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EUROPEAN STANDARD
NORME EUROPÉENNE
EUROPÄISCHE NORM
EN 199311
May 2005
ICS 91.010.30; 91.080.10
Supersedes ENV 199311:1992
Incorporating Corrigenda February 2006
and March 2009
English version
Eurocode 3: Calcul des structures en acier  Partie 11: Règles générales et règles pour les bâtiments  Eurocode 3: Bemessung und Konstruktion von Stahlbauten  Teil 11: 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. Uptodate 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 199311:2005: E
1Page  
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  Throughthickness 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  Groundstructure 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 nonlinearities  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  Crosssection requirements for plastic global analysis  41  
6  Ultimate limit states  45  
6.1  General  45  
6.2  Resistance of crosssections  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 builtup 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 builtup 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 k_{ij} for interaction formula in 6.3.3(4)  76 3  
Annex B [informative] – Method 2: Interaction factors k_{ij} 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 
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 199311.
According to the CENCENELEC 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.
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 agreement^{1} 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.
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 Documents^{2} referred to in Article 12 of the CPD, although they are of a different nature from harmonized product standard^{3}. 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.
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. :
^{2}According 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.
^{3}According to Art. 12 of the CPD the interpretative documents shall :
The Eurocodes, de facto, play a similar role in the field of the ER 1 and a part of ER 2.
There is a need for consistency between the harmonized technical specifications for construction products and the technical rules for works^{4}. 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.
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 19931 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 19932 to EN 19936. It also gives supplementary rules applicable only to buildings.
EN 19931 comprises twelve subparts EN 199311 to EN 1993112 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 19931 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.
7This standard gives values with notes indicating where national choices may have to be made. Therefore the National Standard implementing EN 19931 should have a National Annex containing all Nationally Determined Parameters to be used for the design of steel structures and civil engineering works to be constructed in the relevant country.
National choice is allowed in EN 199311 through the following clauses:
EN 19931  Design of Steel Structures : General rules and rules for buildings. 
EN 19932  Design of Steel Structures: Steel bridges. 
EN 19933  Design of Steel Structures : Towers, masts and chimneys. 
EN 19934  Design of Steel Structures : Silos, tanks and pipelines. 
EN 19935  Design of Steel Structures : Piling. 
EN 19936  Design of Steel Structures: Crane supporting structures. 
EN 199311  Design of Steel Structures : General rules and rules for buildings. 
EN 199312  Design of Steel Structures : Structural fire design. 
EN 199313  Design of Steel Structures: Coldformed members and sheeting . 
EN 199314  Design of Steel Structures : Stainless steels. 
EN 199315  Design of Steel Structures : Plated structural elements. 
EN 199316  Design of Steel Structures : Strength and stability of shell structures. 
EN 199317  Design of Steel Structures : Strength and stability of planar plated structures transversely loaded. 
EN 199318  Design of Steel Structures : Design of joints. 
EN 199319  Design of Steel Structures : Fatigue strength of steel structures. 
EN 1993110  Design of Steel Structures : Selection of steel for fracture toughness and throughthickness properties. 
EN 1993111  Design of Steel Structures : Design of structures with tension components made of steel. 
EN 1993112  Design of Steel Structures : Supplementary rules for high strength steel. 
NOTE For cold formed members and sheeting, see EN 199313 .
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 
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).
EN 1090  Execution of steel structures – Technical requirements 
EN ISO 12944  Paints and varnishes – Corrosion protection of steel structures by protective paint systems 
EN ISO 1461  Hot dip galvanized coatings on fabricated iron and steel articles – specifications and test methods 
EN 100251:2004  Hotrolled products of structural steels  Part 1: General delivery conditions. 
EN 100252:2004  Hotrolled products of structural steels  Part 2: Technical delivery conditions for nonalloy structural steels. 
EN 100253:2004  Hotrolled products of structural steels  Part 3: Technical delivery conditions for normalized / normalized rolled weldable fine grain structural steels. 10 
EN 100254:2004  Hotrolled products of structural steels  Part 4: Technical delivery conditions for thermomechanical rolled weldable fine grain structural steels. 
EN 100255:2004  Hotrolled products of structural steels  Part 5: Technical delivery conditions for structural steels with improved atmospheric corrosion resistance. 
EN 100256:2004  Hotrolled 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 102101:1994  Hot finished structural hollow sections of nonalloy and fine grain structural steels Part 1: Technical delivery requirements. 
EN 102191:1997  Cold formed hollow sections of structural steel  Part 1: Technical delivery requirements. 
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 momentresisting frames and triangulated frames; it covers both plane frames and threedimensional frames
a frame that forms part of a larger frame, but is be treated as an isolated frame in a structural analysis
terms used to distinguish between frames that are either:
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
11distance 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
system length of an otherwise similar member with pinned ends, which has the same critical buckling load as a given member or segment of member
nonuniform stress distribution in wide flanges due to shear deformation; it is taken into account by using a reduced “effective” flange width in safety assessments
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
member with a constant crosssection along its whole length
NOTE Symbols are ordered by appearance in EN 199311. Symbols may have various meanings.
Section 1
xx  axis along a member 
yy  axis of a crosssection 
zz  axis of a crosssection 
uu  major principal axis (where this does not coincide with the yy axis) 
vv  minor principal axis (where this does not coincide with the zz axis) 
b  width of a cross section 
h  depth of a cross section 
d  depth of straight portion of a web 
t_{w}  web thickness 
t_{f}  flange thickness 
r  radius of root fillet 
r_{1}  radius of root fillet 
r_{2}  toe radius 
t  thickness 
Section 2
P_{k}  nominal value of the effect of prestressing imposed during erection 
