Table of Contents

Cover

Series

Title

FOREWORD

PREFACE 2^nd EDITION

PREFACE 1^st EDITION

Chapter 1: INTRODUCTION

1.1. GENERAL OBSERVATIONS
1.2. CODES OF PRACTICE AND NORMALIZATION
1.3. BASIS OF DESIGN
1.4. MATERIALS
1.5. GEOMETRIC CHARACTERISTICS AND TOLERANCES

Chapter 2: STRUCTURAL ANALYSIS

2.1. INTRODUCTION
2.2. STRUCTURAL MODELLING
2.3. GLOBAL ANALYSIS OF STEEL STRUCTURES
2.4. CLASSIFICATION OF CROSS SECTIONS

Chapter 3: DESIGN OF MEMBERS

3.1. INTRODUCTION
3.2. TENSION
3.3. LATERALLY RESTRAINED BEAMS
3.4. TORSION
3.5. COMPRESSION
3.6. LATERALLY UNRESTRAINED BEAMS
3.7. BEAM-COLUMNS

Chapter 4: ELASTIC DESIGN OF STEEL STRUCTURES

4.1. INTRODUCTION
4.2. SIMPLIFIED METHODS OF ANALYSIS
4.3. MEMBER STABILITY OF NON-PRISMATIC MEMBERS AND COMPONENTS
4.4. DESIGN EXAMPLE 1: ELASTIC DESIGN OF BRACED STEEL FRAMED BUILDING

Chapter 5: PLASTIC DESIGN OF STEEL STRUCTURES

5.1. GENERAL PRINCIPLES OF PLASTIC DESIGN
5.2. METHODS OF ANALYSIS
5.3. MEMBER STABILITY AND BUCKLING RESISTANCE
5.4. DESIGN EXAMPLE 2: PLASTIC DESIGN OF INDUSTRIAL BUILDING

REFERENCES

Annex A: FORMULAS FOR COMMON TORSIONAL CASES

A.1. CROSS SECTIONAL PROPERTIES FOR TORSION
A.2. SOLUTION OF DIFFERENTIAL EQUATION FOR TORSION

Annex B: ELASTIC CRITICAL MOMENT

B.1. ABACUS TO CALCULATE THE COEFFICIENTS C_1, C₂ AND C₃
B.2. ALTERNATIVE EQUATIONS FOR THE DETERMINATION OF THE ELASTIC CRITICAL MOMENT

End User License Agreement

List of Tables

Chapter 1: INTRODUCTION

Table 1.1 – Definition of consequence classes
Table 1.2 – Design supervision levels
Table 1.3 – Inspection levels
Table 1.4 – Determination of execution classes in steel structures
Table 1.5 – Limiting values for the vertical displacements in beams (span L)
Table 1.6 – Hot-rolled steel grades and qualities according to EN 10025-2
Table 1.7 – Choice of quality class according EN 10164
Table 1.8 – Dimensional tolerances for structural steel I and H sections (EN 10034)
Table 1.9 – Tolerances on out-of-square and web off-centre of structural steel I and H sections (EN 10034)
Table 1.10 – Tolerances on straightness of structural steel I and H sections (EN 10034)
Table 1.11 – Essential manufacturing tolerances – welded sections (EN 1090-2)

Chapter 2: STRUCTURAL ANALYSIS

Table 2.1 – Comparison of results for c/h₀ = 1
Table 2.2 – Comparison of results for c/h₀ = 2
Table 2.3 – Moments in curved beam
Table 2.4 – Stiffness modification coefficient ?
Table 2.5 – Transformation parameter β
Table 2.6 – Results at the critical cross sections
Table 2.7 – Results at the critical cross sections with eccentricities
Table 2.8 – Results for full-strength joints, evaluated using the secant stiffness
Table 2.9 – Results for partial-strength joints and secant stiffness
Table 2.10 – Results for partial-strength joints and real behaviour
Table 2.11 – Comparative summary of results
Table 2.12 – Iterative process
Table 2.13 – Synthesis of results
Table 2.14 – Bending moments and axial forces at the critical cross sections (beam elements)
Table 2.15 – Displacements δ (mm), at the critical cross sections
Table 2.16 – Moments at the critical cross sections (slab elements)
Table 2.17 – Bending moments in the critical sections (beam and shell elements)
Table 2.18 – Critical buckling loads and buckling modes
Table 2.19 – K_ij stiffness coefficients in beams
Table 2.20 – Initial local bow imperfections
Table 2.21 – Maximum width-to-thickness ratios for internal compression parts
Table 2.22 – Maximum width-to-thickness ratios of outstand flanges
Table 2.23 – Maximum width-to-thickness ratios of angles and tubular sections

