Reinforced Concrete Beams, Columns and Frames

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.

• Preface xiChapter 1. Design at Serviceability Limit State (SLS) 11.1. Nomenclature 11.1.1. Convention with the normal vector orientation 11.1.2. Vectorial notation 11.1.3. Part of the conserved reference section 21.1.4. Frame 21.1.5. Compression stress σc,sup in the most compressed fiber 21.2. Bending behavior of reinforced concrete beams – qualitative analysis 31.2.1. Framework of the study 31.2.2. Classification of cross-sectional behavior 51.2.3. Parameterization of the response curves by the stress σs1 of the most stressed tensile reinforcement 51.2.4. Comparison of σs1 of the tensile reinforcement for a given stress in the most compressed concrete fiber σc,sup 61.2.5. Comparison of the bending moments 81.3. Background on the concept of limit laws 101.3.1. Limit law for material behavior 101.3.2. Example of limit laws in physics, case of the transistor 111.3.3. Design of reinforced concrete beams in bending at the stress Serviceability Limit State 121.4. Limit laws for steel and concrete at Serviceability Limit State 131.4.1. Concrete at the cross-sectional SLS 131.4.2. Steel at the cross-sectional SLS 131.4.3. Equivalent material coefficient 141.5. Pivots notion and equivalent stress diagram 141.5.1. Frame and neutral axis 141.5.2. Conservation of planeity of a cross-section 151.5.3. Planeity conservation law in term of stress 171.5.4. Introduction to pivot concepts 181.5.5. Pivot rules 191.6. Dimensionless coefficients 201.6.1. Goal 201.6.2. Total height of the cross-section 211.6.3. Relative position of the neutral axis 211.6.4. Shape filling coefficient 221.6.5. Dimensionless formulation for the position of the center of pressure 231.7. Equilibrium and resolution methodology 241.7.1. Equilibrium equations 241.7.2. Discussion on the resolution of equations with respect to the number of unknowns 261.7.3 Reduced moments 271.7.4. Case of a rectangular section 291.8. Case of pivot A for a rectangular section 301.8.1. Studied section 301.8.2. Shape filling coefficient 301.8.3. Dimensionless coefficient related to the center of pressure 311.8.4. Equations formulation 321.8.5. Resolution 331.9. Case of pivot B for a rectangular section 351.9.1. Studied section 351.9.2. Shape filling coefficient 351.9.3. Dimensionless coefficient related to the center of pressure 351.9.4. Equations formulation 361.9.5. Resolution 371.9.6. Synthesis 381.10. Examples – bending of reinforced concrete beams with rectangular cross-section 391.10.1. A design problem at SLS – exercise 391.10.2. Resolution in Pivot A – Mser = 225 kN.m 421.10.3. Resolution in Pivot B – Mser = 405 kN.m 451.10.4. Resolution in pivot AB 471.10.5. Design of a reinforced concrete section, an optimization problem 501.10.6. General design at Serviceability Limit State with tensile and compression steel reinforcements 541.11. Reinforced concrete beams with T-cross-section 581.11.1. Introduction 581.11.2. Decomposition of the cross-section 601.11.3. Case of pivot A for a T-cross-section 611.11.4. Case of pivot B for a T-cross-section 631.11.5. Example – design of reinforced concrete beams composed of T-cross-section 65Chapter 2. Verification at Serviceability Limit State (SLS)  692.1. Verification of a given cross-section – control design 692.1.1. Position of the neutral axis 692.1.2. Equation of static moments for the determination of the position of neutral axis 702.1.3. Stress calculation – general case 722.1.4. Rectangular cross-section – verification of a given cross-section 742.1.5. T-cross-section – verification of a given cross-section 762.1.6. Example – verification of a reinforced T-cross-section 792.1.7. Determination of the maximum resisting moment 802.2. Cross-section with continuously varying depth 812.2.1. Triangular or trapezoidal cross-section 812.2.2. Equilibrium equations – normal force resultant 822.2.3. Equilibrium equations – bending resultant moment 842.2.4. Case of pivot A for a triangular cross-section 