sec)

Rd

μ=2μ=3μ=4μ=5μ=6μ=7μ=8

Borzi & Elnashai

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.05 0.1 0.15 0.2 0.25

Period (sec)

Rd

μ=2μ=3μ=4μ=5μ=6

(c)

(d)

(e)

210

Vidic, Fajfar and Fischinger

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.05 0.1 0.15 0.2 0.25 0.3

Period (sec)

Rd

μ=2μ=3μ=4μ=5μ=6μ=7μ=8

Figure 6.1 Relationships between the ductility component (Rd) and the displacement

ductility ratio (μ) at short period ranges proposed by:

(a) Newmark & Hall (b) Riddell et al. (c) Arias & Hidalgo

(d) Nassar & Krawinkler (e) Borzi & Elnashai (f) Vidic et al.

Although the number of tested specimens and variables (such as thickness of the wall)

was too small to justify a general conclusion, the observation during the leakage tests on

the specimens in the current research showed that yielding of the reinforcement can be

considered as the failure of an RC tank to maintain its functionality with respect to

leakage. Therefore, based on the observed evidences, it is reasonable to assume Rd equal

to unity. If future tests show an increase in the reinforcement stress level at leakage, it

may be appropriate to use Figure 6.1 to determine the value of Rd based on the accepted

level of ductility of the specimen.

6.4.3 Determination of R factor

It is reasonable to calculate the response modification factor (R) as the product of the two

determined components equal to 1.29 as determined in Section-6.4.1. It is important to

note that this value indicates the level of the reduction of the design load form the elastic

load (FE). Considering Eqs. 6-1 and 6-2, the elastic loads using Ri =1 become:

(f)

211

2.7 3 3(2.33 ) 51.51F H HE L L= + × = for wall heights of 3, 4, and 5 m 6-3

1.9 3 3(2.08 ) 3.951.51F H HE L L= + × = for wall heights 6 through 10 m 6-4

Dividing the above value for FE by 1.29, the design load level is calculated as:

33.87F HL= for wall heights of 3, 4, and 5 m 6-5

33.06F HL= for wall heights 6 through 10 m 6-6

Incorporating the above values into Eqs. 3.59 and 3.60, the value for Ri is calculated as

1.7 for wall heights of 3, 4, and 5, and 2.0 for wall heights of 6, 7, 8, 9, and 10 m. The

obtained response modification factor is independent of the period of vibration. It is

important to mention that the current edition of the ACI Code prescribes a value of 2 for

response modification factor of the impulsive component for all heights of the tank wall.

Based on the results of the current study the current value in the Code may be slightly

overestimated (unconservative) for tanks with wall heights of 3, 4, and 5 m.

Consider a rectangular tank with the properties such as L (larger side of the tank wall) =

90 m, B (shorter side of the tank wall) = 20 m, tw (thickness of the wall) = 500 mm, HW

(height of the wall) = 5 m, HL (height of the water) = 4.5 m, γL (specific weight of water)

= 9.8 KN/m3, and γC (specific weight of concrete) = 23.6 kN/m3. Ss, S1, Fv, Fa, I, Ri, and

Rc are taken as 150%, 60%, 0.8, 0.8, 1, 2, and 1, respectively. The design force

components are presented in Table 6-1.

The seismic induced moment on the connection of the wall foundation is:

( ) ( ) ( )2 2 2(45.9 1.7 23.6 2.5) 15.9 1.5 4.3 2.25 139 kN.m/mseismicM = × + × + × + × = The static moment on the connection of the wall foundation is:

99.3 1.5 149 kN.m/mstaticM = × =

212

Table 6-1 Design force components

Force component Magnitude

(kN)

Distance to the base from

the centroid of force

(m)

Flexural moment at

base of wall

(kN.m)

Hydrostatic pressure 99.3 1.5 149

Impulsive force 45.9 1.7 78

Convective force 4.3 2.25 9.7

Wall inertia effect 23.6 2.50 59

Vertical ground

motion effect

15.9 1.5 23.8

Considering No.20 bars and normal exposure condition, according to ACI Code the

design load combinations are:

U1 = 1.036 × 1.4 × 149 = 215 kN.m/m

U2 = (1.2× 149) + 139 = 317.8 kN.m/m

The minimum area of reinforcement is calculated as 1506 mm2.

The environmental durability factor was calculated equal to 1.036 based on the bar

spacing of 144 mm.

The required cross section area of steel incorporating the reduction factor for flexure (0.9)

is determined (for a 1000 mm width of the wall) as 2081 mm2, resulting in spacing of 144

mm for No.20 bars.

Incorporating the reasonable value of 1.05 for both Rsize (considered as a reduction in

spacing, and therefore, increase in the cross section area of steel) and Ryield (considered as

the ratio of the actual yield strength to nominal yield strength of the steel) in the

calculation of the flexural moment of resistance of the section and ignoring the reduction

factor for flexure (0.9), the wall section is able to resist 390 kN.m/m at its base prior to

yielding. Assuming a linear elastic behavior of the wall by considering the response

modification factor equal to unity and ignoring the hydrostatic load factor in the second

213

load combination, the flexural moment at the base of the wall is calculated as 424

kN.m/m (the sum of the hydrostatic component and the hydrodynamic component

incorporating the response modification factors of unity). This shows that there is a

likelihood that the reinforcement at the middle of the 20 m wide wall, at the connection to

the foundation, experience yielding due to the probable magnitude of the earthquake.

However, using the proposed value of 1.7 for the response modification factor, the

moment of resistance at the base of the wall is increased to 420 kN.m/m. Therefore, it can

be considered acceptable to assume that the reinforcement will be in the linear elastic

range during the earthquake provided that the design response modification factor is

reduced to 1.7.

214

CHAPTER-7

ANALYTICAL STUDY

7.1 General The cracking behavior of RC members has been historically attributed to the stress/strain

level in the steel reinforcement. The experimental program in the current research also

showed that the level of stress/strain in the reinforcement at the critical section can be

assumed as a good variable to be correlated to the leakage behavior of the specimen.

Although the steel stresses/strain can be determined by concrete section analysis, this

may be quite tedious in case of the actual structure under different types of loadings.

Due to complex interaction between the various components of real structures, their

dynamic characteristics cannot be identified solely from dynamic tests of scale models.

Moreover, the cost of such tests for large specimens is substantial. Historically these

difficulties have been overcome by static tests on components and on reduced scale sub-

assemblages of structures under cyclic load reversals. Results from these tests are then

used in the development of hysteretic models that permits the extrapolation of the

available test data to other cases and to the dynamic response of the complete structures

(Taucer 1991). To simulate the response of the tested specimens in the experimental

program, analytical models are developed using ABAQUS/6.7.1 (Hibbitt et al., 2007)

finite element software and verified against the experimental evidences. Provided that the

behavior of the specimen can be successfully captured under the effects of reversed

cyclic loading, the analytical results may then be extended to the case of a full structure

and under more complex loading condition.

Due to unavailability of successful application of ABAQUS for modeling the cyclic

behavior of RC members in the literature, the software is first verified. The results of a

test on a cantilever RC rectangular column under the effect of reversed cyclic loading

reported in Park (1990) are used for verification purposes. The software is then used to

model a specimen used in the experimental program. Due to complex interaction of the

water-pressure chamber and the specimen, only tests which did not include the water

pressure chamber are evaluated. The same structural geometries, properties, and

215

conditions as those of the experiments are used as input data. In the following the

material models used in ABAQUS software and in the analytical study in the current

research are discussed.

7.2 Material model for concrete in ABAQUS Three different constitutive models are offered in ABAQUS/6.7.1 for the analysis of

concrete: the concrete smeared crack model, the brittle cracking model, and the concrete

damaged plasticity model. Each model is designed to provide a general capability for

modeling plain and reinforced concrete in all types of structures.

7.2.1 Smeared cracking This model is intended for applications in which the concrete is

subjected to essentially monotonic straining at low confining pressures and a material

point exhibits either tensile cracking or compressive crushing. Cracking is assumed to be

the most important aspect of the behavior and the representation of cracking and post-

cracking anisotropic behavior dominates the modeling. Because the model is intended for

application to problems involving relatively monotonic straining, no attempt is made to

include prediction of cyclic response or of the reduction in the elastic stiffness caused by

inelastic straining under predominantly compressive stress.

When concrete is loaded in compression, it initially exhibits elastic response. As the

stress is increased, some non-recoverable (inelastic) straining occurs and the response of

the material softens. After the material reaches the ultimate stress, it loses strength until it

can no longer carry any stress. In real life, if the load is removed at some point after

inelastic straining has occurred, the unloading response is softer than the initial elastic

response and the elasticity is damaged. This effect is ignored in the model, since it is

assumed that the applications involve primarily monotonic straining, with only

occasional, minor unloading.

7.2.2 Brittle cracking

This model is intended for applications in which the concrete behavior is dominated by

tensile cracking, and compressive failure is not important. The compressive behavior is

assumed to be always linear elastic. The crack formation criterion states that a crack

216

forms when the maximum principal tensile stress exceeds the tensile strength of the

brittle material. As soon as the criterion for crack formation has been met, it is assumed

that a first crack has formed. Cracking is irrecoverable in the sense that, once a crack has

occurred at a point, it remains throughout the rest of the calculation. However, crack

closing and reopening may take place along the directions of the crack surface normals.

7.2.3 Concrete damaged plasticity (CDP)

This model uses concept of isotropic damaged elasticity in combination with isotropic

tensile and compressive plasticity to represent the inelastic behavior and is designed for

applications in which the concrete is subjected to arbitrary loading conditions, including

cyclic loading. The model is intended primarily for the analysis of reinforced concrete

structures subjected to monotonic, cyclic, and dynamic loading under low confining

pressures. The CDP model assumes that the two main failure mechanisms are tensile

cracking and compressive crushing of the concrete, and that the uniaxial tensile and

compressive response of concrete is characterized by damaged plasticity. The model

takes into consideration the degradation of the elastic stiffness induced by plastic

straining both in tension and compression. It also accounts for stiffness recovery effects

under cyclic loading. As shown in Figure 7.1(a), under uniaxial tension, the stress-strain

response follows a linear elastic relationship until the failure stress σt0, which represents

the onset of micro-cracking in the concrete material. Beyond this failure stress, the

formation of micro-cracks is represented macroscopically with a softening stress-strain

response.

