PREAMBLE (NOT PART OF THE STANDARD)

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END OF PREAMBLE (NOT PART OF THE STANDARD)

EUROPEAN STANDARD
NORME EUROPÉENNE
EUROPÄISCHE NORM

EN 1993-1-1

May 2005

ICS 91.010.30; 91.080.10

Supersedes ENV 1993-1-1:1992
Incorporating Corrigenda February 2006
and March 2009

English version

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

Eurocode 3: Calcul des structures en acier - Partie 1-1: Règles générales et règles pour les bâtiments Eurocode 3: Bemessung und Konstruktion von Stahlbauten - Teil 1-1: Allgemeine Bemessungsregeln und Regeln für den Hochbau

This European Standard was approved by CEN on 16 April 2004.

CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such national standards may be obtained on application to the Central Secretariat or to any CEN member.

This European Standard exists in three official versions (English, French, German). A version in any other language made by translation under the responsibility of a CEN member into its own language and notified to the Central Secretariat has the same status as the official versions.

CEN members are the national standards bodies of Austria, Belgium, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.

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© 2005 CEN All rights of exploitation in any form and by any means reserved worldwide for CEN national Members.

Ref. No. EN 1993-1-1:2005: E

1

Contents

Page
1 General 9
  1.1 Scope 9
  1.2 Normative references 10
  1.3 Assumptions 11
  1.4 Distinction between principles and application rules 11
  1.5 Terms and definitions 11
  1.6 Symbols 12
  1.7 Conventions for member axes 20
2 Basis of design 22
  2.1 Requirements 22
    2.1.1 Basic Requirements 22
    2.1.2 Reliability management 22
    2.1.3 Design working life, durability and robustness 22
  2.2 Principles of limit state design 23
  2.3 Basic variables 23
    2.3.1 Actions and environmental influences 23
    2.3.2 Material and product properties 23
  2.4 Verification by the partial factor method 23
    2.4.1 Design values of material properties 23
    2.4.2 Design values of geometrical data 23
    2.4.3 Design resistances 24
    2.4.4 Verification of static equilibrium (EQU) 24
  2.5 Design assisted by testing 24
3 Materials 25
  3.1 General 25
  3.2 Structural steel 25
    3.2.1 Material properties 25
    3.2.2 Ductility requirements 25
    3.2.3 Fracture toughness 25
    3.2.4 Through-thickness properties 27
    3.2.5 Tolerances 28
    3.2.6 Design values of material coefficients 28
  3.3 Connecting devices 28
    3.3.1 Fasteners 28
    3.3.2 Welding consumables 28
  3.4 Other prefabricated products in buildings 28
4 Durability 28
5 Structural analysis 29
  5.1 Structural modelling for analysis 29
    5.1.1 Structural modelling and basic assumptions 29 2
    5.1.2 Joint modelling 29
    5.1.3 Ground-structure interaction 29
  5.2 Global analysis 30
    5.2.1 Effects of deformed geometry of the structure 30
    5.2.2 Structural stability of frames 31
  5.3 Imperfections 32
    5.3.1 Basis 32
    5.3.2 Imperfections for global analysis of frames 33
    5.3.3 Imperfection for analysis of bracing systems 36
    5.3.4 Member imperfections 38
  5.4 Methods of analysis considering material non-linearities 38
    5.4.1 General 38
    5.4.2 Elastic global analysis 39
    5.4.3 Plastic global analysis 39
  5.5 Classification of cross sections 40
    5.5.1 Basis 40
    5.5.2 Classification 40
  5.6 Cross-section requirements for plastic global analysis 41
6 Ultimate limit states 45
  6.1 General 45
  6.2 Resistance of cross-sections 45
    6.2.1 General 45
    6.2.2 Section properties 46
    6.2.3 Tension 49
    6.2.4 Compression 49
    6.2.5 Bending moment 50
    6.2.6 Shear 50
    6.2.7 Torsion 52
    6.2.8 Bending and shear 53
    6.2.9 Bending and axial force 54
    6.2.10 Bending shear and axial force 56
  6.3 Buckling resistance of members 56
    6.3.1 Uniform members in compression 56
    6.3.2 Uniform members in bending 60
    6.3.3 Uniform members in bending and axial compression 64
    6.3.4 General method for lateral and lateral torsional buckling of structural components 65
    6.3.5 Lateral torsional buckling of members with plastic hinges 67
  6.4 Uniform built-up compression members 69
    6.4.1 General 69
    6.4.2 Laced compression members 71
    6.4.3 Battened compression members 72
    6.4.4 Closely spaced built-up members 74
7 Serviceability limit states 75
  7.1 General 75
  7.2 Serviceability limit states for buildings 75
    7.2.1 Vertical deflections 75
    7.2.2 Horizontal deflections 75
    7.2.3 Dynamic effects 75
Annex A [informative] – Method 1: Interaction factors kij for interaction formula in 6.3.3(4) 76 3
Annex B [informative] – Method 2: Interaction factors kij for interaction formula in 6.3.3(4) 79
Annex AB [informative] – Additional design provisions 81
Annex BB [informative] – Buckling of components of building structures 82
4

Foreword

This European Standard EN 1993, Eurocode 3: Design of steel structures, has been prepared by Technical Committee CEN/TC250 « Structural Eurocodes », the Secretariat of which is held by BSI. CEN/TC250 is responsible for all Structural Eurocodes.

This European Standard shall be given the status of a National Standard, either by publication of an identical text or by endorsement, at the latest by November 2005, and conflicting National Standards shall be withdrawn at latest by March 2010.

This Eurocode supersedes ENV 1993-1-1.

According to the CEN-CENELEC Internal Regulations, the National Standard Organizations of the following countries are bound to implement these European Standard: Austria, Belgium, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.

Background of the Eurocode programme

In 1975, the Commission of the European Community decided on an action programme in the field of construction, based on article 95 of the Treaty. The objective of the programme was the elimination of technical obstacles to trade and the harmonization of technical specifications.

Within this action programme, the Commission took the initiative to establish a set of harmonized technical rules for the design of construction works which, in a first stage, would serve as an alternative to the national rules in force in the Member States and, ultimately, would replace them.

For fifteen years, the Commission, with the help of a Steering Committee with Representatives of Member States, conducted the development of the Eurocodes programme, which led to the first generation of European codes in the 1980s.

In 1989, the Commission and the Member States of the EU and EFTA decided, on the basis of an agreement1 between the Commission and CEN, to transfer the preparation and the publication of the Eurocodes to the CEN through a series of Mandates, in order to provide them with a future status of European Standard (EN). This links de facto the Eurocodes with the provisions of all the Council’s Directives and/or Commission’s Decisions dealing with European standards (e.g. the Council Directive 89/106/EEC on construction products – CPD – and Council Directives 93/37/EEC, 92/50/EEC and 89/440/EEC on public works and services and equivalent EFTA Directives initiated in pursuit of setting up the internal market).

The Structural Eurocode programme comprises the following standards generally consisting of a number of Parts:

1 Agreement between the Commission of the European Communities and the European Committee for Standardisation (CEN) concerning the work on EUROCODES for the design of building and civil engineering works (BC/CEN/03/89).

EN 1990 Eurocode: Basis of structural design
EN 1991 Eurocode 1: Actions on structures
EN 1992 Eurocode 2: Design of concrete structures
EN 1993 Eurocode 3: Design of steel structures
EN 1994 Eurocode 4: Design of composite steel and concrete structures
EN 1995 Eurocode 5: Design of timber structures
EN 1996 Eurocode 6: Design of masonry structures
EN 1997 Eurocode 7: Geotechnical design
EN 1998 Eurocode 8: Design of structures for earthquake resistance 5
EN 1999 Eurocode 9: Design of aluminium structures

Eurocode standards recognize the responsibility of regulatory authorities in each Member State and have safeguarded their right to determine values related to regulatory safety matters at national level where these continue to vary from State to State.

Status and field of application of Eurocodes

The Member States of the EU and EFTA recognize that Eurocodes serve as reference documents for the following purposes :

The Eurocodes, as far as they concern the construction works themselves, have a direct relationship with the Interpretative Documents2 referred to in Article 12 of the CPD, although they are of a different nature from harmonized product standard3. Therefore, technical aspects arising from the Eurocodes work need to be adequately considered by CEN Technical Committees and/or EOTA Working Groups working on product standards with a view to achieving a full compatibility of these technical specifications with the Eurocodes.

The Eurocode standards provide common structural design rules for everyday use for the design of whole structures and component products of both a traditional and an innovative nature. Unusual forms of construction or design conditions are not specifically covered and additional expert consideration will be required by the designer in such cases.

National Standards implementing Eurocodes

The National Standards implementing Eurocodes will comprise the full text of the Eurocode (including any annexes), as published by CEN, which may be preceded by a National title page and National foreword, and may be followed by a National annex (informative).

The National Annex (informative) may only contain information on those parameters which are left open in the Eurocode for national choice, known as Nationally Determined Parameters, to be used for the design of buildings and civil engineering works to be constructed in the country concerned, i.e. :

2According to Art. 3.3 of the CPD, the essential requirements (ERs) shall be given concrete form in interpretative documents for the creation of the necessary links between the essential requirements and the mandates for hENs and ETAGs/ETAs.

3According to Art. 12 of the CPD the interpretative documents shall :

  1. give concrete form to the essential requirements by harmonizing the terminology and the technical bases and indicating classes or levels for each requirement where necessary ;
  2. indicate methods of correlating these classes or levels of requirement with the technical specifications, e.g. methods of calculation and of proof, technical rules for project design, etc. ;
  3. serve as a reference for the establishment of harmonized standards and guidelines for European technical approvals.

    The Eurocodes, de facto, play a similar role in the field of the ER 1 and a part of ER 2.

Links between Eurocodes and product harmonized technical specifications (ENs and ETAs)

6

There is a need for consistency between the harmonized technical specifications for construction products and the technical rules for works4. Furthermore, all the information accompanying the CE Marking of the construction products which refer to Eurocodes should clearly mention which Nationally Determined Parameters have been taken into account.

Additional information specific to EN 1993-1

EN 1993 is intended to be used with Eurocodes EN 1990 – Basis of Structural Design, EN 1991 – Actions on structures and EN 1992 to EN 1999, when steel structures or steel components are referred to.

EN 1993-1 is the first of six parts of EN 1993 – Design of Steel Structures. It gives generic design rules intended to be used with the other parts EN 1993-2 to EN 1993-6. It also gives supplementary rules applicable only to buildings.

EN 1993-1 comprises twelve subparts EN 1993-1-1 to EN 1993-1-12 each addressing specific steel components, limit states or materials.

It may also be used for design cases not covered by the Eurocodes (other structures, other actions, other materials) serving as a reference document for other CEN TC’s concerning structural matters.

EN 1993-1 is intended for use by

Numerical values for partial factors and other reliability parameters are recommended as basic values that provide an acceptable level of reliability. They have been selected assuming that an appropriate level of workmanship and quality management applies.

4 See Art.3.3 and Art. 12 of the CPD, as well as clauses 4.2, 4.3.1, 4.3.2 and 5.2 of ID 1.

7

National annex for EN 1993-1-1

This standard gives values with notes indicating where national choices may have to be made. Therefore the National Standard implementing EN 1993-1 should have a National Annex containing all Nationally Determined Parameters to be used for the design Image of steel structures and civil engineering works to be constructed Image in the relevant country.

National choice is allowed in EN 1993-1-1 through the following clauses:

8

1 General

1.1 Scope

1.1.1 Scope of Eurocode 3

  1. Eurocode 3 applies to the design of buildings and civil engineering works in steel. It complies with the principles and requirements for the safety and serviceability of structures, the basis of their design and verification that are given in EN 1990 - Basis of structural design.
  2. Eurocode 3 is concerned only with requirements for resistance, serviceability, durability and fire resistance of steel structures. Other requirements, e.g. concerning thermal or sound insulation, are not covered.
  3. Eurocode 3 is intended to be used in conjunction with:
  4. Eurocode 3 is subdivided in various parts:
    EN 1993-1 Design of Steel Structures : General rules and rules for buildings.
    EN 1993-2 Design of Steel Structures: Steel bridges.
    EN 1993-3 Design of Steel Structures : Towers, masts and chimneys.
    EN 1993-4 Design of Steel Structures : Silos, tanks and pipelines.
    EN 1993-5 Design of Steel Structures : Piling.
    EN 1993-6 Design of Steel Structures: Crane supporting structures.
  5. EN 1993-2 to EN 1993-6 refer to the generic rules in EN 1993-1. The rules in parts EN 1993-2 to EN 1993-6 supplement the generic rules in EN 1993-1.
  6. EN 1993-1 “General rules and rules for buildings” comprises:
    EN 1993-1-1 Design of Steel Structures : General rules and rules for buildings.
    EN 1993-1-2 Design of Steel Structures : Structural fire design.
    EN 1993-1-3 Design of Steel Structures: Image Cold-formed members and sheeting Image.
    EN 1993-1-4 Design of Steel Structures : Stainless steels.
    EN 1993-1-5 Design of Steel Structures : Plated structural elements.
    EN 1993-1-6 Design of Steel Structures : Strength and stability of shell structures.
    EN 1993-1-7 Design of Steel Structures : Strength and stability of planar plated structures transversely loaded.
    EN 1993-1-8 Design of Steel Structures : Design of joints.
    EN 1993-1-9 Design of Steel Structures : Fatigue strength of steel structures.
    EN 1993-1-10 Design of Steel Structures : Selection of steel for fracture toughness and through-thickness properties.
    EN 1993-1-11 Design of Steel Structures : Design of structures with tension components made of steel.
    EN 1993-1-12 Design of Steel Structures : Supplementary rules for high strength steel.
9

1.1.2 Scope of Part 1.1 of Eurocode 3

  1. EN 1993-1-1 gives basic design rules for steel structures with material thicknesses t ≥ 3 mm. It also gives supplementary provisions for the structural design of steel buildings. These supplementary provisions are indicated by the letter “B” after the paragraph number, thus ( )B.

    NOTE Image For cold formed members and sheeting, see EN 1993-1-3 Image.

  2. The following subjects are dealt with in EN 1993-1-1:
    Section 1: General
    Section 2: Basis of design
    Section 3: Materials
    Section 4: Durability
    Section 5: Structural analysis
    Section 6: Ultimate limit states
    Section 7: Serviceability limit states
  3. Sections 1 to 2 provide additional clauses to those given in EN 1990 “Basis of structural design”.
  4. Section 3 deals with material properties of products made of low alloy structural steels.
  5. Section 4 gives general rules for durability.
  6. Section 5 refers to the structural analysis of structures, in which the members can be modelled with sufficient accuracy as line elements for global analysis.
  7. Section 6 gives detailed rules for the design of cross sections and members.
  8. Section 7 gives rules for serviceability.

1.2 Normative references

This European Standard incorporates by dated or undated reference, provisions from other publications. These normative references are cited at the appropriate places in the text and the publications are listed hereafter. For dated references, subsequent amendments to or revisions of any of these publications apply to this European Standard only when incorporated in it by amendment or revision. For undated references the latest edition of the publication referred to applies (including amendments).