862.2.5. Case of pivot B for a triangular cross-section 872.2.6. Static moment equation for a triangular cross-section 872.2.7. Design example of a triangular cross-section 882.3. Composed bending with combined axial forces 902.3.1. Steel reinforcement design for a given reinforced concrete section 902.3.2. Determination of the position of the neutral axis – simple bending 912.3.3. Determination of the position of the neutral axis – composed bending with normal force solicitation 922.3.4. Exercises for composed bending with normal force solicitation 962.4. Deflection at Serviceability Limit State 1072.4.1. Effect of crack on the bending curvature relationship 1072.4.2. Simply supported reinforced concrete beam 1122.4.3. Calculation of deflection – safe approach 1132.4.4. Calculation of deflection – a more refined approach; tension stiffening neglected 1142.4.5. Calculation of deflection – a more refined approach; tension stiffening included 1162.4.6. Approximated approach 1182.4.7. Calculation of deflection – a structural example 119Chapter 3. Concepts for the Design at Ultimate Limit State (ULS)  1233.1. Introduction to ultimate limit state 1233.1.1. Yield design 1233.1.2. Application of yield design to the cantilever beam 1253.1.3. Inelastic (plasticity or continuum damage mechanics) bending-curvature constitutive law 1293.2. Postfailure analysis 1333.2.1. Historical perspective 1333.2.2. Wood’s paradox 1353.2.3. Non-local hardening/softening constitutive law, a variational principle 1373.2.4. Non-local softening constitutive law: application to the cantilever beam 1443.2.5. Some other structural cases – the simply supported beam 1493.2.6. Postfailure of reinforced concrete beams under distributed lateral load 1523.3. Constitutive laws for steel and concrete 1563.3.1. Steel behavior 1563.3.2. Concrete behavior 1603.3.3. Dimensionless parameters at ULS 1703.3.4. Calculation of the concrete resultant for the rectangular simplified diagram 1743.3.5. Calculation of the concrete resultant for the bilinear diagram 1743.3.6. Calculation of the concrete resultant for the parabola–rectangle diagram 1793.3.7. Calculation of the concrete resultant for the law of Desayi and Krishnan 1833.3.8. Calculation of the concrete resultant for Sargin’s law of Eurocode 2 1873.3.9. On the use of the reduced moment parameter 191Chapter 4. Bending-Curvature at Ultimate Limit State (ULS)  1934.1. On the bilinear approximation of the moment-curvature relationship of reinforced concrete beams 1934.1.1. Phenomenological approach 1934.1.2. Moment-curvature relationship for concrete – brief overview 1964.1.3. Analytical moment-curvature relationship for concrete 1984.1.4. A model based on the bilinear moment-curvature approximation 2224.2. Postfailure of reinforced concrete beams with the initial bilinear moment-curvature constitutive law 2264.2.1. Elastic-hardening constitutive law 2264.2.2. Plastic hinge approach 2304.2.3. Elastic-hardening constitutive law and local softening collapse: Wood’s paradox 2354.2.4. Elastic-hardening constitutive law and non-local local softening collapse 2384.3. Bending moment-curvature relationship for buckling and postbuckling of reinforced concrete columns 2424.3.1. A continuum damage mechanics-based moment curvature relationship 2424.3.2. Governing equations of the problem and numerical resolution 2454.3.3. Second-order analysis – some analytical arguments 2514.3.4. Postfailure of the non-local continuum damage mechanics column 258Appendix 1. Cardano’s Method 267A1.1. Introduction 267A1.2. Roots of a cubic function – method of resolution 268A1.2.1. Canonical form 268A1.2.2. Resolution – one real and two complex roots 269A1.2.3. Resolution – two real roots 271A1.2.4. Resolution – three real roots 271A1.3. Roots of a cubic function – synthesis 273A1.3.1. Summary of Cardano’s method 273A1.3.2. Resolution of a cubic equation – example 274A1.4. Roots of a quartic function – principle of resolution 275Appendix 2. Steel Reinforcement Table  277Bibliography 279Index 293