Chapter 3: DESIGN OF MEMBERS

Table 3.1 – Reduction factors β₂ and β₃
Table 3.2 – Shear stresses and torsion constant for typical steel cross section shapes
Table 3.3 – Selection of the buckling curve
Table 3.4 – Coefficients C₁ and C₃ for beams with end moments
Table 3.5 – Coefficients C₁, C₂ and C₃ for beams with transverse loads
Table 3.6 – Buckling curves for lateral-torsional buckling (General method)
Table 3.7 – Buckling curves for lateral-torsional buckling (Alternative method)
Table 3.8 – k_c correction factors
Table 3.9 – Summary table
Table 3.10 – Values for the calculation of N_Rk, M_i,Rk and ΔM_i,Ed
Table 3.11 – Interaction factors k_ij according to Method 1
Table 3.12 – Auxiliary terms for the calculation of the interaction factors k_ij of the previous table
Table 3.13 – Equivalent uniform moment factors C_mi_,0
Table 3.14 – Interaction factors k_ij in members not susceptible to torsional deformations according to Method 2
Table 3.15 – Interaction factors k_ij in members susceptible to torsional deformations according to Method 2
Table 3.16 – Equivalent uniform moment factors C_mi

Chapter 4: ELASTIC DESIGN OF STEEL STRUCTURES

Table 4.1 – Bending moments and axial forces at the critical cross sections (ASM)
Table 4.2 – Bending moments and axial forces at the critical cross sections
Table 4.3 – Comparative synthesis of results
Table 4.4 – Elastic critical loads
Table 4.5 – Buckling resistance
Table 4.6 – Equivalent uniform moment factor m_t
Table 4.7 – Equivalent section factor, c₀
Table 4.8 – Comparison of results
Table 4.9 – Geometric characteristics of the beams (1^st floor)
Table 4.10 – Geometric characteristics of the beams (2^nd floor)
Table 4.11 – Geometric characteristics of the beams (3^rd to 8^th floors)
Table 4.12 – Geometric characteristics of the columns
Table 4.13 – External pressure coefficients c_pe
Table 4.14 – External and internal pressures
Table 4.15 – Summary of actions
Table 4.16 – Initial imperfections
Table 4.17 – Equivalent horizontal forces in transversal frames
Table 4.18 – Equivalent horizontal forces in longitudinal frames
Table 4.19 – Reduction factors ψ
Table 4.20 – Elastic critical load factors
Table 4.21 – Comparative results for column E1 (combination 1)
Table 4.22 – Comparative results for beam E1 to E4 (combination 1)
Table 4.23 – Utilization levels for column E1
Table 4.24 – Utilization levels for beam E1-E4
Table 4.25 – Beam E1-E4 LTB ratios
Table 4.26 – Column E1 buckling ratios

Chapter 5: PLASTIC DESIGN OF STEEL STRUCTURES

Table 5.1 – Normative references
Table 5.2 – “Exact” and approximate elastic critical loads
Table 5.3 – Critical loads
Table 5.4 – 2^nd order effects
Table 5.5 – 1^st order elastic-plastic analysis
Table 5.6 – 2^nd order elastic-plastic analysis
Table 5.7 – Approximate 2^nd order elastic-plastic analysis
Table 5.8 – Comparative summary of results
Table 5.9 – Comparative results for 2^nd order elastic-plastic analyses
Table 5.10 – Classification of the cross sections
Table 5.11 – Cross section resistance
Table 5.12 – Applied bending moments and bending resistance
Table 5.13 – Verification of the lateral-torsional buckling resistance
Table 5.14 – External pressure coefficients on the walls (c_pe)
Table 5.15 – External pressure coefficients on the roof (c_pe)
Table 5.16 – Summary of actions
Table 5.17 – Values of imperfections by loading
Table 5.18 – Plastic resistance of the cross sections
Table 5.19 – Results from the 1^st order elastic analyses
Table 5.20 – Critical loads
Table 5.21 – Results of the 2^nd order elastic analysis
Table 5.22 – Results of the 1^st order elastic-plastic analysis (a = 1.0)
Table 5.23 – Results of the 2^nd order elastic-plastic analysis (α= 1.0)
Table 5.24 – Formation of plastic hinges
Table 5.25 – Classification of cross sections for combination 3
Table 5.26 – Lateral-torsional buckling verification

Annex A: FORMULAS FOR COMMON TORSIONAL CASES

Table A.1 – Shear centre, torsion constant and warping constant for typical cross sections
Table A.2 – Warping function and static function with respect to the warping function for typical cross sections
Table A.3 – Formulas for thin-walled elastic cantilevers under a concentrated torsional moment T₀
Table A.4 – Formulas for thin-walled elastic double-supported beams under a concentrated torsional moment
Table A.5 – Formulas for thin-walled elastic cantilevers under uniformly distributed torsional moment
Table A.6 – Formulas for thin-walled elastic double-supported beams under uniformly distributed torsional moment