1.2.1 General reference standards

EN 1090 Execution of steel structures – Technical requirements
EN ISO 12944 Paints and varnishes – Corrosion protection of steel structures by protective paint systems
Image EN ISO 1461 Image Hot dip galvanized coatings on fabricated iron and steel articles – specifications and test methods

1.2.2 Weldable structural steel reference standards

EN 10025-1:2004 Hot-rolled products of structural steels - Part 1: General delivery conditions.
EN 10025-2:2004 Hot-rolled products of structural steels - Part 2: Technical delivery conditions for non-alloy structural steels.
EN 10025-3:2004 Hot-rolled products of structural steels - Part 3: Technical delivery conditions for normalized / normalized rolled weldable fine grain structural steels. 10
EN 10025-4:2004 Hot-rolled products of structural steels - Part 4: Technical delivery conditions for thermomechanical rolled weldable fine grain structural steels.
EN 10025-5:2004 Hot-rolled products of structural steels - Part 5: Technical delivery conditions for structural steels with improved atmospheric corrosion resistance.
EN 10025-6:2004 Hot-rolled products of structural steels - Part 6: Technical delivery conditions for flat products of high yield strength structural steels in the quenched and tempered condition.
EN 10164:1993 Steel products with improved deformation properties perpendicular to the surface of the product - Technical delivery conditions.
EN 10210-1:1994 Hot finished structural hollow sections of non-alloy and fine grain structural steels -Part 1: Technical delivery requirements.
EN 10219-1:1997 Cold formed hollow sections of structural steel - Part 1: Technical delivery requirements.

1.3 Assumptions

  1. In addition to the general assumptions of EN 1990 the following assumptions apply:

1.4 Distinction between principles and application rules

  1. The rules in EN 1990 clause 1.4 apply.

1.5 Terms and definitions

  1. The rules in EN 1990 clause 1.5 apply.
  2. The following terms and definitions are used in EN 1993-1-1 with the following meanings:

1.5.1
frame

the whole or a portion of a structure, comprising an assembly of directly connected structural elements, designed to act together to resist load; this term refers to both moment-resisting frames and triangulated frames; it covers both plane frames and three-dimensional frames

1.5.2
sub-frame

a frame that forms part of a larger frame, but is be treated as an isolated frame in a structural analysis

1.5.3
type of framing

terms used to distinguish between frames that are either:

1.5.4
global analysis

the determination of a consistent set of internal forces and moments in a structure, which are in equilibrium with a particular set of actions on the structure

11

1.5.5
system length

distance in a given plane between two adjacent points at which a member is braced against lateral displacement in this plane, or between one such point and the end of the member

1.5.6
buckling length

system length of an otherwise similar member with pinned ends, which has the same Image critical buckling load Image as a given member or segment of member

1.5.7
shear lag effect

non-uniform stress distribution in wide flanges due to shear deformation; it is taken into account by using a reduced “effective” flange width in safety assessments

1.5.8
capacity design

design method for achieving the plastic deformation capacity of a member by providing additional strength in its connections and in other parts connected to it

1.5.9
uniform member

member with a constant cross-section along its whole length

1.6 Symbols

  1. For the purpose of this standard the following symbols apply.
  2. Additional symbols are defined where they first occur.

    NOTE Symbols are ordered by appearance in EN 1993-1-1. Symbols may have various meanings.

Section 1

x-x axis along a member
y-y axis of a cross-section
z-z axis of a cross-section
u-u major principal axis (where this does not coincide with the y-y axis)
v-v minor principal axis (where this does not coincide with the z-z axis)
b width of a cross section
h depth of a cross section
d depth of straight portion of a web
tw web thickness
tf flange thickness
r radius of root fillet
r1 radius of root fillet
r2 toe radius
t thickness

Section 2

Pk nominal value of the effect of prestressing imposed during erection
Gk nominal value of the effect of permanent actions 12
Image Xk Image characteristic values of material property
Xn nominal values of material property
Rd design value of resistance
Rk characteristic value of resistance
γM general partial factor
γMi particular partial factor
γMf partial factor for fatigue
η conversion factor
ad design value of geometrical data

Section 3

fy yield strength
fu ultimate strength
Image ReH Image yield strength to product standards
Rm ultimate strength to product standards
A0 original cross-section area
εy yield strain
εu ultimate strain
ZEd required design Z-value resulting from the magnitude of strains from restrained metal shrinkage under the weld beads.
ZRd available design Z-value
E modulus of elasticity
G shear modulus
ν Poisson’s ratio in elastic stage
α coefficient of linear thermal expansion

Section 5

αcr factor by which the design loads would have to be increased to cause elastic instability in a global mode
FEd design loading on the structure
Fcr elastic critical buckling load for global instability mode based on initial elastic stiffnesses
Image HEd total design horizontal load, including equivalent forces transferred by the storey (storey shear) Image
Image VEd total design vertical load on the frame transferred by the storey (storey thrust) Image
δH,Ed horizontal displacement at the top of the storey, relative to the bottom of the storey
h storey height
Image non dimensional slenderness
NEd design value of the axial force
ϕ global initial sway imperfection
ϕ0 basic value for global initial sway imperfection
αh reduction factor for height h applicable to columns
h height of the structure 13
αm reduction factor for the number of columns in a row
m number of columns in a row
e0 maximum amplitude of a member imperfection
L member length
ηinit amplitude of elastic critical buckling mode
ηcr shape of elastic critical buckling mode
e0,d design value of maximum amplitude of an imperfection
MRk characteristic moment resistance of the critical cross section
NRk characteristic resistance to normal force of the critical cross section
α imperfection factor
Image bending moment due to ηcr at the critical cross section
χ reduction factor for the relevant buckling curve
Image αult,k minimum load amplifier of the design loads to reach the characteristic resistance of the most critical cross section of the structural component considering its in plane behaviour without taking lateral or lateral torsional buckling into account however accounting for all effects due to in plane geometrical deformation and imperfections, global and local, where relevant Image
αcr minimum force amplifier to reach the Image elastic critical buckling load Image
q equivalent force per unit length
δq in-plane deflection of a bracing system
qd equivalent design force per unit length
MEd design bending moment
k factor for e(0,d)
ε strain
σ stress
σcom,Ed maximum design compressive stress in an element
length
Image ε factor depending on fy Image
c width or depth of a part of a cross section
α portion of a part of a cross section in compression
ψ stress or strain ratio
Image kσ plate buckling factor Image
d outer diameter of circular tubular sections

Section 6

γM0 partial factor for resistance of cross-sections whatever the class is
γM1 partial factor for resistance of members to instability assessed by member checks
γM2 partial factor for resistance of cross-sections in tension to fracture
σx,Ed design value of the local longitudinal stress
σz,Ed design value of the local transverse stress
τEd design value of the local shear stress
NEd design normal force
My,Ed design bending moment, y-y axis
Mz,Ed design bending moment, z-z axis
NRd design values of the resistance to normal forces 14
My,Rd design values of the resistance to bending moments, y-y axis
Mz,Rd design values of the resistance to bending moments, z-z axis
s staggered pitch, the spacing of the centres of two consecutive holes in the chain measured parallel to the member axis
p spacing of the centres of the same two holes measured perpendicular to the member axis
n number of holes extending in any diagonal or zig-zag line progressively across the member or part of the member
d0 diameter of hole
eN shift of the centroid of the effective area Aeff relative to the centre of gravity of the gross cross section
ΔMEd additional moment from shift of the centroid of the effective area Aeff relative to the centre of gravity of the gross cross section
Aeff effective area of a cross section
Nt,Rd design values of the resistance to tension forces
Npl,Rd design plastic resistance to normal forces of the gross cross-section
Nu,Rd design ultimate resistance to normal forces of the net cross-section at holes for fasteners
Anet net area of a cross section
Nnet,Rd design plastic resistance to normal forces of the net cross-section
Nc,Rd design resistance to normal forces of the cross-section for uniform compression
Mc,Rd design resistance for bending about one principal axis of a cross-section
Wpl plastic section modulus
Wel,min minimum elastic section modulus
Weff,min minimum effective section modulus
Af area of the tension flange
Af,net net area of the tension flange
VEd design shear force
Vc,Rd design shear resistance
Image Vpl,Rd design plastic shear resistance Image
Av shear area
η factor for shear area
S first moment of area
I second moment of area
Image A cross-sectional area Image
Aw area of a web
Af area of one flange
TEd design value of total torsional moments
TRd design resistance to torsional moments
Image Tt,Ed design value of internal St. Venant torsional moment Image
Image TW,Ed design value of internal warping torsional moment Image
τt,Ed design shear stresses due to St. Venant torsion
τw,Ed design shear stresses due to warping torsion
σw,Ed design direct stresses due to the bimoment BEd
Image BEd design value of the bimoment Image
Vpl,T,Rd reduced design plastic shear resistance making allowance for the presence of a torsional moment 15
ρ reduction factor to determine reduced design values of the resistance to bending moments making allowance for the presence of shear forces
MV,,Rd reduced design values of the resistance to bending moments making allowance for the presence of shear forces
MN,,Rd reduced design values of the resistance to bending moments making allowance for the presence of normal forces
n ratio of design normal force to design plastic resistance to normal forces of the gross cross-section
a ratio of web area to gross area
α parameter introducing the effect of biaxial bending
β parameter introducing the effect of biaxial bending
eN,y shift of the centroid of the effective area Aeff relative to the centre of gravity of the gross cross section (y-y axis)
eN,z shift of the centroid of the effective area Aeff relative to the centre of gravity of the gross cross section (z-z axis)
Weff,min minimum effective section modulus
Nb,Rd design buckling resistance of a compression member
χ reduction factor for relevant buckling mode
Φ value to determine the reduction factor χ
a0, a, b, c, d class indexes for buckling curves
Ncr elastic critical force for the relevant buckling mode based on the gross cross sectional properties
i radius of gyration about the relevant axis, determined using the properties of the gross cross-section
λl slenderness value to determine the relative slenderness
Image relative slenderness for torsional or torsional-flexural buckling
Ncr,TF elastic torsional-flexural buckling force
Ncr,T elastic torsional buckling force
Mb,Rd design buckling resistance moment
χLT reduction factor for lateral-torsional buckling
ΦLT value to determine the reduction factor χLT
αLT imperfection factor
Image non dimensional slenderness for lateral torsional buckling
Mcr elastic critical moment for lateral-torsional buckling
Image plateau length of the lateral torsional buckling curves Image for rolled and welded sections Image
β correction factor for the lateral torsional buckling curves Image for rolled and welded sections Image
χLT,mod modified reduction factor for lateral-torsional buckling
f modification factor for χLT
kc correction factor for moment distribution
ψ ratio of moments in segment
Lc length between lateral restraints
Image equivalent compression flange slenderness
Image if,zImage radius of gyration of compression flange about the minor axis of the section
Ieff,f effective second moment of area of compression flange about the minor axis of the section 16
Aeff,f effective area of compression flange
Aeff,w,c effective area of compressed part of web
Image slenderness parameter
kf modification factor
Image ΔMy,Ed Image moments due to the shift of the centroidal y-y axis
Image ΔMz,Ed moments due to the shift of the centroidal z-z axis
χy reduction factor due to flexural buckling (y-y axis)
χz reduction factor due to flexural buckling (z-z axis)
kyy interaction factor
kyz interaction factor
kzy interaction factor
kzz interaction factor
Image global non dimensional slenderness of a structural component for out-of-plane buckling
χop reduction factor for the non-dimensional slenderness Image
αult,k minimum load amplifier of the design loads to reach the characteristic resistance of the most critical cross section
αcr,op minimum amplifier for the in plane design loads to reach the elastic Image critical buckling load Image with regard to lateral or lateral torsional buckling
NRk characteristic value of resistance to compression
My,Rk characteristic value of resistance to bending moments about y-y axis
Mz,Rk characteristic value of resistance to bending moments about z-z axis
Qm local force applied at each stabilized member at the plastic hinge locations
Lstable stable length of segment
Lch buckling length of chord
h0 distance of centrelines of chords of a built-up column
a distance between restraints of chords
α angle between axes of chord and lacings
imin minimum radius of gyration of single angles
Ach area of one chord of a built-up column
Nch,Ed design chord force in the middle of a built-up member
Image design value of the Image maximum first order moment Image in the middle of the built-up member
Ieff effective second moment of area of the built-up member
Sv shear stiffness of built-up member from the lacings or battened panel
Image n number of planes of lacings or battens Image
Ad area of one diagonal of a built-up column
d length of a diagonal of a built-up column
AV area of one post (or transverse element) of a built-up column
Ich in plane second moment of area of a chord
Ib in plane second moment of area of a batten
μ efficiency factor 17
iy radius of gyration (y-y axis)

Annex A

Cmy equivalent uniform moment factor
Cmz equivalent uniform moment factor
CmLT equivalent uniform moment factor
μy factor
μz factor
Ncr,y elastic flexural buckling force about the y-y axis
Ncr,z elastic flexural buckling force about the z-z axis
Cyy factor
Cyz factor
Czy factor
Czz factor
wy factor
wz factor
npl factor
Image maximum of Image
bLT factor
cLT factor
dLT factor
eLT factor
ψy ratio of end moments (y-y axis)
Cmy,0 factor
Cmz,0 factor
aLT factor
IT St. Venant torsional constant
Iy second moment of area about y-y axis
Image Cl ratio between the critical bending moment (largest value along the member) and the critical constant bending moment for a member with hinged supports Image
Mi,Ed(X) maximum first order moment
x| maximum member displacement along the member

Annex B

Image αs factor; s = sagging Image
Image αh factor; h = hogging Image
Cm equivalent uniform moment factor

Annex AB

γG partial factor for permanent loads
Gk characteristic value of permanent loads
γQ partial factor for variable loads
Qk characteristic value of variable loads
18

Annex BB

Image effective slenderness ratio for buckling about v-v axis
Image effective slenderness ratio for buckling about y-y axis
Image effective slenderness ratio for buckling about z-z axis
L system length
Lcr buckling length
S shear stiffness provided by sheeting
Iw warping constant
Cϑ,k rotational stiffness provided by stabilizing continuum and connections
Kυ factor for considering the type of analysis
Kϑ factor for considering the moment distribution and the type of restraint
CϑR,k rotational stiffness provided by the stabilizing continuum to the beam assuming a stiff connection to the member
CϑC,k rotational stiffness of the connection between the beam and the stabilizing continuum
CϑD,k rotational stiffness deduced from an analysis of the distorsional deformations of the beam cross sections
Lm stable length between adjacent lateral restraints
Lk stable length between adjacent torsional restraints
Ls stable length between a plastic hinge location and an adjacent torsional restraint
Cl modification factor for moment distribution
Cm modification factor for linear moment gradient
Cn modification factor for non-linear moment gradient
a distance between the centroid of the member with the plastic hinge and the centroid of the restraint members
B0 factor
B1 factor
B2 factor
Image η ratio of elastic critical values of axial forces Image
is radius of gyration related to centroid of restraining member
βt ratio of the algebraically smaller end moment to the larger end moment
R1 moment at a specific location of a member
R2 moment at a specific location of a member
R3 moment at a specific location of a member
R4 moment at a specific location of a member
R5 moment at a specific location of a member
RE maximum of R1 or R5
Rs maximum value of bending moment anywhere in the length Ly
c taper factor
hh additional depth of the haunch or taper
hmax maximum depth of cross-section within the length Ly
hmin minimum depth of cross-section within the length Ly 19
hs vertical depth of the un-haunched section
Lh length of haunch within the length Ly
Ly length between restraints

1.7 Conventions for member axes

  1. The convention for member axes is:
    x-x - along the member
    y-y - axis of the cross-section
    z-z - axis of the cross-section
  2. For steel members, the conventions used for cross-section axes are:
  3. The symbols used for dimensions and axes of rolled steel sections are indicated in Figure 1.1.
  4. The convention used for subscripts that indicate axes for moments is: “Use the axis about which the moment acts.”

