Reinforced Concrete Beams, Columns and Frames

Preface ix

Chapter 1. Advanced Design at Ultimate Limit State (ULS) 1

1.1. Design at ULS - simplified analysis 1

1.1.1. Simplified rectangular behavior - rectangular cross-section 1

1.1.2. Simplified rectangular behavior - T-cross-section 16

1.1.3. Comparison of design between serviceability limit state and ultimate limit state 22

1.1.4. Biaxial bending of a rectangular cross-section 28

1.2. ULS - extended analysis 37

1.2.1. Bilinear constitutive law for concrete - rectangular cross-section 37

1.2.2. Parabola-rectangle constitutive law for concrete - rectangular cross-section 44

1.2.3. T-cross-section - general resolution for bilinear or parabola-rectangle laws for concrete 53

1.2.4. T-cross-section - general equations for composed bending with normal forces 66

1.3. ULS - interaction diagram 82

1.3.1. Theoretical formulation of the interaction diagram 82

1.3.2. Approximation formulations 94

1.3.3. Graphical results for general cross-sections 98

Chapter 2. Slender Compression Members - Mechanics and Design 103

2.1. Introduction 103

2.2. Analysis methods 103

2.2.1. General 103

2.2.2. Requirements to second-order analysis 105

2.3. Member and system instability 105

2.3.1. Elastic critical load and effective (buckling) length 105

2.3.2. System instability principles 110

2.3.3. Concrete column instability - limit load 110

2.4. First- and second-order load effects 112

2.4.1. Global and local second-order effects 112

2.4.2. Single members 113

2.4.3. Frame mechanics - braced and bracing columns 115

2.4.4. Moment equilibrium at joints 119

2.5. Maximum moment formation 120

2.5.1. Maximum first- and second-order moment at the same section 120

2.5.2. Maximum first- and second-order moment at different sections 124

2.5.3. Curvature-based maximum moment expression 136

2.5.4. Unbraced frame application example 141

2.6. Local and global slenderness limits 144

2.6.1. Local, lower slenderness limits - general 144

2.6.2. EC2 - local lower slenderness limits 148

2.6.3. NS-EC2 - Local lower slenderness limits 150

2.6.4. Comparison of the EC2 and NS-EC2 limits 155

2.6.5. Local upper slenderness limit 156

2.6.6. Global lower slenderness limit 159

2.7. Effect of creep deformations 163

2.7.1. General 163

2.7.2. Effects on load and deformation capacity 165

2.7.3. Approximate calculation of creep effects 169

2.8. Geometric imperfections 176

2.8.1. Imperfection inclination 176

2.8.2. Stiffening structural elements 176

2.8.3. Stiffened and isolated structural elements 180

2.9. Elastic analysis methods 181

2.9.1. Principles, equilibrium and compatibility 181

2.9.2. Equilibrium and compatibility at multiple sections 183

2.9.3. Optimization 185

2.10. Practical linear elastic analysis 187

2.10.1. Stiffness assumptions 187

2.10.2. EC2 approach 189

2.10.3. ACI 318 approach 190

2.11. Simplified analysis and design methods 191

2.11.1. General 191

2.11.2. Simplified second-order analysis 192

2.11.3. Method based on nominal stiffness 194

2.11.4. Method based on nominal curvature 200

2.12. ULS design 204

2.12.1. Simplified design methods 204

2.12.2. Alternative design methods 205

2.12.3. Design example - framed column 207

Chapter 3. Approximate Analysis Methods 213

3.1. Effective lengths 213

3.1.1. Definition and exact member analysis 213

3.1.2. EC2 effective length of isolated members 218

3.1.3. Alternative effective length expressions 219

3.1.4. Columns with beam restraints 222

3.2. Method of means 227

3.2.1. General 227

3.2.2. Method of means - typical steps 227

3.2.3. Application of the method of means 230

3.3. Global buckling of unbraced or partially braced systems 236

3.3.1.General considerations 236

3.3.2. Flexibility factors 240

3.3.3. System instability and "system" effective lengths 243

3.3.4. Instability of partially braced column - example 248

3.3.5. Instability of partially braced frame - example 251

3.3.6. Sway buckling of unbraced multistory frames 256

3.4. Story sway and moment magnification 262

3.4.1. General 262

3.4.2. Partially braced column - example 264

3.4.3. Partially braced frame - example 266

3.4.4. Sway magnifier prediction of frames with single curvature regions 268

3.4.5. Iterative elastic analysis method 271

3.4.6. Global magnifiers for sway and moments 272

Appendix 1. Cardano's Method 279

A1.1. Introduction 279

A1.2. Roots of a cubic function - method of resolution 280

A1.2.1. Canonical form 280

A1.2.2. Resolution - one real and two complex roots 281

A1.2.3. Resolution - two real roots 283

A1.2.4. Resolution - three real roots 283

A1.3. Roots of a cubic function - synthesis 285

A1.3.1. Summary of Cardano's method 285

A1.3.2. Resolution of a cubic equation - example 286

A1.4. Roots of a quartic function - principle of resolution 287

Appendix 2. Steel Reinforcement Table 289

Bibliography 291

Index 305