Preface
Information for Students and Instructors
Structural Eurocodes and Product Standards
Units of Measurement
Contents
About the Authors
1 General Structural Design Criteria
1.1 Introduction
1.2 Design Principles for Limit States
1.3 Actions
1.3.1 Permanent Actions (G)
1.3.2 Variable Actions (Q)
1.4 Properties of Materials and Products
1.4.1 Partial Factors for Concrete and Steel
1.5 Combinations of Actions
1.5.1 Combinations of Actions for ULS Verification
1.5.2 Combinations of Actions for SLS Verifications
Reference
2 Materials
2.1 Concrete
2.1.1 Creep
2.1.2 Shrinkage
2.1.3 Stress-Deformation Diagram for Structural Analysis
2.1.4 Compressive and Tensile Design Strength
2.1.5 Flexural Tensile Strength
2.1.6 Triaxial Compressive Strength
2.2 Ordinary Reinforcement
2.3 Prestressing Reinforcement
3 Durability and Cover to Reinforcement
3.1 Introduction
3.2 Concrete Cover
3.3 Minimum Concrete Cover cmin,b for Bond
3.4 Minimum Concrete Cover cmin,dur Due to Environmental Conditions
3.4.1 Environmental Conditions (Exposure Classes Related to Environmental Conditions)
3.4.2 Indicative Strength Classes for Durability
3.4.3 Strength Class for Durability
3.4.4 Values of cmin,dur
3.5 Special Cases for the Choice of cmin
3.5.1 In-Situ Concrete Placed Against Other Concrete Elements
3.5.2 Unbonded Tendons
3.5.3 Uneven Surfaces
3.5.4 Exposure Classes XF and XA
3.6 Allowance in Design for Deviation Δcdev
3.6.1 Reduced Values Δcdev for Some Situations
3.6.2 Uneven Surfaces for Concrete Cast Against Ground
3.7 Examples
3.7.1 Example 1. Floor Beam in a Building with Low Relative Humidity
3.7.2 Example 2. Bridge Slab
3.7.3 Example 3. Platform Roof
3.7.4 Example 4. Beam Inside a Building with Low Relative Humidity
3.7.5 Example 5. Exposed External Beam
3.7.6 Example 6. Exposed External Beam Close to or on the Coast
3.7.7 Example 7. Foundation T-Beam
3.7.8 Example 8. Retaining Wall
3.7.9 Example 9. Retaining Wall in Contact with Chemical Highly Aggressive Ground Soil
3.7.10 Example 10. New Jersey Barrier
3.7.11 Example 11. Drilled Pile in a Slightly Aggressive Ground Soil
3.7.12 Example 12. Prestressed Roofing Beam for a Precast Building Near to or on the Coast
4 Structural Analysis
4.1 The Structural Behaviour at Failure of a Beam in Bending
4.2 General Discussion on the Nonlinear Behaviour of the Structures
4.3 Structural Analysis: General
4.3.1 Linear Elastic Analysis (L)
4.3.2 Linear Elastic Analysis with Limited Redistribution (LR)
4.3.3 Plastic Analysis (P)
4.3.4 Nonlinear Analysis (NL)
4.3.5 Examples
4.3.6 Effective Width of Flanges of T-section Beams
4.3.7 Continuous Beam with Infinite Spans 10 m at ULS-SLS
Reference
5 Analysis of Second Order Effects with Axial Load
5.1 Definitions
5.2 Geometric Imperfections
5.3 Isolated Members and Bracing Systems
5.4 Simplified Criteria for Second Order Effects
5.4.1 Slenderness Criterion for Isolated Members
5.4.2 Global Second-Order Effects in Buildings
5.5 Methods of Analysis
5.5.1 The General Method
5.5.2 The Method Based on Nominal Stiffness
5.5.3 The Method Based on Nominal Curvature
5.5.4 Examples
5.5.5 Synthesis of Developed Examples
References
6 Prestressed Concrete
6.1 Introduction
6.2 Stress–Strain Diagram - for Prestressing Steel
6.3 Maximum Prestress Force
6.3.1 Maximum Prestress Force Applied to a Tendon During Tensioning
6.3.2 Maximum Prestress Force After the Transfer of Prestress to Concrete (After Initial Losses)
6.3.3 Mean Prestress Force in Prestressing Steel at Service Conditions
6.4 Limitation of Concrete Stress
6.4.1 Concrete Stress at the Transfer of Prestress
6.4.2 Limitation of Stresses in Anchorage Zones