Annex B: ELASTIC CRITICAL MOMENT

Table B.1 – Values of C₁ and C₂ for cantilevers with warping prevented at the fixed end
Table B.2 – Values of C₃ for cantilevers with warping prevented at the fixed end
Table B.3 – Values of C₁ and C₂ for cantilevers with warping free at the fixed end
Table B.4 – Values of C₃ for cantilevers with warping free at the fixed end
Table B.5 – Factors for the calculation of the critical moment in spans of beams with length L and doubly symmetric section

List of Illustrations

Chapter 1: INTRODUCTION

Figure 1.1 – European normative structure for the construction sector
Figure 1.2 – Vertical deformations in beams
Figure 1.3 – Limiting values for horizontal displacements in frames
Figure 1.4 – Anti-corrosion details
Figure 1.5 – Location and orientation of samples and pieces for tests
Figure 1.6 – Geometry of proportional flat test samples
Figure 1.7 – Schematic stress-strain curve
Figure 1.8 – Relationship between impact energy and temperature
Figure 1.9 – Lamellar tearing
Figure 1.10 – Rolled sections
Figure 1.11 – Cold formed sections
Figure 1.12 – Conventions for dimensions and axes of steel cross sections

Chapter 2: STRUCTURAL ANALYSIS

Figure 2.1 – Nonlinear analysis
Figure 2.2 – Industrial building
Figure 2.3 – Multi-storey building
Figure 2.4 – Structural model with two-dimensional elements
Figure 2.5 – Influence of the choice of member axis
Figure 2.6 – Influence of load eccentricity in relation to the shear centre
Figure 2.7 – Influence of eccentricities
Figure 2.8 – Influence of discontinuities at intermediate sections
Figure 2.9 – Examples of elements with variable cross section
Figure 2.10 – Alternative discretizations for a beam with parabolic variation of the depth of the cross section
Figure 2.11 – Bending moment diagram and vertical displacements of the beam
Figure 2.12 – Influence of the elements’ curvature
Figure 2.13 – Typical 3D behaviour of bolted end-plate beam-to-column steel joint
Figure 2.14 – Bending-rotation moment curve of a joint
Figure 2.15 – Modelling of joints in structural analysis
Figure 2.16 – Definition of the secant stiffness of a joint
Figure 2.17 – Modelling of joints in the internal node of a frame
Figure 2.18 – Internal node steel beam-to-column joint
Figure 2.19 – Alternative models for the representation of the column web panel in shear
Figure 2.20 – Beam and shell structural model
Figure 2.21 – Modelling of in-plane rigid slab behaviour
Figure 2.22 – Steel frame
Figure 2.23 – Calculation model without eccentricities
Figure 2.24 – Bending moment diagram
Figure 2.25 – Transverse shear diagram
Figure 2.26 – Axial force diagram
Figure 2.27 – Definition of the frame’s eccentricities
Figure 2.28 – Calculation model with eccentricities
Figure 2.29 – Full-strength joint at intermediate beam level
Figure 2.30 – Full-strength joint at top beam level
Figure 2.31 – Steel frame with semi-rigid joints
Figure 2.32 – Partial-strength joint at the intermediate beam level
Figure 2.33 – Partial-strength joint at top beam level
Figure 2.34 – M-ϕ curve of the joints of the intermediate and top beams
Figure 2.35 – Steel frame
Figure 2.36 – Detail of the internal node joint
Figure 2.37 – Modelling of the internal node joint
Figure 2.38 – Bending moment diagram
Figure 2.39 – Two-storey steel framed building
Figure 2.40 – Loads on secondary beams
Figure 2.41 – Critical cross sections in the middle frame
Figure 2.42 – Structural steel model (neglecting the slab)
Figure 2.43 – Structural steel model with horizontal bracing
Figure 2.44 – Bending moment diagram
Figure 2.45 – Shear force diagram
Figure 2.46 – Axial force diagram
Figure 2.47 – Typical displacements Δ and δ
Figure 2.48 – Recommended discretization for the evaluation of critical loads
Figure 2.49 – Definition of the axial loading for the determination of critical load
Figure 2.50 – “Sway mode” (a)) and “Non-sway mode” (b))
Figure 2.51 – Lateral displacements in an unbraced frame
Figure 2.52 – Wood’s equivalent frame
Figure 2.53 – “Equivalent geometric imperfection” in framework structures
Figure 2.54 – Imperfections and corresponding equivalent horizontal forces
Figure 2.55 – Plan view of translational and torsional effects
Figure 2.56 – Imperfections for bracing systems
Figure 2.57 – Bracing forces at splices in compression elements
Figure 2.58 – Steel frame
Figure 2.59 – Loadcase arrangement for combination 1
Figure 2.60 – Loadcase arrangement for combination 2
Figure 2.61 – Bending moment due to vertical loads (comb. 1)
Figure 2.62 – Shear force due to vertical loads (comb. 1)
Figure 2.63 – Axial force due to vertical loads (comb. 1)
Figure 2.64 – Bending moment due to horizontal loads (comb. 1)
Figure 2.65 – Shear force due to horizontal loads (comb. 1)
Figure 2.66 – Axial force due to horizontal loads (comb. 1)
Figure 2.67 – Structural displacements for combination 1 (horizontal loads)
Figure 2.68 – Elastic 2^nd order bending moment diagram
Figure 2.69 – Elastic 2^nd order shear force diagram
Figure 2.70 – Elastic 2^nd order axial force diagram
Figure 2.71 – Bending moments due to vertical loads (comb. 2)
Figure 2.72 – Shear forces due to vertical loads (comb. 2)
Figure 2.73 – Axial forces due to vertical loads (comb. 2)
Figure 2.74 – Bending moments due to horizontal loads (comb. 2)
Figure 2.75 – Shear forces due to horizontal loads (comb. 2)
Figure 2.76 – Axial forces due to horizontal loads (comb. 2)
Figure 2.77 – Structural displacements due to combination 2 (horizontal loads)
Figure 2.78 – Design bending moments
Figure 2.79 – Design shear forces
Figure 2.80 – Design axial forces
Figure 2.81 – Structural displacements for combination 1 (SLS)
Figure 2.82 – Structural displacements for combination 2 (SLS)
Figure 2.83 – Cross section behaviour in bending
Figure 2.84 – Classification of cross sections
Figure 2.85 – Classification of a welded cross section

