PREAMBLE (NOT PART OF THE STANDARD)
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EUROPEAN STANDARD
NORME EUROPÉENNE
EUROPÄISCHE NORM
EN 199315
October 2006
ICS 91.010.30; 91.080.10
Supersedes ENV 199315:1997
Incorporating corrigendum April 2009
English Version
Eurocode 3  Design of steel structures  Part 15: Plated structural elements
Eurocode 3  Calcul des structures en acier  Partie 15: Plaques planes 
Eurocode 3  Bemessung und konstruktion von Stahlbauten  Teil 15: Plattenbeulen 
This European Standard was approved by CEN on 13 January 2006.
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.
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Ref. No. EN 199315:2006: E
1
Content
Page 
1 
Introduction 
5 

1.1 
Scope 
5 

1.2 
Normative references 
5 

1.3 
Terms and definitions 
5 

1.4 
Symbols 
6 
2 
Basis of design and modelling 
7 

2.1 
General 
7 

2.2 
Effective width models for global analysis 
7 

2.3 
Plate buckling effects on uniform members 
7 

2.4 
Reduced stress method 
8 

2.5 
Non uniform members 
8 

2.6 
Members with corrugated webs 
8 
3 
Shear lag in member design 
9 

3.1 
General 
9 

3.2 
Effective^{s} width for elastic shear lag 
9 

3.3 
Shear lag at the ultimate limit state 
12 
4 
Plate buckling effects due to direct stresses at the ultimate limit state 
13 

4.1 
General 
13 

4.2 
Resistance to direct stresses 
13 

4.3 
Effective cross section 
13 

4.4 
Plate elements without longitudinal stiffeners 
15 

4.5 
Stiffened plate elements with longitudinal stiffeners 
18 

4.6 
Verification 
21 
5 
Resistance to shear 
21 

5.1 
Basis 
21 

5.2 
Design resistance 
22 

5.3 
Contribution from the web 
22 

5.4 
Contribution from flanges 
25 

5.5 
Verification 
25 
6 
Resistance to transverse forces 
25 

6.1 
Basis 
25 

6.2 
Design resistance 
26 

6.3 
Length of stiff bearing 
26 

6.4 
Reduction factor χ_{F} for effective length for resistance 
27 

6.5 
Effective loaded length 
27 

6.6 
Verification 
28 
7 
Interaction 
28 

7.1 
Interaction between shear force, bending moment and axial force 
28 

7.2 
Interaction between transverse force, bending moment and axial force 
29 
8 
Flange induced buckling 
29 
9 
Stiffeners and detailing 
30 