6.4.3 Maximum Concrete Stresses at Serviceability Limit State
6.4.4 Design Prestress Force and Concrete Strength in Anchorage Zones of Post-tensioned Tendons
6.5 Local Effects at Anchorage Devices
6.6 Minimum Distance of Pre-tensioned Strands or Ducts of Post-tensioned Tendons from Edges
6.7 Minimum Clear Spacing for Pre-tensioned Strands and Ducts of Post-tensioned Tendons
6.7.1 Minimum Clear Spacing of Pre-tensioned Strands
6.7.2 Minimum Clear Spacing Between Post-tensioned Tendons
6.8 Initial Losses Occurring in Pre-tensioned Beams
6.8.1 Losses Due to the Elastic Shortening of Concrete (Pre-tensioned Strands)
6.9 Initial Losses of Prestress for Post-tensioning
6.9.1 Losses Due to the Shortening of Concrete (Post-tensioned Tendons)
6.9.2 Loss Due to Friction
6.9.3 Loss Due to Wedge Draw-In of the Anchorage Devices
6.10 Time-Dependent Prestress Losses
6.10.1 Loss Due to Shrinkage of Concrete
6.10.2 Loss Due to the Relaxation of Prestressing Steel
6.10.3 Effects of the Heat Curing on the Prestress Loss Due to Steel Relaxation
6.10.4 Thermal Loss ΔPθ
6.10.5 Loss Due to Concrete Creep
6.10.6 Effects of a Thermal Cycle on the Concrete Hardening Age
6.10.7 Long-Term Losses Due to Concrete Creep and Shrinkage and Steel Relaxation
6.11 ULS in Flexure
6.11.1 Dimensionless Calculation of the Failure Type of a Pre-stressed Rectangular Cross-Section Under the Hypothesis of Elastic-Perfectly Plastic Stress–Strain Law for Prestressing Steel
6.11.2 Dimensionless Calculation of the Failure Type for a T or I Pre-stressed Cross-Section Under the Hypothesis of an Elastic-Perfectly Plastic Stress–Strain Law for Prestressing Steel
6.11.3 Example 21. Calculation of the Failure Type of a Pre-stressed Rectangular Cross-Section (Fig. 6.62) Under the Hypothesis of Elastic-Perfectly Plastic Stress–Strain Law for Prestressing Steel
6.11.4 Example 22. Calculation of the Resistant Moment for a Prestressed T Cross-Section (Under the Hypothesis of an Elastic-Perfectly Plastic Stress–Strain Law for Prestressing Steel)
6.11.5 Example 23. Calculation of the Resistant Moment for a Prestressed T Cross-Section Under the Hypothesis of an Elastic/Strain Hardening Stress–Strain Law for Prestressing Steel
6.12 Anchorage Length for Pre-tensioned Tendons
6.12.1 Transfer of Prestress
6.12.2 Anchorage of Pre-tensioned Tendons at ULS
6.12.3 Example 24. Calculation of Transmission Length, Dispersion Length and Anchorage Length for Pre-tensioned Tendons
6.13 Anchorage Zones of Post-tensioned Members
6.13.1 “Bursting” Stresses and Anti-burst Reinforcement
6.14 Shear Strength
6.14.1 Beneficial Effects of Prestressing on the Shear Strength
Reference
7 Ultimate Limit State for Bending with or Without Axial Force
7.1 Introduction
7.2 Main Hypotheses
7.3 Resultant Compressive Force in Case of Rectangular Cross-Section
7.4 Equilibrium Configurations of a Rectangular Cross-Section Under Axial Force Combined with Bending
7.5 Reinforcement Design for Uniaxial Bending and Axial Force Combined with Bending
7.6 Rectangular Cross-Section in Uniaxial Bending and Axial Force Combined with Bending
7.6.1 Rectangular Cross-Section: Generalized Parabola-Rectangle Diagrams; Bilinear Diagram
7.6.2 Rectangular Cross-Section: Rectangular Stress Diagram
7.6.3 Examples of Application for Rectangular Cross-Sections in Uniaxial Bending (7.6.1.1.1 and 7.6.2)
7.6.4 Examples of Application for Rectangular Cross-Section with Axial Force Combined with Uniaxial Bending (7.6.1.1.2)
7.7 T-shaped Cross-Section in Bending with or Without Axial Force
7.7.1 T-shaped Cross-Section in Bending