Chapter 3: DESIGN OF MEMBERS

Figure 3.1 – Definition of the net area of a cross section
Figure 3.2 – Angle with holes in both legs
Figure 3.3 – Effective class 2 web
Figure 3.4 – Class 4 cross section submitted to a compressive axial force
Figure 3.5 – Class 4 cross section submitted to bending moment
Figure 3.6 – Structures with some members in tension
Figure 3.7 – Typical cross sections of members in tension
Figure 3.8 – Concentration of tension next to a hole
Figure 3.9 – Eccentric connections
Figure 3.10 – Welded connections between hollow sections
Figure 3.11 – Angles connected by one leg
Figure 3.12 – Net area of a plate
Figure 3.13 – Steel truss
Figure 3.14 – A_net in the bolted connection
Figure 3.15 – Lattice girder
Figure 3.16 – Actions and internal forces on the structure
Figure 3.17 – Connection of the diagonal bars of the truss
Figure 3.18 – Elastic distributions of normal stresses and shear stresses
Figure 3.19 – Elastic and plastic bending moment cross sectional resistance
Figure 3.20 – Shear area for a I cross section
Figure 3.21 – Model for bending moment - shear force interaction in a I or H section
Figure 3.22 – Bending moment – shear force interaction diagrams for I or H sections
Figure 3.23 – Simply supported steel beam
Figure 3.24 – Diagrams of internal forces
Figure 3.25 – Continuous beam
Figure 3.26 – Load combination 1
Figure 3.27 – Load combination 2
Figure 3.28 – Load combination 3
Figure 3.29 – Diagrams of elastic internal forces for load combination 1
Figure 3.30 – Redistribution of bending moments
Figure 3.31 – Design internal forces after redistribution
Figure 3.32 – Deformation of the beam for the characteristic combination of actions
Figure 3.33 – Cantilever beam
Figure 3.34 – Member under uniform torsion
Figure 3.35 – Member with I section under non-uniform torsion
Figure 3.36 – Shear stresses due to uniform torsion for typical steel cross section shapes
Figure 3.37 – Deformation in an I section under non-uniform torsion
Figure 3.38 – Stresses due to warping torsion T_w
Figure 3.39 – Thin-walled closed rectangular hollow section
Figure 3.40 – Plastic stress distribution in thin-walled open cross section (Gjelsvik, 1981)
Figure 3.41 – Plastic normal warping stress distribution in I-section
Figure 3.42 – Thin-walled steel cross sections
Figure 3.43 – Cantilever beam
Figure 3.44 – Distributions of stresses at section A
Figure 3.45 – Cantilever overhang structure
Figure 3.46 – Design internal forces
Figure 3.47 – Bimoment, warping torsion and St. Venant’s torsion diagrams
Figure 3.48 – Buckling in a pinned member (Euler’s column)
Figure 3.49 – Buckling length L_E as a function of the real length L of the column
Figure 3.50 – σ - λ relationship of a compressed member
Figure 3.51 – Torsional buckling and flexural-torsional buckling
Figure 3.52 – Column with initial geometrical imperfection
Figure 3.53 – Typical residual stresses in a rolled member with a I cross section
Figure 3.54 – Results of experimental tests in real members
Figure 3.55 – Behaviour of metals
Figure 3.56 – Buckling curves according to EC3-1-1
Figure 3.57 – Closely spaced built-up members
Figure 3.58 – Design of a column
Figure 3.59 – Actions and internal forces on the lattice girder of example 3.3
Figure 3.60 – Detail in the diagonals of the truss
Figure 3.61 – Lateral-torsional buckling of a cantilever beam
Figure 3.62 – Lateral-torsional buckling in a doubly symmetric I section under constant bending moment
Figure 3.63 – Effect of the point of load’s application
Figure 3.64 – Sections mono-symmetric about the weak axis
Figure 3.65 – Beams with end moments and transverse loads
Figure 3.66 – Cantilever beam at the end of a continuous beam
Figure 3.67 – Simply supported beam with two concentrated loads
Figure 3.68 – Diagrams of internal forces
Figure 3.69 – Laterally braced beam
Figure 3.70 – Continuous beam with a cantilever span
Figure 3.71 – Diagrams of internal forces
Figure 3.72 – Frame of a catwalk (vertical cut)
Figure 3.73 – Design internal forces for beam A-B with 10 m span
Figure 3.74 – Steel members subjected to bending and axial force
Figure 3.75 – Normal elastic stresses in a section with bi-axial bending combined with axial force
Figure 3.76 – Complete plastification of an I or H section with bi-axial bending combined with axial force
Figure 3.77 – Classification of the sections based on plastic stress diagrams
Figure 3.78 – Concept of utilization factor U_F
Figure 3.79 – Bending moment-axial force plastic interaction
Figure 3.80 – M + N interaction diagram in a rectangular solid section
Figure 3.81 – Behaviour of a member subjected to bending and compression
Figure 3.82 – Hot-finished rectangular hollow section structure
Figure 3.83 – Internal force diagrams
Figure 3.84 – Column subjected to major-axis bending and compression
Figure 3.85 – Member under bi-axial bending and compression
Figure 3.86 – Diagrams of internal forces
Figure 3.87 – Variation of utilization factor along the member