9.1 
General 
30 

9.2 
Direct stresses 
30 

9.3 
Shear 
34 

9.4 
Transverse loads 
35 
10 
Reduced stress method 
36 
Annex A (informative) Calculation of critical stresses for stiffened plates 
38 2 
Annex B (informative) Non uniform members 
43 
Annex C (informative) Finite Element Methods of Analysis (FEM) 
45 
Annex D (informative) Plate girders with corrugated web 
50 
Annex E (normative) Alternative methods for determining effective cross sections 
53 
3
Foreword
This European Standard EN 199315,, Eurocode 3: Design of steel structures Part 1.5: Plated structural elements, 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 April 2007 and conflicting National Standards shall be withdrawn at latest by March 2010.
This Eurocode supersedes ENV 199315.
According to the CENCENELEC Internal Regulations, the National Standard Organizations of the following countries are bound to implement this 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, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.
National annex for EN 199315
This standard gives alternative procedures, values and recommendations with notes indicating where national choices may have to be made. The National Standard implementing EN 199315 should have a National Annex containing all Nationally Determined Parameters to be used for the design of steel structures to be constructed in the relevant country.
National choice is allowed in EN 199315 through:
 – 2.2(5)
 – 3.3(1)
 – 4.3(6)
 – 5.1(2)
 – 6.4(2)
 – 8(2)
 – 9.1(1)
 – 9.2.1(9)
 – 10(1)
 – 10(5)
 – C.2(1)
 – C.5(2)
 – C.8(1)
 – C.9(3)
 – D.2.2(2)
4
1 Introduction
1.1 Scope
 EN 199315 gives design requirements of stiffened and unstiffened plates which are subject to inplane forces.
 Effects due to shear lag, inplane load introduction and plate buckling for Isection girders and box girders are covered. Also covered are plated structural components subject to inplane loads as in tanks and silos. The effects of outofplane loading are outside the scope of this document.
NOTE 1: The rules in this part complement the rules for class 1,2,3 and 4 sections, see EN 19931 1.
NOTE 2: For the design of slender plates which are subject to repeated direct stress and/or shear and also fatigue due to outofplane bending of plate elements (breathing) see EN 19932 and EN 19936.
NOTE 3: For the effects of outofplane loading and for the combination of inplane effects and outofplane loading effects see EN 19932 and EN 199317.
NOTE 4: Single plate elements may be considered as flat where the curvature radius r satisfies:
where
a 
is the panel width 
t 
is the plate thickness 
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.
EN 199311 
Eurocode 3: Design of steel structures: Part 11: General rules and rules for buildings 
1.3 Terms and definitions
For the purpose of this standard, the following terms and definitions apply:
1.3.1
elastic critical stress
stress in a component at which the component becomes unstable when using small deflection elastic theory of a perfect structure
1.3.2
membrane stress
stress at midplane of the plate
1.3.3
gross crosssection
the total crosssectional area of a member but excluding discontinuous longitudinal stiffeners
1.3.4
effective crosssection and effective width
the gross crosssection or width reduced for the effects of plate buckling or shear lag or both; to distinguish between their effects the word “effective” is clarified as follows:
“effective^{P}“ denotes effects of plate buckling
5
“effective^{s}“ denotes effects of shear lag
“effective“ denotes effects of plate buckling and shear lag
1.3.5
plated structure
a structure built up from nominally flat plates which are connected together; the plates may be stiffened or un stiffened
1.3.6
stiffener
a plate or section attached to a plate to resist buckling or to strengthen the plate; a stiffener is denoted:
 – longitudinal if its direction is parallel to the member;
 – transverse if its direction is perpendicular to the member.
1.3.7
stiffened plate
plate with transverse or longitudinal stiffeners or both
1.3.8
subpanel
unstiffened plate portion surrounded by flanges and/or stiffeners
1.3.9
hybrid girder
girder with flanges and web made of different steel grades; this standard assumes higher steel grade in flanges compared to webs
1.3.10
sign convention
unless otherwise stated compression is taken as positive
1.4 Symbols
 In addition to those given in EN 1990 and EN 19931 1, the following symbols are used:
A_{sℓ} 
total area of all the longitudinal stiffeners of a stiffened plate; 
A_{st} 
gross cross sectional area of one transverse stiffener; 
A_{eff} 
effective cross sectional area; 
A_{c,eff} 
effective^{P} cross sectional area; 
A_{c,eff,loc} 
effective^{P} cross sectional area for local buckling; 
a 
length of a stiffened or unstiffened plate; 
b 
width of a stiffened or unstiffened plate; 
b_{w} 
clear width between welds for welded sections or between ends of radii for rolled sections; 
b_{eff} 
effective^{s} width for elastic shear lag; 
F_{Ed} 
design transverse force; 
h_{w} 
clear web depth between flanges; 
L_{eff} 
effective length for resistance to transverse forces, see 6; 
M_{f.Rd} 
design plastic moment of resistance of a crosssection consisting of the flanges only; 
M_{pl.Rd} 
design plastic moment of resistance of the crosssection (irrespective of crosssection class); 
M_{Ed} 
design bending moment; 
N_{Ed} 
design axial force; 
t 
thickness of the plate; 6 
V_{Ed} 
design shear force including shear from torque; 
W_{eff} 
effective elastic section modulus; 
β 
effective^{s} width factor for elastic shear lag; 
 Additional symbols are defined where they first occur.
2 Basis of design and modelling
2.1 General
 P The effects of shear lag and plate buckling shall be taken into account at the ultimate, serviceability or fatigue limit states.
NOTE: Partial factors γ_{M0} and γ_{M1} used in this part are defined for different applications in the National Annexes of EN 19931 to EN 19936.
2.2 Effective width models for global analysis
 P The effects of shear lag and of plate buckling on the stiffness of members and joints shall be taken into account in the global analysis.
 The effects of shear lag of flanges in global analysis may be taken into account by the use of an effective^{s} width. For simplicity this effective^{s} width may be assumed to be uniform over the length of the span.