    NOTE All rules in this Eurocode relate to principal axis properties, which are generally defined by the axes y-y and z-z but for sections such as angles are defined by the axes u-u and v-v.

20

Figure 1:1: Dimensions and axes of sections

Figure 1.1: Dimensions and axes of sections

21

2 Basis of design

2.1 Requirements

2.1.1 Basic requirements

  1. Image P The design of steel structures shall be in accordance with the general rules given in EN 1990. Image
  2. The supplementary provisions for steel structures given in this section should also be applied.
  3. The basic requirements of EN 1990 section 2 should be deemed be satisfied where limit state design is used in conjunction with the partial factor method and the load combinations given in EN 1990 together with the actions given in EN 1991.
  4. The rules for resistances, serviceability and durability given in the various parts of EN 1993 should be applied.

2.1.2 Reliability management

  1. Where different levels of reliability are required, these levels should preferably be achieved by an appropriate choice of quality management in design and execution, according to EN 1990 Annex C and EN 1090.

2.1.3 Design working life, durability and robustness

2.1.3.1 General
  1. Image P Depending upon the type of action affecting durability and the design working life (see EN 1990) steel structures shall be Image
2.1.3.2 Design working life for buildings
  1. Image P,B The design working life shall be taken as the period for which a building structure is expected to be used for its intended purpose. Image
  2. B For the specification of the intended design working life of a permanent building see Table 2.1 of EN 1990.
  3. B For structural elements that cannot be designed for the total design life of the building, see 2.1.3.3(3)B.
2.1.3.3 Durability for buildings
  1. Image P,B To ensure durability, buildings and their components shall either be designed for environmental actions and fatigue if relevant or else protected from them. Image 22
  2. Image P,B The effects of deterioration of material, corrosion or fatigue where relevant shall be taken into account by appropriate choice of material, see EN 1993-1-4 and EN 1993-1-10, and details, see EN 1993-1-9, or by structural redundancy and by the choice of an appropriate corrosion protection system. Image
  3. B If a building includes components that need to be replaceable (e.g. bearings in zones of soil settlement), the possibility of their safe replacement should be verified as a transient design situation.

2.2 Principles of limit state design

  1. The resistance of cross-sections and members specified in this Eurocode 3 for the ultimate limit states as defined Image in the clause 3.3 of EN 1990 are Image based on tests in which the material exhibited sufficient ductility to apply simplified design models.
  2. The resistances specified in this Eurocode Part may therefore be used where the conditions for materials in section 3 are met.

2.3 Basic variables

2.3.1 Actions and environmental influences

  1. Actions for the design of steel structures should be taken from EN 1991. For the combination of actions and partial factors of actions see Annex A to EN 1990.

    NOTE 1 The National Annex may define actions for particular regional or climatic or accidental situations.

    NOTE 2B For proportional loading for incremental approach, see Annex AB.1.

    NOTE 3B For simplified load arrangement, see Annex AB.2.

  2. The actions to be considered in the erection stage should be obtained from EN 1991-1-6.
  3. Where the effects of predicted absolute and differential settlements need to be considered, best estimates of imposed deformations should be used.
  4. The effects of uneven settlements or imposed deformations or other forms of prestressing imposed during erection should be taken into account by their nominal value Pk as permanent actions and grouped with other permanent actions Image Gk to form a single action Image (Gk + Pk).
  5. Fatigue actions not defined in EN 1991 should be determined according to Annex A of EN 1993-1-9.

2.3.2 Material and product properties

  1. Material properties for steels and other construction products and the geometrical data to be used for design should be those specified in the relevant ENs, ETAGs or ETAs unless otherwise indicated in this standard.

2.4 Verification by the partial factor method

2.4.1 Design values of material properties

  1. Image P For the design of steel structures characteristic values Image Xk Image or nominal values Xn of material properties shall be used as indicated in this Eurocode. Image

2.4.2 Design values of geometrical data

  1. Geometrical data for cross-sections and systems may be taken from product standards hEN or drawings for the execution to EN 1090 and treated as nominal values. 23
  2. Design values of geometrical imperfections specified in this standard are equivalent geometric imperfections that take into account the effects of:

2.4.3 Design resistances

  1. For steel structures equation (6.6c) or equation (6.6d) of EN 1990 applies:

    Image

    where

    Rk is the characteristic value of the particular resistance determined with characteristic or nominal values for the material properties and dimensions
    γM is the global partial factor for the particular resistance

    NOTE For the definitions of ηl, ηi, Xkl, Xki and ad see EN 1990.

2.4.4 Verification of static equilibrium (EQU)

  1. The reliability format for the verification of static equilibrium in Table 1.2 (A) in Annex A of EN 1990 also applies to design situations equivalent to (EQU), e.g. for the design of holding down anchors or the verification of uplift of bearings of continuous beams.

2.5 Design assisted by testing

  1. The resistances Rk in this standard have been determined using Annex D of EN 1990.
  2. In recommending classes of constant partial factors γMi the characteristic values Rk were obtained from

    Rk = Rd γMi     (2.2)

    where

    Rd are design values according to Annex D of EN 1990
    γMi are recommended partial factors.

    NOTE 1 The numerical values of the recommended partial factors γMi have been determined such that Rk represents approximately the 5 %-fractile for an infinite number of tests.

    NOTE 2 For characteristic values of fatigue strength and partial factors γMf for fatigue see EN 1993-1-9.

    NOTE 3 For characteristic values of toughness resistance and safety elements for the toughness verification see EN 1993-1-10.

  3. Where resistances Rk for prefabricated products should be determined from tests, the procedure in (2)should be followed.
24

3 Materials

3.1 General

  1. The nominal values of material properties given in this section should be adopted as characteristic values in design calculations.
  2. This Part of EN 1993 covers the design of steel structures fabricated from steel material conforming to the steel grades listed in Table 3.1.

    NOTE For other steel material and products see National Annex.

3.2 Structural steel

3.2.1 Material properties

  1. The nominal values of the yield strength fy and the ultimate strength fu for structural steel should be obtained
    1. either by adopting the values Image fy = ReH Image and fu = Rm direct from the product standard
    2. or by using the simplification given in Table 3.1

      NOTE The National Annex may give the choice.

3.2.2 Ductility requirements

  1. For steels a minimum ductility is required that should be expressed in terms of limits for:
  2. Steel conforming with one of the steel grades listed in Table 3.1 should be accepted as satisfying these requirements.

3.2.3 Fracture toughness

  1. Image P The material shall have sufficient fracture toughness to avoid brittle fracture of tension elements at the lowest service temperature expected to occur within the intended design life of the structure. Image

    NOTE The lowest service temperature to be adopted in design may be given in the National Annex.

  2. No further check against brittle fracture need to be made if the conditions given in EN 1993-1-10 are satisfied for the lowest temperature. 25
  3. B For building components under compression a minimum toughness property should be selected.

    NOTE B The National Annex may give information on the selection of toughness properties for members in compression. The use of Table 2.1 of EN 1993-1-10 for σEd = 0,25 fy(t) is recommended.

  4. For selecting steels for members with hot dip galvanized coatings see Image EN ISO 1461 Image.
Table 3.1: Nominal values of yield strength fy and ultimate tensile strength fu for hot rolled structural steel
Standard and steel grade Nominal thickness of the element t [mm]
t ≤ 40 mm 40 mm < t ≤ 80 mm
fy [N/mm2] fu [N/mm2] fy [N/mm2] fu [N/mm2]
EN 10025-2        
S 235 235 360 215 360
S 275 275 430 255 410
S 355 355 Image 490 Image 335 470
S 450 440 550 410 550
EN 10025-3        
S 275 N/NL 275 390 255 370
S 355 N/NL 355 490 335 470
S 420 N/NL 420 520 390 520
S 460 N/NL 460 540 430 540
EN 10025-4        
S 275 M/ML 275 370 255 360
S 355 M/ML 355 470 335 450
S 420 M/ML 420 520 390 500
S 460 M/ML 460 540 430 530
EN 10025-5        
S 235 W 235 360 215 340
S 355 W 355 Image 490 Image 335 490
EN 10025-6        
S 460 Q/QL/QL1 460 570 440 550 26
EN 10210-1        
S 235 H 235 360 215 340
S 275 H 275 430 255 410
S 355 H 355 510 335 490
S 275 NH/NLH 275 390 255 370
S 355 NH/NLH 355 490 335 470
Image S 420 NH/NLH Image 420 540 390 520
S 460 NH/NLH 460 560 430 550
EN 10219-1        
S 235 H 235 360    
S 275 H 275 430    
S 355 H 355 510    
S 275 NH/NLH 275 370    
S 355 NH/NLH 355 470    
S 460 NH/NLH 460 550    
S 275 MH/MLH 275 360    
S 355 MH/MLH 355 470    
S 420 MH/MLH 420 500    
S 460 MH/MLH 460 530    

3.2.4 Through-thickness properties

  1. Where steel with improved through-thickness properties is necessary according to EN 1993-1-10, steel according to the required quality class in EN 10164 should be used.

    NOTE 1 Guidance on the choice of through-thickness properties is given in EN 1993-1-10.

    NOTE 2B Particular care should be given to welded beam to column connections and welded end plates with tension in the through-thickness direction.

    NOTE 3B The National Annex may give the relevant allocation of target values ZEd according to 3.2(2) of EN 1993-1-10 to the quality class in EN 10164. The allocation in Table 3.2 is recommended for buildings:

Table 3.2: Choice of quality class according to EN 10164
Target value of ZEd according to EN 1993-1-10 Required value of ZRd expressed in terms of design Z-values according to EN 10164
ZEd ≤ 10
10 < ZEd ≤ 20 Z 15
20 < ZEd ≤ 30 Z 25
ZEd > 30 Z 35
27

3.2.5 Tolerances

  1. The dimensional and mass tolerances of rolled steel sections, structural hollow sections and plates should conform with the relevant product standard, ETAG or ETA unless more severe tolerances are specified.
  2. For welded components the tolerances given in EN 1090 should be applied.
  3. For structural analysis and design the nominal values of dimensions should be used.

3.2.6 Design values of material coefficients

  1. The material coefficients to be adopted in calculations for the structural steels covered by this Eurocode Part should be taken as follows:
    modulus of elasticity E = 210 000 N / mm2
    shear modulus Image
    Poisson’s ratio in elastic stage v = 0,3
    coefficient of linear thermal expansion α = 12×10−6 perK (for T ≤ 100 °C)

    NOTE For calculating the structural effects of unequal temperatures in composite concrete-steel structures to EN 1994 the coefficient of linear thermal expansion is taken as α = 10 × 10−6 per K.

3.3 Connecting devices

3.3.1 Fasteners

  1. Requirements for fasteners are given in EN 1993-1-8.

3.3.2 Welding consumables

  1. Requirements for welding consumables are given in EN 1993-1-8.

3.4 Other prefabricated products in buildings

  1. B Any semi-finished or finished structural product used in the structural design of buildings should comply with the relevant EN Product Standard or ETAG or ETA.

4 Durability

  1. The basic requirements for durability are set out in EN 1990.
  2. Image P The means of executing the protective treatment undertaken off-site and on-site shall be in accordance with EN 1090. Image

    NOTE EN 1090 lists the factors affecting execution that need to be specified during design.

  3. Parts susceptible to corrosion, mechanical wear or fatigue should be designed such that inspection, maintenance and reconstruction can be carried out satisfactorily and access is available for in-service inspection and maintenance. 28
  4. B For building structures no fatigue assessment is normally required except as follows:
    1. Members supporting lifting appliances or rolling loads
    2. Members subject to repeated stress cycles from vibrating machinery
    3. Members subject to wind-induced vibrations
    4. Members subject to crowd-induced oscillations
  5. Image P For elements that cannot be inspected an appropriate corrosion allowance shall be included. Image
  6. B Corrosion protection does not need to be applied to internal building structures, if the internal relative humidity does not exceed 80%.

5 Structural analysis

5.1 Structural modelling for analysis

5.1.1 Structural modelling and basic assumptions

  1. Image P Analysis shall be based upon calculation models of the structure that are appropriate for the limit state under consideration. Image
  2. The calculation model and basic assumptions for the calculations should reflect the structural behaviour at the relevant limit state with appropriate accuracy and reflect the anticipated type of behaviour of the cross sections, members, joints and bearings.
  3. Image P The method used for the analysis shall be consistent with the design assumptions. Image
  4. B For the structural modelling and basic assumptions for components of buildings see also EN 1993-1-5 and EN 1993-1-11.

5.1.2 Joint modelling

  1. The effects of the behaviour of the joints on the distribution of internal forces and moments within a structure, and on the overall deformations of the structure, may generally be neglected, but where such effects are significant (such as in the case of semi-continuous joints) they should be taken into account, see EN 1993-1-8.
  2. To identify whether the effects of joint behaviour on the analysis need be taken into account, a distinction may be made between three joint models as follows, see EN 1993-1-8, 5.1.1:
  3. The requirements of the various types of joints are given in EN 1993-1-8.

5.1.3 Ground-structure interaction

  1. Account should be taken of the deformation characteristics of the supports where significant.

    NOTE EN 1997 gives guidance for calculation of soil-structure interaction.

29

5.2 Global analysis

5.2.1 Effects of deformed geometry of the structure

  1. The internal forces and moments may generally be determined using either:
  2. The effects of the deformed geometry (second-order effects) should be considered if they increase the action effects significantly or modify significantly the structural behaviour.
  3. First order analysis may be used for the structure, if the increase of the relevant internal forces or moments or any other change of structural behaviour caused by deformations can be neglected. This condition may be assumed to be fulfilled, if the following criterion is satisfied:

    Image

    where

    αcr is the factor by which the design loading would have to be increased to cause elastic instability in a global mode
    FEd is the design loading on the structure
    Fcr is the elastic critical buckling load for global instability mode based on initial elastic stiffnesses

    NOTE A greater limit for αcr for plastic analysis is given in equation (5.1) because structural behaviour may be significantly influenced by non linear material properties in the ultimate limit state (e.g. where a frame forms plastic hinges with moment redistributions or where significant non linear deformations from semi-rigid joints occur). Where substantiated by more accurate approaches the National Annex may give a lower limit for αcr for certain types of frames.

  4. B Portal frames with shallow roof slopes and beam-and-column type plane frames in buildings may be checked for sway mode failure with first order analysis if the criterion (5.1) is satisfied for each storey. In these structures αcr Image should Image be calculated using the following approximative formula, provided that the axial compression in the beams or rafters is not significant:

    Image

    Image where

    HEd is the total design horizontal load, including equivalent forces according to 5.3.2(7), transferred by the storey (storey shear)
    VEd is the total design vertical load on the frame transferred by the storey (storey thrust) Image
    δH,Ed is the horizontal displacement at the top of the storey, relative to the bottom of the storey, when the frame is loaded with horizontal loads (e.g. wind) and fictitious horizontal loads which are applied at each floor level
    h is the storey height
    30

    Image

    Image Figure 5.1: Notations for 5.2.1(4) Image

    NOTE 1B For the application of (4)B in the absence of more detailed information a roof slope may be taken to be shallow if it is not steeper that 1:2 (26°).