Chapter 4: ELASTIC DESIGN OF STEEL STRUCTURES

Figure 4.1 – Braced and unbraced frames
Figure 4.2 – Amplified sway moment method
Figure 4.3 – Steel frame
Figure 4.4 – Critical cross sections
Figure 4.5 – Modified no-sway frame and load arrangement for load combination 1
Figure 4.6 – Original sway frame subjected to the horizontal reactions (load comb. 1)
Figure 4.7 – Design bending moment diagram (ASM)
Figure 4.8 – Design shear force diagram (ASM)
Figure 4.9 – Design axial force diagram (ASM)
Figure 4.10 – Load arrangement for load combination 1
Figure 4.11 – Original frame subjected to the horizontal reactions (load comb. 1)
Figure 4.12 – Design bending moment diagram for load combination 1 (SMBL)
Figure 4.13 – Design shear force diagrams for load combination 1 (SMBL)
Figure 4.14 – Design axial force diagrams for load combination 1 (SMBL)
Figure 4.15 – Lowest sway buckling mode
Figure 4.16 – Double-symmetric tapered I section beam
Figure 4.17 – Variation of the elastic critical loads with reference cross section
Figure 4.18 – Utilization ratio α (%) for the relevant modes
Figure 4.19 – Non-prismatic I-section columns (b, t_f and t_w are all constant)
Figure 4.20 – Partial and total bracing
Figure 4.21 – Reference and bracing axes
Figure 4.22 – Value of β_t
Figure 4.23 – Definition of the cross sections for non linear moment variation
Figure 4.24 – Flowchart for the application of the General Method
Figure 4.25 – Double-symmetric tapered I section beam
Figure 4.26 – Design internal force diagrams
Figure 4.27 – Cross section classification
Figure 4.28 – Cross section plastic interaction diagram
Figure 4.29 – Variation of the utilization factor along the length of the member
Figure 4.30 – Elastic critical load multiplier
Figure 4.31 – Side elevation view
Figure 4.32 – Plan of the 3^rd to 7^th floor
Figure 4.33 – Plan of the 1^st floor
Figure 4.34 – Plan of the 2^nd floor
Figure 4.35 – Plan of the 8^th floor
Figure 4.36 – Velocity pressure distribution on face D (θ = 0°)
Figure 4.37 – Velocity pressure distribution on face D (θ = 90°)
Figure 4.38 – Pressure zones for wind direction θ = 0°
Figure 4.39 – Pressure zones for wind direction θ = 90°
Figure 4.40 – External and internal coefficients
Figure 4.41 – Wind pressures (kN/m²) on walls, θ = 0°
Figure 4.42 – Wind pressures (kN/m²) on walls, θ = 90°
Figure 4.43 – Load arrangement for the analysis of the shadow areas
Figure 4.44 – Load distribution in the secondary beams
Figure 4.45 – 3D structural model
Figure 4.46 – 1^st buckling mode for combination 1 – frontal view
Figure 4.47 – Internal forces on column E1 for load combination 1 (local axes)
Figure 4.48 – Internal forces on beam E1-E4 in the 4^th floor for load combination 1