 For each span of a member the effective^{s} width of flanges should be taken as the lesser of the full width and L/8 per side of the web, where L is the span or twice the distance from the support to the end of a cantilever.
 The effects of plate buckling in elastic global analysis may be taken into account by effective^{s} cross sectional areas of the elements in compression, see 4.3.
 For global analysis the effect of plate buckling on the stiffness may be ignored when the effective^{s} crosssectional area of an element in compression is larger than ρ_{lim} times the gross crosssectional area of the same element.
NOTE 1: The parameter ρ_{lim} may be given in the National Annex. The value ρ_{lim} = 0,5 is recommended.
NOTE 2: For determining the stiffness when (5) is not fulfilled, see Annex E.
2.3 Plate buckling effects on uniform members
 Effective^{P} width models for direct stresses, resistance models for shear buckling and buckling due to transverse loads as well as interactions between these models for determining the resistance of uniform members at the ultimate limit state may be used when the following conditions apply:
  panels are rectangular and flanges are parallel;
  the diameter of any unstiffened open hole or cut out does not exceed 0,05b, where b is the width of the
NOTE: The rules may apply to non rectangular panels provided the angle α_{limit} (see Figure 2.1) is not greater than 10 degrees. If α_{limit} exceeds 10, panels may be assessed assuming it to be a rectangular panel based on the larger of b_{1} and b_{2} of the panel.
7
Figure 2.1: Definition of angle α
 For the calculation of stresses at the serviceability and fatigue limit state the effective^{s} area may be used if the condition in 2.2(5) is fulfilled . For ultimate limit states the effective area according to 3.3 should be used β replaced by β_{ult}.
2.4 Reduced stress method
 As an alternative to the use of the effective^{P} width models for direct stresses given in sections 4 to 7, the cross sections may be assumed to be class 3 sections provided that the stresses in each panel do not exceed the limits specified in section 10.
NOTE: The reduced stress method is analogous to the effective^{P} width method (see 2.3) for single plated elements. However, in verifying the stress limitations no load shedding has been assumed between the plated elements of the cross section.
2.5 Non uniform members
 Non uniform members (e.g. haunched members, non rectangular panels) or members with regular or irregular large openings may be analysed using Finite Element (FE) methods.
NOTE 1: See Annex B for non uniform members.
NOTE 2: For FEcalculations see Annex C.
2.6 Members with corrugated webs
 For members with corrugated webs, the bending stiffness should be based on the flanges only and webs should be considered to transfer shear and transverse loads.
NOTE: For text deleted buckling resistance of flanges in compression and the shear resistance of webs see Annex D.
8
3 Shear lag in member design
3.1 General
 Shear lag in flanges may be neglected if b_{0} < L_{e}/50 where b_{0} is taken as the flange outstand or half the width of an internal element and L_{e} is the length between points of zero bending moment, see 3.2.1(2).
 Where the above limit for b_{0} is exceeded the effects due to shear lag in flanges should be considered at serviceability and fatigue limit state verifications by the use of an effective^{s} width according to 3.2.1 and a stress distribution according to 3.2.2. For the ultimate limit state verification an effective area according to 3.3 may be used.
 Stresses due to patch loading in the web applied at the flange level should be determined from 3.2.3.
3.2 Effective^{s} width for elastic shear lag
3.2.1 Effective^{s} width
 The effective^{s} width b_{eff} for shear lag under elastic conditions should be determined from:
b_{eff} = β b_{0} (3.1)
where the effective^{s} factor β is given in Table 3.1.
This effective^{s} width may be relevant for serviceability and fatigue limit states.
 Provided adjacent spans do not differ more than 50% and any cantilever span is not larger than half the adjacent span the effective lengths L_{e} may be determined from Figure 3.1. For all other cases L_{e} should be taken as the distance between adjacent points of zero bending moment.
Figure 3.1 : Effective length L_{e} for continuous beam and distribution of effective^{s} width
9
Figure 3.2: Notations for shear lag
10
3.2.2 Stress distribution due to shear lag
 The distribution of longitudinal stresses across the flange plate due to shear lag should be obtained
Figure 3.3: Distribution of stresses due to shear lag
3.2.3 Inplane load effects
 The elastic stress distribution in a stiffened or unstiffened plate due to the local introduction of inplane forces (patch loads), see Figure 3.4, should be determined from:
where
a_{st,1} 
is the gross crosssectional area of the directly loaded stiffeners divided over the length s_{e}. This may be taken as the area of a stiffener smeared over the length of the spacing s_{st}; 
t_{w} 
is the web thickness; 
z 
is the distance to flange. 
S_{e} 
is the length of the stiff bearing; 
S_{st} 
is the spacing of stiffeners; 
NOTE: The equation (3.2) is valid when s_{st}/s_{e} ≤ 0,5; otherwise the contribution of stiffeners should be neglected.
11
Figure 3.4: Inplane load introduction
NOTE: The above stress distribution may also be used for the fatigue verification.
3.3 Shear lag at the ultimate limit state
 At the ultimate limit state shear lag effects may be determined as follows:
 elastic shear lag effects as determined for serviceability and fatigue limit states,
 combined effects of shear lag and of plate buckling,
 elasticplastic shear lag effects allowing for limited plastic strains.
NOTE 1: The National Annex may choose the method to be applied. Unless specified otherwise in EN 19932 to EN 19936, the method in NOTE 3 is recommended.
NOTE 2: The combined effects of plate buckling and shear lag may be taken into account by using A_{eff} as given by:
A_{eff} = A_{c,eff} β_{ult} (3.3)
where
A_{c,eff} 
is the effective^{P} area of the compression flange due to plate buckling (see 4.4 and 4.5); 
β_{ult} 
is the effective^{s} width factor for the effect of shear lag at the ultimate limit state, which may be taken as β determined from Table 3.1 with α_{0} replaced by