As shown in Figure 7.1(b), under uniaxial compression the response is linear until the

value of initial yield, σc0 is reached. In the plastic regime the response is typically

characterized by stress hardening followed by strain softening beyond the ultimate stress,

σcu. When the concrete specimen is unloaded from any point on the strain softening

branch of the stress-strain curves, the unloading response is weakened and the elastic

stiffness of the material is damaged.

217

Figure 7.1 Uniaxial behaviour of plain concrete (Hibbitt et al., 2007)

(a) in tension (b) in compression

The degradation of the elastic stiffness is characterized by two damage variables, dt, and

dc (0≤ dt, dc ≤1) which are assumed to be dependent on the plastic strains. The damage

variables can take values from zero, representing the undamaged material, to one, which

218

represents total loss of strength. If E0 is the initial (undamaged) elastic stiffness of the

material, the stress-strain relations under uniaxial tension and compression loading are,

respectively:

(1 ) ( )Pld Et t o t tσ ε ε= − − % 7-1

(1 ) ( )Pld Ec c o c cσ ε ε= − − % 7-2

where E0 is the initial (undamaged) modulus of the material. Other parameters are

illustrated in Figure 7.1.

Under cyclic loading conditions the degradation mechanisms are quite complex,

involving the opening and closing of previously formed micro-cracks and their

interaction. Experimentally, it is observed that there is some recovery of the elastic

stiffness as the load changes sign during a uniaxial cyclic test. The stiffness recovery

effect is an important aspect of the concrete behavior under cyclic loading. The effect is

usually more pronounced as the load changes from tension to compression, causing

tensile cracks to close, which results in the recovery of the compressive stiffness.

In ABAQUS, the damage variables are treated as non-decreasing material point

quantities. At any increment during the analysis, the new value of each damage variable

is obtained as the maximum between the value at the end of the previous increment and

the value corresponding to the current state (interpolated from the user provided tabular

data). The choice of the damage properties is important since excessive damage may have

a critical effect on the rate of convergence.

The concrete damaged plasticity model assumes that the reduction of the elastic modulus

is given in terms of a scalar degradation variable d as:

(1 )E d Eo= − 7-3

The stiffness degradation variable, d, is a function of the stress state and the uniaxial

damage variables, dt and dc. For the uniaxial cyclic conditions ABAQUS assumes that

(1 ) (1 )(1 )d s d s dt c c t− = − − 7-4

219

where st and sc are functions of the stress state that are introduced to model stiffness

recovery effects associated with stress reversals. In the compressive side of the cycle the

values of st and sc are equal to 1 and (1- wc), respectively. In the tensile side of the cycle

the values of st and sc are equal to (1- wt) and (1), respectively.

The weight factors, wt and wc, both ranging between zero and one, control the recovery

of the tensile and compressive stiffness upon load reversal, respectively. The effect of

assignment of values of wt and wc on the behavior of the concrete material is shown in

Figure 7.2. Stiffness recovery is an important aspect of the mechanical response of

concrete under cyclic loading. ABAQUS allows direct user specification of the stiffness

recovery factors wt and wc. “COMPRESSION RECOVERY” parameter is used in the

“CONCRETE TENSION DAMAGE” option to specify the compression stiffness

recovery factor, wc; and the “TENSION RECOVERY” parameter is used in the

“CONCRETE COMPRESSION DAMAGE” option to specify the tension stiffness

recovery factor, wt.

Figure 7.2 Effect of assignment of values of wt and wc on the behaviour of concrete

7.2.3.1 Behavior of concrete in tension The post-failure behavior for direct straining is modeled with the “CONCRETE

TENSION STIFFENING” option, which allows the user to define the strain-softening

behavior for cracked concrete. This option also allows for the effects of the reinforcement

220

interaction with concrete to be simulated in a simple manner. The “CONCRETE

TENSION STIFFENING” option is required in the concrete damaged plasticity model to

specify tension stiffening by means of a post-failure stress-strain relation. For the tension

stiffening effect, “CONRETE TENSION STIFFENING TYPE=STRAIN” option (more

suitable for concrete with reinforcement) is used. In reinforced concrete the specification

of post-failure behavior generally means presenting the post-failure stress as a function of

cracking strain, cktε% . The cracking strain shown in Figure 7.3, is defined as the total

strain minus the elastic strain corresponding to the undamaged material; that is,

ck tt t Eo

σε ε= −% 7-5

Figure 7.3 The postfailure behaviour of concrete in tension (Hibbitt et al., 2007)

In cases with little or no reinforcement, the specification of a post-failure stress-strain

relation introduces mesh sensitivity in the results, in the sense that the finite element

predictions do not converge to a unique solution as the mesh is refined. The interaction

221

between the rebars and the concrete tends to reduce the mesh sensitivity, provided that a

reasonable amount of tension stiffening is introduced in the concrete model to simulate

this interaction. This requires an estimate of the tension stiffening effect, which depends

on factors such as the density of reinforcement, the quality of the bond between the rebar

and the concrete, the relative size of the concrete aggregate compared to the rebar

diameter, and the mesh.

7.2.3.2 Behavior of concrete in compression The “CONCRETE COMPRESSION HARDENING” option is used to define the stress-

strain behavior of plain concrete in uniaxial compression outside the elastic range as

shown in Figure 7.4. Compressive stress data are provided as a tabular function of

inelastic strain, incε% . The stress-strain curve can be defined beyond the ultimate stress,

into the strain-softening regime. Hardening data are given in terms of an inelastic

strain incε% , instead of plastic strain, pl

cε% . The compressive inelastic strain is defined as the

total strain minus the elastic strain corresponding to the undamaged material

in cc c Eo

σε ε= −% 7-6

7.3 Reinforcement

In ABAQUS reinforcement in RC structures is typically provided by means of rebars,

which are one-dimensional members that can be defined singly or embedded in the

concrete members. Rebars are defined with the “REBAR” option. They are typically used

with metal plasticity models to describe the behavior of the rebar material and are

superposed on a mesh of the elements that are used to model the concrete. With this

modeling approach, the concrete behavior is considered independent of the rebar. Effects

associated with the rebar/concrete interface, such as bond slip is approximated by

222

introducing “tension stiffening” into the concrete modeling to simulate load transfer

across cracks through the rebar.

Figure 7.4 The postfailure behaviour of concrete in compression (Hibbitt et al., 2007)

The classical metal plasticity models use perfect plasticity or isotropic hardening

behavior. In ABAQUS a perfectly plastic material (with no hardening) can be defined, or

work hardening can be specified. Work hardening materials, in contrast to perfectly

plastic materials, may change their response during yielding. These changes are

accomplished by altering the shape or size of the yield surface as plastic flow occurs.

Yielding is accompanied by an apparent increase in strength called hardening. For the

work hardening case an initial yield stress exists similar to the perfectly plastic case, but

as the strain increases the yield stress also increases. If the material yields by tension,

there are two possible options for consideration of the yield stress in compression as

shown in Figure 7.5. In the figure the plane strain Tresca hexagon is shown in the σ1- σ3

plane. The dashed hexagon is the initial yield surface and the two solid hexagons are

possible subsequent surfaces after hardening has occurred. A simple tension stress

223

trajectory is the horizontal line beginning at the origin and ending at b. Point a is the

initial yield point and the space between a and b corresponds to hardening.

Figure 7.5 Comparison of isotropic and kinematic hardening

In the isotropic hardening option, the yield surface has been inflated by the hardening

process so that if the stress is reversed after this initial loading the yield point will

increase. In the kinematic hardening option, the size of the yield surface has not changed

and hardening has simply translated the initial surface to the right by the amount ab. A

reversal of stress here would encounter the yield surface much earlier than in the isotropic

hardening case. Isotropic hardening means that the yield surface changes size uniformly

in all directions such that the yield stress increases (or decreases) in all stress directions

as plastic straining occurs (Davis and Selvadurai 2002).

When most metals are tested in simple tension, post-yield hardening is observed. If the

stress is then reversed, the reverse yielding usually occurs near the point predicted by the

kinematically translated surface. This occurs despite the fact that the yield point in

compression for an unhardened specimen would be the same as that in tension. Despite

the fact that kinematic hardening appears to better represent actual test data, isotropic

hardening is still widely used. The reason lies in the ease with which isotropic hardening

may be implemented mathematically. However, in applications where stress reversals are

not anticipated, the choice of model makes little difference.

224

In ABAQUS Isotropic hardening and kinematic hardening for materials subjected to

cyclic loading are available. ABAQUS provides an isotropic hardening model, which is

useful for cases involving gross plastic straining or in cases where the straining at each

point is essentially in the same direction in strain space throughout the analysis. For

kinematic hardening, two models are provided in ABAQUS to model the cyclic loading

of metals. The first one, the linear kinematic model, approximates the hardening behavior

with a constant rate of hardening using two pairs of data. The second model, the

nonlinear isotropic/kinematic model, will give better predictions but requires more

detailed calibration. The linear kinematic hardening model is selected by specifying the

“PLASTIC”, and “HARDENING=KINEMATIC” option, and the nonlinear

isotropic/kinematic hardening model is selected by specifying the “PLASTIC”, and

“HARDENING=COMBINED” option. The kinematic hardening models are suited for

most metals subjected to cyclic loading conditions, except voided metals.