G_{k}  nominal value of the effect of permanent actions 12 
X_{k}  characteristic values of material property 
X_{n}  nominal values of material property 
R_{d}  design value of resistance 
R_{k}  characteristic value of resistance 
γ_{M}  general partial factor 
γ_{Mi}  particular partial factor 
γ_{Mf}  partial factor for fatigue 
η  conversion factor 
a_{d}  design value of geometrical data 
Section 3
f_{y}  yield strength 
f_{u}  ultimate strength 
R_{eH}  yield strength to product standards 
R_{m}  ultimate strength to product standards 
A_{0}  original crosssection area 
ε_{y}  yield strain 
ε_{u}  ultimate strain 
Z_{Ed}  required design Zvalue resulting from the magnitude of strains from restrained metal shrinkage under the weld beads. 
Z_{Rd}  available design Zvalue 
E  modulus of elasticity 
G  shear modulus 
ν  Poisson’s ratio in elastic stage 
α  coefficient of linear thermal expansion 
Section 5
α_{cr}  factor by which the design loads would have to be increased to cause elastic instability in a global mode 
F_{Ed}  design loading on the structure 
F_{cr}  elastic critical buckling load for global instability mode based on initial elastic stiffnesses 
H_{Ed}  total design horizontal load, including equivalent forces transferred by the storey (storey shear) 
V_{Ed}  total design vertical load on the frame transferred by the storey (storey thrust) 
δ_{H,Ed}  horizontal displacement at the top of the storey, relative to the bottom of the storey 
h  storey height 
non dimensional slenderness  
N_{Ed}  design value of the axial force 
ϕ  global initial sway imperfection 
ϕ_{0}  basic value for global initial sway imperfection 
α_{h}  reduction factor for height h applicable to columns 
h  height of the structure 13 
α_{m}  reduction factor for the number of columns in a row 
m  number of columns in a row 
e_{0}  maximum amplitude of a member imperfection 
L  member length 
η_{init}  amplitude of elastic critical buckling mode 
η_{cr}  shape of elastic critical buckling mode 
e_{0,d}  design value of maximum amplitude of an imperfection 
M_{Rk}  characteristic moment resistance of the critical cross section 
N_{Rk}  characteristic resistance to normal force of the critical cross section 
α  imperfection factor 
bending moment due to η_{cr} at the critical cross section  
χ  reduction factor for the relevant buckling curve 
α_{ult,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 
α_{cr}  minimum force amplifier to reach the elastic critical buckling load 
q  equivalent force per unit length 
δ_{q}  inplane deflection of a bracing system 
q_{d}  equivalent design force per unit length 
M_{Ed}  design bending moment 
k  factor for e_{(0,d)} 
ε  strain 
σ  stress 
σ_{com,Ed}  maximum design compressive stress in an element 
ℓ  length 
ε  factor depending on f_{y} 
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 
k_{σ}  plate buckling factor 
d  outer diameter of circular tubular sections 
Section 6
γ_{M0}  partial factor for resistance of crosssections whatever the class is 
γ_{M1}  partial factor for resistance of members to instability assessed by member checks 
γ_{M2}  partial factor for resistance of crosssections in tension to fracture 
σ_{x,Ed}  design value of the local longitudinal stress 
σ_{z,Ed}  design value of the local transverse stress 
τ_{Ed}  design value of the local shear stress 
N_{Ed}  design normal force 
M_{y,Ed}  design bending moment, yy axis 
M_{z,Ed}  design bending moment, zz axis 
N_{Rd}  design values of the resistance to normal forces 14 
M_{y,Rd}  design values of the resistance to bending moments, yy axis 
M_{z,Rd}  design values of the resistance to bending moments, zz 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 zigzag line progressively across the member or part of the member 
d_{0}  diameter of hole 
e_{N}  shift of the centroid of the effective area A_{eff} relative to the centre of gravity of the gross cross section 
ΔM_{Ed}  additional moment from shift of the centroid of the effective area A_{eff} relative to the centre of gravity of the gross cross section 
A_{eff}  effective area of a cross section 
N_{t,Rd}  design values of the resistance to tension forces 
N_{pl,Rd}  design plastic resistance to normal forces of the gross crosssection 
N_{u,Rd}  design ultimate resistance to normal forces of the net crosssection at holes for fasteners 
A_{net}  net area of a cross section 
N_{net,Rd}  design plastic resistance to normal forces of the net crosssection 
N_{c,Rd}  design resistance to normal forces of the crosssection for uniform compression 
M_{c,Rd}  design resistance for bending about one principal axis of a crosssection 
W_{pl}  plastic section modulus 
W_{el,min}  minimum elastic section modulus 
W_{eff,min}  minimum effective section modulus 
A_{f}  area of the tension flange 
A_{f,net}  net area of the tension flange 
V_{Ed}  design shear force 
V_{c,Rd}  design shear resistance 
V_{pl,Rd}  design plastic shear resistance 
A_{v}  shear area 
η  factor for shear area 
S  first moment of area 
I  second moment of area 
A  crosssectional area 
A_{w}  area of a web 
A_{f}  area of one flange 
T_{Ed}  design value of total torsional moments 
T_{Rd}  design resistance to torsional moments 
T_{t,Ed}  design value of internal St. Venant torsional moment 
T_{W,Ed}  design value of internal warping torsional moment 
τ_{t,Ed}  design shear stresses due to St. Venant torsion 
τ_{w,Ed}  design shear stresses due to warping torsion 
σ_{w,Ed}  design direct stresses due to the bimoment B_{Ed} 
B_{Ed}  design value of the bimoment 
V_{pl,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 
M_{V,,Rd}  reduced design values of the resistance to bending moments making allowance for the presence of shear forces 
M_{N,,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 crosssection 
a  ratio of web area to gross area 
α  parameter introducing the effect of biaxial bending 
β  parameter introducing the effect of biaxial bending 
e_{N,y}  shift of the centroid of the effective area A_{eff} relative to the centre of gravity of the gross cross section (yy axis) 
e_{N,z}  shift of the centroid of the effective area A_{eff} relative to the centre of gravity of the gross cross section (zz axis) 
W_{eff,min}  minimum effective section modulus 
N_{b,Rd}  design buckling resistance of a compression member 
χ  reduction factor for relevant buckling mode 
Φ  value to determine the reduction factor χ 
a_{0}, a, b, c, d  class indexes for buckling curves 
N_{cr}  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 crosssection 
λ_{l}  slenderness value to determine the relative slenderness 
relative slenderness for torsional or torsionalflexural buckling  
N_{cr,TF}  elastic torsionalflexural buckling force 
N_{cr,T}  elastic torsional buckling force 
M_{b,Rd}  design buckling resistance moment 
χ_{LT}  reduction factor for lateraltorsional buckling 
Φ_{LT}  value to determine the reduction factor χ_{LT} 
α_{LT}  imperfection factor 
non dimensional slenderness for lateral torsional buckling  
M_{cr}  elastic critical moment for lateraltorsional buckling 
plateau length of the lateral torsional buckling curves for rolled and welded sections  
β  correction factor for the lateral torsional buckling curves for rolled and welded sections 
χ_{LT,mod}  modified reduction factor for lateraltorsional buckling 
f  modification factor for χ_{LT} 
k_{c}  correction factor for moment distribution 
