Reinforced Concrete Beams, Columns and Frames: Mechanics and Design

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). 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). A second book, entitled Reinforced Concrete Beams, Columns and Frames – Section and Slender Member Analysis, 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. Design at Serviceability Limit State (SLS). 2. Verification at Serviceability Limit State (SLS). 3. Concepts for the Design at Ultimate Limit State (ULS). 4. Bending-Curvature at Ultimate Limit State (ULS). Appendix 1. Cardano’s Method. Appendix 2. Steel Reinforcement Table. About the Authors 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. 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). 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. 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.

Table of Contents:
Preface xi Chapter 1. Design at Serviceability Limit State (SLS) 1 1.1. Nomenclature 1 1.1.1. Convention with the normal vector orientation 1 1.1.2. Vectorial notation 1 1.1.3. Part of the conserved reference section 2 1.1.4. Frame 2 1.1.5. Compression stress σc,sup in the most compressed fiber 2 1.2. Bending behavior of reinforced concrete beams – qualitative analysis 3 1.2.1. Framework of the study 3 1.2.2. Classification of cross-sectional behavior 5 1.2.3. Parameterization of the response curves by the stress σs1 of the most stressed tensile reinforcement 5 1.2.4. Comparison of σs1 of the tensile reinforcement for a given stress in the most compressed concrete fiber σc,sup 6 1.2.5. Comparison of the bending moments 8 1.3. Background on the concept of limit laws 10 1.3.1. Limit law for material behavior 10 1.3.2. Example of limit laws in physics, case of the transistor 11 1.3.3. Design of reinforced concrete beams in bending at the stress Serviceability Limit State 12 1.4. Limit laws for steel and concrete at Serviceability Limit State 13 1.4.1. Concrete at the cross-sectional SLS 13 1.4.2. Steel at the cross-sectional SLS 13 1.4.3. Equivalent material coefficient 14 1.5. Pivots notion and equivalent stress diagram 14 1.5.1. Frame and neutral axis 14 1.5.2. Conservation of planeity of a cross-section 15 1.5.3. Planeity conservation law in term of stress 17 1.5.4. Introduction to pivot concepts 18 1.5.5. Pivot rules 19 1.6. Dimensionless coefficients 20 1.6.1. Goal 20 1.6.2. Total height of the cross-section 21 1.6.3. Relative position of the neutral axis 21 1.6.4. Shape filling coefficient 22 1.6.5. Dimensionless formulation for the position of the center of pressure 23 1.7. Equilibrium and resolution methodology 24 1.7.1. Equilibrium equations 24 1.7.2. Discussion on the resolution of equations with respect to the number of unknowns 26 1.7.3 Reduced moments 27 1.7.4. Case of a rectangular section 29 1.8. Case of pivot A for a rectangular section 30 1.8.1. Studied section 30 1.8.2. Shape filling coefficient 30 1.8.3. Dimensionless coefficient related to the center of pressure 31 1.8.4. Equations formulation 32 1.8.5. Resolution 33 1.9. Case of pivot B for a rectangular section 35 1.9.1. Studied section 35 1.9.2. Shape filling coefficient 35 1.9.3. Dimensionless coefficient related to the center of pressure 35 1.9.4. Equations formulation 36 1.9.5. Resolution 37 1.9.6. Synthesis 38 1.10. Examples – bending of reinforced concrete beams with rectangular cross-section 39 1.10.1. A design problem at SLS – exercise 39 1.10.2. Resolution in Pivot A – Mser = 225 kN.m 42 1.10.3. Resolution in Pivot B – Mser = 405 kN.m 45 1.10.4. Resolution in pivot AB 47 1.10.5. Design of a reinforced concrete section, an optimization problem 50 1.10.6. General design at Serviceability Limit State with tensile and compression steel reinforcements 54 1.11. Reinforced concrete beams with T-cross-section 58 1.11.1. Introduction 58 1.11.2. Decomposition of the cross-section 60 1.11.3. Case of pivot A for a T-cross-section 61 1.11.4. Case of pivot B for a T-cross-section 63 1.11.5. Example – design of reinforced concrete beams composed of T-cross-section 65 Chapter 2. Verification at Serviceability Limit State (SLS)  69 2.1. Verification of a given cross-section – control design 69 2.1.1. Position of the neutral axis 69 2.1.2. Equation of static moments for the determination of the position of neutral axis 70 2.1.3. Stress calculation – general case 72 2.1.4. Rectangular cross-section – verification of a given cross-section 74 2.1.5. T-cross-section – verification of a given cross-section 76 2.1.6. Example – verification of a reinforced T-cross-section 79 2.1.7. Determination of the maximum resisting moment 80 2.2. Cross-section with