Chapter 5: PLASTIC DESIGN OF STEEL STRUCTURES

Figure 5.1 – Elementary mechanisms
Figure 5.2 – Pitched-roof portal frame
Figure 5.3 – Mechanism 1
Figure 5.4 – Mechanism 2
Figure 5.5 – Mechanism 3
Figure 5.6 – Bending moment diagram at collapse
Figure 5.7 – Bilinear stress-strain relation
Figure 5.8 – Structural model
Figure 5.9 – Equivalent statically determinate structure
Figure 5.10 – Internal force diagrams: a) real loading; b) statically redundant internal forces
Figure 5.11 – Potential mechanisms
Figure 5.12 – Structural model
Figure 5.13 – Potential locations of plastic hinges
Figure 5.14 – Load combination where the leading variable action is the imposed loading
Figure 5.15 – Bending moment diagrams: a) real loading; b) statically indeterminate unknown internal forces
Figure 5.16 – Bending moment diagram at collapse
Figure 5.17 – Load combination where the leading variable action is wind
Figure 5.18 – Bending moment diagrams: a) real loading; b) statically indeterminate unknown internal forces
Figure 5.19 – Incremental non linear procedure
Figure 5.20 – Plastic analysis of a hinged fixed-ended beam
Figure 5.21 – Alternative implementations of yield criteria
Figure 5.22 – Behaviour of plastic hinge in the unloading
Figure 5.23 – Continuous beam with transitory plastic hinge
Figure 5.24 – Load factor-vertical displacement diagram
Figure 5.25 – Unstable behaviour
Figure 5.26 – Symmetric and anti-symmetric buckling modes in pitched-roof portal frames
Figure 5.27 – Amplification in the symmetric and anti-symmetric modes
Figure 5.28 – Pitched-roof portal frame
Figure 5.29 – Bending moment diagram, linear elastic analysis
Figure 5.30 – Shear force diagram, linear elastic analysis
Figure 5.31 – Axial force diagram, linear elastic analysis
Figure 5.32 – Displacements in the structure, linear elastic analysis
Figure 5.33a – 1^st (ASM) and 2^nd (SM) buckling modes
Figure 5.33b – 3^rd and 4^th buckling modes
Figure 5.34 – Bending moment diagram; 2^nd order “exact” elastic analysis
Figure 5.35 – Amplified bending moment diagram (equations (2.17))
Figure 5.36 – 1^st order bending moment diagram,
Figure 5.37 –1^st order bending moment diagram,
Figure 5.38 – Bending moment diagram that corresponds to the formation of the first plastic hinge (α = 1.00), 1^st order elastic-plastic analysis
Figure 5.39 – Collapse mechanism and sequence of formation of the plastic hinges
Figure 5.40 –Bending moment diagram that corresponds to the formation of the first plastic hinge (α = 0.965), 2^nd order elastic-plastic analysis
Figure 5.41 – Collapse mechanism with the indication of the load factors that correspond to formation of the plastic hinges
Figure 5.42 – Collapse mechanism and load factors corresponding to the formation of the plastic hinges
Figure 5.43 – Load factor-horizontal displacement diagram at the apex
Figure 5.44 – Equivalent local bow imperfections
Figure 5.45 – Generic member containing one plastic hinge and bracings
Figure 5.46 – Typical bracing
Figure 5.47 – Typical lateral and torsional bracing of the compression flange by a slab
Figure 5.48 – Definition of stable length with partial bracings
Figure 5.49 – Three flange haunch
Figure 5.50 – Two flange haunch
Figure 5.51 – Value of β_t
Figure 5.52 – Definition of the cross sections for a non linear moment variation
Figure 5.53 – Dimensions for the definition of the taper factor
Figure 5.54 – Rafter
Figure 5.55 – Effective class 2 cross section
Figure 5.56 – Bending moment diagram in segment BB₁
Figure 5.57 – Detail of the column
Figure 5.58 – Portal frame
Figure 5.59 – Pressure zones on the walls
Figure 5.60 – Pressure zones on the roof
Figure 5.61 – External pressure coefficients on the walls and roofs: (a) wind in transverse direction (θ = 0°); (b) wind in longitudinal direction (θ = 90°).
Figure 5.62 – Final pressure coefficients: (a) wind in transverse direction (θ = 0°); (b) wind in longitudinal direction (θ = 90°)
Figure 5.63 – Wind force: (a) in transverse direction (θ = 0°); (b) in longitudinal direction (θ = 90°)
Figure 5.64 – Load combination 1
Figure 5.65 – Load combination 2
Figure 5.66 – Load combination 3
Figure 5.67 – Diagram of collapse moments
Figure 5.68 – Detailing of column AB, rafter BC and position of additional effective restraints
Figure 5.69 – Bending moment diagram (LEA) for C1
Figure 5.70 – Bending moment diagram (LEA) for C2
Figure 5.71 – Bending moment diagram (LEA) for C3
Figure 5.72 – Bending moment diagram (LEA) for C4
Figure 5.73 – Displacements (LEA) for C1
Figure 5.74 – Displacements (LEA) for C2
Figure 5.75 – Displacements (LEA) for C3
Figure 5.76 – Displacements (LEA) for C4
Figure 5.77 – Bending diagram in the rafter for Combination 3
Figure 5.78 – Final detailing of rafter BC