Finally it should be noted that the inelastic material properties must be input into

ABAQUS in the form of true (Cauchy) stress (σtrue) and true (logarithmic) strain (εtrue),

which can be calculated from the engineering stress (σeng) and engineering strain (ε eng)

using the following relationships:

(1 )true eng engσ σ ε= + 7-7

ln(1 )true engε ε= + 7-8

7.4 Validation To investigate the ability of ABAQUS in modeling the cyclic behavior of RC members,

the result of an experiment reported in Park (1990) on a rectangular cantilever column is

used for evaluation of the software. The simulation of the experimental test was

performed using fiber element model incorporated in DRAIN-2DX and reported in

Sadjadi and Kianoush (2007). The RC column has a height of 1784 mm and is

constructed by concrete with compressive strength of 26.9 MPa. It is reinforced with 10

longitudinal bars of Grade 380 (fy = 432 MPa) with a diameter of 24 mm corresponding

to a longitudinal reinforcement ratio of 1.9%. The transverse reinforcement has a yield

225

strength of 305 MPa and diameter of 12 mm placed at spacing of 80 mm corresponding

to a transverse reinforcement ratio of 1.2%. The clear cover to the transverse

reinforcement is 24 mm. The column is under constant axial load of 646 kN while lateral

cyclic load is applied on the top of the column.

In the ABAQUS analysis, the column consisted of a concrete “Part” with the dimensions

of the column and a rebar “Part” with the dimensions of the steel rebars in the actual

member. The material behavior for both parts will be described later in this section. For

creating all the 10 rebars in their appropriate position in the cross section the “Linear

Pattern” and “Translate” sub-options in the “Instance” option in the “Assembly” module

were used. The rebars were then selected as “Embedded Region” in the “Host Region”

(the concrete part) using the “Constraint” option in the “Interaction” module. The “Step”

module was then used to create necessary steps for the application of the cyclic load. The

axial load on the column was applied using an equivalent uniform pressure on top of the

column, while the cyclic load was applied using displacement loading in the “BC

(Boundary Condition)” option using appropriate steps and incorporating smooth step

“Amplitude”. The bottom of the column was fixed using the “Encastre” sub-option in the

“BC” option which should be activated from the initial step.

Before execution of the analysis in the “Job” module, the RC member should be meshed

into a number of elements of appropriate type. It is important to note that the element

type for both constituent parts should correspond to “Explicit” type since the analysis is

performed in ABAQUS/Explicit. For meshing the parts, it is recommended that the

aspect ratio of the size of the element be close to unity. Therefore, the “Seed Instance”

option (using an approximate size) was selected rather that the other options such as

“Seed Edge by Size”, “Seed Edge by Number”, and “Seed Edge Biased”. The

approximate seed size was chosen as 75 mm with the result that the height of the column,

the larger side and the shorter side of the cross section were divided into 22, 8, and 5

elements, respectively. The meshed member is shown in Figure 7.6.

226

Figure 7.6 Meshing scheme for the column (approximate seed size of 75 mm)

7.4.1 Reinforcement Steel rebars in the RC member were modeled as one-directional strain elements (rods)

and were simulated by “REBAR” option. The embedded element technique was used to

specify steel reinforcement elements that were embedded in host concrete elements.

ABAQUS checks the position of nodes of the embedded elements in host elements. If a

node of an embedded element lies within a host element, its translational degrees of

freedom are constrained to the interpolated values of the corresponding degrees of

freedom of the host element. The concrete behavior will be considered independent of the

reinforcing bars. The effects associated with the rebar- concrete interface, such as bond

slip is approximated by introducing some “tension stiffening” into the concrete modeling.

Two options are available for defining the nonlinear behavior of reinforcement steel in

ABAQUS, namely, “Isotropic Hardening” and “Kinematic Hardening”. There is no

major difference in modeling the monotonic behavior of the member using either of the

two options; however, for the case of cyclic behavior, the difference is significant. For

investigation of this difference, a cantilever steel member was evaluated where cyclic

displacement was applied on the free end of the member. The member length was chosen

as 100 mm and the square cross section had a dimension of 5 mm. The cross section was

divided into 4 equal elements and the length of the member was divided into 40 equal

227

elements. For defining the nonlinear behavior of steel, three options were considered,

namely, “Isotropic”, “Kinematic (linear kinematic)”, and “Combined (nonlinear

isotropic/kinematic)” options. First, a simple bilinear behavior as indicated in the first

two columns of Table 7-1 was used for the three mentioned options. Then a more refined

multilinear behavior as indicated in the last two columns of Table 7-1 was used only for

the “Isotropic”, and “Combined” options, as the “Kinematic” option was linear and

required only two data pairs. The data are given as tabular pairs corresponding to the

stress-strain data obtained from the fist half cycle of a unidirectional tension experiment.

This approach is usually adequate when the simulation involves only a few cycles of

loading (Hibbitt et al., 2007).

Table 7-1 Material model information for steel

Bilinear behavior Multilinear behavior

True stress (MPa)

True plastic strain

True stress (MPa)

True plastic strain

430 0 430 0. 600 0.09 502 0.023

574 0.037 624 0.056 645 0.084 617 0.1 574 0.116

Figure 7.7(a) shows the comparison of the behavior of the specimen in terms of the

lateral load-lateral displacement of the free end of the member. It is observed that the

“Kinematic”, and “Combined” options are identical and the difference between these two

options and the “Isotropic” option is as expected and explained in Section-7.3 in the

current Chapter and illustrated in Figure 7.5. Figure 7.7(b) shows the same comparison

between the “Isotropic”, and “Combined” models for the more refined material behavior.

The refined material behavior included more stress-strain data pairs compared to the

former case (two data pairs), therefore, the linear kinematic option could not be used. As

shown in Figure 7.7(b), the same difference as in Figure 7.7(a) can be observed.

228

Comparison of steel models

-250-200-150-100

-500

50100150200250

-60 -40 -20 0 20 40 60


Late

ral f

orce

(kN)

IsotropicKinematicCombined

Comparison of steel models

-250-200-150-100

-500

50100150200250

-60 -40 -20 0 20 40 60Top displacement (mm)

Late

ral f

orce

(kN

)

IsotropicCombined

Figure 7.7 Comparison of the effect of modelling the steel behaviour

(a) Isotropic, kinematic and combined (linear) (b) Isotropic and combined (multilinear)

To investigate the behavior of the mentioned RC column, the longitudinal rebars using

T3D2 “2-node linear 3-D truss” elements are embedded in the host (concrete) element

using the “Constraint” option. The material behavior for steel includes general material

information such as the weight, the elastic stiffness, and Poisson ratio which were taken

as 7.8 E-09 tonne/mm3, 200000 MPa, and 0.2, respectively. For the nonlinear material

behavior the “Combined” option was used with parameters in the form of true (Cauchy)

stress (σtrue) and true (logarithmic) strain (εtrue) as defined in Table 7-2.

(a)

(b)

229

Table 7-2 Stress-strain relationships for steel rebar

Nominal stress (MPa)

Nominal strain

True stress (MPa)

True strain

True plastic strain

0 0 0 0.000 0

430 0.002 431 0.002 0

502 0.025 515 0.025 0.023

574 0.04 597 0.039 0.037

624 0.06 662 0.058 0.056

645 0.09 704 0.086 0.084

617 0.105 682 0.100 0.097

574 0.125 646 0.118 0.116

7.4.2 Concrete The concrete member was meshed using C3D8R (8-node linear brick, reduced

integration, hourglass control) elements. The general material information of concrete

includes the specific weight, the elastic stiffness, and Poisson ratio which were taken as

2.4 E-09 tonne/mm3, 25000 MPa, and 0.2, respectively. The CDP option was used to

define the behavior of concrete in the nonlinear range since it was the only appropriate

model for the case of cyclic loading. For plasticity part of the behavior, the default values

suggested by the ABAQUS manual were applied. The precise determination of these

values requires triaxial test information for the concrete sample. Therefore, in the absence

of the required information, the “Dilation Angle”, “Eccentricity”, “fbo/fco”, “K”, and

“Viscosity Parameter” were taken as the default values of 36, 0.1, 1.16, 0.666, and 0,

respectively. The variation in the values of the mentioned parameter was observed to

have negligible effect on the results of the analysis of the RC column case under study.

For members with flexural dominant mode of behavior, the three-dimensional strain field

developed in the RC member can be decomposed into a one-dimensional condition over

the cross-section (Maekawa et al. 2003). With accepted level of accuracy, the three

dimensional behavior of concrete and steel can be simplified into uniaxial stress-strain

relationships. The effect of concrete confinement by reinforcement is considered by using

appropriate uniaxial monotonic envelope of concrete which incorporates such effect

including arrangement and mechanical properties of the transverse reinforcement. This is

230

of great importance since it is assumed that the concrete monotonic stress-strain curve

represents the envelope for the cyclic stress strain branches.

For confined concrete behavior in compression, the material model of Hoshikuma et al.

(1997) which were successfully verified for the aforementioned RC column in Sadjadi

and Kianoush (2007) was used. Incorporation of other material models for concrete

behavior in compression showed to have minimal change in the response. Similar

observation that the compressive concrete constitutive law appears to have minor effects

on the overall results was reported elsewhere (ASCE 1993, Cope and Rao 1981). To

avoid significant time consumption and complexity of the analysis, the effect of the cover

concrete is ignored and the confined concrete behavior is assumed to define the property

of the concrete in compression for the entire cross-section. Table 7-3 shows the input for

definition of the concrete behavior in compression in ABAQUS.

Table 7-3 Concrete behaviour in compression

Stress (MPa) Plastic strain

13.4 0

30.8 0.003

28.5 0.005

26.5 0.006

24.0 0.008

21.5 0.01

15.4 0.015

15.3 0.1

Similar to the example of the “Gravity Dam” discussed in the ABAQUS documentation,

the effect of concrete compression damage was ignored to prevent more complexity in

the analysis, although it was observed to have minimal effects. The behavior of concrete

in tension is extremely important as any slight variation in the defined stress-strain

relationship was observed to influence the results drastically. The behavior in tension is

mesh-sensitive as the stress-strain values were observed to depend on the size of the

element. In this study, for definition of the behavior of concrete in tension, it was

231

assumed that the nonlinear behavior (cracking in tension) would start at a tensile stress

equal to 10% of the compressive strength of the concrete as suggested in the ABAQUS

User’s Manual.