    NOTE 2B For the application of (4)B in the absence of more detailed information the axial compression in the beams or rafters Image should Image be assumed to be significant if

    Image

    where

    NEd is the design value of the compression force,
    Image is the inplane non dimensional slenderness calculated for the beam or rafters considered as hinged at its ends of the system length measured along the beams of rafters.
  5. The effects of shear lag and of local buckling on the stiffness should be taken into account if this significantly influences the global analysis, see EN 1993-1-5.

    NOTE For rolled sections and welded sections with similar dimensions shear lag effects may be neglected.

  6. The effects on the global analysis of the slip in bolt holes and similar deformations of connection devices like studs and anchor bolts on action effects should be taken into account, where relevant and significant.

5.2.2 Structural stability of frames

  1. If according to 5.2.1 the influence of the deformation of the structure has to be taken into account (2) to (6) should be applied to consider these effects and to verify the structural stability.
  2. The verification of the stability of frames or their parts should be carried out considering imperfections and second order effects.
  3. According to the type of frame and the global analysis, second order effects and imperfections may be accounted for by one of the following methods:
    1. both totally by the global analysis,
    2. partially by the global analysis and partially through individual stability checks of members according to 6.3,
    3. for basic cases by individual stability checks of equivalent members according to 6.3 using appropriate buckling lengths according to the global buckling mode of the structure.
    31
  4. Second order effects may be calculated by using an analysis appropriate to the structure (including step-by-step or other iterative procedures). For frames where the first sway buckling mode is predominant first order elastic analysis should be carried out with subsequent amplification of relevant action effects (e.g. bending moments) by appropriate factors.
  5. B For single storey frames designed on the basis of elastic global analysis second order sway effects due to vertical loads may be calculated by increasing the horizontal loads HEd (e.g. wind) and equivalent loads VEd ϕ due to imperfections (see 5.3.2(7)) and other possible sway effects according to first order theory by the factor:

    Image

    provided that αcr ≥ 3,0,

    where

    αcr may be calculated according to (5.2) in 5.2.1(4)B, provided that the roof slope is shallow and that the axial compression in the beams or rafters is not significant as defined in 5.2.1(4)B.

    NOTE B For αcr < 3,0 a more accurate second order analysis applies.

  6. B For multi-storey frames second order sway effects may be calculated by means of the method given in (5)B provided that all storeys have a similar

    NOTE B For the limitation of the method see also 5.2.1(4)B.

  7. In accordance with (3) the stability of individual members should be checked according to the following:
    1. If second order effects in individual members and relevant member imperfections (see 5.3.4) are totally accounted for in the global analysis of the structure, no individual stability check for the members according to 6.3 is necessary.
    2. If second order effects in individual members or certain individual member imperfections (e.g. member imperfections for flexural and/or lateral torsional buckling, see 5.3.4) are not totally accounted for in the global analysis, the individual stability of members should be checked according to the relevant criteria in 6.3 for the effects not included in the global analysis. This verification should take account of end moments and forces from the global analysis of the structure, including global second order effects and global imperfections (see 5.3.2) when relevant and may be based on a buckling length equal to the system length
  8. Where the stability of a frame is assessed by a check with the equivalent column method according to 6.3 the buckling length values should be based on a global buckling mode of the frame accounting for the stiffness behaviour of members and joints, the presence of plastic hinges and the distribution of compressive forces under the design loads. In this case internal forces to be used in resistance checks are calculated according to first order theory without considering imperfections.

    NOTE The National Annex may give information on the scope of application.

5.3 Imperfections

5.3.1 Basis

  1. Appropriate allowances should be incorporated in the structural analysis to cover the effects of imperfections, including residual stresses and geometrical imperfections such as lack of verticality, lack of 32straightness, lack of flatness, lack of fit Image eccentricities greater than the essential tolerances give in EN 1090-2 Image present in joints of the unloaded structure.
  2. Equivalent geometric imperfections, see 5.3.2 and 5.3.3, should be used, with values which reflect the possible effects of all type of imperfections unless these effects are included in the resistance formulae for member design, see section 5.3.4.
  3. The following imperfections should be taken into account:
    1. global imperfections for frames and bracing systems
    2. local imperfections for individual members

5.3.2 Imperfections for global analysis of frames

  1. The assumed shape of global imperfections and local imperfections may be derived from the elastic buckling mode of a structure in the plane of buckling considered.
  2. Both in and out of plane buckling including torsional buckling with symmetric and asymmetric buckling shapes should be taken into account in the most unfavourable direction and form.
  3. For frames sensitive to buckling in a sway mode the effect of imperfections should be allowed for in frame analysis by means of an equivalent imperfection in the form of an initial sway imperfection and individual bow imperfections of members. The imperfections may be determined from:
    1. global initial sway imperfections, see Figure 5.2:

      ϕ = ϕ0 αh αm     (5.5)

      where

      ϕ0 is the basic value: ϕ0 = 1/200
      α is the reduction factor for height h applicable to columns:

      Image

      h is the height of the structure in meters
      αm is the reduction factor for the number of columns in a row: Image
      m is the number of columns in a row including only those columns which carry a vertical load NEd not less than 50% of the average value of the column in the vertical plane considered

      Figure 5.2: Equivalent sway imperfections

      Figure 5.2: Equivalent sway imperfections

    2. relative initial local bow imperfections of members for flexural buckling

      e0/L     (5.6)

      where L is the member length

      NOTE The values e0 / L may be chosen in the National Annex. Recommended values are given in Table 5.1.

    33
    Image Table 5.1: Design value of initial local bow imperfections e0/L for members Image
    Image Buckling curve according to Table 6.2 Image elastic analysis plastic analysis
    e0 / L e0 / L
    a0 1 / 350 1 / 300
    a 1 / 300 1 / 250
    b 1 / 250 1 / 200
    c 1 / 200 1 / 150
    d 1 / 150 1 / 100
  4. B For building frames sway imperfections may be disregarded where

    HEd ≥ 0,15 VEd     (5.7)

  5. B For the determination of horizontal forces to floor diaphragms the configuration of imperfections as given in Figure 5.3 should be applied, where ϕ is a sway imperfection obtained from (5.5) assuming a single storey with height h, see (3) a).

    Figure 5.3: Configuration of sway imperfections ϕ for horizontal forces on floor diaphragms

    Figure 5.3: Configuration of sway imperfections ϕ for horizontal forces on floor diaphragms

  6. When performing the global analysis for determining end forces and end moments to be used in member checks according to 6.3 local bow imperfections may be neglected. However for frames sensitive to second order effects local bow imperfections of members additionally to global sway imperfections (see 5.2.1(3)) should be introduced in the structural analysis of the frame for each compressed member where the following conditions are met:
    where NEd is the design value of the compression force
    and Image is the in-plane non-dimensional slenderness calculated for the member considered as hinged at its ends

    NOTE Local bow imperfections are taken into account in member checks, see 5.2.2 (3) and 5.3.4.

    34
  7. The effects of initial sway imperfection and local bow imperfections may be replaced by systems of equivalent horizontal forces, introduced for each column, see Figure 5.3 and Figure 5.4.

    Figure 5.4: Replacement of initial imperfections by equivalent horizontal forces

    Figure 5.4: Replacement of initial imperfections by equivalent horizontal forces

  8. These initial sway imperfections should apply in all relevant horizontal directions, but need only be considered in one direction at a time.
  9. B Where, in multi-storey beam-and-column building frames, equivalent forces are used they should be applied at each floor and roof level.
  10. The possible torsional effects on a structure caused by anti-symmetric sways at the two opposite faces, should also be considered, see Figure 5.5.

    Figure 5.5: Translational and torsional effects (plan view)

    Figure 5.5: Translational and torsional effects (plan view)

    35
  11. As an alternative to (3) and (6) the shape of the elastic critical buckling mode ηcr of the structure may be applied as a unique global and local imperfection. The amplitude of this imperfection may be determined from:

    Image

    where:

    Image

    and

    Image
    α is the imperfection factor for the relevant buckling curve, see Table 6.1 and Table 6.2;
    χ is the reduction factor for the relevant buckling curve depending on the relevant cross-section, see 6.3.1;
    αult,k is the minimum force amplifier for the axial force configuration NEd in members to reach the characteristic resistance NRk of the most axially stressed cross section without taking buckling into account
    αcr is the minimum force amplifier for the axial force configuration NEd in members to reach Image the elastic critical buckling load Image
    MRk is the characteristic moments resistance of the critical cross section, e.g. Mel,Rk or Mpl,Rk as relevant
    NRk is the characteristic resistance to normal force of the critical cross section, i.e. Npl,Rk
    Image is the bending moment due to ηcr at the critical cross section
    ηcr is the shape of elastic critical buckling mode

    NOTE 1 For calculating the amplifiers αult,k and αcr the members of the structure may be considered to be loaded by axial forces NEd only that result from the first order elastic analysis of the structure for the design loads. Image In case of elastic global calculation and plastic cross-section check the linear formula Image should be used. Image

    NOTE 2 The National Annex may give information for the scope of application of (11).

5.3.3 Imperfection for analysis of bracing systems

  1. In the analysis of bracing systems which are required to provide lateral stability within the length of beams or compression members the effects of imperfections should be included by means of an equivalent geometric imperfection of the members to be restrained, in the form of an initial bow imperfection:

    e0 = αm L / 500     (5.12)

    where L is the span of the bracing system

    and Image

    in which m is the number of members to be restrained.

  2. For convenience, the effects of the initial bow imperfections of the members to be restrained by a bracing system, may be replaced by the equivalent stabilizing force as shown in Figure 5.6:

    Image

    36

    where

    δq is the inplane deflection of the bracing system due to q plus any external loads calculated from first order analysis

    NOTE δq may be taken as 0 if second order theory is used.

  3. Where the bracing system is required to stabilize the compression flange of a beam of constant height, the force NEd in Figure 5.6 may be obtained from:

    NEd = MEd/h     (5.14)

    where MEd is the maximum moment in the beam
    and h is the overall depth of the beam.

    NOTE Where a beam is subjected to external compression NEd should include a part of the compression force.

  4. At points where beams or compression members are spliced, it should also be verified that the bracing system is able to resist a local force equal to αmNEd / 100 applied to it by each beam or compression member which is spliced at that point, and to transmit this force to the adjacent points at which that beam or compression member is restrained, see Figure 5.7.
  5. For checking for the local force according to clause (4), any external loads acting on bracing systems should also be included, but the forces arising from the imperfection given in (1) may be omitted.

    Figure 5.6: Equivalent stabilizing force

    Figure 5.6: Equivalent stabilizing force

    37

    Figure 5.7: Bracing forces at splices in compression elements

    Figure 5.7: Bracing forces at splices in compression elements

5.3.4 Member imperfections

  1. The effects of local bow imperfections of members are incorporated within the formulas given for buckling resistance for members, see section 6.3.
  2. Where the stability of members is accounted for by second order analysis according to 5.2.2(7)a) for compression members imperfections e0 according to 5.3.2(3)b), 5.3.2(5) or 5.3.2(6) should be considered.
  3. For a second order analysis taking account of lateral torsional buckling of a member in bending the imperfections may be adopted as ke0,d, Image where e0 is Image the equivalent initial bow imperfection of the weak axis of the profile considered. In general an additional torsional imperfection need not to be allowed for.

    NOTE The National Annex may choose the value of k. The value k = 0,5 is recommended.

5.4 Methods of analysis considering material non-linearities

5.4.1 General

  1. The internal forces and moments may be determined using either
    1. elastic global analysis
    2. plastic global analysis.

      NOTE For finite element model (FEM) analysis see EN 1993-1-5.

  2. Elastic global analysis may be used in all cases. 38
  3. Plastic global analysis may be used only where the structure has sufficient rotation capacity at the actual locations of the plastic hinges, whether this is in the members or in the joints. Where a plastic hinge occurs in a member, the member cross sections should be double symmetric or single symmetric with a plane of symmetry in the same plane as the rotation of the plastic hinge and it should satisfy the requirements specified in 5.6. Where a plastic hinge occurs in a joint the joint should either have sufficient strength to ensure the hinge remains in the member or should be able to sustain the plastic resistance for a sufficient rotation, see EN 1993-1-8.
  4. B As a simplified method for a limited plastic redistribution of moments in continuous beams where following an elastic analysis some peak moments exceed the plastic bending resistance of 15 % maximum, the parts in excess of these peak moments may be redistributed in any member, provided, that:
    1. the internal forces and moments in the frame remain in equilibrium with the applied loads, and
    2. all the members in which the moments are reduced have Class 1 or Class 2 cross-sections (see 5.5), and
    3. lateral torsional buckling of the members is prevented.

5.4.2 Elastic global analysis

  1. Elastic global analysis should be based on the assumption that the stress-strain behaviour of the material is linear, whatever the stress level is.

    NOTE For the choice of a semi-continuous joint model Image see 5.1.2 Image.

  2. Internal forces and moments may be calculated according to elastic global analysis even if the resistance of a cross section is based on its plastic resistance, see 6.2.
  3. Elastic global analysis may also be used for cross sections the resistances of which are limited by local buckling, see 6.2.

5.4.3 Plastic global analysis

  1. Plastic global analysis allows for the effects of material non-linearity in calculating the action effects of a structural system. The behaviour should be modelled by one of the following methods:
  2. Plastic global analysis may be used where the members are capable of sufficient rotation capacity to enable the required redistributions of bending moments to develop, see 5.5 and 5.6.
  3. Plastic global analysis should only be used where the stability of members at plastic hinges can be assured, see 6.3.5.
  4. The bi-linear stress-strain relationship indicated in Figure 5.8 may be used for the grades of structural steel specified in section 3. Alternatively, a more precise relationship may be adopted, see EN 1993-1-5.

    Figure 5.8: Bi-linear stress-strain relationship

    Figure 5.8: Bi-linear stress-strain relationship

    39
  5. Rigid plastic analysis may be applied if no effects of the deformed geometry (e.g. second-order effects) have to be considered. In this case joints are classified only by strength, see EN 1993-1-8.
  6. The effects of deformed geometry of the structure and the structural stability of the frame should be verified according to the principles in 5.2.

    NOTE The maximum resistance of a frame with significantly deformed geometry may occur before all hinges of the first order collapse mechanism have formed.

5.5 Classification of cross sections

5.5.1 Basis

  1. The role of cross section classification is to identify the extent to which the resistance and rotation capacity of cross sections is limited by its local buckling resistance.

5.5.2 Classification

  1. Four classes of cross-sections are defined, as follows:
  2. In Class 4 cross sections effective widths may be used to make the necessary allowances for reductions in resistance due to the effects of local buckling, see Image EN 1993-1-5, 4.4 Image.
  3. The classification of a cross-section depends on the width to thickness ratio of the parts subject to compression.
  4. Compression parts include every part of a cross-section which is either totally or partially in compression under the load combination considered.
  5. The various compression parts in a cross-section (such as a web or flange) can, in general, be in different classes.
  6. A cross-section is classified according to the highest (least favourable) class of its compression parts. Exceptions are specified in 6.2.1(10) and 6.2.2.4(1).
  7. Alternatively the classification of a cross-section may be defined by quoting both the flange classification and the web classification.
  8. The limiting proportions for Class 1, 2, and 3 compression parts should be obtained from Table 5.2. A part which fails to satisfy the limits for Class 3 should be taken as Class 4.
  9. Except as given in (10) Class 4 sections may be treated as Class 3 sections if the width to thickness ratios are less than the limiting proportions for Class 3 obtained from Table 5.2 when ε is increased by Image, where σcom,Ed is the maximum design compressive stress in the part taken from first order or where necessary second order analysis. 40
  10. However, when verifying the design buckling resistance of a member using section 6.3, the limiting proportions for Class 3 should always be obtained from Table 5.2.
  11. Cross-sections with a Class 3 web and Class 1 or 2 flanges may be classified as class 2 cross sections with an effective web in accordance with 6.2.2.4.
  12. Where the web is considered to resist shear forces only and is assumed not to contribute to the bending and normal force resistance of the cross section, the cross section may be designed as Class 2, 3 or 4 sections, depending only on the flange class.