ψ  ratio of moments in segment 
L_{c}  length between lateral restraints 
equivalent compression flange slenderness  
i_{f,z}  radius of gyration of compression flange about the minor axis of the section 
I_{eff,f}  effective second moment of area of compression flange about the minor axis of the section 16 
A_{eff,f}  effective area of compression flange 
A_{eff,w,c}  effective area of compressed part of web 
slenderness parameter  
k_{fℓ}  modification factor 
ΔM_{y,Ed}  moments due to the shift of the centroidal yy axis 
ΔM_{z,Ed}  moments due to the shift of the centroidal zz axis 
χ_{y}  reduction factor due to flexural buckling (yy axis) 
χ_{z}  reduction factor due to flexural buckling (zz axis) 
k_{yy}  interaction factor 
k_{yz}  interaction factor 
k_{zy}  interaction factor 
k_{zz}  interaction factor 
global non dimensional slenderness of a structural component for outofplane buckling  
χ_{op}  reduction factor for the nondimensional slenderness 
α_{ult,k}  minimum load amplifier of the design loads to reach the characteristic resistance of the most critical cross section 
α_{cr,op}  minimum amplifier for the in plane design loads to reach the elastic critical buckling load with regard to lateral or lateral torsional buckling 
N_{Rk}  characteristic value of resistance to compression 
M_{y,Rk}  characteristic value of resistance to bending moments about yy axis 
M_{z,Rk}  characteristic value of resistance to bending moments about zz axis 
Q_{m}  local force applied at each stabilized member at the plastic hinge locations 
L_{stable}  stable length of segment 
L_{ch}  buckling length of chord 
h_{0}  distance of centrelines of chords of a builtup column 
a  distance between restraints of chords 
α  angle between axes of chord and lacings 
i_{min}  minimum radius of gyration of single angles 
A_{ch}  area of one chord of a builtup column 
N_{ch,Ed}  design chord force in the middle of a builtup member 
design value of the maximum first order moment in the middle of the builtup member  
I_{eff}  effective second moment of area of the builtup member 
S_{v}  shear stiffness of builtup member from the lacings or battened panel 
n  number of planes of lacings or battens 
A_{d}  area of one diagonal of a builtup column 
d  length of a diagonal of a builtup column 
A_{V}  area of one post (or transverse element) of a builtup column 
I_{ch}  in plane second moment of area of a chord 
I_{b}  in plane second moment of area of a batten 
μ  efficiency factor 17 
i_{y}  radius of gyration (yy axis) 
Annex A
C_{my}  equivalent uniform moment factor 
C_{mz}  equivalent uniform moment factor 
C_{mLT}  equivalent uniform moment factor 
μ_{y}  factor 
μ_{z}  factor 
N_{cr,y}  elastic flexural buckling force about the yy axis 
N_{cr,z}  elastic flexural buckling force about the zz axis 
C_{yy}  factor 
C_{yz}  factor 
C_{zy}  factor 
C_{zz}  factor 
w_{y}  factor 
w_{z}  factor 
n_{pl}  factor 
maximum of  
b_{LT}  factor 
c_{LT}  factor 
d_{LT}  factor 
e_{LT}  factor 
ψ_{y}  ratio of end moments (yy axis) 
C_{my,0}  factor 
C_{mz,0}  factor 
a_{LT}  factor 
I_{T}  St. Venant torsional constant 
I_{y}  second moment of area about yy axis 
C_{l}  ratio between the critical bending moment (largest value along the member) and the critical constant bending moment for a member with hinged supports 
M_{i,Ed}(X)  maximum first order moment 
δ_{x}  maximum member displacement along the member 
Annex B
α_{s}  factor; s = sagging 
α_{h}  factor; h = hogging 
C_{m}  equivalent uniform moment factor 
Annex AB
γ_{G}  partial factor for permanent loads 
G_{k}  characteristic value of permanent loads 
γ_{Q}  partial factor for variable loads 
Q_{k}  characteristic value of variable loads 
Annex BB
effective slenderness ratio for buckling about vv axis  
effective slenderness ratio for buckling about yy axis  
effective slenderness ratio for buckling about zz axis  
L  system length 
L_{cr}  buckling length 
S  shear stiffness provided by sheeting 
I_{w}  warping constant 
C_{ϑ,k}  rotational stiffness provided by stabilizing continuum and connections 
K_{υ}  factor for considering the type of analysis 
K_{ϑ}  factor for considering the moment distribution and the type of restraint 
C_{ϑR,k}  rotational stiffness provided by the stabilizing continuum to the beam assuming a stiff connection to the member 
C_{ϑC,k}  rotational stiffness of the connection between the beam and the stabilizing continuum 
C_{ϑD,k}  rotational stiffness deduced from an analysis of the distorsional deformations of the beam cross sections 
L_{m}  stable length between adjacent lateral restraints 
L_{k}  stable length between adjacent torsional restraints 
L_{s}  stable length between a plastic hinge location and an adjacent torsional restraint 
C_{l}  modification factor for moment distribution 
C_{m}  modification factor for linear moment gradient 
C_{n}  modification factor for nonlinear moment gradient 
a  distance between the centroid of the member with the plastic hinge and the centroid of the restraint members 
B_{0}  factor 
B_{1}  factor 
B_{2}  factor 
η  ratio of elastic critical values of axial forces 
i_{s}  radius of gyration related to centroid of restraining member 
β_{t}  ratio of the algebraically smaller end moment to the larger end moment 
R_{1}  moment at a specific location of a member 
R_{2}  moment at a specific location of a member 
R_{3}  moment at a specific location of a member 
R_{4}  moment at a specific location of a member 
R_{5}  moment at a specific location of a member 
R_{E}  maximum of R_{1} or R_{5} 
R_{s}  maximum value of bending moment anywhere in the length L_{y} 
c  taper factor 
h_{h}  additional depth of the haunch or taper 
h_{max}  maximum depth of crosssection within the length L_{y} 
h_{min}  minimum depth of crosssection within the length L_{y} 19 
h_{s}  vertical depth of the unhaunched section 
L_{h}  length of haunch within the length L_{y} 
L_{y}  length between restraints 
xx   along the member 
yy   axis of the crosssection 
zz   axis of the crosssection 
yy   crosssection axis parallel to the flanges 
zz   crosssection axis perpendicular to the flanges 
yy   axis parallel to the smaller leg 
zz   axis perpendicular to the smaller leg 
uu   major principal axis (where this does not coincide with the yy axis) 
vv   minor principal axis (where this does not coincide with the zz axis) 
NOTE All rules in this Eurocode relate to principal axis properties, which are generally defined by the axes yy and zz but for sections such as angles are defined by the axes uu and vv.
Figure 1.1: Dimensions and axes of sections
21NOTE 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.
where
R_{k}  is the characteristic value of the particular resistance determined with characteristic or nominal values for the material properties and dimensions 
γ_{M}  is the global partial factor for the particular resistance 
NOTE For the definitions of η_{l}, η_{i}, X_{kl}, X_{ki} and a_{d} see EN 1990.
R_{k} = R_{d} γ_{Mi} (2.2)
where
R_{d}  are design values according to Annex D of EN 1990 
γ_{Mi}  are recommended partial factors. 
NOTE 1 The numerical values of the recommended partial factors γ_{Mi} have been determined such that R_{k} represents approximately the 5 %fractile for an infinite number of tests.
NOTE 2 For characteristic values of fatigue strength and partial factors γ_{Mf} for fatigue see EN 199319.
NOTE 3 For characteristic values of toughness resistance and safety elements for the toughness verification see EN 1993110.
