Reinforced Concrete Beams, Columns and Frames: Casandjian/Reinforced Concrete Beams, Columns and Frames

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. Content: Chapter 1 Design at Serviceability Limit State (SLS) (pages 1–68): Charles Casandjian, Noel Challamel, Christophe Lanos and Jostein Hellesland Chapter 2 Verification at Serviceability Limit State (SLS) (pages 69–122): Charles Casandjian, Noel Challamel, Christophe Lanos and Jostein Hellesland Chapter 3 Concepts for the Design at Ultimate Limit State (ULS) (pages 123–192): Charles Casandjian, Noel Challamel, Christophe Lanos and Jostein Hellesland Chapter 4 Bending?Curvature at Ultimate Limit State (ULS) (pages 193–266): Charles Casandjian, Noel Challamel, Christophe Lanos and Jostein Hellesland Reinforced Concrete Beams, Columns and Frames......Page 2 Copyright......Page 3 Table of Contents......Page 4 Preface......Page 9 1.1.2. Vectorial notation......Page 15 1.1.5. Compression stress σc,sup in the most compressed fiber......Page 16 1.2.1. Framework of the study......Page 17 1.2.3. Parameterization of the response curves by the stress σs1 of the most stressed tensile reinforcement......Page 19 1.2.4. Comparison of σs1 of the tensile reinforcement for a given stress inthe most compressed concrete fiber σc,sup......Page 20 1.2.5. Comparison of the bending moments......Page 22 1.3.1. Limit law for material behavior......Page 24 1.3.2. Example of limit laws in physics, case of the transistor......Page 25 1.3.3. Design of reinforced concrete beams in bending at the stress Serviceability Limit State......Page 26 1.4.2. Steel at the cross-sectional SLS......Page 27 1.5.1. Frame and neutral axis......Page 28 1.5.2. Conservation of planeity of a cross-section......Page 29 1.5.3. Planeity conservation law in term of stress......Page 31 1.5.4. Introduction to pivot concepts......Page 32 1.5.5. Pivot rules......Page 33 1.6.1. Goal......Page 34 1.6.3. Relative position of the neutral axis......Page 35 1.6.4. Shape filling coefficient......Page 36 1.6.5. Dimensionless formulation for the position of the center of pressure......Page 37 1.7.1. Equilibrium equations......Page 38 1.7.2. Discussion on the resolution of equations with respect to the number of unknowns......Page 40 1.7.3 Reduced moments......Page 41 1.7.4. Case of a rectangular section......Page 43 1.8.2. Shape filling coefficient......Page 44 1.8.3. Dimensionless coefficient related to the center of pressure......Page 45 1.8.4. Equations formulation......Page 46 1.8.5. Resolution......Page 47 1.9.3. Dimensionless coefficient related to the center of pressure......Page 49 1.9.4. Equations formulation......Page 50 1.9.5. Resolution......Page 51 1.9.6. Synthesis......Page 52 1.10.1. A design problem at SLS ? exercise......Page 53 1.10.2. Resolution in Pivot A ? Mser = 225 kN.m......Page 56 1.10.3. Resolution in Pivot B ? Mser = 405 kN.m......Page 59 1.10.4. Resolution in pivot AB......Page 61 1.10.5. Design of a reinforced concrete section, an optimization problem......Page 64 1.10.6. General design at Serviceability Limit State with tensile and compression steel reinforcements......Page 68 1.11.1. Introduction......Page 72 1.11.2. Decomposition of the cross-section......Page 74 1.11.3. Case of pivot A for a T-cross-section......Page 75 1.11.4. Case of pivot B for a T-cross-section......Page 77 1.11.5. Example ? design of reinforced concrete beams composed of T-cross-section......Page 79 2.1.1. Position of the neutral axis......Page 83 2.1.2. Equation of static moments for the determination of the position of neutral axis......Page 84 2.1.3. Stress calculation ? general case......Page 86 2.1.4. Rectangular cross-section ? verification of a given cross-section......Page 88 2.1.5. T-cross-section ? verification of a given cross-section......Page 90 2.1.6. Example ? verification of a reinforced T-cross-section......Page 93 2.1.7. Determination of the maximum resisting moment......Page 94 2.2.1. Triangular or trapezoidal cross-section......Page 95 2.2.2. Equilibrium equations ? normal force resultant......Page 96 2.2.3. Equilibrium equations ? bending