    NOTE For flange induced web buckling see EN 1993-1-5.

5.6 Cross-section requirements for plastic global analysis

  1. At plastic hinge locations, the cross-section of the member which contains the plastic hinge should have a rotation capacity of not less than the required at the plastic hinge location.
  2. In a uniform member sufficient rotation capacity may be assumed at a plastic hinge if both the following requirements are satisfied:
    1. the member has Class 1 cross-sections at the plastic hinge location;
    2. where a transverse force that exceeds 10 % of the shear resistance of the cross section, see 6.2.6, is applied to the web at the plastic hinge location, web stiffeners should be provided within a distance along the member of h/2 from the plastic hinge location, Image where h is the height of the cross section Image.
  3. Where the cross-section of the member Image vary along its length Image, the following additional criteria should be satisfied:
    1. Adjacent to plastic hinge locations, the thickness of the web should not be reduced for a distance each way along the member from the plastic hinge location of at least 2d, where d is the clear depth of the web at the plastic hinge location.
    2. Adjacent to plastic hinge locations, the compression flange should be Class 1 for a distance each way along the member from the plastic hinge location of not less than the greater of:
      • – 2d, where d is as defined in (3)a)
      • – the distance to the adjacent point at which the moment in the member has fallen to 0,8 times the plastic moment resistance at the point concerned.
    3. Elsewhere in the member the compression flange should be class 1 or class 2 and the web should be class1, class 2 or class 3.
  4. Adjacent to plastic hinge locations, any fastener holes in tension should satisfy 6.2.5(4) for a distance such as defined in (3)b) each way along the member from the plastic hinge location.
  5. For plastic design of a frame, regarding cross section requirements, the capacity of plastic redistribution of moments may be assumed sufficient if the requirements in (2) to (4) are satisfied for all members where plastic hinges exist, may occur or have occurred under design loads.
  6. In cases where methods of plastic global analysis are used which consider the real stress and strain behaviour along the member including the combined effect of local, member and global buckling the requirements (2) to (5) need not be applied.
41
Table 5.2 (sheet 1 of 3): Maximum width-to-thickness ratios for compression parts
Internal compression parts
Image
Class Part subject
to bending
Part subject to
compression
Part subject to
bending and
compression
Stress distribution
in parts
(compression positive)
Image Image Image
1 c/t ≤ 72ε c/t ≤ 33ε Image
2 c/t ≤ 83ε c/t ≤ 38ε Image
Stress distribution
in parts
(compression positive)
Image Image Image
3 c/t ≤ 124ε c/t ≤ 42ε Image
Image fy 235 275 355 420 460
ε 1,00 0,92 0,81 0,75 0,71
*) ψ ≤ −1 applies where either the compression stress σ ≤ fy or the tensile strain εy > fy/E
42
Table 5.2 (sheet 2 of 3): Maximum width-to-thickness ratios for compression parts
Outstand flanges
Image
Class Part subject to compression Part subject to bending and compression
Tip in compression Tip in tension
Stress distribution
in parts
(compression positive)
Image Image Image
1 c/t ≤ 9ε Image Image
2 c/t ≤ 10ε Image Image
Stress distribution
in parts
(compression positive)
Image Image Image
3 c/t ≤ 14ε Image For kσ see EN 1993-1-5
Image fy 235 275 355 420 460
ε 1,00 0,92 0,81 0,75 0,71
43
Table 5.2 (sheet 3 of 3): Maximum width-to-thickness ratios for compression parts
Angles
Image
Class Section in compression
Stress distribution
across section
(compression positive)
Image
3 Image
Tubular sections
Image
Class Section in bending and/or compression
1 d/t ≤ 50ε2
2 d/t ≤ 70ε2
3 d/t ≤ 90ε2

NOTE For d/t > 90ε2 see EN 1993-1-6.

Image fy 235 275 355 420 460
ε 1,00 0,92 0,81 0,75 0,71
ε2 1,00 0,85 0,66 0,56 0,51
44

6 Ultimate limit states

6.1 General

  1. The partial factors γM as defined in 2.4.3 should be applied to the various characteristic values of resistance in this section as follows:
    resistance of cross-sections whatever the class is: γM0
    resistance of members to instability assessed by member checks: γM1
    resistance of cross-sections in tension to fracture: γM2
    resistance of joints: see EN 1993-1-8

    NOTE 1 For other recommended numerical values see EN 1993 Part 2 to Part 6. For structures not covered by EN 1993 Part 2 to Part 6 the National Annex may define the partial factors γMi; it is recommended to take the partial factors γMi from EN 1993-2.

    NOTE 2B Partial factors γMi for buildings may be defined in the National Annex. The following numerical values are recommended for buildings:

    γM0 = 1,00

    γM1 = 1,00

    γM2 = 1,25

6.2 Resistance of cross-sections

6.2.1 General

  1. Image P The design value of an action effect in each cross section shall not exceed the corresponding design resistance and if several action effects act simultaneously the combined effect shall not exceed the resistance for that combination. Image
  2. Shear lag effects and local buckling effects should be included by an effective width according to EN 1993-1-5. Shear buckling effects should also be considered according to EN 1993-1-5.
  3. The design values of resistance should depend on the classification of the cross-section.
  4. Elastic verification according to the elastic resistance may be carried out for all cross sectional classes provided the effective cross sectional properties are used for the verification of class 4 cross sections.
  5. For the elastic verification the following yield criterion for a critical point of the cross section may be used unless other interaction formulae apply, see 6.2.8 to 6.2.10.

    Image

    where

    σx,Ed is the design value of the Image text deleted Image longitudinal stress at the point of consideration
    σz,Ed is the design value of the Image text deleted Image transverse stress at the point of consideration
    τEd is the design value of the Image text deleted Image shear stress at the point of consideration

    NOTE The verification according to (5) can be conservative as it excludes partial plastic stress distribution, which is permitted in elastic design. Therefore it should only be performed where the interaction of on the basis of resistances NRd, MRd, VRd cannot be performed.

    45
  6. The plastic resistance of cross sections should be verified by finding a stress distribution which is in equilibrium with the internal forces and moments without exceeding the yield strength. This stress distribution should be compatible with the associated plastic deformations.
  7. As a conservative approximation for all cross section classes a linear summation of the utilization ratios for each stress resultant may be used. For class 1, class 2 or class 3 cross sections subjected to the combination of NEd, My,Ed and Mz,Ed this method may be applied by using the following criteria:

    Image

    where NRd, My,Rd and Mz,Rd are the design values of the resistance depending on the cross sectional classification and including any reduction that may be caused by shear effects, see 6.2.8.

    NOTE For class 4 cross sections see 6.2.9.3(2).

  8. Where all the compression parts of a cross-section are Image Class 1 or Class 2 Image, the cross-section may be taken as capable of developing its full plastic resistance in bending.
  9. Where all the compression parts of a cross-section are Class 3, its resistance should be based on an elastic distribution of strains across the cross-section. Compressive stresses should be limited to the yield strength at the extreme fibres.

    NOTE The extreme fibres may be assumed at the midplane of the flanges for ULS checks. For fatigue see EN 1993-1-9.

  10. Where yielding first occurs on the tension side of the cross section, the plastic reserves of the tension zone may be utilized by accounting for partial plastification when determining the resistance of a Class 3cross-section.

6.2.2 Section properties

6.2.2.1 Gross cross-section
  1. The properties of the gross cross-section should be determined using the nominal dimensions. Holes for fasteners need not be deducted, but allowance should be made for larger openings. Splice materials should not be included.
6.2.2.2 Net area
  1. The net area of a cross-section should be taken as its gross area less appropriate deductions for all holes and other openings.
  2. For calculating net section properties, the deduction for a single fastener hole should be the gross cross-sectional area of the hole in the plane of its axis. For countersunk holes, appropriate allowance should be made for the countersunk portion.
  3. Provided that the fastener holes are not staggered, the total area to be deducted for fastener holes should be the maximum sum of the sectional areas of the holes in any cross-section perpendicular to the member axis (see failure plane in Figure 6.1).

    NOTE The maximum sum denotes the position of the critical fracture line.

    46
  4. Where the fastener holes are staggered, the total area to be deducted for fasteners should be the greater of:
    1. the deduction for non-staggered holes given in (3)
    2. Image

    where

    s is the staggered pitch, the spacing of the centres of two consecutive holes in the chain measured parallel to the member axis;
    p is the spacing of the centres of the same two holes measured perpendicular to the member axis;
    t is the thickness;
    n is the number of holes extending in any diagonal or zig-zag line progressively across the member or part of the member, see Figure 6.1.
    d0 is the diameter of hole
  5. In an angle or other member with holes in more then one plane, the spacing p should be measured along the centre of thickness of the material (see Figure 6.2).

    Figure 6.1: Staggered holes and critical fracture lines 1 and 2

    Figure 6.1: Staggered holes and critical fracture lines 1 and 2

    Figure 6.2: Angles with holes in both legs

    Figure 6.2: Angles with holes in both legs

6.2.2.3 Shear lag effects
  1. The calculation of the effective widths is covered in EN 1993-1-5.
  2. In class 4 sections the interaction between shear lag and local buckling should be considered according to EN 1993-1-5.

    NOTE Image For cold formed members see Image EN 1993-1-3.

47
6.2.2.4 Effective properties of cross sections with class 3 webs and class 1 or 2 flanges
  1. Where cross-sections with a class 3 web and class 1 or 2 flanges are classified as effective Class 2 cross-sections, see 5.5.2(11), the proportion of the web in compression should be replaced by a part of 20εtw adjacent to the compression flange, with another part of 20εtw adjacent to the plastic neutral axis of the effective cross-section in accordance with Figure 6.3.

    Figure 6.3: Effective class 2 web

    Figure 6.3: Effective class 2 web

6.2.2.5 Effective cross-section properties of Class 4 cross-sections
  1. The effective cross-section properties of Class 4 cross-sections should be based on the effective widths of the compression parts.
  2. For Image cold formed sections Image see 1.1.2(1) and EN 1993-1-3.
  3. The effective widths of planar compression parts should be obtained from EN 1993-1-5.
  4. Where a class 4 cross section is subjected to an axial compression force, the method given in EN 1993-1-5 should be used to determine the possible shift eN of the centroid of the effective area Aeff relative to the centre of gravity of the gross cross section and the resulting additional moment:

    ΔMEd = NEdeN     (6.4)

    NOTE The sign of the additional moment depends on the effect in the combination of internal forces and moments, see 6.2.9.3(2).

  5. For circular hollow sections with class 4 cross sections see EN 1993-1-6.
48

6.2.3 Tension

  1. ImageP The design value of the tension force NEd at each cross section shall satisfy: Image

    Image

  2. For sections with holes the design tension resistance Nt,Rd should be taken as the smaller of:
    1. the design plastic resistance of the cross-section

      Image

    2. the design ultimate resistance of the net cross-section at holes for fasteners

      Image

  3. Where capacity design is requested, see EN 1998, the design plastic resistance Npl,Rd (as given in 6.2.3(2) a)) should be less than the design ultimate resistance of the net section at fasteners holes Nu,Rd (as given in 6.2.3(2) b)).
  4. In category C connections Image (see EN 1993-1-8, 3.4.1(1)) Image, the design tension resistance Nt,Rd in 6.2.3(1) of the net section at holes for fasteners should be taken as Nnet,Rd, where:

    Image

  5. For angles connected through one leg, Image see also EN 1993-1-8, 3.10.3 Image. Similar consideration should also be given to other type of sections connected through outstands.

6.2.4 Compression

  1. Image P The design value of the compression force Ned at each cross-section shall satisfy: Image

    Image

  2. The design resistance of the cross-section for uniform compression Nc,Rd should be determined as follows:

    Image

    Image

  3. Fastener holes except for oversize and slotted holes as defined in EN 1090 need not be allowed for in compression members, provided that they are filled by fasteners.
  4. In the case of unsymmetrical Class 4 sections, the method given in 6.2.9.3 should be used to allow for the additional moment ΔMEd due to the eccentricity of the centroidal axis of the effective section, see 6.2.2.5(4).
49

6.2.5 Bending moment

  1. ImageP The design value of the bending moment MEd at each cross-section shall satisfy: Image

    Image

    where

    Mc,Rd is determined considering fastener holes, see (4) to (6).
  2. The design resistance for bending about one principal axis of a cross-section is determined as follows:

    Image

    Image

    Image

    where Wel,min and Weff,min corresponds to the fibre with the maximum elastic stress.

  3. For bending about both axes, the methods given in 6.2.9 should be used.
  4. Fastener holes in the tension flange may be ignored provided that for the tension flange:

    Image

    where Af is the area of the tension flange.

    NOTE The criterion in (4) provides capacity design (see 1.5.8) Image text deleted Image.

  5. Fastener holes in tension zone of the web need not be allowed for, provided that the limit given in (4) is satisfied for the complete tension zone comprising the tension flange plus the tension zone of the web.
  6. Fastener holes except for oversize and slotted holes in compression zone of the cross-section need not be allowed for, provided that they are filled by fasteners.

6.2.6 Shear

  1. Image P The design value of the shear force VEd at each cross section shall satisfy: Image

    Image

    where Vc,Rd is the design shear resistance. For plastic design Vc,Rd is the design plastic shear resistance Vpl,Rd as given in (2). For elastic design Vc,Rd is the design elastic shear resistance calculated using (4) and (5).

  2. In the absence of torsion the design plastic shear resistance is given by:

    Image

    where Av is the shear area.

    50
  3. The shear area Av may be taken as follows:
    1. rolled I and H sections, load parallel to web     A − 2btf + (tw + 2r)tf but not less than ηhwtw
    2. rolled channel sections, load parallel to web     A − 2btf + (tw + r)tf
    3. Image rolled T-section, load parallel to web
      • – for rolled T-sections: Image
      • – for welded T-sections: Image Image
    4. welded I, H, channel and box sections, load parallel to web Image
    5. welded I, H, channel and box sections, load parallel to flanges Image
    6. rolled rectangular hollow sections of uniform thickness:
      load parallel to depth Ah/(b+h)
      load parallel to width Ab/(b+h)
    7. circular hollow sections and tubes of uniform thickness      2A/π

    where

    A is the crosssectional area;
    b is the overall breadth;
    h is the overall depth;
    hw is the depth of the web;
    r is the root radius;
    tf is the flange thickness;
    tw is the web thickness (If the web thickness in not constant, tw should be taken as the minimum thickness.).
    η see EN 1993-1-5.

    NOTE η may be conservatively taken equal 1,0.