t_{f} 
is the flange thickness. 
12
NOTE 3: Elasticplastic shear lag effects allowing for limited plastic strains may be taken into account using A_{eff} as follows:
A_{eff} = A_{c,eff} β^{κ} ≥ A_{c,eff} β (3.5)
where β and κ are taken from Table 3.1.
The expressions in NOTE 2 and NOTE 3 may also be applied for flanges in tension in which case A_{c,eff} should be replaced by the gross area of the tension flange.
4 Plate buckling effects due to direct stresses at the ultimate limit state
4.1 General
 This section gives rules to account for plate buckling effects from direct stresses at the ultimate limit state when the following criteria are met:
 The panels are rectangular and flanges are parallel or nearly parallel (see 2.3);
 Stiffeners, if any, are provided in the longitudinal or transverse direction or both;
 Open holes and cut outs are small (see 2.3);
 Members are of uniform cross section;
 No flange induced web buckling occurs.
NOTE 1: For compression flange buckling in the plane of the web see section 8.
NOTE 2: For stiffeners and detailing of plated members subject to plate buckling see section 9.
4.2 Resistance to direct stresses
 The resistance of plated members may be determined using the effective^{P} areas of plate elements in compression for class 4 sections using cross sectional data (A_{eff}, l_{eff}, W_{eff} for cross sectional verifications and member verifications for column buckling and lateral torsional buckling according to EN 19931 1.
 Effective^{P} areas should be determined on the basis of the linear strain distributions with the attainment of yield strain in the mid plane of the compression plate.
4.3 Effective cross section
 In calculating longitudinal stresses, account should be taken of the combined effect of shear lag and plate buckling using the effective areas given in 3.3.
 The effective cross sectional properties of members should be based on the effective areas of the compression elements and on the effective^{s} area of the tension elements due to shear lag.
 The effective area A_{eff} should be determined assuming that the cross section is subject only to stresses due to uniform axial compression. For nonsymmetrical cross sections the possible shift e_{N} of the centroid of the effective area A_{eff} relative to the centre of gravity of the gross crosssection, see Figure 4.1, gives an additional moment which should be taken into account in the cross section verification using 4.6.
 The effective section modulus W_{eff} should be determined assuming the cross section is subject only to bending stresses, see Figure 4.2. For biaxial bending effective section moduli should be determined about both main axes.
NOTE: As an alternative to 4.3(3) and (4) a single effective section may be determined from N_{Ed} and M_{Ed} acting simultaneously. The effects of e_{N} should be taken into account as in 4.3(3). This requires an iterative procedure.
13
 The stress in a flange should be calculated using the elastic section modulus with reference to the midplane of the flange.
 Hybrid girders may have flange material with yield strength f_{yf} up to ϕ_{h} × f_{yw} provided that:
 the increase of flange stresses caused by yielding of the web is taken into account by limiting the stresses in the web to f_{yw} ;
 f_{yf} text deleted is used in determining the effective area of the web.
NOTE: The National Annex may specify the value ϕ_{h}. A value of ϕ_{h} = 2,0 is recommended.
 The increase of deformations and of stresses at serviceability and fatigue limit states may be ignored for hybrid girders complying with 4.3(6) including the NOTE.
 For hybrid girders complying with 4.3(6) the stress range limit in EN 199319 may be taken as 1,5 f_{yf}.
Figure 4.1: Class 4 crosssections  axial force
Figure 4.2: Class 4 crosssections  bending moment
14
4.4 Plate elements without longitudinal stiffeners
 The effective^{P} areas of flat compression elements should be obtained using Table 4.1 for internal elements and Table 4.2 for outstand elements. The effective^{P} area of the compression zone of a plate with the gross crosssectional area A_{c} should be obtained from:
A_{c,eff} = ρ A_{c} (4.1)
where ρ is the reduction factor for plate buckling.
 The reduction factor ρ may be taken as follows:
  internal compression elements:
  outstand compression elements:
ψ 
is the stress ratio determined in accordance with 4.4(3) and 4.4(4) 