NOTE For other steel material and products see National Annex.
NOTE The National Annex may give the choice.
NOTE The limiting values of the ratio f_{u} / f_{y}, the elongation at failure and the ultimate strain ε_{u} may be defined in the National Annex. The following values are recommended:
NOTE The lowest service temperature to be adopted in design may be given in the National Annex.
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 1993110 for σ_{Ed} = 0,25 f_{y}(t) is recommended.
Standard and steel grade  Nominal thickness of the element t [mm]  
t ≤ 40 mm  40 mm < t ≤ 80 mm  
f_{y} [N/mm^{2}]  f_{u} [N/mm^{2}]  f_{y} [N/mm^{2}]  f_{u} [N/mm^{2}]  
EN 100252  
S 235  235  360  215  360 
S 275  275  430  255  410 
S 355  355  490  335  470 
S 450  440  550  410  550 
EN 100253  
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 100254  
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 100255  
S 235 W  235  360  215  340 
S 355 W  355  490  335  490 
EN 100256  
S 460 Q/QL/QL1  460  570  440  550 26 
EN 102101  
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 
S 420 NH/NLH  420  540  390  520 
S 460 NH/NLH  460  560  430  550 
EN 102191  
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 
NOTE 1 Guidance on the choice of throughthickness properties is given in EN 1993110.
NOTE 2B Particular care should be given to welded beam to column connections and welded end plates with tension in the throughthickness direction.
NOTE 3B The National Annex may give the relevant allocation of target values Z_{Ed} according to 3.2(2) of EN 1993110 to the quality class in EN 10164. The allocation in Table 3.2 is recommended for buildings:
Target value of Z_{Ed} according to EN 1993110  Required value of Z_{Rd} expressed in terms of design Zvalues according to EN 10164 
Z_{Ed} ≤ 10  — 
10 < Z_{Ed} ≤ 20  Z 15 
20 < Z_{Ed} ≤ 30  Z 25 
Z_{Ed} > 30  Z 35 
–  modulus of elasticity  E = 210 000 N / mm^{2} 
–  shear modulus  
–  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 concretesteel structures to EN 1994 the coefficient of linear thermal expansion is taken as α = 10 × 10^{−6} per K.
NOTE EN 1090 lists the factors affecting execution that need to be specified during design.
NOTE EN 1997 gives guidance for calculation of soilstructure interaction.
where
α_{cr}  is the factor by which the design loading would have to be increased to cause elastic instability in a global mode 
F_{Ed}  is the design loading on the structure 
F_{cr}  is the elastic critical buckling load for global instability mode based on initial elastic stiffnesses 
NOTE A greater limit for α_{cr} 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 semirigid joints occur). Where substantiated by more accurate approaches the National Annex may give a lower limit for α_{cr} for certain types of frames.
where
H_{Ed}  is the total design horizontal load, including equivalent forces according to 5.3.2(7), transferred by the storey (storey shear) 
V_{Ed}  is the total design vertical load on the frame transferred by the storey (storey thrust) 
δ_{H,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 
Figure 5.1: Notations for 5.2.1(4)
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 should be assumed to be significant if
where
N_{Ed} is the design value of the compression force, 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.
NOTE For rolled sections and welded sections with similar dimensions shear lag effects may be neglected.
provided that α_{cr} ≥ 3,0,
where
α_{cr}  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 α_{cr} < 3,0 a more accurate second order analysis applies.
NOTE B For the limitation of the method see also 5.2.1(4)B.
NOTE The National Annex may give information on the scope of application.
ϕ = ϕ_{0} α_{h} α_{m} (5.5)
where
ϕ_{0}  is the basic value: ϕ_{0} = 1/200 
α  is the reduction factor for height h applicable to columns: 
h  is the height of the structure in meters 
α_{m}  is the reduction factor for the number of columns in a row: 
m  is the number of columns in a row including only those columns which carry a vertical load N_{Ed} not less than 50% of the average value of the column in the vertical plane considered 
Figure 5.2: Equivalent sway imperfections
e_{0}/L (5.6)
where L is the member length
NOTE The values e_{0} / L may be chosen in the National Annex. Recommended values are given in Table 5.1.
Buckling curve according to Table 6.2  elastic analysis  plastic analysis 
e_{0} / L  e_{0} / L  
a_{0}  1 / 350  1 / 300 
a  1 / 300  1 / 250 
b  1 / 250  1 / 200 
c  1 / 200  1 / 150 
d  1 / 150  1 / 100 
H_{Ed} ≥ 0,15 V_{Ed} (5.7)
Figure 5.3: Configuration of sway imperfections ϕ for horizontal forces on floor diaphragms
where  N_{Ed}  is the design value of the compression force 
and  is the inplane nondimensional slenderness calculated for the member considered as hinged at its ends 
34NOTE Local bow imperfections are taken into account in member checks, see 5.2.2 (3) and 5.3.4.
Figure 5.4: Replacement of initial imperfections by equivalent horizontal forces
Figure 5.5: Translational and torsional effects (plan view)
35where:
and
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 crosssection, see 6.3.1; 
α_{ult,k}  is the minimum force amplifier for the axial force configuration N_{Ed} in members to reach the characteristic resistance N_{Rk} of the most axially stressed cross section without taking buckling into account 
α_{cr}  is the minimum force amplifier for the axial force configuration N_{Ed} in members to reach the elastic critical buckling load 
M_{Rk}  is the characteristic moments resistance of the critical cross section, e.g. M_{el,Rk} or M_{pl,Rk} as relevant 
N_{Rk}  is the characteristic resistance to normal force of the critical cross section, i.e. N_{pl,Rk} 
is the bending moment due to η_{cr} at the critical cross section  
η_{cr}  is the shape of elastic critical buckling mode 
NOTE 1 For calculating the amplifiers α_{ult,k} and α_{cr} the members of the structure may be considered to be loaded by axial forces N_{Ed} only that result from the first order elastic analysis of the structure for the design loads. In case of elastic global calculation and plastic crosssection check the linear formula should be used.
NOTE 2 The National Annex may give information for the scope of application of (11).
e_{0} = α_{m} L / 500 (5.12)
where L is the span of the bracing system
and
in which m is the number of members to be restrained.
where
δ_{q}  is the inplane deflection of the bracing system due to q plus any external loads calculated from first order analysis 
NOTE δ_{q} may be taken as 0 if second order theory is used.
N_{Ed} = M_{Ed}/h (5.14)
where  M_{Ed}  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 N_{Ed} should include a part of the compression force.
Figure 5.6: Equivalent stabilizing force
37Figure 5.7: Bracing forces at splices in compression elements
NOTE The National Annex may choose the value of k. The value k = 0,5 is recommended.
NOTE For finite element model (FEM) analysis see EN 199315.
NOTE For the choice of a semicontinuous joint model see 5.1.2 .
Figure 5.8: Bilinear stressstrain relationship
39NOTE The maximum resistance of a frame with significantly deformed geometry may occur before all hinges of the first order collapse mechanism have formed.
NOTE For flange induced web buckling see EN 199315.
Internal compression parts 

