Reinforced Concrete Beams, Columns and Frames

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