is the appropriate width to be taken as follows (for definitions, see Table 5.2 of EN 19931 1)
b_{w} 
for webs; 
b 
for internal flange elements (except RHS); 
b  3 t 
for flanges of RHS; 
c 
for outstand flanges; 
h 
for equalleg angles; 
h 
for unequalleg angles; 

k_{σ} 
is the buckling factor corresponding to the stress ratio ψ and boundary conditions. For long plates k_{σ} is given in Table 4.1 or Table 4.2 as appropriate; 
t 
is the thickness; 
σ_{cr} 
is the elastic critical plate buckling stress see equation (A.1) in Annex A.1(2) and Table 4.1 and Table 4.2; 
 For flange elements of Isections and box girders the stress ratio ψ used in Table 4.1 and Table 4.2 should be based on the properties of the gross crosssectional area, due allowance being made for shear lag in the flanges if relevant. For web elements the stress ratio ψ used in Table 4.1 should be obtained using a stress distribution based on the effective area of the compression flange and the gross area of the web.
NOTE: If the stress distribution results from different stages of construction (as e.g. in a composite bridge) the stresses from the various stages may first be calculated with a cross section consisting of effective flanges and
15
gross web and these stresses are added together. This resulting stress distribution determines an effective web section that can be used for all stages to calculate the final stress distribution for stress analysis.
 Except as given in 4.4(5), the plate slenderness of an element may be replaced by:
where
σ_{com,Ed} 
is the maximum design compressive stress in the element determined using the effective^{P} area of the section caused by all simultaneous actions. 
NOTE 1: The above procedure is conservative and requires an iterative calculation in which the stress ratio ψ (see Table 4.1 and Table 4.2) is determined at each step from the stresses calculated on the effective^{P} crosssection defined at the end of the previous step.
NOTE 2: See also alternative procedure in Annex E.
 For the verification of the design buckling resistance of a class 4 member using 6.3.1, 6.3.2 or 6.3.4 of EN 199311, either the plate slenderness or with σ_{com,Ed} based on second order analysis with global imperfections should be used.
 For aspect ratios a/b < 1 a column type of buckling may occur and the check should be performed according to 4.5.4 using the reduction factor ρ_{c}.
NOTE: This applies e.g. for flat elements between transverse stiffeners where plate buckling could be columnlike and require a reduction factor ρ_{c} close to χ_{c} as for column buckling, see Figure 4.3 a) and b). For plates with longitudinal stiffeners column type buckling may also occur for a/b ≥ 1, see Figure 4.3 c).
Figure 4.3: Columnlike behaviour
16
Table 4.1: Internal compression elements
Stress distribution (compression positive) 
Effective^{P} width b_{eff} 