Concrete in RC members can still support part of the applied tension even after cracking,

which is known as tension stiffening effect. Due to bond transfer at the interface of

concrete and reinforcement, concrete between cracks can develop local tensile stresses

after cracks occur. Due to bonding, local stresses in concrete and reinforcement are not

uniform but vary along the bar axis. The steel stress reaches the minimum value at the

center of an uncracked segment where the concrete stress is at its maximum, and

increases to the maximum value at the crack locations. This local stress distribution must

be taken into account in modeling the average post-cracking tensile behavior. Since

concrete continues to bear part of the applied tension, tensile stress of concrete in RC

gradually decreases when the average strain is more than 500-1000 με even though rapid

tensile stress release is similarly experienced just after cracking. This differs from the

stress release behavior of plain concrete, which exhibits a sharp drop immediately after

cracking and no recovery of the stress transfer mechanism (Maekawa et al. 2003). The

fundamental difference in post-cracking tensile behavior of plain concrete and reinforced

concrete is observed to be extremely important for the analytical simulations in the

current study. Plain concrete resists tension through bridging of aggregates at the crack

surface only, whereas concrete in an RC member resists tension mainly through the bond

stress transfer from the reinforcing bar. Analytical studies showed that modeling the

softening behavior of concrete in tension by a curve which approaches zero stress at the

failure strain resulted in premature failure of the RC member.

In the analytical studies in the current research, the modeling of tensile behavior of

concrete after cracking is based on the results of uniaxial tension and pull-out tests of a

one-dimensional RC member conducted by Shima and Okamura (1987). The proposed

model is expressed as the explicit relation between average stress and average strain as

shown by Eq. 7-9.

.( )ctuftε

σε

= 7-9

232

where σ is the average tensile stress, ε is the average tensile strain, ft it is uniaxial tensile

strength, εtu is cracking strain and c is a stiffening parameter.

The formulation of the average stress-average strain relation is dependent on the element

size to maintain the same energy release. Parameter c is dependent on the size of the

element and the resulting average tensile stress-strain relation varies according to the

element size. Through the appropriate determination of parameter c, the envelope of post-

cracking tensile stress defined by Eq. 7-9 is applicable to both plain concrete and RC.

Incorporating the proposed equation for defining the behavior of concrete in tension

requires the accurate value of c. However, by comparison of the experimental results of

the lateral force-displacement for the top of the column and those obtained by analytical

simulation obtained by trial and error process, it is observed that a value of c = 0.2 results

in more comparable results than other values of c. Table 7-4 illustrates the values of

tensile stress, corresponding plastic strain, and tension damage used to define the

behavior of concrete material in tension.

Table 7-4 Concrete “Tension Stiffening” and “Tension Damage”

Stress (MPa) Plastic strain Tension damage

3.1 0 0

3.1 0.00015 0.35

2.55 0.00025 0.40

2.22 0.00065 0.50

2.05 0.00105 0.65

1.85 0.00185 0.70

1.61 0.00385 0.70

1.48 0.00585 0.70

1.4 0.00785 0.70

1.34 0.00985 0.70

1.25 0.03 0.70

233

Figure 7.8 compares the results of the analysis using values in Table 7-4 for the behavior

of concrete in tension, to the experimental results of the column cyclic loading test

reported in Park (1990). The analytical results correlated well with the experimental

results in terms of the maximum loads at each level of displacement.

Load-displacement

-500-400-300-200-100

0100200300400500

-100 -50 0 50 100 150


Late

ral F

orce

(kN

)

Experimental

ABAQUS(c=0.2)

Figure 7.8 Comparison of the experimental and analytical results

It is important to note that the execution of each of the complete cyclic analysis

mentioned above, took a considerable amount of time (around two days). The time is

mainly dependent on the size of the mesh and the number of element. Due to the

limitation of the computer facility, it was not possible to perform any analysis with the

column discretized into finer meshes to check the possible improvements in the results.

Therefore, no sensitivity study was performed in this study. Instead, for minimizing the

analysis time, a coarser mesh was used and investigated for accuracy of the results.

Therefore, the column was meshed into coarser mesh with approximate seed size of 150

mm, with the result that the height of the column, the larger side and the shorter side of

the cross section were divided into 10, 4, and 3 elements respectively, as shown in Figure

7.9. The previously described procedures were repeated for the new case and it was

234

interestingly observed that no change in the material behavior from the column with finer

mesh, discussed above, was necessary except for the concrete behavior in tension. The

strain stress relationship for concrete in tension for the case of the column with coarser

mesh corresponds to Eq. 7-9 using the value of c equal to 0.175. The ABAQUS input file

for this analysis is included in Appendix B; however, the sections corresponding to the

definition of the nodes and elements are excluded due to the excessive required space.

The result of the analysis of the column with coarser mesh with element size equal to 150

mm is compared to the experimental results in Figure 7.10. As expected the execution

time was significantly reduced with no compromise in the accuracy of the results.

Figure 7.9 The column with coarser mesh (approximate seed size of 150 mm)

It is observed that the above procedure was able to provide results with acceptable

accuracy in terms of the load displacement relationships as compared to the experimental

observation, however, the model needs improvement. The improvement includes the

representation of the area enclosed by the hysteretic loops which have shown to

overestimate the actual response as shown in Figures 7.8 and 7.10. This problem was

235

addressed in Sadjadi and Kianoush (2007) when experimental results of the columns

were compared to the analytical results obtained using Fiber Element concept

incorporated in DRAIN-2DX.

It was reported that using a steel tri-linear model with properties defined in Figure 7.11

provides a better representation of the energy dissipation in comparison to conventional

bilinear stress strain model for steel. It was decided to assess the improvement of

incorporating the proposed trilinear behavior of the steel into the finite element analysis

of the mentioned cantilever column. The results of the analysis are compared with the

experimental results in Figure 7.12. Better agreement between the experimental and

analytical results can be observed.

Load-displacement

-500-400-300-200-100

0100200300400500

-100 -50 0 50 100 150


Late

ral F

orce

(kN

)

Experimental

ABAQUS(c=0.175)


236

Figure 7.11 The proposed tri-linear material behaviour for steel

Load-displacement

-500-400-300-200-100

0100200300400500

-100 -50 0 50 100 150


Late

ral l

oad

(kN

)

Experimental

ABAQUS(Trilinear)


7.5 Application of the analytical study in the current research The application of the FE modeling using ABAQUS was evaluated with respect to the

experiments in the current research. As was mentioned earlier, due to considerable

237

complexity, the experimental tests involving the interaction of the water-pressure

chamber and the wall-foundation specimen were not evaluated in the analytical studies.

Therefore, the pretest, and the test until failure of a specimen (Specimen-4) which were

reported in Chapter-5 were simulated as discussed in the following.

The model consisted of three “Parts”, namely, specimen, rebar, and floor. The floor

“Part” was considered as a linear elastic concrete member with thickness of 1000 mm and

dimension of 5000×5000 mm as shown in Figure 7.13.

The rebar “Part” was defined using the properties of the actual reinforcement used in the

construction of the specimen. The rebars for the wall and foundation were then modeled

and arranged similar to those of the actual specimen as shown in Figure 7.14. These

rebars were embedded inside the host element using the same procedure that was

implemented in the verification analysis of the RC column discussed in Section-7.4.

Figure 7.13 Modelling of Specimen-4 and the floor

238

Figure 7.14 Modelling the reinforcement for the wall and foundation

The wall-foundation specimen was defined with the same dimensions as those of the

actual specimen. In order to simulate the experimental test as close as possible, the wall-

foundation was defined to be in rigid contact with the floor using “Surface to Surface

Contact” option and four special spring elements were used to simulate the four threaded

rods which were used to anchor the specimen to the floor as shown in Figure 7.15. The

stiffness of theses springs were determined based on the length, diameter and modulus of

elasticity of the actual threaded rods.

Figure 7.15 Special springs for anchoring the specimen to the floor

239

The material properties for the concrete were defined in a similar method to those defined

in the verification case of the RC column considering the compressive strength of the

concrete cylinder sample used for construction of Specimen-4. The important point was

the definition of the tensile property of the concrete which showed to be of great

significance. In order to minimize the analysis time, the specimen was first divided into

different regions using “Partition” tool. Then the critical regions such as the region near

the base of the wall were discretized into finer mesh than the mesh used for less critical

regions such as front and back sides of the foundation. This process was evaluated to be

not appropriate because the tensile properties of the concrete for each element were

dependent on the size of the element. This requires a mesh dependent definition for

tensile properties of concrete for different element sizes used for meshing the specimen.

To avoid this extra complexity, the specimen was discretized into equal elements. The

approximate size of the elements was chosen as 75 mm resulting in the thickness of the

wall divided into four elements as shown in Figure 7.16. It is important to mention that in

the verification analysis of the RC column, the element size was approximately 75 mm.

Therefore, it might be possible to extend the information such as the value of c in Eq. 7-9

used for the verification analysis of the RC column to the analysis of Specimen-4.

The floor part was then meshed into elements of approximately 1000 mm and the

boundaries for the sides were fixed to simulate the rigid floor in the Structural

Laboratory. The bottom part of the foundation at the back and front side were also

restrained against horizontal movement to simulate the floor brakes used in the actual

specimen to prevent the horizontal movement of the foundation as shown in Figure 7.16.

The load was then applied at the top of the wall using displacement control loading

defined in the “Boundary” option in the “Load” module.

The first part of the analysis of Specimen-4 was the pretest which involved a monotonic

loading until the first visible crack was observed at the wall-foundation interface. The

experimental results of this part of the test in terms of top load displacement and also the

strain condition of both reinforcement layers at the front and back of the wall at the

interface of the wall-foundation were reported in Chapter-5.

240

Figure 7.16 Boundary conditions defined for the model

Figure 7.17 compares the experimental and analytical results for lateral load-

displacement of the top of the wall where the displacement of 6 mm was reported due to

lateral load of 90 kN (experimental) and 207 kN (analytical). Figure 7.18 compares the

experimental and analytical results for steel strain condition for the reinforcement at the

front side of the wall at the wall-foundation interface. The strain values of 915 με and 925

με were reported for experimental and analytical results, respectively, due to top of the

wall displacement of 6 mm.