Class  Part subject to bending 
Part subject to compression 
Part subject to bending and compression 

Stress distribution in parts (compression positive) 

1  c/t ≤ 72ε  c/t ≤ 33ε  
2  c/t ≤ 83ε  c/t ≤ 38ε  
Stress distribution in parts (compression positive) 

3  c/t ≤ 124ε  c/t ≤ 42ε  
f_{y}  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 ≤ f_{y} or the tensile strain ε_{y} > f_{y}/E 
Outstand flanges 

Class  Part subject to compression  Part subject to bending and compression  
Tip in compression  Tip in tension  
Stress distribution in parts (compression positive) 

1  c/t ≤ 9ε  
2  c/t ≤ 10ε  
Stress distribution in parts (compression positive) 

3  c/t ≤ 14ε  For k_{σ} see EN 199315  
f_{y}  235  275  355  420  460  
ε  1,00  0,92  0,81  0,75  0,71 
Angles 

Class  Section in compression  
Stress distribution across section (compression positive) 

3  
Tubular sections 

Class  Section in bending and/or compression  
1  d/t ≤ 50ε^{2}  
2  d/t ≤ 70ε^{2}  
3  d/t ≤ 90ε^{2}
NOTE For d/t > 90ε^{2} see EN 199316. 

f_{y}  235  275  355  420  460  
ε  1,00  0,92  0,81  0,75  0,71  
ε^{2}  1,00  0,85  0,66  0,56  0,51 
–  resistance of crosssections whatever the class is:  γ_{M0} 
–  resistance of members to instability assessed by member checks:  γ_{M1} 
–  resistance of crosssections in tension to fracture:  γ_{M2} 
–  resistance of joints:  see EN 199318 
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 γ_{Mi}; it is recommended to take the partial factors γ_{Mi} from EN 19932.
NOTE 2B Partial factors γ_{Mi} for buildings may be defined in the National Annex. The following numerical values are recommended for buildings:
γ_{M0} = 1,00
γ_{M1} = 1,00
γ_{M2} = 1,25
where
σ_{x,Ed}  is the design value of the text deleted longitudinal stress at the point of consideration 
σ_{z,Ed}  is the design value of the text deleted transverse stress at the point of consideration 
τ_{Ed}  is the design value of the text deleted shear stress at the point of consideration 
45NOTE 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 N_{Rd}, M_{Rd}, V_{Rd} cannot be performed.
where N_{Rd}, M_{y,Rd} and M_{z,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).
NOTE The extreme fibres may be assumed at the midplane of the flanges for ULS checks. For fatigue see EN 199319.
46NOTE The maximum sum denotes the position of the critical fracture line.
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 zigzag line progressively across the member or part of the member, see Figure 6.1. 
d_{0}  is the diameter of hole 
Figure 6.1: Staggered holes and critical fracture lines 1 and 2
Figure 6.2: Angles with holes in both legs
NOTE For cold formed members see EN 199313.
Figure 6.3: Effective class 2 web
ΔM_{Ed} = N_{Ed}e_{N} (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).
where
M_{c,Rd}  is determined considering fastener holes, see (4) to (6). 
where W_{el,min} and W_{eff,min} corresponds to the fibre with the maximum elastic stress.
where A_{f} is the area of the tension flange.
NOTE The criterion in (4) provides capacity design (see 1.5.8) text deleted .
where V_{c,Rd} is the design shear resistance. For plastic design V_{c,Rd} is the design plastic shear resistance V_{pl,Rd} as given in (2). For elastic design V_{c,Rd} is the design elastic shear resistance calculated using (4) and (5).
where A_{v} is the shear area.
50load parallel to depth  Ah/(b+h) 
load parallel to width  Ab/(b+h) 
where
A  is the crosssectional area; 
b  is the overall breadth; 
h  is the overall depth; 
h_{w}  is the depth of the web; 
r  is the root radius; 
t_{f}  is the flange thickness; 
t_{w}  is the web thickness (If the web thickness in not constant, t_{w} should be taken as the minimum thickness.). 
η  see EN 199315. 
NOTE η may be conservatively taken equal 1,0.
where τ_{Ed} may be obtained from:
V_{Ed}  is the design value of the shear force 
S  is the first moment of area about the centroidal axis of that portion of the crosssection between the point at which the shear is required and the boundary of the crosssection 
I  is second moment of area of the whole cross section 
t  is the thickness at the examined point 
51NOTE 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 V_{c,Rd} according to equation (6.17) cannot be performed.
where
A_{f}  is the area of one flange; 
A_{w}  is the area of the web: A_{w} = h_{w} t_{w}. 
For η see section 5 of EN 199315.
NOTE η may be conservatively taken equal to 1,0.
where T_{Rd} is the design torsional resistance of the cross section.
T_{Ed} = T_{t,Ed} + T_{w,Ed} (6.24)
where
T_{t,Ed}  is the design value of the internal St. Venant torsion moment ; 
T_{w,Ed}  is the design value of the internal warping torsional moment . 
in which V_{pl,T,Rd} may be derived as follows:
where V_{pl,Rd} is given in 6.2.6.
(1 − ρ) f_{y} (6.29)
for the shear area,
where and V_{pl,Rd} is obtained from 6.2.6(2).
NOTE See also 6.2.10(3).
where M_{y,c,Rd} is obtained from 6.2.5(2)
and A_{w} = h_{w} t_{w}
M_{Ed} ≤ M_{N,Rd} (6.31)
where M_{N,Rd} is the design plastic moment resistance reduced due to the axial force N_{Ed}.
N_{Ed} ≤ 0,25 N_{pl,Rd} and (6.33)
For doubly symmetrical I and Hsections, allowance need not be made for the effect of the axial force on the plastic resistance moment about the zz axis when:
M_{N,y,Rd} = M_{Pl,y,Rd}(1n)/(10,5a) but M_{N,y,Rd} ≤ M_{pl,y,Rd} (6.36)
for n ≤ a: M_{N,z,Rd} = M_{pl,z,Rd} (6.37)
where
n  =  N_{Ed} / N_{pl,Rd} 
a  =  (A2bt_{f})/A but a ≤ 0,5 
For crosssections 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:
M_{N,y,Rd} = M_{pl,y,Rd}(1  n)/(1  0,5a_{w}) but M_{N,y,Rd} ≤ M_{pl,y,Rd} (6.39)
M_{N,z,Rd} = M_{pl,z,Rd} (1  n)/(1  0,5a_{f}) but M_{N,z,Rd} ≤ M_{pl,z,Rd} (6.40)
where
a_{w}  =  (A  2bt)/A  but  a_{w} ≤ 0,5  for hollow sections 
a_{w}  =  (A2bt_{f})/A  but  a_{w} ≤ 0,5  for welded box sections 
a_{f}  =  (A  2ht)/A  but  a_{f} ≤ 0,5  for hollow sections 
a_{f}  =  (A2ht_{w})/A  but  a_{f} ≤ 0,5  for welded box sections 
in which α and β are constants, which may conservatively be taken as unity, otherwise as follows:
α = 2; β = 5n but β ≥ l
α = 2; β = 2
M_{N,y,Rd} = M_{N,z,Rd} = M_{pl,Rd} (1n^{1,7})
where n = N_{Ed} / N_{pl,Rd}.
where σ_{x,Ed} is the design value of the longitudinal stress due to moment and axial force taking account of fastener holes where relevant, see 6.2.3, 6.2.4 and 6.2.5
where σ_{x,Ed} is the design value of the local longitudinal stress due to moment and axial force taking account of fastener holes where relevant, see 6.2.3, 6.2.4 and 6.2.5
55where
A_{eff}  is the effective area of the crosssection when subjected to uniform compression 
W_{eff,min}  is the effective section modulus (corresponding to the fibre with the maximum elastic stress) of the crosssection when subjected only to moment about the relevant axis 
e_{N}  is the shift of the relevant centroidal axis when the crosssection is subjected to compression only, see 6.2.2.5(4) 
NOTE The signs of N_{Ed}, M_{y,Ed}, M_{z,Ed} and ΔM_{i} = N_{Ed} e_{Ni} depend on the combination of the respective direct stresses.
(1ρ)f_{y} (6.45)
for the shear area
where ρ = (2V_{Ed} / V_{pl,Rd}1)^{2} and V_{pl,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.
where
N_{Ed}  is the design value of the compression force; 
N_{b,Rd}  is the design buckling resistance of the compression member. 
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 nonuniform distribution of the compression force second order analysis according to 5.3.4(2) may be performed. For outofplane buckling see also 6.3.4.
where
for Class 1,2 and 3 crosssection  
for Class 4 crosssections 
α  is an imperfection factor 
N_{cr}  is the elastic force for the appropriate bucking curve should be obtained from Table 6.1 and Table 6.2. 
Buckling curve  a_{0}  a  b  c  d 
Imperfection factor α  0,13  0,21  0,34  0,49  0,76 
Cross section  Limits  Buckling about axis 
Buckling curve  
S 235 S 275 S 355 S 420 
S 460  
Rolled sections  h/b > 1,2  t_{f} ≤ 40 mm  y – y z – z 
a b 
a_{0} a_{0} 