ψ = 1 :
b_{eff} = ρ
b_{e1} = 0,5 b_{eff} b_{e2} = 0,5 b_{eff} 

1 > ψ ≥ 0: b_{eff} = ρ
b_{e2} = b_{eff}  b_{e1} 

ψ < 0:
b_{eff} = ρ b_{c} = ρ (1ψ)
b_{e1} = 0,4 b_{eff} b_{e2} = 0,6 b_{eff} 
ψ = σ_{2}/σ_{1} 
1 
1 > ψ > 0 
0 
0 > ψ > 1 
1 
1 > ψ ≥ 3 
Buckling factor k_{σ} 
4,0 
8,2/ (1,05 + ψ 
7,81 
7,81  6,29ψ + 9,78ψ^{2} 
23,9 
5,98 (1  ψ)^{2} 
Table 4.2: Outstand compression elements
Stress distribution (compression positive) 
Effective^{P} width b_{eff} 

1 > ψ ≥ 0:
b_{eff} = ρ c 

ψ < 0:
b_{eff} = ρ b_{c} = ρ c / (1ψ) 
ψ = σ_{2}/σ_{1} 
1 
0 
1 
1 ≥ ψ ≥ 3 
Buckling factor k_{σ} 
0,43 
0,57 
0,85 
0.57  0,21 ψ + 0,07ψ^{2} 

1 > ψ ≥ 0:
b_{eff} = ρ c 

ψ < 0:
b_{eff} = ρ b_{c} = ρ c / (1ψ) 
ψ = σ_{2}/σ_{1} 
1 
1 > ψ > 0 
0 
0 > ψ > 1 
1 
Buckling factor k_{σ} 
0,43 
0,578 / (ψ + 0,34) 
1,70 
1,7  5ψ + 17,1ψ^{2} 
23,8 
17
4.5 Stiffened plate elements with longitudinal stiffeners
4.5.1 General
 For plates with longitudinal stiffeners the effective^{P} areas from local buckling of the various subpanels between the stiffeners and the effective^{P} areas from the global buckling of the stiffened panel should be accounted for.
 effective^{P} section area of each subpanel should be determined by a reduction factor in accordance with 4.4 to account for local plate buckling. The stiffened plate with effective^{P} section areas for the stiffeners should be checked for global plate buckling (by modelling it as an equivalent orthotropic plate) and a reduction factor ρ_{c} should be determined for overall plate buckling.
 effective^{P} area of the compression zone of the stiffened plate should be taken as:
where A_{c,eff,loc} is effective^{P} section area of all the stiffeners and subpanels that are fully or partially in the compression zone except the effective parts supported by an adjacent plate element with the width bedge,eir, see example in Figure 4.4.
 The area A_{c,eff,loc} should be obtained from:
where

applies to the part of the stiffened panel width that is in compression except the parts b_{edge,eff}, see Figure 4.4; 
A_{sℓ,eff} 
is the sum of effective^{P} sections according to 4.4 of all longitudinal stiffeners with gross area A_{sℓ} located in the compression zone; 
b_{c,loc} 
is the width of the compressed part of each subpanel; 
ρ_{loC} 
is the reduction factor from 4.4(2) for each subpanel. 
Figure 4.4: Stiffened plate under uniform compression
NOTE: For nonuniform compression see Figure A. 1.
18
 In determining the reduction factor ρ_{c} for overall buckling, the reduction factor for columntype buckling, which is more severe than the reduction factor than for plate buckling, should be considered.
 Interpolation should be carried out in accordance with 4.5.4(1) between the reduction factor ρ for plate buckling and the reduction factor χ_{c} for column buckling to determine ρ_{c} see 4.5.4.
 The reduction of the compressed area A_{c,eff,loc} through ρ_{c} may be taken as a uniform reduction across the whole cross section.
 If shear lag is relevant (see 3.3), the effective crosssectional area A_{c.eff} of the compression zone of the stiffened plate should then be taken as accounting not only for local plate buckling effects but also for shear lag effects.
 The effective crosssectional area of the tension zone of the stiffened plate should be taken as the gross area of the tension zone reduced for shear lag if relevant, see 3.3.
 The effective section modulus W_{eff} should be taken as the second moment of area of the effective cross section divided by the distance from its centroid to the mid depth of the flange plate.
4.5.2 Plate type behaviour
 The relative plate slenderness of the equivalent plate is defined as:
with 