Annex A: FORMULAS FOR COMMON TORSIONAL CASES

Figure A.1 – Generic open cross section beam subject to concentrated torsional moment
Figure A.2 – Generic open cross section beam subject to a uniformly distributed torsional moment

Annex B: ELASTIC CRITICAL MOMENT

Figure B1 – C₁ coefficient for end moment and uniform load, with µ > 0
Figure B2 – C₁ coefficient for end moment and uniform load, with µ < 0
Figure B3 – C₂ coefficient for end moment and uniform load, with µ > 0
Figure B4 – C₂ coefficient for end moment and uniform load, with µ < 0
Figure B5 – C₁ coefficient for end moment and point load at mid span, with µ > 0
Figure B6 – C₁ coefficient for end moment and point load at mid span, with µ < 0
Figure B7 – C₂ coefficient for end moment and point load at mid span, with µ > 0
Figure B8 – C₂ coefficient for end moment and point load at mid span, with µ < 0

Design of Steel Structures

2^nd Edition, 2016

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ECCS – European Convention for Constructional Steelwork

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ISBN (ECCS): 978-92-9147-134-8

ISBN (Ernst & Sohn): 978-3-433-0316-36

ECCS EUROCODE DESIGN MANUALS

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Luís Simões da Silva (ECCS)

António Lamas (Portugal)

Jean-Pierre Jaspart (Belgium)

Reidar Bjorhovde (USA)

Ulrike Kuhlmann (Germany)

DESIGN OF STEEL STRUCTURES – 2^ND EDITION

Luís Simões da Silva, Rui Simões and Helena Gervásio

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Jean-Marc Franssen and Paulo Vila Real

DESIGN OF PLATED STRUCTURES

Darko Beg, Ulrike Kuhlmann, Laurence Davaine and Benjamin Braun

FATIGUE DESIGN OD STEEL AND COMPOSITE STRUCTURES

Alain Nussbaumer, Luís Borges and Laurence Davaine

DESIGN OF COLD-FORMED STEEL STRUCTURES

Dan Dubina, Viorel Ungureanu and Raffaele Landolfo

DESIGN OF JOINTS IN STEEL AND COMPOSITE STRUCTURES

Jean-Pierre Jaspart and Klaus Weynand

DESIGN OF STEEL STRUCTURES FOR BUILDINGS IN SEISMIC AREAS

Raffaele Landolfo, Federico Mazzolani, Dan Dubina, Luís Simões da Silva and Mario d'Aniello

FOREWORD

The development program for the design manuals of the European Convention for Constructional Steelwork (ECCS) represents a major effort for the steel construction industry and the engineering profession in Europe. Conceived by the ECCS Technical Activities Board under the leadership of its chairman, Professor Luis Simões da Silva, the manuals are being prepared in close agreement with the final stages of Eurocode 3 and its national Annexes. The scope of the development effort is vast, and reflects a unique undertaking in the world.

The publication of the first of the manuals, Design of Steel Structures, is a signal achievement which heralds the successful completion of the Eurocode 3 work and brings it directly to the designers who will implement the actual use of the code. As such, the book is more than a manual – it is a major textbook that details the fundamental concepts of the code and their practical application. It is a unique publication for a major construction market.

Following a discussion of the Eurocode 3 basis of design, including the principles of reliability management and the limit state approach, the steel material standards and their use under Eurocode 3 are detailed. Structural analysis and modeling are presented in a chapter that will assist the design engineer in the first stages of a design project. This is followed by a major chapter that provides the design criteria and approaches for the various types of structural members. The theories of behavior and strength are closely tied to the Eurocode requirements, making for a unique presentation of theory into practice. The following chapters expand on the principles and applications of elastic and plastic design of steel structures.

The many design examples that are presented throughout the book represent a significant part of the manual. These will be especially well received by the design profession. Without a doubt, the examples will facilitate the acceptance of the code and provide for a smooth transition from earlier national codes to the Eurocode.

Reidar Bjorhovde
Member, ECCS Editorial Board

PREFACE 2^nd EDITION

The first edition of Design of Steel Structures was published by ECCS as a paperback in 2010. Since 2012, this publication is also available in electronic format as an e-book. The first edition was sold in over 100 countries and the interest for this publication was so high that a second edition would have to be printed.