40mm < t_{f} ≤ 100  y – y z – z 
b c 
a a 

h/b ≤ 1,2  t_{f} ≤ 100 mm  y – y z – z 
b c 
a a 

t_{f} > 100 mm  y – y z – z 
d d 
c c 

Welded Isections 
t_{f} ≤ 40 mm  y – y z – z 
b c 
b c 

t_{f} > 40 mm  y – y z – z 
c d 
c d 

Hollow sections 
hot finished  any  a  a_{0}  
cold formed  any  c  c  
Welded box sections 
generally (except as below) 
any  b  b  
thick welds: a > 0,5t_{f} b/t_{f} < 30 h/t_{w} < 30 
any  c  c  
U, T  and solid sections 
any  c  c  
L  sections  any  b  b 
Figure 6.4: Buckling curves
where
L_{cr}  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 crosssection 
NOTE B For elastic buckling of components of building structures see Annex BB.
where
N_{cr} = N_{cr,TF} but N_{cr} < N_{cr,T}  
N_{cr,TF}  is the elastic torsionalflexural buckling force; 
N_{cr,T}  is the elastic torsional buckling force. 
where
M_{Ed}  is the design value of the moment 
M_{b,Rd}  is the design buckling resistance moment. 
where W_{y} is the appropriate section modulus as follows:
–  W_{y} = W_{pl,y}  for Class 1 or 2 crosssections 
–  W_{y} = W_{el,y}  for Class 3 crosssections 
–  W_{y} = W_{eff,y}  for Class 4 crosssections 
60χ_{LT} is the reduction factor for lateraltorsional 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 outofplane buckling see also 6.3.4.
NOTE 2B For buckling of components of building structures see also Annex BB.
where
α_{LT} is an Imperfection factor
M_{cr} is the elastic critical moment for lateraltorsional buckling
NOTE The imperfection factor α_{LT} corresponding to the appropriate buckling curve may be obtained from the National Annex. The recommended values α_{LT} are given in Table 6.3.
Buckling curve  a  b  c  d 
Imperfection factor α_{LT}  0,21  0,34  0,49  0,76 
The recommendations for buckling curves are given in Table 6.4.
Crosssection  Limits  Buckling curve 
Rolled Isections  h/b ≤ 2 h/b > 2 
a b 
Welded Isections  h/b ≤ 2 h/b > 2 
c d 
Other crosssections    d 
NOTE The parameters 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:
= 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) Crosssection Limits Buckling curve Rolled Isections h/b ≤ 2
h/b > 2b
cWelded Isections h/b ≤ 2
h/b > 2c
d
62NOTE The values f may be defined in the National Annex. The following minimum values are recommended:
k_{c} is a correction factor according to Table 6.6
Moment distribution  k_{c} 