where
A_{c} 
is the gross area of the compression zone of the stiffened plate except the parts of subpanels supported by an adjacent plate, see Figure 4.4 (to be multiplied by the shear lag factor if shear lag is relevant, see 3.3); 
A_{c.eff.loc} 
is the effective area of the same part of the plate (including shear lag effect, if relevant) with due allowance made for possible plate buckling of subpanels and/or stiffeners. 
 The reduction factor ρ for the equivalent orthotropic plate is obtained from 4.4(2) provided is calculated from equation (4.7).
NOTE: For calculation of σ_{cr,p} see Annex A.
4.5.3 Column type buckling behaviour
 The elastic critical column buckling stress σ_{cr,c} of an unstiffened (see 4.4) or stiffened (see 4.5) plate should be taken as the buckling stress with the supports along the longitudinal edges removed.
 For an unstiffened plate the elastic critical column buckling stress σ_{cr,c} may be obtained from
 For a stiffened plate σ_{cr,c} may be determined from the elastic critical column buckling stress σ_{cr,sl} of the stiffener closest to the panel edge with the highest compressive stress as follows:
19
where
I_{sℓ,1} 
is the second moment of area of the gross cross section of the stiffener and the adjacent parts of the plate, relative to the outofplane bending of the plate; 
A_{sℓ,1} 
is the gross crosssectional area of the stiffener and the adjacent parts of the plate according to Figure A. 1. 
NOTE: σ_{cr,c} may be obtained from where σ_{cr,c} is related to the compressed edge of the plate, and, b_{sℓ,1} and b_{c} are geometric values from the stress distribution used for the extrapolation, see Figure A. 1.
 The relative column slenderness is defined as follows:
where

A_{sℓ,1} 
is defined in 4.5.3(3); 
A_{sℓ,1,eff} 
is the effective crosssectional area of the stiffener and the adjacent parts of the plate with due allowance for plate buckling, see Figure A.1. 
 The reduction factor χ_{c} should be obtained from 6.3.1.2 of EN 199311. For unstiffened plates α = 0,21 corresponding to buckling curve a should be used. For stiffened plates its value should be increased to:
with

e 
= max (e_{1}, e_{2}) is the largest distance from the respective centroids of the plating and the onesided stiffener (or of the centroids of either set of stiffeners when present on both sides) to the neutral axis of the effective column, see Figure A.1; 
α 
= 0,34 (curve b) for closed section stiffeners; = 0,49 (curve c) for open section stiffeners. 
4.5.4 Interaction between plate and column buckling
 The final reduction factor ρ_{c} should be obtained by interpolation between χ_{c} and ρ as follows:
ρ_{c} = (ρ_{c} – χ_{c}) ξ (2 – ξ) + χ_{c} (4.13)
where but 0 ≤ ξ ≤ 1
σ_{cr,p} 
is the elastic critical plate buckling stress, see Annex A. 1(2); 
σ_{cr,c} 
is the elastic critical column buckling stress according to 4.5.3(2) and (3), respectively; 20 
χ_{c} 
is the reduction factor due to column buckling. 
ρ 
is the reduction factor due to plate buckling, see 4.4(1). 
4.6 Verification
 Member verification for compression and uniaxial bending should be performed as follows:
where
A_{eff} 
is the effective crosssection area in accordance with 4.3(3); 
e_{N} 
is the shift in the position of neutral axis, see 4.3(3); 
M_{Ed} 
is the design bending moment; 
N_{Ed} 
is the design axial force; 
W_{eff} 
is the effective elastic section modulus, see 4.3(4); 
γ_{M0} 
is the partial factor, see application parts EN 19932 to 6. 
NOTE: For members subject to compression and biaxial bending the above equation (4.14) may be modified as follows:
M_{y,Ed}, M_{z,Ed} 
are the design bending moments with respect to yy and zz axes respectively; 
e_{y,N}, e_{z,N} 
are the eccentricities with respect to the neutral axis. 
 Action effects M_{Ed} and N_{Ed} should include global second order effects where relevant.
 The plate buckling verification of the panel should be carried out for the stress resultants at a distance 0,4a or 0,5b, whichever is the smallest, from the panel end where the stresses are the greater. In this case the gross sectional resistance needs to be checked at the end of the panel.
5 Resistance to shear
5.1 Basis
 This section gives rules for shear resistance of plates considering shear buckling at the ultimate limit state where the following criteria are met:
 the panels are rectangular within the angle limit stated in 2.3;
 stiffeners, if any, are provided in the longitudinal or transverse direction or both;
 all holes and cut outs are small (see 2.3);
 members are of uniform cross section.
 Plates with h_{w}/t greater than for an unstiffened web, or for a stiffened web, should be checked for resistance to shear buckling and should be provided with transverse stiffeners at the supports, where
21
NOTE 1: h_{w} see Figure 5.1 and for K_{τ} see 5.3(3).
NOTE 2: The National Annex will define η. The value η = 1,20 is recommended for steel grades up to and including S460. For higher steel grades η = 1,00 is recommended.
5.2 Design resistance
 For unstiffened or stiffened webs the design resistance for shear should be taken as:
in which the contribution from the web is given by:
and the contribution from the flanges V_{bf,Rd} is according to 5.4.
 Stiffeners should comply with the requirements in 9.3 and welds should fulfil the requirement given in 9.3.5.
Figure 5.1 : End supports
5.3 Contribution from the web
 For webs with transverse stiffeners at supports only and for webs with either intermediate transverse stiffeners or longitudinal stiffeners or both, the factor χ_{w}, for the contribution of the web to the shear buckling resistance should be obtained from Table 5.1 or Figure 5.2.
Table 5.1 : Contribution from the web χ_{w} to shear buckling resistance