0

50

100

150

200

250

0 2 4 6 8


Late

ral l

oad

(kN

)

ExperimentalAnalytical


241

-200

0

200

400

600

800

1000

0 2 4 6 8


Stra

in (×

10-6

)


Figure 7.18 Steel strain value at the front side of the wall at the wall-foundation interface

Figure 7.19 compares the experimental and analytical results for steel strain condition for

the reinforcement at the back side of the wall at the wall-foundation interface. The strain

values of 254 με and 256 με were reported for experimental and analytical results,

respectively, due to top of the wall displacement of 6 mm.

-300

-250

-200

-150

-100

-50

0

50

0 2 4 6 8


Stra

in (×

10-6

)


Figure 7.19 Steel strain value at the back side of the wall at the wall-foundation interface

242

It is observed that the experimental and analytical values of steel strain correlated well

during the monotonic loading stage of the pretest. However, there is a considerable

difference between the load-displacement relationships in the beginning of the test as can

be observed from Figure 7.17. For the top displacement of 6 mm, the experimental value

for lateral load was 90 kN while the corresponding analytical value was 207 kN

indicating that the stiffness for the experimental results was 0.43 of that of the analysis.

An important aspect of the comparison between the experimental and analytical behavior

of an RC member is the initial stiffness of the member. RC members tend to experience

microcracking due to several phenomena such as shrinkage which reduces the initial

stiffness of the member. It is important to note that the initial stiffness properties in the

RC structures are smaller than the properties of the uncracked sections and the reduced

values of stiffness should be considered for the analysis purposes (Bracci et al.1992,

Filiatrault et al. 1998, Li et al. 2005).

In the analytical simulation of the cyclic behavior of a retrofitted shear wall using the

ABAQUS software reported in (Li et al. 2005), the difference in the values of analytical

and experimental flexural stiffness were taken into account. In the tested shear wall

specimen, as a result of existence of micro-cracks due to shrinkage effects of concrete,

the initial stiffness and the stiffness in the process of loading, of the tested specimen was

decreased. In ABAQUS, such micro-cracks could not be modeled, and thus when

comparing the FEA results with the test results, the displacement of their FEA was

amplified by a factor of 4, which was obtained by dividing the ultimate stiffness of test

results by that of the FE analysis.

7.5.1 Response under cyclic loading The next part of the analytical investigation in the current study is the simulation of the

cyclic loading part of the pretest. Figure 7.20(a) illustrates the lateral load-displacement

of the top of the wall where the top displacement of 11 mm and 12 mm were reported due

to actuator lateral load of 185 kN in push and pull directions, respectively. For analytical

simulation of the test, since it was not feasible to apply the loading with the same scheme

as used for the experimental results (three cycles of push and pull with increments of 10

kN), only few cycles of displacement loading was applied on top of the wall. Figure

7.20(b) illustrates the analytical results indicating the lateral load of 251 kN and 268 kN

243

corresponding to the top displacements of 11 mm in the pull and 12 mm in the push

directions, respectively. Same reasoning as the effect of reduced stiffness can be

attributed to the cause of difference. However, the difference is not as significant as the

case of initial monotonic loading stage. This is probably due to the reduced stiffness of

the concrete member due to cracking under the applied cyclic loads and the resulting

stiffness reduction in the analytical model.

Top load-displacement

-250-200-150-100

-500

50100150200250

-15 -10 -5 0 5 10 15


Late

ral f

orce

(kN

)


-300

-200

-100

0

100

200

300

-15 -10 -5 0 5 10 15


Late

ral l

oad

(kN

)


(a) Experimental (b) Analytical

It was not possible to obtain the direct relationship between the steel strain values and the

top displacement for the cyclic part of the test as the program would terminate with an

error message (due to different output time steps considered for displacement and strain

by ABAQUS). Therefore, the top displacement of the wall was obtained during the time

(a)

(b)

244

of the analytical simulation as shown in Figure 7.21; and the steel strains are reported

with respect to the analysis time. To follow the same loading scheme as that of the

experiment, the monotonic part of the pretest was first performed and the cyclic loading

part was followed with fewer numbers of cycles than that of the experiment. Figures 7.22

and 7.23 compare the experimental and analytical values for steel strain at the wall-

foundation interface at the front side and back side of the wall, respectively. As shown in

Figure 7.22(a), the front side steel strain fluctuated between 1832 με corresponding to the

top displacement of 11 mm in the push direction, and -423 με corresponding to the top

displacement of 12 mm in the pull direction. Figure 7.22(b), shows the analytical results

for the steel strain which fluctuated between 1728 με corresponding to the top

displacement of 11 mm in the push direction, and -358 με corresponding to the top

displacement of 12 mm in the pull direction.

Top displacement

-15

-10

-5

0

5

10

15

0 5 10 15 20 25

Time (s)

Top

disp

lace

men

t (m

m)

Figure 7.21 Top displacement of the wall during the ABAQUS analysis

As shown in Figure 7.23(a), the back side steel strain fluctuated between 1716 με

corresponding to the top displacement of 11 mm in the push direction, and -308 με

corresponding to the top displacement of 12 mm in the pull direction. Figure 7.23(b),

shows the analytical results for the steel strain which fluctuated between 1920 με

corresponding to the top displacement of 11 mm in the push direction, and -367 με

corresponding to the top displacement of 12 mm in the pull direction. It is important to

mention that the steel bar and concrete interaction in ABAQUS can only be modeled by

245

using tension stiffening which is a simplification of an actual complex phenomenon that

happens in the experiment.

Figure 7.24 compares the experimental and analytical results of the test of Specimen-4 up

to failure. Again the number of cycles was reduced in the analytical simulations

compared to the number of cycles in the experiments. For the analytical simulation the

specimen was loaded from its initial undamaged condition, while in the experimental test,

the specimen was damaged due to previous experimental tests. Good agreement can be

observed between the two results in terms of load-displacement relationships after

yielding stage, especially at the maximum lateral load. The specimen resisted the

maximum lateral load of 338 kN at the top displacement of 66.7 mm while the

corresponding analytical result was 340 kN at top displacement of 67 mm.

Front side steel strains

-1000

-500

0

500

1000

1500

2000

-15 -10 -5 0 5 10 15


Stra

ins

(×10

-6)

Front side of the wall

-500

0

500

1000

1500

2000

0 5 10 15 20 25

Time (s)

Stra

in (×

10-6)

Figure 7.22 Steel strain value at the front side of the wall at the wall-foundation interface


(a)

(b)

246

7.5.1 Pushover analysis

A pushover analysis was also carried out in which the top of the wall was subjected to a

monotonic displacement controlled loading until the analysis stopped at the lateral load

level of 291 kN corresponding to top displacement of 84.6 mm as shown in Figure 7.25.

The maximum lateral load the specimen was able to resist was 388 kN corresponding to

top displacement 65 mm. As shown in Figure 7.25, the pushover curve overestimated the

response and also indicated the failure at a smaller displacement compared to the

experimental evidences.

Back side steel strains

-500

0

500

1000

1500

2000

-15 -10 -5 0 5 10 15


Stra

ins

(×10

-6)


-1000-500

0500

1000150020002500

0 5 10 15 20 25

Time (s)

Stra

in (×

10-6)

Figure 7.23 Steel strain value at the back side of the wall at the wall-foundation interface


(a)

(b)

247

Figures 7.26(a) and (b) illustrate the analytical results of the pushover analysis for steel

strain values at the interface of the wall-foundation for the reinforcement layer at the

front side and back side of the wall, respectively. According to Figure 7.26(a), the

reinforcement at the front side of the wall experienced tensile strain of 2450 με at a top

wall displacement of 17.9 mm.


-300-200-100

0100200300400

-50 0 50 100 150


Late

ral l

oad

(kN

)

Experimental

Analytical



-300-200-100

0100200300400500

-50 0 50 100 150


Late

ral l

oad

(kN

)

Experimental

Pushover

Figure 7.25 Comparison of the response of the wall and the pushover analysis

248

This is comparable to the experimental observations that the front side strain gauges


displacement of 17.9 mm in the push direction during the cyclic loading. The pushover

analysis results indicate that the front side reinforcement yielded at a top of wall

displacement of 13 mm corresponding to a lateral load of 270 kN on top of the wall. By

comparison of the analytical results to the experimental values of 15 mm top

displacement and 210 kN lateral load, it is observed that the yield displacement obtained

by the pushover analysis correlated reasonably well with the experimental data. However,

the lateral load obtained by the pushover analysis overestimated the experimental value

by about 30%. This may be due to difference between the initial stiffness of the member

in the analysis and experiment.

Front side of the wall

02000400060008000

100001200014000

0 20 40 60 80 100


Stra

in (×

10-6)


-500

0

500

1000

1500

2000

0 20 40 60 80 100


Stra

in (×

10-6)

Figure 7.26 Analytical results for steel strain value at the interface of the wall-foundation

(a) Front side of the wall (b) Back side of the wall

(a)

(b)

249

More experimental and analytical tests are needed to enable a conclusion regarding the

initial stiffness. In addition, more experimental information is needed to determine the

appropriate values of c in Eq. 7-9 for other cases with different conditions such as

element size and concrete properties.

250

CHAPTER-8

SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS

8.1 Summary

Reinforced Concrete (RC) Liquid containing structures (LCS) are designed not only to

have functionality during the normal life cycle, but also to withstand the earthquake

loading without any extensive cracking that causes leakage. It is possible to design a RC

tank for forces smaller than the elastic forces and safely survive the ground motion

excitation. In RC tanks, cracking which leads to leakage can be regarded as a possible

mode of failure. Therefore a thorough understanding of the cracking and possible leakage

phenomena in concrete tanks, especially during the earthquake loading, is of main

importance. An important aspect of the design of RC tanks for the earthquake loading is

the response modification factor (R factor), which was fully described in Chapter-3. The

response modification factor for the impulsive component Ri affects several components

of the hydrodynamic force, and the seismic load is not linearly proportional to Ri.

Equations were proposed to facilitate the calculation of the seismic induced vertical

flexural moment at the base of the wall as a function of Ri.

This study showed that the portion of a cantilever wall near the base is the most critical

region with respect to leakage. In the current study the cracking and possible leakage

behavior of such critical region under different loading conditions were studied.