The authors took the opportunity of this second edition to revise their manuscript. The standard that constitutes the object of this book, namely EN 1993-1-1, is still in application in the same versions as those that prevailed at the time of writing the first edition except for a minor amendment published in 2013. However, many comments were received by readers that resulted in the correction of some small mistakes and the rephrasing of some sentences or sections and the addition of some new material.

The new material comprises:

– A revised section dealing with the design for torsion of steel members, including a new worked example illustrating an open cross section beam subject to bending and torsional moments;
– A revised section on the elastic critical moment of beams;
– An improved explanation on the classification of cross sections subject to bending and axial force;
– An additional worked example of a beam-column with transversal loads and end moments;
– A new Annex containing formulas for common torsional cases;
– A revised and expanded Annex with formulas for elastic critical moment calculation.

The authors are indebted to Profs. E. Mirambell and K. Rasmussen for their thorough revision of the section on torsion.

Luís Simões da Silva
Rui Simões
Helena Gervásio
Coimbra, 2016

PREFACE 1^st EDITION

The General rules and rules for buildings of part 1-1 of Eurocode 3 constitute the core of the code procedures for the design of steel structures. They contain the basic guidance for structural modeling and analysis of steel frameworks and the rules for the evaluation of the resistance of structural members and components subject to different loading conditions.

According to the objectives of the ECCS Eurocode Design Manuals, it is the objective of this book to provide mix of “light” theoretical background, explanation of the code prescriptions and detailed design examples. Consequently, this book is more than a manual: it provides an all-in-one source for an explanation of the theoretical concepts behind the code and detailed design examples that try to reproduce real design situations instead of the usually simplified examples that are found in most textbooks.

This book evolved from the experience of teaching Steel Structures according to ENV 1993-1-1 since 1993. It further benefited from the participation in Technical Committees TC8 and TC10 of ECCS where the background and the applicability of the various clauses of EN 1993-1-1 was continuously questioned. This book covers exclusively part 1-1 of Eurocode 3 because of the required level of detail. Forthcoming volumes discuss and apply most of the additional parts of Eurocode 3 using a consistent format.

Chapter 1 introduces general aspects such as the basis of design, material properties and geometric characteristics and tolerances, corresponding to chapters 1 to 4 and chapter 7 of EN 1993-1-1. It highlights the important topics that are required in the design of steel structures. Structural analysis is discussed in chapter 2, including structural modelling, global analysis and classification of cross sections, covering chapter 5 of EN 1993-1-1. The design of steel members subjected to various types of internal force (tension, bending and shear, compression and torsion) and their combinations is described in chapter 3, corresponding to chapter 6 of EN 1993-1-1. Chapter 4 presents the design of steel structures using 3D elastic analysis based on the case study of a real building. Finally, chapter 5 discusses plastic design, using a pitched-roof industrial building to exemplify all relevant aspects.

Furthermore, the design examples provided in this book are chosen from real design cases. Two complete design examples are presented: i) a braced steel-framed building; and ii) a pitched-roof industrial building. The chosen design approach tries to reproduce, as much as possible, real design practice instead of more academic approaches that often only deal with parts of the design process. This means that the design examples start by quantifying the actions. They then progress in a detailed step-by-step manner to global analysis and individual member verifications. The design tools currently available and adopted in most design offices are based on software for 3D analysis. Consequently, the design example for multi-storey buildings is analysed as a 3D structure, all subsequent checks being consistent with this approach. This is by no means a straightforward implementation, since most global stability verifications were developed and validated for 2D structures.

The authors are indebted to Prof. Reidar Bjorhovde who carried out a detailed technical review of the manuscript and provided many valuable comments and suggestions. Warm thanks to Prof. David Anderson who carried out an additional detailed revision of the book and also made sure that the English language was properly used. Further thanks to Liliana Marques and José Alexandre Henriques, PhD students at the University of Coimbra, for the help with the design examples of chapter 4. Additional thanks to Prof. Tiago Abecasis who spotted innumerous “bugs” in the text. Finally, thanks to Filipe Dias and the staff of cmm and ECCS for all the editorial and typesetting work, making it possible to bring to an end two years of work in this project.

Luís Simões da Silva
Rui Simões
Helena Gervásio
Coimbra, 2010

DESIGN OF STEEL STRUCTURES

Eurocode 3: Design of steel structures
Part 1-1 – General rules and rules for buildings

ECCS EUROCODE DESIGN MANUALS

ECCS – SCI EUROCODE DESIGN MANUALS

ECCS EUROCODE DESIGN MANUALS – BRAZILIAN EDITIONS

FOREWORD

PREFACE 2^nd EDITION

PREFACE 1^st EDITION

Eurocode 3: Design of steel structuresPart 1-1 – General rules and rules for buildings

ECCS EUROCODE DESIGN MANUALS

ECCS – SCI EUROCODE DESIGN MANUALS

ECCS EUROCODE DESIGN MANUALS – BRAZILIAN EDITIONS

Eurocode 3: Design of steel structures
Part 1-1 – General rules and rules for buildings