  4. For verifying the design elastic shear resistance Vc,Rd the following criterion for a critical point of the cross section may be used unless the buckling verification in section 5 of EN 1993-1-5 applies:

    Image

    where τEd may be obtained from: Image

    VEd is the design value of the shear force
    S is the first moment of area about the centroidal axis of that portion of the cross-section between the point at which the shear is required and the boundary of the cross-section
    I is second moment of area of the whole cross section
    t is the thickness at the examined point

    NOTE The verification according to (4) is conservative as it excludes partial plastic shear distribution, which is permitted in elastic design, see (5). Therefore it should only be carried out where the verification on the basis of Vc,Rd according to equation (6.17) cannot be performed.

    51
  5. For I- or H-sections the shear stress in the web may be taken as:

    Image

    where

    Af is the area of one flange;
    Aw is the area of the web: Aw = hw tw.
  6. In addition the shear buckling resistance for webs without intermediate stiffeners should be according to section 5 of EN 1993-1-5, if

    Image

    For η see section 5 of EN 1993-1-5.

    NOTE η may be conservatively taken equal to 1,0.

  7. Fastener holes need not be allowed for in the shear verification except in verifying the design shear resistance at connection zones as given in EN 1993-1-8.
  8. Where the shear force is combined with a torsional moment, the plastic shear resistance Vpl,Rd should be reduced as specified in 6.2.7(9).

6.2.7 Torsion

  1. For members subject to torsion for which distortional deformations may be disregarded the design value of the torsional moment TEd at each cross-section should satisfy:

    Image

    where TRd is the design torsional resistance of the cross section.

  2. The total torsional moment TEd at any cross- section should be considered as the sum of two internal effects:

    TEd = Tt,Ed + Tw,Ed     (6.24)

    where

    Tt,Ed is Image the design value of the internal St. Venant torsion moment Image;
    Tw,Ed is Image the design value of the internal warping torsional moment Image.
  3. The values of Tt,Ed and Tw,Ed at any cross-section may be determined from TEd by elastic analysis, taking account of the section properties of the member, the conditions of restraint at the supports and the distribution of the actions along the member.
  4. The following stresses due to torsion should be taken into account:
  5. For the elastic verification the yield criterion in 6.2.1(5) may be applied.
  6. For determining the plastic moment resistance of a cross section due to bending and torsion only torsion effects BEd should be derived from elastic analysis, see (3).
  7. As a simplification, in the case of a member with a closed hollow cross-section, such as a structural hollow section, it may be assumed that the effects of torsional warping can be neglected. Also as a simplification, in the case of a member with open cross section, such as I or H, it may be assumed that the effects of St. Venant torsion can be neglected. 52
  8. For the calculation of the resistance TRd of closed hollow sections the design shear strength of the individual parts of the cross section according to EN 1993-1-5 should be taken into account.
  9. For combined shear force and torsional moment the plastic shear resistance accounting for torsional effects should be reduced from Vpl,Rd to Vpl,T,Rd and the design shear force should satisfy:

    Image

    in which Vpl,T,Rd may be derived as follows:

    where Vpl,Rd is given in 6.2.6.

6.2.8 Bending and shear

  1. Where the shear force is present allowance should be made for its effect on the moment resistance.
  2. Where the shear force is less than half the plastic shear resistance its effect on the moment resistance may be neglected except where shear buckling reduces the section resistance, see EN 1993-1-5.
  3. Otherwise the reduced moment resistance should be taken as the design resistance of the cross-section, calculated using a reduced yield strength

    (1 − ρ) fy     (6.29)

    for the shear area,

    where Image and Vpl,Rd is obtained from 6.2.6(2).

    NOTE See also 6.2.10(3).

  4. When torsion is present ρ should be obtained from Image, see 6.2.7, but should be taken as 0 for VEd ≤ 0,5Vpl,T,Rd. 53
  5. The reduced design plastic resistance moment allowing for the shear force may alternatively be obtained for I-cross-sections with equal flanges and bending about the major axis as follows:

    Image

    where My,c,Rd is obtained from 6.2.5(2)

    and Aw = hw tw

  6. For the interaction of bending, shear and transverse loads see section 7 of EN 1993-1-5.

6.2.9 Bending and axial force

6.2.9.1 Class 1 and 2 cross-sections
  1. Where an axial force is present, allowance should be made for its effect on the plastic moment resistance.
  2. Image P For class 1 and 2 cross sections, the following criterion shall be satisfied:Image

    MEd ≤ MN,Rd     (6.31)

    where MN,Rd is the design plastic moment resistance reduced due to the axial force NEd.

  3. For a rectangular solid section without fastener holes MN,Rd should be taken as:

    Image

  4. For doubly symmetrical I- and H-sections or other flanges sections, allowance need not be made for the effect of the axial force on the plastic resistance moment about the y-y axis when both the following criteria are satisfied:

    NEd ≤ 0,25 Npl,Rd and     (6.33)

    Image

    For doubly symmetrical I- and H-sections, allowance need not be made for the effect of the axial force on the plastic resistance moment about the z-z axis when:

    Image

  5. For cross-sections where fastener holes are not to be accounted for, the following approximations may be used for standard rolled I or H sections and for welded I or H sections with equal flanges:

    MN,y,Rd = MPl,y,Rd(1-n)/(1-0,5a) but MN,y,Rd ≤ Mpl,y,Rd     (6.36)

    for n ≤ a: MN,z,Rd = Mpl,z,Rd     (6.37)

    Image

    where

    n = NEd / Npl,Rd
    a = (A-2btf)/A but a ≤ 0,5
    54

    For cross-sections where fastener holes are not to be accounted for, the following approximations may be used for rectangular structural hollow sections of uniform thickness and for welded box sections with equal flanges and equal webs:

    MN,y,Rd = Mpl,y,Rd(1 - n)/(1 - 0,5aw) but MN,y,Rd ≤ Mpl,y,Rd     (6.39)

    MN,z,Rd = Mpl,z,Rd (1 - n)/(1 - 0,5af) but MN,z,Rd ≤ Mpl,z,Rd     (6.40)

    where

    aw = (A - 2bt)/A but aw ≤ 0,5 for hollow sections
    aw = (A-2btf)/A but aw ≤ 0,5 for welded box sections
    af = (A - 2ht)/A but af ≤ 0,5 for hollow sections
    af = (A-2htw)/A but af ≤ 0,5 for welded box sections
  6. For bi-axial bending the following criterion may be used:

    Image

    in which α and β are constants, which may conservatively be taken as unity, otherwise as follows:

6.2.9.2 Class 3 cross-sections
  1. Image P In the absence of shear force, for Class 3 cross-sections the maximum longitudinal stress shall satisfy the criterion: Image

    Image

    where σx,Ed is the design value of Image the longitudinal stress Image due to moment and axial force taking account of fastener holes where relevant, see 6.2.3, 6.2.4 and 6.2.5

6.2.9.3 Class 4 cross-sections
  1. Image P In the absence of shear force, for Class 4 cross-sections the maximum longitudinal stress σx,Ed calculated using the effective cross sections (see 5.5.2(2)) shall satisfy the criterion: Image

    Image

    where σx,Ed is the design value of Image the local longitudinal stress Image due to moment and axial force taking account of fastener holes where relevant, see 6.2.3, 6.2.4 and 6.2.5

    55
  2. Image As an alternative to the criterion in (1) the following simplified criterion may be used: Image

    Image

    where

    Aeff is the effective area of the cross-section when subjected to uniform compression
    Weff,min is the effective section modulus (corresponding to the fibre with the maximum elastic stress) of the cross-section when subjected only to moment about the relevant axis
    eN is the shift of the relevant centroidal axis when the cross-section is subjected to compression only, see 6.2.2.5(4)

    NOTE The signs of NEd, My,Ed, Mz,Ed and ΔMi = NEd eNi depend on the combination of the respective direct stresses.

6.2.10 Bending, shear and axial force

  1. Where shear and axial force are present, allowance should be made for the effect of both shear force and axial force on the resistance moment.
  2. Provided that the design value of the shear force VEd does not exceed 50% of the design plastic shear resistance Vpl,Rd no reduction of the resistances defined for bending and axial force in 6.2.9 need be made, except where shear buckling reduces the section resistance, see EN 1993-1-5.
  3. Where VEd exceeds 50% of Vpl,Rd the design resistance of the cross-section to combinations of moment and axial force should be calculated using a reduced yield strength

    (1-ρ)fy     (6.45)

    for the shear area

    where ρ = (2VEd / Vpl,Rd-1)2 and Vpl,Rd is obtained from 6.2.6(2).

    NOTE Instead of reducing the yield strength also the plate thickness of the relevant part of the cross section may be reduced.

6.3 Buckling resistance of members

6.3.1 Uniform members in compression

6.3.1.1 Buckling resistance
  1. A compression member should be verified against buckling as follows:

    Image

    where

    NEd is the design value of the compression force;
    Nb,Rd is the design buckling resistance of the compression member.
  2. For members with non-symmetric Class 4 sections allowance should be made for the additional moment ΔMEd due to the eccentricity of the centroidal axis of the effective section, see also 6.2.2.5(4), and the interaction should be carried out to 6.3.4 or 6.3.3. 56
  3. The design buckling resistance of a compression member should be taken as:

    Image

    Image

    where χ is the reduction factor for the relevant buckling mode.

    NOTE For determining the buckling resistance of members with tapered sections along the member or for non-uniform distribution of the compression force second order analysis according to 5.3.4(2) may be performed. For out-of-plane buckling see also 6.3.4.

  4. In determining A and Aeff holes for fasteners at the column ends need not to be taken into account.
6.3.1.2 Buckling curves
  1. For axial compression in members the value of χ for the appropriate non-dimensional slenderness Image should be determined from the relevant buckling curve according to:

    Image

    where

    Image

    Image for Class 1,2 and 3 cross-section
    Image for Class 4 cross-sections
    α is an imperfection factor
    Ncr is the elastic force for the appropriate bucking curve should be obtained from Table 6.1 and Table 6.2.
  2. The imperfection factor α corresponding to the appropriate buckling curve should be obtained from Table 6.1 and Table 6.2.
    Table 6.1: imperfection factors for buckling curves
    Buckling curve a0 a b c d
    Imperfection factor α 0,13 0,21 0,34 0,49 0,76
  3. Values of the reduction factor χ for the appropriate non-dimensional slenderness Image may be obtained from Figure 6.4.
  4. For slenderness Image or for Image the buckling effects may be ignored and only cross sectional checks apply.
57
Table 6.2: Selection of buckling curve for a cross-section
Cross section Limits Buckling
about axis
Buckling curve
S 235
S 275
S 355
S 420
S 460
Rolled sections Image h/b > 1,2 tf ≤ 40 mm y – y
z – z
a
b
a0
a0
40mm < tf ≤ 100 y – y
z – z
b
c
a
a
h/b ≤ 1,2 tf ≤ 100 mm y – y
z – z
b
c
a
a
tf > 100 mm y – y
z – z
d
d
c
c
Welded
I-sections
Image tf ≤ 40 mm y – y
z – z
b
c
b
c
tf > 40 mm y – y
z – z
c
d
c
d
Hollow
sections
Image hot finished any a a0
cold formed any c c
Welded box
sections
Image generally (except
as below)
any b b
thick welds:
a > 0,5tf
b/tf < 30
h/tw < 30
any c c
U-, T - and
solid sections
Image any c c
L - sections Image any b b
58

Figure 6.4: Buckling curves

Figure 6.4: Buckling curves

6.3.1.3 Slenderness for flexural buckling
  1. The non-dimensional slenderness Image is given by:

    Image

    Image

    where

    Lcr is the buckling length in the buckling plane considered
    i is the radius of gyration about the relevant axis, determined using the properties of the gross cross-section

    Image

    NOTE B For elastic buckling of components of building structures see Annex BB.

  2. For flexural buckling the appropriate buckling curve should be determined from Table 6.2. 59
6.3.1.4 Slenderness for torsional and torsional-flexural buckling
  1. For members with open cross-sections account should be taken of the possibility that the resistance of the member to either torsional or torsional-flexural buckling could be less than its resistance to flexural buckling.
  2. The non-dimensional slenderness Image for torsional or torsional-flexural buckling should be taken as:

    Image

    Image

    where

    Ncr = Ncr,TF but Ncr < Ncr,T
    Ncr,TF is the elastic torsional-flexural buckling force;
    Ncr,T is the elastic torsional buckling force.
  3. For torsional or torsional-flexural buckling the appropriate buckling curve may be determined from Table 6.2 considering the one related to the z-axis.

6.3.2 Uniform members in bending

6.3.2.1 Buckling resistance
  1. A laterally unrestrained member subject to major axis bending should be verified against lateral torsional buckling as follows:

    Image

    where

    MEd is the design value of the moment
    Mb,Rd is the design buckling resistance moment.
  2. Beams with sufficient restraint to the compression flange are not susceptible to lateral-torsional buckling. In addition, beams with certain types of cross-sections, such as square or circular hollow sections, fabricated circular tubes or square box sections are not susceptible to lateral-torsional buckling.
  3. The design buckling resistance moment of a laterally unrestrained beam should be taken as:

    Image

    where Wy is the appropriate section modulus as follows:

    Wy = Wpl,y for Class 1 or 2 cross-sections
    Wy = Wel,y for Class 3 cross-sections
    Wy = Weff,y for Class 4 cross-sections

    χLT is the reduction factor for lateral-torsional buckling.

    NOTE 1 For determining the buckling resistance of beams with tapered sections second order analysis according to 5.3.4(3) may be performed. For out-of-plane buckling see also 6.3.4.

    NOTE 2B For buckling of components of building structures see also Annex BB.

    60
  4. In determining Wy holes for fasteners at the beam end need not to be taken into account.
6.3.2.2 Lateral torsional buckling curves - General case
  1. Unless otherwise specified, see 6.3.2.3, for bending members of constant cross-section, the value of χLT for the appropriate non-dimensional slenderness Image, should be determined from:

    Image

    where

    Image

    αLT is an Imperfection factor

    Image

    Mcr is the elastic critical moment for lateral-torsional buckling

  2. Mcr is based on gross cross sectional properties and takes into account the loading conditions, the real moment distribution and the lateral restraints.

    NOTE The imperfection factor αLT corresponding to the appropriate buckling curve may be obtained from the National Annex. The recommended values αLT are given in Table 6.3.

    Table 6.3: Recommended values for imperfection factors for lateral torsional buckling curves
    Buckling curve a b c d
    Imperfection factor αLT 0,21 0,34 0,49 0,76

    The recommendations for buckling curves are given in Table 6.4.

    Table 6.4: Recommended values for lateral torsional buckling curves for cross-sections using equation (6.56)
    Cross-section Limits Buckling curve
    Rolled I-sections h/b ≤ 2
    h/b > 2
    a
    b
    Welded I-sections h/b ≤ 2
    h/b > 2
    c
    d
    Other cross-sections - d
  3. Values of the reduction factor χLT for the appropriate non-dimensional slenderness Image may be obtained from Figure 6.4.
  4. For slendernesses Image (see 6.3.2.3) or for Image (see 6.3.2.3) lateral torsional buckling effects may be ignored and only cross sectional checks apply.
61
6.3.2.3 Lateral torsional buckling curves for rolled sections or equivalent welded sections
  1. For rolled or equivalent welded sections in bending the values of χLT for the appropriate non-dimensional slenderness may be determined from

    Image

    NOTE The parameters Image and β and any limitation of validity concerning the beam depth or h/b ratio may be given in the National Annex. The following values are recommended for rolled sections or equivalent welded sections:

    Image = 0,4 (maximum value)

    β = 0,75 (minimum value)

    The recommendations for buckling curves are given in Table 6.5.