7.7.2 Examples of Application for T-shaped Cross-Section in Bending
7.7.3 T-shaped Cross-Section with Axial Force and Bending
7.7.4 Examples of Application for a T-shaped Cross-Section Under Axial Force Combined with Uniaxial Bending
7.8 Interaction Diagrams for Axial Force and Bending at ULS
7.8.1 Examples of Application of the Interaction Diagrams v - µ
7.9 “Rose” Shaped Diagrams for Axial Force Combined with Uniaxial or Biaxial Bending
7.9.1 Example 1: Axial Force Combined with Biaxial Bending
Reference
8 Shear and Torsion at Ultimate Limit State
8.1 Shear
8.1.1 Symbols and Definitions
8.1.2 Members Without Transverse Reinforcements
8.1.3 Members with Transverse Reinforcements
8.1.4 Examples of Verification of Beams Provided with Transverse Reinforcements (Theme 1)
8.1.5 Examples of Reinforcement Design (Theme 2)
8.1.6 Shear Strength in Case of Loads Near to Supports
8.1.7 Shear Between Web and Flanges for T-beams
8.2 Torsion
8.2.1 General
8.2.2 Calculation Procedure
8.2.3 General and Practical Rules of EC2
8.2.4 Verification and Design in Case of Pure Torsion
8.2.5 Shear-Torsion Interaction Diagrams
Reference
9 Punching Shear
9.1 Introduction
9.2 The Failure Mechanism Due to Punching Shear
9.2.1 Contributions to Punching Shear Strength
9.2.2 Size Effect
9.2.3 Types of Punching Shear Reinforcement
9.3 Phases of Punching Shear Verification
9.4 Punching Shear Strength
9.5 Design Value of the Shear/Punching Stress
9.6 Perimeters u0 and u1 for Rectangular Columns
9.6.1 Internal Column
9.6.2 Edge Column
9.6.3 Corner Column
9.7 The Coefficient β
9.7.1 Values of the Coefficient k
9.7.2 Calculation of the Coefficient β for Rectangular or Circular Columns
9.7.3 Example No. 1—Evaluation of the Coefficient β for an Internal Rectangular Column
9.8 Punching Shear Calculation on the Perimeter of the Column or Loaded Area
9.8.1 Maximum Punching Shear Resistance vRd,max
9.8.2 Design Value of the Punching Shear Resistance
9.8.3 Maximum Punching Shear Resistance for Slabs on Circular Columns
9.8.4 Maximum Punching Shear Force for Slabs on Rectangular Columns
9.8.5 Minimum Value of the Slab Effective Depth
9.9 Columns with Enlarged Heads
9.9.1 Enlarged Column Head with lH ≤ 2hH
9.9.2 Enlarged Column Head with lH > 2hH
9.10 Punching Shear Verification Along the Control Perimeter u1
9.10.1 Punching Shear Resistance of Slabs Without Shear Reinforcement vRd,c
9.10.2 Verification Problem: Calculation of vRd,c Using Tabulated Values
9.11 Comparison of the Shear Forces VRd,max and VRd,c
9.11.1 Example No. 8—Maximum Value of the Effective Depth to Have VRd,c ≤ VRd,max
9.12 Punching Shear Resistance of Slabs with Punching Shear Reinforcement
9.13 Arrangement of the Punching Shear Reinforcement
9.13.1 Studs
9.13.2 Bent-Up Bars
9.14 Maximum Area of Transverse Reinforcement
9.14.1 Case A: Studs on Two Perimeters
9.14.2 Case B: Bent-Up Bars on One Perimeter
9.15 Foundations
9.15.1 Examples
Appendix 1: Tables for Rapid Calculation of vRd,c (N/mm2) with Varying Diameter and Spacing of Flexural Reinforcement
Concrete Class: C25/30
Concrete Class: C28/35
Concrete Class: C32/40
Concrete Class: C35/45
Concrete Class: C40/50
Concrete Class: C45/55
Appendix 2: Tables with the Maximum Area of Studs Within Each Perimeter for Slabs on Rectangular Columns Equipped with Two Reinforcement Perimeters
Concrete Class: C28/35
Concrete Class: C32/40
Concrete Class: C35/45
Concrete Class: C40/50
Concrete Class: C45/55
References
10 Strut-And-Tie Models
10.1 Introduction
10.1.1 Strut and Tie Method as an Application of the Lower Bound (Static) Theorem of Limit Analysis
10.2 Identification of the Geometry of the S&T Model