Several full scale RC specimens representing the aforementioned critical region of a

rectangular tank were constructed and tested. All specimens were designed based on ACI

Code (ACI 350 2006). The specimens were subjected to reverse cyclic loading using an

actuator while a water pressure chamber was installed in the critical region to simulate

the effect of the pressurized water at the bottom portion of a tank wall. Two different

configurations for the shear key were used and examined with respect to leakage.

Additionally, the effect of retrofitting of the cracked wall with Glass Fiber Reinforcing

Polymer (GFRP) sheet with respect to leakage was investigated. The study was limited to

251

rectangular open top tank in which the dimensions of the walls promoted one-way

behavior. Also the effect of soil-structure interaction was not considered.

In the analytical phase of this study, the FE computer program ABAQUS/6.7.1 was used.

This program was first verified to assess its ability for simulating the cyclic behavior of

RC members. Parts of the actual tests conducted during the experimental part of the

research were simulated. The results showed reasonably well correlations between the

experimental and analytical observation especially in terms of the reinforcement strain

values at the critical region. The analytical values of lateral load in the analysis

procedures overestimated the actual values observed in the experiment especially at the

initial stages of loading. This was mainly attributed to the difference between the initial

stiffness of the member in the analysis and experiments.

8.2 Conclusions

Based on the results of the experimental and analytical study conducted on five wall-

foundation specimens, the following conclusions were made.

1- Cracking at both faces of the wall is necessary for leakage to occur. Specimen-1

was subject to a cyclic loading condition in such a way that the front face would

experience severe cracking, while the back face would not crack. This was due to

the fact that the concrete at back face of the wall did not experience cracking

tensile stress/strain. Although the specimen was additionally subjected to more

number of cycles at the near failure loading, no leakage was observed at the back

of the wall. It is postulated that the presence of compression zone at the back side

of the wall was able to prevent the leakage. The tensile steel stress/strain values

were far beyond the current Code maximum permissible limits for environmental

engineering structures.

2- For the specimen with inverted shear key configuration without a water-stop,

leakage at the back side of the wall was observed at the initial stage of cyclic

loading where the steel strain level was below 15% of yield value. The ensuing

252

monotonic loading stage did not show any apparent increase in the rate of leakage

indicating that the leakage was due to the effects of cyclic loading. Therefore,

application of such configuration for seismic regions is discouraged.

3- For Specimens-3, 4, and 5 the conventional shear key incorporated a water stop at

the joint region. The loading condition was in such a way that both faces of the

wall experienced cracking during the cyclic loading stage. Although the base of

the wall was the first region that experienced cracking and the crack width

showed to be wider in this region, no leakage was observed at the back face of the

wall. It is postulated that the water stop was effective in preventing the leakage at

the base of the wall.

4- For Specimens-3, 4, and 5, the level of steel strain was at or beyond the yield

level. Therefore, it may be appropriate to assume that leakage occurs soon after

the yielding of the reinforcement.

5- Retrofitting of the cracked wall with Glass Fiber Reinforcing Polymer (GFRP)

sheet with respect to leakage was effective only for monotonic loading situation

and also for very low level of cyclic loading. The cyclic straining of the GFRP

and its matrix in tension and compression might be the cause for leakage of water

past the GFRP layer(s). Although the GFRP layer is very effective in tension, its

performance when subjected to compression is questionable with respect to

leakage. In the test of the retrofitted Specimen-1, the cyclic load put the GFRP

layer in repeated states of tension and neural stress (no compression stress). The

GFRP was able to prevent the leakage up to the load levels corresponding to

about twice the hydrostatic force. However, the loading scheme for Specimen-3

subjected the GFRP layers to repeated cycles of tensile and compressive

stress/strain, resulting in leakage at relatively low levels of cyclic loading.

6- It is believed that under cyclic loading, the crack opening may start as the load is

increased; but when the load direction is reversed and the part of section

253

containing the crack, experience compressive stress/strain, the crack closes and

the compression block is able to prevent the leakage. This crack opening and

closing would continue while the reinforcement is in the linear elastic range and

retrieves its initial condition when the load is removed. The reinforcing bars at

both sides would start to deform linearly until yielding during cyclic loading.

After the reinforcement experiences plastic deformation as a result of yielding in

tension, it requires more compressive force to retrieve its original condition and to

close the crack(s). If opening of the crack(s) at the side which is not in contact

with the liquid, happens before complete closing of the crack(s) at the other side

(in contact with the liquid), leakage may occur.

7- The response modification factor was considered as a product of overstrength and

ductility components. The overstrength component was calculated as 1.29 and the

ductility component was assumed as unity. Using calculations as presented in

Chapter-6, the response modification factor for the impulsive component, Ri, was

calculated to be equal to 1.7 for wall heights of 3, 4, and 5 m, and equal to 2 for

wall heights of 6 through 10 m. The code prescribed value of the response

modification factor for the impulsive component, Ri is equal to 2, which might be

overestimated as illustrated by a numerical example in Chapter-6.

8- Finite element simulation of the experiments indicated reasonably well prediction

of the results in terms of the strain values in the reinforcement at critical section

of the wall with respect to the displacement of the top of the wall. The initial

stiffness of wall during the analytical simulation was higher than that of the

experiment. This was considered as the reason for the difference in the lateral load

values at the initial stage (before yielding) obtained from the analysis and the

experiments.

8.3 Recommendations for future work During the experimental program in the current study, five specimens were constructed

and tested. More specimens with more variables especially in terms of the wall height

254

and wall thickness are needed to be tested. Another important parameter to be evaluated

is the volume and spacing of reinforcement and their effect on the possible leakage.

Considering the reinforcement strain as the most important parameter with respect to

leakage of the wall, such findings may assist in the generalization of the results of the

current study to a broader range. More analytical simulations are needed for

determination of the accurate relationships for the material models incorporated in the

finite element software.

It is suggested that a full scale model of rectangular tanks be analyzed considering

appropriate interactions with the contained liquid, and subjected to different earthquakes

excitations (time history analysis). Using the appropriate value of response modification

factor results in the volume and spacing of reinforcement such that the reinforcement will

remain in the linear elastic range during and at the end of such time history analyses.

Another subject to be investigated can be considered to be the determination of the

response modification factor for other tanks with different boundary conditions such as

flexible and non-flexible base tanks and aspect ratio of the wall.

255

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268

APPENDIX A

DESIGN PARAMETERS

The design parameters for Specimen-1 are outlined in Table A-1.

Table A-1 Design parameters for Specimen-1

B (Width of a rectangular tank, perpendicular to the direction

of the ground motion) 40.0 m

L (Length of a rectangular tank, parallel to the direction of the

ground motion) 20.0 m

tw (Average wall thickness) 400 mm

HL (Designed depth of stored liquid) 3.5 m

HW (Wall height) 4.00 m

γL (Density of the liquid) 9.8 kN/m3

γC (Density of concrete) 23.6 kN/m3

ρL (Mass density of the liquid) 1 kN.S2/m4

ρC (Mass density of concrete) 2.4 kN.S2/m4

f'c (28 days compressive strength of concrete) 30 MPa

Ss (mapped maximum considered earthquake 5% damped

spectral response acceleration parameter at short periods,

expressed as a fraction of the acceleration due to gravity 1.5

S1 (mapped maximum considered earthquake 5% damped

spectral response acceleration parameter at a period of 1 s,

expressed as a fraction of the acceleration due to gravity) 0.6

Fa (short-period site coefficient (at 0.2 s period)) 0.8

Fv (long-period site coefficient (at 1.0 s period)) 0.8

I (Importance factor) 1

Ri (Response modification factor for impulsive component) 2

Rc (Response modification factor for convective component) 1

269

Ss and S1(mapped maximum considered earthquake 5% damped spectral response

acceleration parameter at short periods, and period of 1 s, respectively, expressed as a

fraction of the acceleration due to gravity, g) are taken as 150%, and 60% respectively as

obtained from ASCE 7-05, Fig. 22-1 through 22-14

Fa and Fv (short period (at 0.2 s period) and long period (at 1.0 s period) site coefficient),

respectively from ASCE 7-05, Table 11.4-1) are both taken as 0.8.

The importance factor, I is taken as 1.0 as the tank is assumed to be in Category I in

Table 4.1.1 (a).

The Response Modification Factors, Ri (impulsive component) and Rc (convective

component) are taken as 2, and 1, respectively.

Checking the freeboard

The maximum vertical displacement, dmax, to be accommodated shall be calculated from

the following expressions:

Rectangular tanks:

ICLc2

=maxd (A-1)

Where Cc is the seismic response coefficient and shall be determined For Tc ≤ 1.6/Ts s

DSS1.5cTD1S1.5

cC ≤= (A-2)

For Tc > 1.6/Ts s

2cT

DSS2.42cT

DS0.4S6cC ==

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ (A-3)

Tc is the natural period of the first (convective) mode of sloshing, s, and is obtained as:

Lλ

2π

cω2π

cT ⎟⎠⎞

⎜⎝⎛== (A-4)

270

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛=

LLH

3.16tanhg3.16λ (A-5)

3.53.16 32.17 tanh[3.16 ] 7.1520

λ ⎛ ⎞= × × =⎜ ⎟⎝ ⎠

2 3.14 20( ) 7.11 7.15 0.305CT s×

= × =

SDS and SD1(design spectral response acceleration 5% damped spectral response

acceleration parameter at short periods, and period of 1 s, respectively, as defined in 9.4.1,

expressed as a fraction of the acceleration due to gravity, g.

aFSS32

DSS = (A-6)

2 1.50 0.8 0.803DSS g= × × =

vF1S32

D1S = (A-7)

12 0.60 0.8 0.323DS g= × × =

DSSD1S

ST = (A-8)

0.32 0.400.80S

gTg

= =

Therefore Tc > 1.6 Ts , and

271

2

2.4 0.8 0.0387.11cC ×

= =

And the required freeboard height is obtained as:

max20 0.038 1 0.382

d m= × × =

Therefore the maximum height of water for a safe design would be 3.5 +0.38 = 3.88m

The dynamic model: W'W = γCxtwxHw = 23.6×0.4×4.0 = 37.76 kN/m WL = γLxHLxL =9.8×3.5×20.0 = 686 kN/m