continuously varying depth 81 2.2.1. Triangular or trapezoidal cross-section 81 2.2.2. Equilibrium equations – normal force resultant 82 2.2.3. Equilibrium equations – bending resultant moment 84 2.2.4. Case of pivot A for a triangular cross-section 86 2.2.5. Case of pivot B for a triangular cross-section 87 2.2.6. Static moment equation for a triangular cross-section 87 2.2.7. Design example of a triangular cross-section 88 2.3. Composed bending with combined axial forces 90 2.3.1. Steel reinforcement design for a given reinforced concrete section 90 2.3.2. Determination of the position of the neutral axis – simple bending 91 2.3.3. Determination of the position of the neutral axis – composed bending with normal force solicitation 92 2.3.4. Exercises for composed bending with normal force solicitation 96 2.4. Deflection at Serviceability Limit State 107 2.4.1. Effect of crack on the bending curvature relationship 107 2.4.2. Simply supported reinforced concrete beam 112 2.4.3. Calculation of deflection – safe approach 113 2.4.4. Calculation of deflection – a more refined approach; tension stiffening neglected 114 2.4.5. Calculation of deflection – a more refined approach; tension stiffening included 116 2.4.6. Approximated approach 118 2.4.7. Calculation of deflection – a structural example 119 Chapter 3. Concepts for the Design at Ultimate Limit State (ULS)  123 3.1. Introduction to ultimate limit state 123 3.1.1. Yield design 123 3.1.2. Application of yield design to the cantilever beam 125 3.1.3. Inelastic (plasticity or continuum damage mechanics) bending-curvature constitutive law 129 3.2. Postfailure analysis 133 3.2.1. Historical perspective 133 3.2.2. Wood’s paradox 135 3.2.3. Non-local hardening/softening constitutive law, a variational principle 137 3.2.4. Non-local softening constitutive law: application to the cantilever beam 144 3.2.5. Some other structural cases – the simply supported beam 149 3.2.6. Postfailure of reinforced concrete beams under distributed lateral load 152 3.3. Constitutive laws for steel and concrete 156 3.3.1. Steel behavior 156 3.3.2. Concrete behavior 160 3.3.3. Dimensionless parameters at ULS 170 3.3.4. Calculation of the concrete resultant for the rectangular simplified diagram 174 3.3.5. Calculation of the concrete resultant for the bilinear diagram 174 3.3.6. Calculation of the concrete resultant for the parabola–rectangle diagram 179 3.3.7. Calculation of the concrete resultant for the law of Desayi and Krishnan 183 3.3.8. Calculation of the concrete resultant for Sargin’s law of Eurocode 2 187 3.3.9. On the use of the reduced moment parameter 191 Chapter 4. Bending-Curvature at Ultimate Limit State (ULS)  193 4.1. On the bilinear approximation of the moment-curvature relationship of reinforced concrete beams 193 4.1.1. Phenomenological approach 193 4.1.2. Moment-curvature relationship for concrete – brief overview 196 4.1.3. Analytical moment-curvature relationship for concrete 198 4.1.4. A model based on the bilinear moment-curvature approximation 222 4.2. Postfailure of reinforced concrete beams with the initial bilinear moment-curvature constitutive law 226 4.2.1. Elastic-hardening constitutive law 226 4.2.2. Plastic hinge approach 230 4.2.3. Elastic-hardening constitutive law and local softening collapse: Wood’s paradox 235 4.2.4. Elastic-hardening constitutive law and non-local local softening collapse 238 4.3. Bending moment-curvature relationship for buckling and postbuckling of reinforced concrete columns 242 4.3.1. A continuum damage mechanics-based moment curvature relationship 242 4.3.2. Governing equations of the problem and numerical resolution 245 4.3.3. Second-order analysis – some analytical arguments 251 4.3.4. Postfailure of the non-local continuum damage mechanics column 258 Appendix 1. Cardano’s Method 267 A1.1. Introduction 267 A1.2. Roots of a cubic function – method of resolution 268 A1.2.1. Canonical form 268 A1.2.2. Resolution – one real and two complex roots 269 A1.2.3. Resolution – two real roots 271 A1.2.4. Resolution – three real roots 271 A1.3. Roots of a cubic function – synthesis 273 A1.3.1. Summary of Cardano’s method 273 A1.3.2. Resolution of a cubic equation – example 274 A1.4. Roots of a quartic function – principle of resolution 275 Appendix 2. Steel Reinforcement Table  277 Bibliography 279 Index 293

About the Author :
Charles CASANDJIAN, retired from INSA (National Institute of Applied Sciences) of Rennes, France. Noël CHALLAMEL, Professor at University of South Brittany, France. Christophe LANOS, Professor at University of Rennes 1, France. Jostein HELLESLAND, Professor at University of Oslo, Norway.