10.2.1 Position and Extension of “D” Regions
10.2.2 Evaluation of the Stress Field and Design of Reinforcement in “B” Regions
10.2.3 Forces at the Boundary of “D” Regions
10.3 Choice of the S&T Model
10.4 Kinematically Unstable S&T Models
10.5 Practical Rules for the Identification of the S&T Model
10.6 Common S&T Models
10.6.1 Spread of a Concentrated Load Within a Strut (D1)
10.6.2 Spread of a Concentrated Eccentric Load (D2)
10.6.3 Single-Span Deep Beam Uniformly Loaded on the Top Edge (D3)
10.7 Verification of Members and Nodes of the S&T Model
10.8 Reinforcement Design
10.9 Verification of Struts
10.9.1 Transverse Reinforcing Bars
10.10 Verification of Nodes
10.10.1 Types of Nodes
10.10.2 Strength of Nodes
10.10.3 Compression Nodes (CCC)
10.10.4 Compression-Tension Nodes with Anchored Ties Provided in One Direction (CCT)
10.10.5 Compression-Tension Nodes with Ties Arranged in Two Directions (CTT)
10.10.6 Conditions for Increasing the Strength of Nodes and Confined Nodes
10.11 Frame Corners
10.11.1 Frame Corner with Closing Moments
10.11.2 Frame Corners with Opening Moments
10.12 Examples
10.12.1 Example No. 1—Simply Supported Deep Beam Under a Uniformly Distributed Load of 280 kN/m
10.12.2 Example No. 2—Simply Supported Deep Beam Under a Uniformly Distributed Load of 420 kN/m
10.12.3 Example No. 3—Rigid Spread Footing
10.12.4 Example No. 4—Isolated Footing on Four Piles
10.12.5 Example No. 5—Gerber Hinge
10.12.6 Example No. 6—Abrupt Change of the Height of a Slender Beam
10.12.7 Example No. 7—Corbel
10.12.8 Example No. 8—Design of the Corbel Secondary Reinforcement
10.13 Corbel Subjected to a Concentrated Load at the Bottom
References
11 Serviceability Limit States (SLS)
11.1 General
11.2 Stress Limitation
11.2.1 Bending–Solving Formulas
11.2.2 Axial Force Combined with Bending–Rectangular Cross-Section–Solving Formulas
11.2.3 Service Interaction Diagrams v - µ for Rectangular Cross-Sections with Double Symmetrical Reinforcement
11.2.4 Cases When SLS Stress Verifications Are Implicitly Satisfied by ULS Verifications
11.2.5 Application Examples
11.3 Crack Control
11.3.1 General Considerations
11.3.2 Calculation of Crack Widths
11.3.3 Minimum Reinforcement Areas
11.3.4 Surface Reinforcements in High Beams
11.4 Deflection Control
11.4.1 Application Examples
11.4.2 Deflection Calculation Due to Shrinkage
11.5 Further Verifications at SLS
References
12 Detailing of Reinforcement and Structural Members for Buildings
12.1 Detailing of Reinforcement
12.1.1 Spacing of Bars
12.1.2 Permissible Mandrel Diameters for Bent Bars
12.1.3 Anchorage of Longitudinal Reinforcement
12.1.4 Ultimate Bond Stress
12.1.5 Anchorage Length
12.1.6 Anchorage of Links and Shear Reinforcement
12.1.7 Anchorage by Welded Transverse Bars
12.1.8 Laps and Mechanical Couplers
12.1.9 Bundles of Bars
12.1.10 Rules for Prestressing Reinforcement
12.2 Detailing of Beams, Columns, Slabs and Walls
12.2.1 Beams
12.2.2 Solid Slabs
12.2.3 Flat Slabs
12.2.4 Columns
12.2.5 Walls
Appendix Tables and Diagrams
Ultimate Limit State
Tables µ-ω-ξ for Rectangular Cross-Sections 2d/d = 0.10
T-Section Tables
U.10–U.22—Interaction Diagrams ν- for Rectangular and Circular Cross-Sections
U.23–U.27—Interaction Diagrams ν- of Rectangular Cross-Section with Reinforcement in the Four Corners Under Axial Force and Biaxial Bending Moments
Serviceability Limit State
Tables for Rectangular Cross-Sections: ρ-ξ-i, Dʹ/d = 0.10, αe = 15 and αe = 9 (αe Notional Modular Ratio)
Tables for T Sections: ρ-ξ-i
Interaction Diagrams ν-µ of Rectangular Cross-Sections with Double Symmetrical Reinforcement