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

=

LHL

LHL

0.866

0.866tanh

LWiW

(A-9)

20tanh[0.866 ( )]3.5686 138.7 /200.866 ( )

3.5

iW kN m×

= × =×

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= L

LH 3.16tanh

LHL0.264

LWcW

(A-10)

20 3.5686 0.264 ( ) tanh[3.16 ( )] 520.5 /3.5 20CW kN m= × × × × =

20 5.71 1.333.5L

LH

= = > , therefore:

0.375LHih

= → hi = 3.5×0.375 = 1.3125 m

272

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

LLH

3.16sinhLLH

3.16

1LLH

3.16cosh

1LHch

(A-11)

3.5cosh[3.16 ( )]203.5 [1 ] 1.793.5 3.53.16 ( ) sinh[3.16 ( )]

20 20

Ch m×

= × − =× × ×

20 5.71 0.753.5L

LH

= = >

81

- = LH

'ih

tanh2

0.866

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

LHL0.866

LHL

(A-12)

'

200.866 ( ) 13.53.5 [ ] 8.2220 82 tanh[0.866 ( )]3.5

ih m×

= × − =× ×

For all tanks,

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

LLH

3.16sinh3.16

2.01- cosh

-1= LH

'ch

LLH

LLH

3.16

(A-13)

273

'

3.5cosh[3.16 ( )] 2.01203.5 [1 ] 12.783.5 3.5[3.16 ( )] sinh[3.16 ( )]

20 20

ch m× −

= × − =× × ×

2

2

138.7 .7.12 2 9.8

ii

W kN smg m

= = =× ×

' 2

2

37.7 .3.849.8

WW

W kN smg m

= = = , and 4 22 2

WW

Hh m= = =

i i W W

i W

h m h mhm m

× + ×=

+

1.3125 7.1 2 3.84 1.557.1 3.84

h m× + ×= =

+

2

2

.7.1 3.84 10.94i wkN sm m m

m= + = + =

3( )4

c wE tkh

= ×

325000 0.4( ) 1074 1.55

k MPa= × =

2imTk

π= × ×

10.942 0.06 107 1000iT sπ= × × =

×

Ci shall be determined as follows:

For TI ≤ TS Ci = SDS (A-14)

For TI > TS DSSiT

D1SiC ≤= (A-15)

274

Where: DSSD1S

ST = (A-16)

0.32 0.400.80s

gT sg

= =

TI ≤ TS → Ci = 0.80

Effective mass coefficient, ε

01.1.021L

H

L0.1908

2

LH

L0.0151ε ≤+−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ (A-17)

220 20[0.0151( ) 0.1908( ) 1.021] 0.423.5 3.5

ε = − + =

The dynamic lateral forces above the base shall be determined as follows:

iRwWε

IiCwP ×= (A-18)

0.42 37.76 20.8 1 12.8 /2WP kN m× ×

= × × = (Considering 2 accelerating walls)

iR

'wWε

IiC'wP ×= (A-19)

' 0.42 37.760.8 1 6.4 /2W

P kN m×= × × =

iRiW

IiCiP ×= (A-20)

138.70.8 1 55.48 /2iP kN m= × × =

275

cRcW

Ic

Cc

P ×= (A-21)

520.50.038 1 19.77 /1cP kN m= × × =

Vertical acceleration

In the absence of a site-specific response spectrum, the ratio b of the vertical to horizontal

acceleration shall not be less than 2/3. The hydrostatic load qhy from the tank contents, at

level y above the base, shall be multiplied by the spectral acceleration üv to account for

the effect of the vertical acceleration. The resulting hydrodynamic pressure pvy shall be

computed as follows:

hyqvuvyp ×= && (A-22)

Where

DSS.20≥=

iRb

It

Cv

u&& (A-23)

For rectangular tanks, Ct = 0.4SDS.

2 / 30.4 0.8 1 0.012

= × × × =&&Vu

0.2 SDS = 0.2×0.8 = 0.16 → 0.16Vu =&&

3.5 9.8 34.3HLq kPa= × =

0.16 34.3 5.49HLP kPa= × =

Magnitude and location of the resultant loads applied on the accelerating wall

Hydrostatic pressure:

1 3.5 9.8 3.5 60.0 /2hydrostaticP kN m= × × × = hh = 1.17 m

276

Impulsive force:

1 55.48 27.74 /2impulsiveP kN m= × = hi = 1.31 m

Convective force:

1 19.77 9.885 /2convectiveP kN m= × = hc = 1.79 m

Wall inertia effect

6.4 /wallP kN m= hw = 2.0 m

Vertical ground motion effect

1 3.5 5.49 9.60 /2verticalP kN m= × × = hv = 1.17 m

The Horizontal seismic induced shear force on the connection of the wall foundation

( ) ( ) ( )2 2 2(27.74 6.4) 9.60 9.885 36.8 /seismicV kN m= + + + =

The seismic induced moment on the connection of the wall foundation

( ) ( ) ( )2 2 2(27.74 1.31 6.4 2.0) 9.60 1.17 9.885 1.79 53.42 . /seismicM kN m m= × + × + × + × =

The Horizontal static shear force on the connection of the wall foundation

60 /staticV kN m=

The static moment on the connection of the wall foundation

1.17 60 70.2 . /staticM kN m m= × =

It is desired to observe the behavior of the connection of the wall-foundation for the

effect of the loading that it might experience

While the hydro static shear force and moment of 60 /staticV kN m= , and

70.2 . /staticM kN m m= is always applied when the height of the liquid is 3.5 m, the

277

earthquake can exert a maximum seismic shear force and moment of 36.8 /seismicV kN m= ,

and 53.42 . /seismicM kN m m= .

For the application of a quasi-static cyclic load, the point of application of the load is

calculated as

70.2 53.42 128060 36.8

H mm+= =

+

This height will ensure that the effect of the extreme loading will be correctly simulated

Design of the reinforced concrete section

In ACI 350-06 two load combinations should be considered for seismic design of liquid

containing structures when only hydrostatic and hydrodynamics loads are applied on the

structure. These combinations are U1= [1.4 × (D + F)], and U2= [(1.2 × F) + (1 × E)]. D

and E correspond to the dead load and earthquake load effects, respectively. F denotes the

loads due to the liquid pressure

The required strength, U, shall be multiplied by the environmental durability factor (Sd)

when durability, liquid-tightness, or similar serviceability are of main concern. This

durability factor shall not be used for designs using service loads and permissible service

load stresses.

1.0yd

s

fS

fφγ

= ≥

where

where fs is the permissible tensile stress in reinforcement.

In normal environmental exposure areas, the calculated maximum stress fsmax in

reinforcement closest to a surface in tension at service loads shall not exceed that given

by Eq. 2-1 and shall not exceed a maximum of o.6fy:

factored loadunfactored load

γ =

(3-55)

278

320, max

2 2( ) 4(2 )25 50

f s ds bβ

=

+ +

where s and db are the spacing and diameter of the wall reinforcement, respectively. β is

defined as the ratio of distances to the neutral axis from the extreme tension fiber and

from the centroid of the main reinforcement.

For the design of Specimen-1, it was assumed that earthquake induced load in the pull-

direction only involves the flexural moment related to the concrete wall inertia of 6.4

kN.m, and the load in the push direction includes seismic and static flexural moments of

53.42 kN.m and 70.2 kN.m, respectively. In the design of Specimen-1, the effect of

pulling load of wall inertia during the earthquake was ignored due to its relative small

value and the loading scheme considered only pushing the specimen and pulling it back

to the initial position.

The two load combinations are

1 (1.4 ) = 1.3 1.4 70.1 127.5 .u d FM S M kN m= × × × =

2 (1.2 ) =(1.2 70.1) 53.5 137.6 .u F EM M M kN m= + × + =

Therefore, the governing design load combination value is 137.6 kN.m. The section

properties for the design are as follows:

' 30

25000400

10004000.9

Cover to main bars= 50Selected main bars= No.20

(400 50 10) 3400.85

c

c

y

f MPaE MPaf MPa

b mmh mm

mm

d mm

ϕ

β

===

===

= − − ==

( - )2u caM C d= '0.85 c cC f c bβ=

(2-1)

279

0.85 30 0.85 1000 21675cC c c= × × × × =

0.85 121675 (340 ) 137.6 . /2 0.9

21.3

cc kN m m

c mm

× − = ×

=

461.8 cC kN= → 2461800 1155 400

cs

y y

CTA mmf f

= = = =

For Specimen-1, the ACI Code prescribed minimum area of reinforcement based on 0.3

percent of gross cross section area (1164 mm2) exceeded the amount of reinforcement

based on both load combinations stated above. The spacing of No.20 rebars based on

1155 mm2 for 1000 mm width of the section is calculated as 250 mm which will result in

increase in the value of the durability factor (Sd) from 1.3 to 1.7. This leads to an increase

in the value of the first load combination from 127.5 kN.m to to 167 kN.m, resulting in

spacing of 200 mm for the rebars at the front face of the wall. The rebars at the back face

of the wall are designed based on the minimum area of reinforcement mentioned equal to

1164 mm2. Therefore, eight No.20 rebars and seven No.20 rebars are provided at the

front face and the back face of the wall, respectively.