1,0 

0,94 0,90 0,91 

0,86 0,77 0,82 
where
M_{y,Ed}  is the maximum design value of the bending moment within the restraint spacing 
W_{y}  is the appropriate section modulus corresponding to the compression flange 
k_{c}  is a slenderness correction factor for moment distribution between restraints, see Table 6.6 
i_{f,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 
is a slenderness limit of the equivalent compression flange defined above 
NOTE 1B For Class 4 crosssections i_{f,z} may be taken as
where
I_{eff,f}  is the effective second moment of area of the compression flange about the minor axis of the section 
A_{eff,f}  is the effective area of the compression flange 
A_{eff,w,c}  is the effective area of the compressed part of the web 
NOTE 2B The slenderness limit may be given in the National Annex. A limit value is recommended, see 6.3.2.3.
M_{b,Rd} = k_{fℓ} χ M_{b,Rd} but M_{b,Rd} ≤ M_{c,Rd} (6.60)
where
χ  is the reduction factor of the equivalent compression flange determined with 
k_{fℓ}  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 k_{fℓ} = 1,10 is recommended.
curve d for welded sections provided that:
curve c for all other sections
where
h  is the overall depth of the crosssection 
t_{f}  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.
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.
64NOTE 2 In case the conditions of application expressed in (1) and (2) are not fulfilled, see 6.3.4.
where
N_{Ed}, M_{y,Ed} and M_{z,Ed}  are the design values of the compression force and the maximum moments about the yy and zz axis along the member, respectively 
ΔM_{y,Ed}, ΔM_{z,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, 
χ_{y} and χ_{z}  are the reduction factors due to flexural buckling from 6.3.1 
χ_{LT}  is the reduction factor due to lateral torsional buckling from 6.3.2 
k_{yy}, k_{yz}, k_{zy}, k_{zz}  are the interaction factors 
Class  1  2  3  4 
A_{i}  A  A  A  A_{eff} 
W_{y}  W_{pl,y}  W_{pl,y}  W_{el,y}  W_{eff,y} 
W_{z}  W_{pl,z}  W_{pl,z}  W_{el,z}  W_{eff,z} 
ΔM_{y,Ed}  0  0  0  e_{N,y} N_{Ed} 
ΔM_{z,Ed}  0  0  0  e_{N,z} N_{Ed} 
NOTE For members not susceptible to torsional deformation χ_{LT} would be χ_{LT} = 1,0.
NOTE 1 The interaction factors k_{yy}, k_{yz}, k_{zy} and k_{zz} 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.
which are subject to compression and/or monoaxial 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.
where
α_{ult,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; 
χ_{op}  is the reduction factor for the nondimensional slenderness , see (3), to take account of lateral and lateral torsional buckling. 
where
α_{ult,k}  is defined in (2) 
α_{cr,op}  is the minimum amplifier for the in plane design loads to reach the elastic critical load of the structural component with regards to lateral or lateral torsional buckling without accounting for in plane flexural buckling 
NOTE In determining α_{cr,op} and α_{ult,k} Finite Element analysis may be used.
χ  for lateral buckling according to 6.3.1 
χ_{LT}  for lateral torsional buckling according to 6.3.2 
each calculated for the global non dimensional slenderness .
NOTE For example where α_{ult,k} is determined by the cross section check this method leads to:
66NOTE For example where α_{ult,k} is determined by the cross section check this method leads to:
Figure 6.5: Typical stiff torsional restraint
Figure 6.6: Typical lateral and torsional restraint by a slab to the compression flange
67where
N_{f,Ed}  is the axial force in the compressed flange of the stabilized member at the plastic hinge location; 
α_{m}  is according to 5.3.3(1). 
NOTE For combination with external loads see also 5.3.3(5).
For uniform beam segments with I or H cross sections with under linear moment and without significant axial compression the stable length may be taken from
where
ε  
ψ 
NOTE B For the stable length of a segment see also Annex BB.3.
NOTE B For more information see Annex BB.3.
NOTE For other end conditions appropriate modifications may be performed.
NOTE This assumption allows the structure to be regular and smearing the discrete structure to a continuum.
Figure 6.7: Uniform builtup columns with lacings and battens
69Figure 6.8: Lacings on four sides and buckling length L_{ch} of chords
where
is the effective critical force of the builtup member  
N_{Ed}  is the design value of the compression force to the builtup member 
M_{Ed}  is the design value of the maximum moment in the middle of the builtup member considering second order effects 
is the design value of the maximum moment in the middle of the builtup member without second order effects  
h_{0}  is the distance between the centroids of chords 
A_{ch}  is the crosssectional area of one chord 
I_{eff}  is the effective second moment of area of the builtup member, see 6.4.2 and 6.4.3 
S_{V}  is the shear stiffness of the lacings or battened panel, see 6.4.2 and 6.4.3. 
NOTE Secondary moments may be neglected.
where  N_{ch,Ed}  is the design compression force in the chord at midlength of the builtup member according to 6.4.1(6) 
and  N_{b,Rd}  is the design value of the buckling resistance of the chord taking the buckling length L_{ch} from Figure 6.8. 
Figure 6.9: Shear stiffness of lacings of builtup members
Figure 6.10: Single lacing system on opposite faces of a builtup member with two parallel laced planes
72NOTE For simplicity the maximum chord forces N_{ch,Ed} may be combined with the maximum shear force V_{Ed}.
Figure 6.11: Moments and forces in an end panel of a battened builtup member
where
I_{ch}  =  in plane second moment of area of one chord 
I_{b}  =  in plane second moment of area of one batten 
μ  =  efficiency factor from Table 6.8 
n  =  number of planes of battens 
Criterion  Efficiency factor μ 
λ ≥ 150  0 
75 < λ < 150  
λ ≤ 75  1,0 
Figure 6.12: Closely spaced builtup members
Type of builtup member  Maximum spacing between interconnections *) 
Members according to Figure 6.12 connected by bolts or welds  15 i_{min} 
Members according to Figure 6.13 connected by pair of battens  70 i_{min} 
*) centretocentre distance of interconnections i_{min} is the minimum radius of gyration of one chord or one angle 
where i_{0} is the minimum radius of gyration of the builtup member.
74Figure 6.13: Starbattened angle members
NOTE B The National Annex may specify the limits.
NOTE B The National Annex may specify the limits.
NOTE B The National Annex may specify limits for vibration of floors.
[informative]
Table A.1: interaction factors k_{jj} (6.3.3(4))
76 77Moment diagram  C_{mi,0} 
M_{i,Ed} (x) is the maximum moment M_{y,Ed} or M_{z,Ed} according δ_{X} is the maximum member deflection along the member 

[informative]
Interaction factors  Type of sections  Design assumptions  
elastic crosssectional properties class 3, class 4 
plastic crosssectional properties class 1, class 2 

k_{yy}  Isections RHSsections  
k_{yz}  Isections RHSsections  k_{zz}  0,6 k_{zz} 
k_{zy}  Isections RHSsections  0,8 k_{yy}  0,6 k_{yy} 
k_{zz}  Isections  
RHSsections  
For I and Hsections and rectangular hollow sections under axial compression and uniaxial bending M_{y,Ed} the coefficient k_{zy} may be k_{zy} = 0. 
Interaction factors  Design assumptions  
elastic crosssectional properties class 3, class 4 
plastic crosssectional properties class 1, class 2 