resultant moment......Page 98 2.2.4. Case of pivot A for a triangular cross-section......Page 100 2.2.6. Static moment equation for a triangular cross-section......Page 101 2.2.7. Design example of a triangular cross-section......Page 102 2.3.1. Steel reinforcement design for a given reinforced concrete section......Page 104 2.3.2. Determination of the position of the neutral axis ? simple bending......Page 105 2.3.3. Determination of the position of the neutral axis ? composed bending with normal force solicitation......Page 106 2.3.4. Exercises for composed bending with normal force solicitation......Page 110 2.4.1. Effect of crack on the bending curvature relationship......Page 121 2.4.2. Simply supported reinforced concrete beam......Page 126 2.4.3. Calculation of deflection ? safe approach......Page 127 2.4.4. Calculation of deflection ? a more refined approach; tension stiffening neglected......Page 128 2.4.5. Calculation of deflection ? a more refined approach; tension stiffening included......Page 130 2.4.6. Approximated approach......Page 132 2.4.7. Calculation of deflection ? a structural example......Page 133 3.1.1. Yield design......Page 137 3.1.2. Application of yield design to the cantilever beam......Page 139 3.1.3. Inelastic plasticity or continuum damage mechanics bending curvature constitutive law......Page 143 3.2.1. Historical perspective......Page 147 3.2.2. Wood’s paradox......Page 149 3.2.3. Non-local hardening/softening constitutive law, a variational principle......Page 151 3.2.4. Non-local softening constitutive law: application to the cantilever beam......Page 158 3.2.5. Some other structural cases ? the simply supported beam......Page 163 3.2.6. Postfailure of reinforced concrete beams under distributed lateral load......Page 166 3.3.1. Steel behavior......Page 170 3.3.2. Concrete behavior......Page 174 3.3.3. Dimensionless parameters at ULS......Page 184 3.3.5. Calculation of the concrete resultant for the bilinear diagram......Page 188 3.3.6. Calculation of the concrete resultant for the parabola?rectangle diagram......Page 193 3.3.7. Calculation of the concrete resultant for the law of Desayi and Krishnan......Page 197 3.3.8. Calculation of the concrete resultant for Sargin’s law of Eurocode 2......Page 201 3.3.9. On the use of the reduced moment parameter......Page 205 4.1.1. Phenomenological approach......Page 207 4.1.2. Moment-curvature relationship for concrete ? brief overview......Page 210 4.1.3. Analytical moment-curvature relationship for concrete......Page 212 4.1.4. A model based on the bilinear moment-curvature approximation......Page 236 4.2.1. Elastic-hardening constitutive law......Page 240 4.2.2. Plastic hinge approach......Page 244 4.2.3. Elastic-hardening constitutive law and local softening collapse:Wood’s paradox......Page 249 4.2.4. Elastic-hardening constitutive law and non-local local softening collapse......Page 252 4.3.1. A continuum damage mechanics-based moment curvature relationship......Page 256 4.3.2. Governing equations of the problem and numerical resolution......Page 259 4.3.3. Second-order analysis ? some analytical arguments......Page 265 4.3.4. Postfailure of the non-local continuum damage mechanics column......Page 272 A1.1. Introduction......Page 280 A1.2.1. Canonical form......Page 281 A1.2.2 Resolution ? one real and two complex roots......Page 282 A1.2.4. Resolution ? three real roots......Page 284 A1.3.1. Summary of Cardano’s method......Page 286 A1.3.2. Resolution of a cubic equation ? example......Page 287 A1.4. Roots of a quartic function ? principle of resolution......Page 288 Appendix 2 Steel Reinforcement Table......Page 290 Bibliography......Page 291 Index......Page 305 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.

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 focuses on the theoretical and practical design of reinforced concrete beams, columns, and frame structures. Its analytical approach for the design of normal reinforced concrete structural elements is compatible with most international design rules for reinforced concrete structures, including Europe's Eurocode 2.