This book is focused on the theoretical and practical design of reinforced concrete beams, columns and frame structures. It is based on an analytical approach of designing normal reinforced concrete structural elements that are compatible with most international design rules, including for instance the European design rules – Eurocode 2 – for reinforced concrete structures. The book tries to distinguish between what belongs to the structural design philosophy of such structural elements (related to strength of materials arguments) and what belongs to the design rule aspects associated with specific characteristic data (for the material or loading parameters). A previous book, entitled Reinforced Concrete Beams, Columns and Frames – Mechanics and Design, deals with the fundamental aspects of the mechanics and design of reinforced concrete in general, both related to the Serviceability Limit State (SLS) and the Ultimate Limit State (ULS), whereas the current book deals with more advanced ULS aspects, along with instability and second-order analysis aspects. Some recent research results including the use of non-local mechanics are also presented. This book is aimed at Masters-level students, engineers, researchers and teachers in the field of reinforced concrete design. Most of the books in this area are very practical or code-oriented, whereas this book is more theoretically based, using rigorous mathematics and mechanics tools.

Contents

1. Advanced Design at Ultimate Limit State (ULS).
2. Slender Compression Members – Mechanics and Design.
3. Approximate Analysis Methods.
Appendix 1. Cardano’s Method.
Appendix 2. Steel Reinforcement Table.

About the Authors

Jostein Hellesland has been Professor of Structural Mechanics at the University of Oslo, Norway since January 1988. His contribution to the field of stability has been recognized and magnified by many high-quality papers in famous international journals such as Engineering Structures, Thin-Walled Structures, Journal of Constructional Steel Research and Journal of Structural Engineering.
Noël Challamel is Professor in Civil Engineering at UBS, University of South Brittany in France and chairman of the EMI-ASCE Stability committee. His contributions mainly concern the dynamics, stability and inelastic behavior of structural components, with special emphasis on Continuum Damage Mechanics (more than 70 publications in International peer-reviewed journals).
Charles Casandjian was formerly Associate Professor at INSA (French National Institute of Applied Sciences), Rennes, France and the chairman of the course on reinforced concrete design. He has published work on the mechanics of concrete and is also involved in creating a web experience for teaching reinforced concrete design – BA-CORTEX.
Christophe Lanos is Professor in Civil Engineering at the University of Rennes 1 in France. He has mainly published work on the mechanics of concrete, as well as other related subjects. He is also involved in creating a web experience for teaching reinforced concrete design – BA-CORTEX.

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