K_{yy}  k_{yy} from Table B.1  k_{yy} from Table B.1 
k_{yz}  k_{yz} from Table B.1  k_{yz} from Table B.1 
k_{zy}  79 

k_{zz}  k_{zz} from Table B.1  k_{zz} from Table B. 
Moment diagram  range  C_{my} and C_{mz} and C_{mLT}  
uniform loading  concentrated load  
−1 ≤ Ψ ≤ 1  0,6 + 0,4Ψ ≥ 0,4  
0 ≤ α_{s} ≤ 1  −1 ≤ Ψ ≤ 1  0,2 + 0,8α_{s} ≥ 0,4  0,2 + 0,8α_{s} ≥ 0,4  
−1 ≤ α_{s} < 0  0 ≤ Ψ ≤ 1  0,1 – 0,8α_{s} ≥ 0,4  −0,8α_{s} ≥ 0,4  
−1 ≤ Ψ < 0  0,l(1 – Ψ) – 0,8α_{s} ≥ 0,4  0,2(–Ψ) – 0,8α_{s} ≥ 0,4  
0 ≤ α_{h} ≤ 1  −1 ≤ Ψ ≤ 1  0,95 + 0,05α_{h}  0,90 + 0,10α_{h}  
−1 ≤ α_{h} < 0  0 ≤ Ψ ≤ 1  0,95 + 0,05α_{h}  0,90 + 0,10α_{h}  
−1 ≤ Ψ < 0  0,95 + 0,05α_{h}(1 + 2Ψ)  0,90 + 0,10α_{h}(1 + 2Ψ)  
For members with sway buckling mode the equivalent uniform moment factor should be taken C_{my} = 0,9 or C_{mz} = 0,9 respectively.  
C_{my}, C_{mz} and C_{mLT} should be obtained according to the bending moment diagram between the relevant braced points as follows:

[informative]
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.
[informative]
where is as defined in 6.3.1.2.
NOTE The National Annex may give more information on buckling lengths.
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 to be connected to the beam at the bottom at each rib . 
I_{w}  is the warping constant 
I_{T}  is the torsion constant 
I_{z}  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 Formula (BB.2) 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.
where
C_{ϑ,k}  =  rotational stiffness (per unit of beam length) provided to the beam by the stabilizing continuum (e.g. roof structure) and the connections 
K_{υ}  =  0,35 for elastic analysis 
K_{υ}  =  1,00 for plastic analysis 
K_{ϑ}  =  factor for considering the moment distribution see Table BB.1 and the type of restraint 
M_{pl,k}  =  characteristic value of the plastic moment of the beam 
Case  Moment distribution  without translational restraint  with translational restraint 
1  4,0  0  
2a  3,5  0,12  
2b  0,23  
3  2,8  0  
4  1,6  1,0  
5  1,0  0,7 
where
C_{ϑR,k}  =  rotational stiffness (per unit of the beam length) provided by the stabilizing continuum to the beam assuming a stiff connection to the member 
C_{ϑC,k}  =  rotational stiffness (per unit of the beam length) of the connection between the beam and the stabilizing continuum 
C_{ϑD,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)C_{ϑD,k} = ∞ 
NOTE For more information see EN 199313.
where
N_{Ed}  is the design value of the compression force [N] in the member 
A  is the cross section area [mm^{2}] of the member 
W_{pl,y}  is the plastic section modulus of the member 
I_{T}  is the torsion constant of the member 
f_{y}  is the yield strength in [N/mm^{2}] 
C_{l}  is a factor depending on the loading and end conditions and may be taken as where k_{c} is to be taken from Table 6.6. 
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 L_{s} is greater than L_{m}.
Figure BB.1: Checks in a member without a haunch
85Figure BB.2: Checks in a member with a three flange haunch
Figure BB.3: Checks in a member with a two flange haunch
where
where
C_{m}  is the modification factor for linear moment gradient, see BB.3.3.1; 
a  is the distance between the centroid of the member with the plastic hinge and the centroid of the restraint members; 
M_{pl,y,Rk}  is the characteristic plastic moment resistance of the cross section about the yy axis 
M_{N,y,Rk}  is the characteristic plastic moment resistance of the cross section about the yy axis with reduction due to the axial force N_{Ed} 
where
C_{n} is the modification factor for nonlinear moment gradient, see BB.3.3.2,
see Figure BB.1, Figure BB.2 and Figure BB.3.
where
N_{Ed}  is the design value of the compression force [N] in the member 
is the maximum value in the segment  
A  is the cross sectional area [mm^{2}] at the location where is a maximum of the tapered member 
C_{l}  is a factor depending on the loading and end conditions and may be taken as where k_{c} is to be taken from Table 6.6. 
W_{pl,y}  is the plastic section modulus of the member 
I_{T}  is the torsional constant of the member 
f_{y}  is the yield strength in [N/mm^{2}] 
i_{z}  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
where
where
L_{k}  is the length derived for a uniform member with a crosssection equal to the shallowest section, see BB.3.1.2 
C_{n}  seeBB.3.3.2 
c  is the taper factor defined in BB.3.3.3 
in which
L_{t} is the distance between the torsional restraints
is the elastic critical torsional buckling force for an Isection between restraints to both flanges at spacing L_{t} with intermediate lateral restraints to the tension flange.
where
a  is the distance between the centroid of the member and the centroid of the restraining members, such as purlins restraining rafters 89 
β_{t}  is the ratio of the algebraically smaller end moment to the larger end moment. Moments that produce compression in the nonrestrained flange should be taken as positive. If the ratio is less than −1,0 the value of β_{t} should be taken as −1,0, see Figure BB.4. 
Figure BB.4: Value of β_{t}
in which R_{1} to R_{5} are the values of R according to (2)B at the ends, quarter points and midlength, see Figure BB.5, and only positive values of R should be included.
In addition, only positive values of (R_{s} – R_{E}) should be included, where
Figure BB.5: Moment values
where
a  is the distance between the centroid of the member and the centroid of the restraining members, such as purlins restraining rafters. 
where
h_{h}  is the additional depth of the haunch or taper, see Figure BB.6; 
h_{max}  is the maximum depth of crosssection within the length L_{y}, see Figure BB.6; 
h_{min}  is the minimum depth of crosssection within the length L_{y}, see Figure BB.6; 
h_{s}  is the vertical depth of the unhaunched section, see Figure BB.6; 
L_{h}  is the length of haunch within the length L_{y}, see Figure BB.6; 
L_{y}  is the length between points at which the compression flange is laterally restrained. 
(h/t_{f}) is to be derived from the shallowest section.
Figure BB.6: Dimensions defining taper factor