    Table 6.5: Recommendation for the selection of lateral torsional buckling curve for cross sections using equation (6.57)
    Cross-section Limits Buckling curve
    Rolled I-sections h/b ≤ 2
    h/b > 2
    b
    c
    Welded I-sections h/b ≤ 2
    h/b > 2
    c
    d
  2. For taking into account the moment distribution between the lateral restraints of members the reduction factor χLT may be modified as follows:

    Image

    NOTE The values f may be defined in the National Annex. The following minimum values are recommended:

    Image

    kc is a correction factor according to Table 6.6

    62
    Table 6.6: Correction factors kc
    Moment distribution kc
    Image 1,0


    Image
    Image
    0,94

    0,90

    0,91
    Image
    0,86

    0,77

    0,82
6.3.2.4 Simplified assessment methods for beams with restraints in buildings
  1. B Members with discrete lateral restraint to the compression flange are not susceptible to lateraltorsional buckling if the length Lc between restraints or the resulting slenderness Image of the equivalent compression flange satisfies:

    Image

    where

    My,Ed is the maximum design value of the bending moment within the restraint spacing

    Image

    Wy is the appropriate section modulus corresponding to the compression flange
    kc is a slenderness correction factor for moment distribution between restraints, see Table 6.6
    if,z is the radius of gyration of the equivalent compression flange composed of the compression flange plus 1/3 of the compressed part of the web area, about the minor axis of the section
    Image is a slenderness limit of the equivalent compression flange defined above

    Image

    63

    NOTE 1B For Class 4 cross-sections if,z may be taken as

    Image

    where

    Ieff,f is the effective second moment of area of the compression flange about the minor axis of the section
    Aeff,f is the effective area of the compression flange
    Aeff,w,c is the effective Image area Image of the compressed part of the web

    NOTE 2B The slenderness limit Image may be given in the National Annex. A limit value Image is recommended, see 6.3.2.3.

  2. B If the slenderness of the compression flange Image exceeds the limit given in (1)B, the design buckling resistance moment may be taken as:

    Mb,Rd = kfℓ χ Mb,Rd but Mb,Rd ≤ Mc,Rd     (6.60)

    where

    χ is the reduction factor of the equivalent compression flange determined with Image
    kfℓ is the modification factor accounting for the conservatism of the equivalent compression flange method

    NOTE B The modification factor may be given in the National Annex. A value kfℓ = 1,10 is recommended.

  3. B The buckling curves to be used in (2)B should be taken as follows:

    curve d for welded sections provided that: Image

    curve c for all other sections

    where

    h is the overall depth of the cross-section
    tf is the thickness of the compression flange

    NOTE B For lateral torsional buckling of components of building structures with restraints see also Annex BB.3.

6.3.3 Uniform members in bending and axial compression

  1. Unless second order analysis is earned out using the imperfections as given in 5.3.2, the stability of uniform members with double symmetric cross sections for sections not susceptible to distortional deformations should be checked as given in the following clauses, where a distinction is made for:
  2. In addition, the resistance of the cross-sections at each end of the member should satisfy the requirements given in 6.2.

    NOTE 1 The interaction formulae are based on the modelling of simply supported single span members with end fork conditions and with or without continuous lateral restraints, which are subjected to compression forces, end moments and/or transverse loads.

    64

    NOTE 2 In case the conditions of application expressed in (1) and (2) are not fulfilled, see 6.3.4.

  3. For members of structural systems the resistance check may be carried out on the basis of the individual single span members regarded as cut out of the system. Second order effects of the sway system (P-Δ-effects) have to be taken into account, either by the end moments of the member or by means of appropriate buckling lengths respectively, see 5.2.2(3)c) and 5.2.2(8).
  4. Members which are subjected to combined bending and axial compression should satisfy:

    Image

    Image

    where

    NEd, My,Ed and Mz,Ed are the design values of the compression force and the maximum moments about the y-y and z-z axis along the member, respectively
    ΔMy,Ed, ΔMz,Ed are the moments due to the shift of the centroidal axis according to 6.2.9.3 for class 4 sections, see Table 6.7,
    χy and χz are the reduction factors due to flexural buckling from 6.3.1
    χLT is the reduction factor due to lateral torsional buckling from 6.3.2
    kyy, kyz, kzy, kzz are the interaction factors
    Table 6.7: Values for NRK = fyAi, Mi,RK = fyWi and ΔMi,Ed
    Class 1 2 3 4
    Ai A A A Aeff
    Wy Wpl,y Wpl,y Wel,y Weff,y
    Wz Wpl,z Wpl,z Wel,z Weff,z
    ΔMy,Ed 0 0 0 eN,y NEd
    ΔMz,Ed 0 0 0 eN,z NEd

    NOTE For members not susceptible to torsional deformation χLT would be χLT = 1,0.

  5. The interaction factors kyy, kyz, kzy, kzz depend on the method which is chosen.

    NOTE 1 The interaction factors kyy, kyz, kzy and kzz have been derived from two alternative approaches. Values of these factors may be obtained from Annex A (alternative method 1) or from Annex B (alternative method 2).

    NOTE 2 The National Annex may give a choice from alternative method 1 or alternative method 2.

    NOTE 3 For simplicity verifications may be performed in the elastic range only.

6.3.4 General method for lateral and lateral torsional buckling of structural components

  1. The following method may be used where the methods given in 6.3.1, 6.3.2 and 6.3.3 do not apply. It allows the verification of the resistance to lateral and lateral torsional buckling for structural components such as 65

    which are subject to compression and/or mono-axial bending in the plane, but which do not contain rotative plastic hinges.

    NOTE The National Annex may specify the field and limits of application of this method.

  2. Overall resistance to out-of-plane buckling for any structural component conforming to the scope in(1) can be verified by ensuring that:

    Image

    where

    αult,k is the minimum load amplifier of the design loads to reach the characteristic resistance of the most critical cross section of the structural component considering its in plane behaviour without taking lateral or lateral torsional buckling into account however accounting for all effects due to in plane geometrical deformation and imperfections, global and local, where relevant;
    χop is the reduction factor for the non-dimensional slenderness Image, see (3), to take account of lateral and lateral torsional buckling.
  3. The global non dimensional slenderness Image for the structural component should be determined from

    Image

    where

    αult,k is defined in (2)
    αcr,op is the minimum amplifier for the in plane design loads to reach the Image elastic critical load Image of the structural component with regards to lateral or lateral torsional buckling without accounting for in plane flexural buckling

    NOTE In determining αcr,op and αult,k Finite Element analysis may be used.

  4. The reduction factor χop may be determined from either of the following methods:
    1. the minimum value of
      χ for lateral buckling according to 6.3.1
      χLT for lateral torsional buckling according to 6.3.2

      each calculated for the global non dimensional slenderness Image.

      NOTE For example where αult,k is determined by the cross section check Image this method leads to:

      Image

    2. a value interpolated between the values χ and χLT as determined in a) by using the formula for αult,k corresponding to the critical cross section

      NOTE For example where αult,k is determined by the cross section check Image this method leads to:

      66

      Image

6.3.5 Lateral torsional buckling of members with plastic hinges

6.3.5.1 General
  1. B Structures may be designed with plastic analysis provided lateral torsional buckling in the frame is prevented by the following means:
    1. restraints at locations of “rotated” plastic hinges, see 6.3.5.2, and
    2. verification of stable length of segment between such restraints and other lateral restraints, see 6.3.5.3
  2. B Where under all ultimate limit state load combinations, the plastic hinge is “not-rotated” no restraints are necessary for such a plastic hinge.
6.3.5.2 Restraints at rotated plastic hinges
  1. B At each rotated plastic hinge location the cross section should have an effective lateral and torsional restraint with appropriate resistance to lateral forces and torsion induced by local plastic deformations of the member at this location.
  2. B Effective restraint should be provided
  3. B At each plastic hinge location, the connection (e.g. bolts) of the compression flange to the resisting element at that point (e.g. purlin), and any intermediate element (e.g. diagonal brace) should be designed to resist to a local force of at least 2,5% of Nf,Ed (defined in 6.3.5.2(5)B) transmitted by the flange in its plane and perpendicular to the web plane, without any combination with other loads.
  4. B Where it is not practicable providing such a restraint directly at the hinge location, it should be provided within a distance of h/2 along the length of the member, where h is its overall depth at the plastic hinge location.
  5. B For the design of bracing systems, see 5.3.3, it should be verified by a check in addition to the check for imperfection according to 5.3.3 that the bracing system is able to resist the effects of local forces Qm applied at each stabilized member at the plastic hinge locations, where;

    Image

    where

    Nf,Ed is the axial force in the compressed flange of the stabilized member at the plastic hinge location;
    αm is according to 5.3.3(1).

    NOTE For combination with external loads see also 5.3.3(5).

6.3.5.3 Verification of stable length of segment
  1. B The lateral torsional buckling verification of segments between restraints may be performed by checking that the length between restraints is not greater than the stable length.

    For uniform beam segments with I or H cross sections with Image under linear moment and without significant axial compression the stable length may be taken from

    Image

    where

    ε Image
    ψ Image

    NOTE B For the stable length of a segment see also Annex BB.3.

  2. B Where a rotated plastic hinge location occurs immediately adjacent to one end of a haunch, the tapered segment need not be treated as a segment adjacent to a plastic hinge location if the following criteria are satisfied:
    1. the restraint at the plastic hinge location should be within a distance h/2 along the length of the tapered segment, not the uniform segment;
    2. the compression flange of the haunch remains elastic throughout its length.

      NOTE B For more information see Annex BB.3.

68

6.4 Uniform built-up compression members

6.4.1 General

  1. Uniform built-up compression members with hinged ends that are laterally supported should be designed with the following model, see Figure 6.7.
    1. The member may be considered as a column with a bow imperfection Image
    2. The elastic deformations of lacings or Image battens Image, see Figure 6.7, may be considered by a continuous (smeared) shear stiffness SV of the column.

      NOTE For other end conditions appropriate modifications may be performed.

  2. The model of a uniform built-up compression member applies when
    1. the lacings or Image battens Image consist of equal modules with parallel chords
    2. the minimum numbers of modules in a member is three.

      NOTE This assumption allows the structure to be regular and smearing the discrete structure to a continuum.

  3. The design procedure is applicable to built-up members with lacings in two planes, see Figure 6.8.
  4. The chords may be solid members or may themselves be laced or battened in the perpendicular plane.

    Figure 6.7: Uniform built-up columns with lacings and battens

    Figure 6.7: Uniform built-up columns with lacings and Image battens Image

    69

    Figure 6.8: Lacings on four sides and buckling length Lch of chords

    Figure 6.8: Lacings on four sides and buckling length Lch of chords

  5. Checks should be performed for chords using the design chord forces Nch,Ed from compression forces NEd and moments MEd at mid span of the built-up member.
  6. For a member with two identical chords the design force Nch,Ed should be determined from:

    Image

    where

    Image
    Image is the effective critical force of the built-up member
    NEd is the design value of the compression force to the built-up member
    MEd is the design value of the maximum moment in the middle of the built-up member considering second order effects
    Image is the design value of the maximum moment in the middle of the built-up member without second order effects
    h0 is the distance between the centroids of chords
    Ach is the cross-sectional area of one chord
    Ieff is the effective second moment of area of the built-up member, see 6.4.2 and 6.4.3
    SV is the shear stiffness of the lacings or battened panel, see 6.4.2 and 6.4.3.
    70
  7. The checks for the lacings of laced built-up members or for the frame moments and shear forces of the battened panels of battened built-up members should be performed for the end panel taking account of the shear force in the built-up member:

    Image

6.4.2 Laced compression members

6.4.2.1 Resistance of components of laced compression members
  1. The chords and diagonal lacings subject to compression should be designed for buckling.

    NOTE Secondary moments may be neglected.

  2. For chords the buckling verification should be performed as follows:

    Image

    where Nch,Ed is the design compression force in the chord at mid-length of the built-up member according to 6.4.1(6)
    and Nb,Rd is the design value of the buckling resistance of the chord taking the buckling length Lch from Figure 6.8.
  3. The shear stiffness SV of the lacings should be taken from Figure 6.9.
  4. The effective second order moment of area of laced built-up members may be taken as:

    Image

    Figure 6.9: Shear stiffness of lacings of built-up members

    Figure 6.9: Shear stiffness of lacings of built-up members

6.4.2.2 Constructional details
  1. Single lacing systems in opposite faces of the built-up member with two parallel laced planes should be corresponding systems as shown in Figure 6.10(a), arranged so that one is the shadow of the other. 71
  2. When the single lacing systems on opposite faces of a built-up member with two parallel laced planes are mutually opposed in direction as shown in Figure 6.10(b), the resulting torsional effects in the member should be taken into account.
  3. Tie panels should be provided at the ends of lacing systems, at points where the lacing is interrupted and at joints with other members.

    Figure 6.10: Single lacing system on opposite faces of a built-up member with two parallel laced planes

    Figure 6.10: Single lacing system on opposite faces of a built-up member with two parallel laced planes

6.4.3 Battened compression members

6.4.3.1 Resistance of components of battened compression members
  1. The chords and the battens and their joints to the chords should be checked for the actual moments and forces in an end panel and at mid-span as indicated in Figure 6.11.

    NOTE For simplicity the maximum chord forces Nch,Ed may be combined with the maximum shear force VEd.

    72

    Figure 6.11: Moments and forces in an end panel of a battened built-up member

    Figure 6.11: Moments and forces in an end panel of a battened built-up member

  2. The shear stiffness SV should be taken as follows:

    Image

  3. The effective second moments of area of battened built-up members may be taken as:

    Image

    where

    Ich = in plane second moment of area of one chord
    Ib = in plane second moment of area of one batten
    μ = efficiency factor from Table 6.8
    Image n = number of planes of battens Image
    Table 6.8: Efficiency factor μ.
    Criterion Efficiency factor μ
    λ ≥ 150 0
    75 < λ < 150 Image
    λ ≤ 75 1,0
    Image
73
6.4.3.2 Design details
  1. Battens should be provided at each end of a member.
  2. Where parallel planes of battens are provided, the battens in each plane should be arranged opposite each other.
  3. Battens should also be provided at intermediate points where loads are applied or lateral restraint is supplied.

6.4.4 Closely spaced built-up members

  1. Built-up compression members with chords in contact or closely spaced and connected through packing plates, see Figure 6.12, or star battened angle members connected by pairs of battens in two perpendicular planes, see Figure 6.1.3 should be checked for buckling as a single integral member ignoring the effect of shear stiffness (SV = ∞), when the conditions in Table 6.9 are met.

    Figure 6.12: Closely spaced built-up members

    Figure 6.12: Closely spaced built-up members

    Table 6.9: Maximum spacings for interconnections in closely spaced built-up or star battened angle members
    Type of built-up member Maximum spacing between
    interconnections *)
    Members according to Figure 6.12 connected by bolts or welds 15 imin
    Members according to Figure 6.13 connected by pair of battens 70 imin

    *) centre-to-centre distance of interconnections

    imin is the minimum radius of gyration of one chord or one angle

  2. The shear forces to be transmitted by the battens should be determined from 6.4.3.1(1).
  3. In the case of unequal-leg angles, see Figure 6.13, buckling about the y-y axis may be verified with:

    Image

    where i0 is the minimum radius of gyration of the built-up member.