280

APPENDIX B

ABAQUS INPUT FILE ** Section: Column *Solid Section, elset=_PickedSet2, material=Concrete 1., *End Part ** *Part, name=Rebar *Node 1, 350., -892., 0. 2, 350., -810.909119, 0. 3, 350., -729.818176, 0. 4, 350., -648.727295, 0. 5, 350., -567.636353, 0. 6, 350., -486.545441, 0. 7, 350., -405.454559, 0. 8, 350., -324.363647, 0. 9, 350., -243.27272, 0. 10, 350., -162.181824, 0. 11, 350., -81.0909119, 0. 12, 350., 0., 0. 13, 350., 81.0909119, 0. 14, 350., 162.181824, 0. 15, 350., 243.27272, 0. 16, 350., 324.363647, 0. 17, 350., 405.454559, 0. 18, 350., 486.545441, 0. 19, 350., 567.636353, 0. 20, 350., 648.727295, 0. 21, 350., 729.818176, 0. 22, 350., 810.909119, 0. 23, 350., 892., 0. *Element, type=T3D2 1, 1, 2 2, 2, 3 3, 3, 4 4, 4, 5 5, 5, 6 6, 6, 7 7, 7, 8 8, 8, 9 9, 9, 10 10, 10, 11 11, 11, 12

281

12, 12, 13 13, 13, 14 14, 14, 15 15, 15, 16 16, 16, 17 17, 17, 18 18, 18, 19 19, 19, 20 20, 20, 21 21, 21, 22 22, 22, 23 *Nset, nset=_PickedSet2, internal, generate 1, 23, 1 *Elset, elset=_PickedSet2, internal, generate 1, 22, 1 ** Section: Rebar *Solid Section, elset=_PickedSet2, material=Steel 452., *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=Column-1, part=Column *End Instance ** *Instance, name=Rebar-1, part=Rebar -100., 0., 50. *End Instance ** *Instance, name=Rebar-2, part=Rebar -100., 0., 200. *End Instance ** *Instance, name=Rebar-3, part=Rebar -100., 0., 350. *End Instance ** *Instance, name=Rebar-4, part=Rebar -600., 0., 50. *End Instance ** *Instance, name=Rebar-5, part=Rebar -600., 0., 200.

282

*End Instance ** *Instance, name=Rebar-6, part=Rebar -600., 0., 350. *End Instance ** *Instance, name=Rebar-7, part=Rebar -433.33, 0., 50. *End Instance ** *Instance, name=Rebar-8, part=Rebar -266.66, 0., 50. *End Instance ** *Instance, name=Rebar-9, part=Rebar -266.66, 0., 350. *End Instance ** *Instance, name=Rebar-10, part=Rebar -433.33, 0., 350. *End Instance ** *Nset, nset=_PickedSet24, internal, instance=Rebar-1, generate 1, 23, 1 *Nset, nset=_PickedSet24, internal, instance=Rebar-2, generate 1, 23, 1 *Nset, nset=_PickedSet24, internal, instance=Rebar-3, generate 1, 23, 1 *Nset, nset=_PickedSet24, internal, instance=Rebar-7, generate 1, 23, 1 *Nset, nset=_PickedSet24, internal, instance=Rebar-5, generate 1, 23, 1 *Nset, nset=_PickedSet24, internal, instance=Rebar-9, generate 1, 23, 1 *Nset, nset=_PickedSet24, internal, instance=Rebar-4, generate 1, 23, 1 *Nset, nset=_PickedSet24, internal, instance=Rebar-8, generate 1, 23, 1 *Nset, nset=_PickedSet24, internal, instance=Rebar-6, generate 1, 23, 1 *Nset, nset=_PickedSet24, internal, instance=Rebar-10, generate 1, 23, 1 *Elset, elset=_PickedSet24, internal, instance=Rebar-1, generate 1, 22, 1 *Elset, elset=_PickedSet24, internal, instance=Rebar-2, generate 1, 22, 1

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*Elset, elset=_PickedSet24, internal, instance=Rebar-3, generate 1, 22, 1 *Elset, elset=_PickedSet24, internal, instance=Rebar-7, generate 1, 22, 1 *Elset, elset=_PickedSet24, internal, instance=Rebar-5, generate 1, 22, 1 *Elset, elset=_PickedSet24, internal, instance=Rebar-9, generate 1, 22, 1 *Elset, elset=_PickedSet24, internal, instance=Rebar-4, generate 1, 22, 1 *Elset, elset=_PickedSet24, internal, instance=Rebar-8, generate 1, 22, 1 *Elset, elset=_PickedSet24, internal, instance=Rebar-6, generate 1, 22, 1 *Elset, elset=_PickedSet24, internal, instance=Rebar-10, generate 1, 22, 1 *Nset, nset=_PickedSet25, internal, instance=Column-1, generate 1, 1242, 1 *Elset, elset=_PickedSet25, internal, instance=Column-1, generate 1, 880, 1 *Nset, nset=_PickedSet26, internal, instance=Column-1, generate 23, 1242, 23 *Elset, elset=_PickedSet26, internal, instance=Column-1, generate 22, 880, 22 *Nset, nset=_PickedSet27, internal, instance=Column-1, generate 1, 1220, 23 *Elset, elset=_PickedSet27, internal, instance=Column-1, generate 1, 859, 22 *Nset, nset=Support, instance=Column-1, generate 23, 1242, 23 *Elset, elset=Support, instance=Column-1, generate 22, 880, 22 *Elset, elset=__PickedSurf29_S6, internal, instance=Column-1, generate 1, 859, 22 *Surface, type=ELEMENT, name=_PickedSurf29, internal __PickedSurf29_S6, S6 ** Constraint: Constraint-1 *Embedded Element, host elset=_PickedSet25 _PickedSet24 *End Assembly *Amplitude, name=Amp-1, definition=SMOOTH STEP 0., 0., 2., 1. *Amplitude, name=Amp-2, definition=SMOOTH STEP 0., 0., 4., 1. *Amplitude, name=Amp-3, definition=SMOOTH STEP 0., 0., 6., 1.

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** ** MATERIALS ** *Material, name=Concrete *Density 2.4e-09, *Elastic 28000., 0.17 *Concrete Damaged Plasticity 30., 0.1, 1.16, 0.666, 0. *Concrete Compression Hardening 13.4, 0. 30.8, 0.00317 28.5, 0.005 26.5, 0.006 24., 0.008 21.5, 0.01 15.4, 0.01408 15.3, 0.1 *Concrete Tension Stiffening 3.1, 0. 3.1 , 0.0003 2.55, 0.0005 2.22, 0.0013 2.05, 0.0021 1.85, 0.0037 1.61, 0.0077 1.48, 0.0117 1.4 , 0.0157 1.34, 0.0197 1.25, 0.03 *Concrete Tension Damage, compression recovery=0.25 0., 0. 0.4, 0.0003 0.65, 0.0009 0.75, 0.0012 0.75, 0.0015 0.75, 0.015 0.75, 0.03 *Material, name=Steel *Density 7.8e-09, *Elastic 200000., 0.2 *Plastic, hardening=COMBINED 430., 0.

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502., 0.023 574., 0.037 624, 0.056 645., 0.084 617, 0.1 574., 0.116 ** ** BOUNDARY CONDITIONS ** ** Name: BC-1 Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet26, ENCASTRE ** ---------------------------------------------------------------- ** ** STEP: Step-1 ** *Step, name=Step-1, nlgeom=NO *Dynamic, Explicit , 2. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-1 _PickedSet27, 1, 1, 7. ** ** LOADS ** ** Name: Load-1 Type: Pressure *Dsload _PickedSurf29, P, 2.69 ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step

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** ---------------------------------------------------------------- ** ** STEP: Step-2 ** *Step, name=Step-2, nlgeom=NO *Dynamic, Explicit , 2. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-1 _PickedSet27, 1, 1, -7. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-3 ** *Step, name=Step-3, nlgeom=NO *Dynamic, Explicit , 2. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-1 _PickedSet27, 1, 1, -7. ** ** OUTPUT REQUESTS **

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*Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-4 ** *Step, name=Step-4, nlgeom=NO *Dynamic, Explicit , 4. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-2 _PickedSet27, 1, 1, 28. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-5 ** *Step, name=Step-5, nlgeom=NO *Dynamic, Explicit , 4. *Bulk Viscosity

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0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-2 _PickedSet27, 1, 1, -21. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-6 ** *Step, name=Step-6, nlgeom=NO *Dynamic, Explicit , 4. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-2 _PickedSet27, 1, 1, -21. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 **

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*Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-7 ** *Step, name=Step-7, nlgeom=NO *Dynamic, Explicit , 4. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-2 _PickedSet27, 1, 1, 21. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-8 ** *Step, name=Step-8, nlgeom=NO *Dynamic, Explicit , 6. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-3 _PickedSet27, 1, 1, 42. **

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** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-9 ** *Step, name=Step-9, nlgeom=NO *Dynamic, Explicit , 6. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-3 _PickedSet27, 1, 1, -42. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-10 ** *Step, name=Step-10, nlgeom=NO *Dynamic, Explicit

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, 6. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-3 _PickedSet27, 1, 1, -42. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-11 ** *Step, name=Step-11, nlgeom=NO *Dynamic, Explicit , 6. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-3 _PickedSet27, 1, 1, 42. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT **

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** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-12 ** *Step, name=Step-12, nlgeom=NO *Dynamic, Explicit , 6. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-3 _PickedSet27, 1, 1, 63. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-13 ** *Step, name=Step-13, nlgeom=NO *Dynamic, Explicit , 6. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-3

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_PickedSet27, 1, 1, -63. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-14 ** *Step, name=Step-14, nlgeom=NO *Dynamic, Explicit , 6. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-3 _PickedSet27, 1, 1, -63. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-15 **

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*Step, name=Step-15, nlgeom=NO *Dynamic, Explicit , 6. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-3 _PickedSet27, 1, 1, 63. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-16 ** *Step, name=Step-16, nlgeom=NO *Dynamic, Explicit , 6. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-3 _PickedSet27, 1, 1, 84. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 **

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*Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-17 ** *Step, name=Step-17, nlgeom=NO *Dynamic, Explicit , 6. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-3 _PickedSet27, 1, 1, -84. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-18 ** *Step, name=Step-18, nlgeom=NO *Dynamic, Explicit , 6. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS **

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** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-3 _PickedSet27, 1, 1, -84. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- ** ** STEP: Step-19 ** *Step, name=Step-19, nlgeom=NO *Dynamic, Explicit , 6. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-3 _PickedSet27, 1, 1, 100. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step ** ---------------------------------------------------------------- **

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** STEP: Step-20 ** *Step, name=Step-20, nlgeom=NO *Dynamic, Explicit , 6. *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-2 Type: Displacement/Rotation *Boundary, amplitude=Amp-3 _PickedSet27, 1, 1, 90. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step

response of reinforced concrete rectangular liquid containing

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Transcript of response of reinforced concrete rectangular liquid containing