    74

    Figure 6.13: Star-battened angle members

    Figure 6.13: Star-battened angle members

7 Serviceability limit states

7.1 General

  1. A steel structure should be designed and constructed such that all relevant serviceability criteria are satisfied.
  2. The basic requirements for serviceability limit states are given in 3.4 of EN 1990.
  3. Any serviceability limit state and the associated loading and analysis model should be specified for a project.
  4. Where plastic global analysis is used for the ultimate limit state, plastic redistribution of forces and moments at the serviceability limit state may occur. If so, the effects should be considered.

7.2 Serviceability limit states for buildings

7.2.1 Vertical deflections

  1. B With reference to EN 1990 – Annex A 1.4 limits for vertical deflections according to Figure A 1.1 should be specified for each project and agreed with the client.

    NOTE B The National Annex may specify the limits.

7.2.2 Horizontal deflections

  1. B With reference to EN 1990 – Annex A1.4 limits for horizontal deflections according to Figure A1.2 should be specified for each project and agreed with the client.

    NOTE B The National Annex may specify the limits.

7.2.3 Dynamic effects

  1. B With reference to EN 1990 – Annex A 1.4.4 the vibrations of structures on which the public can walk should be limited to avoid significant discomfort to users, and limits should be specified for each project and agreed with the client.

    NOTE B The National Annex may specify limits for vibration of floors.

75

Annex A – Method 1: Interaction factors kij for interaction formula in 6.3.3(4)

[informative]

Table A.1: interaction factors kjj (6.3.3(4))

Table A.1: interaction factors kjj (6.3.3(4))

76

Image

77
Table A.2: Equivalent uniform moment factors Cmi,0
Moment diagram Cmi,0
Image Image
Image Image

Mi,Ed (x) is the maximum moment My,Ed or Mz,Ed Image according
to the first order analyses Image

X| is the maximum member Image deflection Image along the member

Image

Image

78

Annex B – Method 2: Interaction factors kij for interaction formula in 6.3.3(4)

[informative]

Table B.1: interaction factors kij for members not susceptible to torsional deformations
Interaction factors Type of sections Design assumptions
elastic cross-sectional properties
class 3, class 4
plastic cross-sectional properties
class 1, class 2
kyy I-sections RHS-sections Image Image
kyz I-sections RHS-sections kzz 0,6 kzz
kzy I-sections RHS-sections 0,8 kyy 0,6 kyy
kzz I-sections Image Image
RHS-sections Image
For I- and H-sections and rectangular hollow sections under axial compression and uniaxial bending My,Ed the coefficient kzy may be kzy = 0.
Table B.2: interaction factors kij for members susceptible to torsional deformations
Interaction factors Design assumptions
elastic cross-sectional properties
class 3, class 4
plastic cross-sectional properties
class 1, class 2
Kyy kyy from Table B.1 kyy from Table B.1
kyz kyz from Table B.1 kyz from Table B.1
kzy Image Image
Image 79
kzz kzz from Table B.1 kzz from Table B.
Table B.3: Equivalent uniform moment factors Cm in Tables B.1 and B.2
Moment diagram range Cmy and Cmz and CmLT
uniform loading concentrated load
Image −1 ≤ Ψ ≤ 1 0,6 + 0,4Ψ ≥ 0,4
Image 0 ≤ αs ≤ 1 −1 ≤ Ψ ≤ 1 0,2 + 0,8αs ≥ 0,4 0,2 + 0,8αs ≥ 0,4
−1 ≤ αs < 0 0 ≤ Ψ ≤ 1 0,1 – 0,8αs ≥ 0,4 −0,8αs ≥ 0,4
−1 ≤ Ψ < 0 0,l(1 – Ψ) – 0,8αs ≥ 0,4 0,2(–Ψ) – 0,8αs ≥ 0,4
Image 0 ≤ αh ≤ 1 −1 ≤ Ψ ≤ 1 0,95 + 0,05αh 0,90 + 0,10αh
−1 ≤ αh < 0 0 ≤ Ψ ≤ 1 0,95 + 0,05αh 0,90 + 0,10αh
−1 ≤ Ψ < 0 0,95 + 0,05αh(1 + 2Ψ) Image 0,90 + 0,10αh(1 + 2Ψ) Image
For members with sway buckling mode the equivalent uniform moment factor should be taken Cmy = 0,9 or Image Cmz Image = 0,9 respectively.
Cmy, Cmz and CmLT should be obtained according to the bending moment diagram between the relevant braced points as follows:
moment factor bending axis points braced in direction
Cmy y-y z-z
Cmz z-z y-y
CmLT y-y y-y
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Annex AB – Additional design provisions

[informative]

AB.1 structural analysis taking account of material non-linearities

  1. B In case of material non-linearities the action effects in a structure may be determined by incremental approach to the design loads to be considered for the relevant design situation.
  2. B In this incremental approach each permanent or variable action should be increased proportionally.

AB.2 simplified provisions for the design of continuous floor beams

  1. B For continuous beams with slabs in buildings without cantilevers on which uniformly distributed loads are dominant, it is sufficient to consider only the following load arrangements:
    1. alternative spans carrying the design permanent and variable load (γG Gk + γQ Qk), all other spans carrying only the design permanent load γG Gk
    2. any two adjacent spans carrying the design permanent and variable loads (γG Gk + γQ Qk), all other spans carrying only the design permanent load γG Gk

      NOTE 1 a) applies to sagging moments, b) to hogging moments.

      NOTE 2 This annex is intended to be transferred to EN 1990 in a later stage.

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Annex BB – Buckling of components of building structures

[informative]

BB.1 Flexural buckling of members in triangulated and lattice structures

BB.1.1 General

  1. B For chord members generally and for out-of-plane buckling of web members, the buckling length Lcr may be taken as equal to the system length L, see BB.1.3(1)B, unless a smaller value can be justified by analysis.
  2. B The buckling length Lcr of an I or H section chord member may be taken as 0,9L for in-plane buckling and 1,0L for out-of-plane buckling, unless a smaller value is justified by analysis.
  3. B Web members may be designed for in-plane buckling using a buckling length smaller than the system length, provided the chords supply appropriate end restraint and the end connections supply appropriate fixity (at least 2 bolts if bolted).
  4. B Under these conditions, in normal triangulated structures the buckling length Lcr of web members for in-plane buckling may be taken as 0,9L, except for angle sections, sec BB.1.2.

BB.1.2 Angles as web members

  1. B Provided that the chords supply appropriate end restraint to web members made of angles and the end connections of such web members supply appropriate fixity (at least two bolts if bolted), the eccentricities may be neglected and end fixities allowed for in the design of angles as web members in compression. The effective slenderness ratio Image may be obtained as follows:

    Image

    where Image is as defined in 6.3.1.2.

  2. B When only one bolt is used for end connections of angle web members the eccentricity should be taken into account using 6.2.9 and the buckling length Lcr should be taken as equal to the system length L.

BB.1.3 Hollow sections as members

  1. B The buckling length Lcr of a hollow section chord member may be taken as 0,9L for both in-plane and out-of-plane buckling, where L is the system length for the relevant plane. The in-plane system length is the distance between the joints. The out-of-plane system length is the distance between the lateral supports, unless a smaller value is justified by analysis.
  2. B The buckling length Lcr of a hollow section brace member (web member) with bolted connections may be taken as 1,0L for both in-plane and out-of-plane buckling.
  3. Image B The buckling length Lcr of a hollow section brace member without cropping or flattening, welded around its perimeter to hollow section chords, may be generally taken as 0,75L for both in-plane and out-of-plane buckling. Lower buckling lengths may be used based on testing or calculations. In this case the buckling length of the cord may not be reduced. Image

    NOTE The National Annex may give more information on buckling lengths.

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BB.2 Continuous restraints

BB.2.1 Continuous lateral restraints

  1. B If trapezoidal sheeting according to EN 1993-1-3 is connected to a beam and the condition expressed Image by formula (BB.2) Image is met, the beam at the connection may be regarded as being laterally restrained in the plane of the sheeting.

    Image

    where

    S is the shear stiffness (per unit of beam length) provided by the sheeting to the beam regarding its deformation in the plane of the sheeting Image to be connected to the beam at the bottom at each rib Image.
    Iw is the warping constant
    Image IT Image is the torsion constant
    Iz is the second moment of area of the cross section about the minor axis of the cross section
    L is the beam length
    h is the depth of the beam

    If the sheeting is connected to a beam at every second rib only, S should be substituted by 0,20S.

    NOTE Image Formula (BB.2) Image may also be used to determine the lateral stability of beam flanges used in combination with other types of cladding than trapezoidal sheeting, provided that the connections are of suitable design.

BB.2.2 Continuous torsional restraints

  1. B A beam may be considered as sufficiently restraint from torsional deformations if

    Image

    where

    Cϑ,k = rotational stiffness (per unit of beam length) provided to the beam by the stabilizing continuum (e.g. roof structure) and the connections
    Kυ = 0,35 for elastic analysis
    Kυ = 1,00 for plastic analysis
    Kϑ = factor for considering the moment distribution see Table BB.1 and the type of restraint
    Mpl,k = characteristic value of the plastic moment of the beam
    83
    Table BB.1: Factor kϑ for considering the moment distribution and the type of restraint
    Case Moment distribution without translational restraint with translational restraint
    1 Image 4,0 0
    2a Image 3,5 0,12
    2b Image 0,23
    3 Image 2,8 0
    4 Image 1,6 1,0
    5 Image 1,0 0,7
  2. B The rotational stiffness provided by the stabilizing continuum to the beam may be calculated from

    Image

    where

    CϑR,k = rotational stiffness (per unit of the beam length) provided by the stabilizing continuum to the beam assuming a stiff connection to the member
    CϑC,k = rotational stiffness (per unit of the beam length) of the connection between the beam and the stabilizing continuum
    CϑD,k = rotational stiffness (per unit of the beam length) deduced from an analysis of the distorsional deformations of the beam cross sections, where the flange in compression is the free one; where the compression flange is the connected one or where distorsional deformations of the cross sections may be neglected (e.g. for usual rolled profiles)CϑD,k = ∞

    NOTE For more information see EN 1993-1-3.

BB.3 Stable lengths of segment containing plastic hinges for out-of-plane buckling

BB.3.1 Uniform members made of rolled sections or equivalent welded I-sections

BB.3.1.1 Stable lengths between adjacent lateral restraints
  1. B Lateral torsional buckling effects may be ignored where the length L of the segment of a member between the restrained section at a plastic hinge location and the adjacent lateral restraint is not greater than Lm, where: 84

    Image

    where

    NEd is the design value of the compression force [N] in the member
    A is the cross section area [mm2] of the member
    Wpl,y is the plastic section modulus of the member
    Image IT Image is the torsion constant of the member
    fy is the yield strength in [N/mm2]
    Image Cl is a factor depending on the loading and end conditions and may be taken as Image where kc is to be taken from Table 6.6.Image

    provided that the member is restrained at the hinge as required by 6.3.5 and that the other end of the segment is restrained

    see Figure BB.1, Figure BB.2 and Figure BB3.

    NOTE In general Ls is greater than Lm.

    Figure BB.1: Checks in a member without a haunch

    Figure BB.1: Checks in a member without a haunch

    85

    Figure BB.2: Checks in a member with a three flange haunch

    Figure BB.2: Checks in a member with a three flange haunch

    Figure BB.3: Checks in a member with a two flange haunch

    Figure BB.3: Checks in a member with a two flange haunch

86
BB.3.1.2 Stable length between torsional restraints
  1. B Lateral torsional buckling effects may be ignored where the length L of the segment of a member between the restrained section at a plastic hinge location and the adjacent torsional restraint subject to a constant moment is not greater than Lk, provided that

    where

    Image

  2. B Lateral torsional buckling effects may be ignored where the length L of the segment of a member between the restrained section at a plastic hinge location and the adjacent torsional restraint subject to a linear moment gradient and axial compression is not greater than Ls, provided that
  3. B Lateral torsional buckling effects may be ignored where the length L of a segment of a member between the restrained section at a plastic hinge location and the adjacent torsional restraint subject to a non linear moment gradient and axial compression is not greater than Ls, provided that
87

BB.3.2 Haunched or tapered members made of rolled sections or equivalent welded I-sections

BB.3.2.1 Stable length between adjacent lateral restraints
  1. B Lateral torsional buckling effects may be ignored where the length L of the segment of a member between the restrained section at a plastic hinge location and the adjacent lateral restraint is not greater than Lm, where

    where

    NEd is the design value of the compression force [N] in the member
    Image is the maximum value in the segment
    A is the cross sectional area [mm2] at the location where Image is a maximum of the tapered member
    Image Cl is a factor depending on the loading and end conditions and may be taken as Image where kc is to be taken from Table 6.6. Image
    Wpl,y is the plastic section modulus of the member
    Image IT Image is the torsional constant of the member
    fy is the yield strength in [N/mm2]
    iz is the minimum value of the radius of gyration in the segment

    provided that the member is restrained at the hinge as required by 6.3.5 and that the other end of segment is restrained

BB.3.2.2 Stable length between torsional restraints
  1. B For non uniform members with constant flanges under linear or non-linear moment gradient and axial compression, lateral torsional buckling effects may be ignored where the length L of the segment of a member between the restrained section at a plastic hinge location and the adjacent torsional restraint is not greater than Ls, provided that 88

    where

    where

    Lk is the length derived for a uniform member with a cross-section equal to the shallowest section, see BB.3.1.2
    Cn seeBB.3.3.2
    c is the taper factor defined in BB.3.3.3

BB.3.3 Modification factors for moment gradients in members laterally restrained along the tension flange

BB.3.3.1 Linear moment gradients
  1. B The modification factor Cm may be determined from

    Image

    in which

    Image

    Image

    Image

    Image

    Image

    Lt is the distance between the torsional restraints

    Image is the elastic critical torsional buckling force for an I-section between restraints to both flanges at spacing Lt with intermediate lateral restraints to the tension flange.

    Image

    where

    a is the distance between the centroid of the member and the centroid of the restraining members, such as purlins restraining rafters 89
    βt is the ratio of the algebraically smaller end moment to the larger end moment. Moments that produce compression in the non-restrained flange should be taken as positive. If the ratio is less than −1,0 the value of βt should be taken as −1,0, see Figure BB.4.

    Figure BB.4: Value of βt

    Figure BB.4: Value of βt

BB.3.3.2 Non linear moment gradients
  1. B The modification factor Cn may be determined from

    Image

    in which R1 to R5 are the values of R according to (2)B at the ends, quarter points and mid-length, see Figure BB.5, and only positive values of R should be included.

    In addition, only positive values of (Rs – RE) should be included, where

  2. B The value of R should be obtained from:

    Image

    90

    where

    a is the distance between the centroid of the member and the centroid of the restraining members, such as purlins restraining rafters.
BB.3.3.3 Taper factor
  1. B For a non uniform member with constant flanges, for which h ≥ 1,2b and h/tf ≥ 20 the taper factor c should be obtained as follows:

    where

    hh is the additional depth of the haunch or taper, see Figure BB.6;
    hmax is the maximum depth of cross-section within the length Ly, see Figure BB.6;
    hmin is the minimum depth of cross-section within the length Ly, see Figure BB.6;
    hs is the vertical depth of the un-haunched section, see Figure BB.6;
    Lh is the length of haunch within the length Ly, see Figure BB.6;
    Ly is the length between points at which the compression flange is laterally restrained.

    (h/tf) is to be derived from the shallowest section.

    Figure BB.6: Dimensions defining taper factor

    Figure BB.6: Dimensions defining taper factor

91