Rigid end post 
Nonrigid end post 
< 0,83/η 
η 
η 
0,83/η ≤ < 1,08 
0,83/ 
0,83/ 
≥ 1,08 
1,37/(0,7+ 
0,83/ 
NOTE: See 6.2.6 in EN 199311.
22
 Figure 5.1 shows various end supports for girders:
 No end post, see 6.1 (2), type c);
 Rigid end posts, see 9.3.1; this case is also applicable for panels at an intermediate support of a continuous girder;
 Non rigid end posts see 9.3.2.
 The modified slenderness in Table 5.1 and Figure 5.2 should be taken as:
where τ_{cr} = k_{τ} σ_{E} (5.4)
NOTE 1: Values for σ_{E} and k_{τ} may be taken from Annex A.
NOTE 2: The modified slenderness may be taken as follows:
 transverse stiffeners at supports only:
 transverse stiffeners at supports and intermediate transverse or longitudinal stiffeners or both:
in which k_{τ} is the minimum shear buckling coefficient for the web panel.
NOTE 3: Where nonrigid transverse stiffeners are also used in addition to rigid transverse stiffeners, k_{τ} is taken as the minimum of the values from the web panels between any two transverse stiffeners (e.g. a_{2} × h_{w} and a_{3} × h_{w}) and that between two rigid stiffeners containing nonrigid transverse stiffeners (e.g. a_{4} × h_{w}).
NOTE 4: Rigid boundaries may be assumed for panels bordered by flanges and rigid transverse stiffeners. The web buckling analysis can then be based on the panels between two adjacent transverse stiffeners (e.g. a_{1} × h_{w} in Figure 5.3).
NOTE 5: For nonrigid transverse stiffeners the minimum value k_{τ} may be obtained from the buckling analysis of the following:
 a combination of two adjacent web panels with one flexible transverse stiffener
 a combination of three adjacent web panels with two flexible transverse stiffeners
For procedure to determine k_{τ} see Annex A.3.
 The second moment of area of a longitudinal stiffener should be reduced to 1/3 of its actual value when calculating k_{τ}. Formulae for k_{τ} taking this reduction into account in A.3 may be used. 23
Figure 5.2: Shear buckling factor χ_{w}
 For webs with longitudinal stiffeners the modified slenderness in (3) should not be taken as less than
where h_{wi} and k_{τi}, refer to the subpanel with the largest modified slenderness of all subpanels within the web panel under consideration.
NOTE: To calculate k_{τi} the expression given in A.3 may be used with k_{τst} = 0.
24
Figure 5.3: Web with transverse and longitudinal stiffeners
5.4 Contribution from flanges
 When the flange resistance is not completely utilized in resisting the bending moment (M_{Ed} < M_{f,Rd}) the contribution from the fiances should be obtained as follows:
b_{f} and t_{f} 
are taken for the flange which provides the least axial resistance, 
b_{f} 
being taken as not larger than 15εt_{f} on each side of the web, 

is the moment of resistance of the cross section consisting of the effective area of the flanges only, 
 When an axial force N_{Ed} is present, the value of M_{f,Rd} should be reduced by multiplying it by the
where A_{f1} and A_{f2} are the areas of the top and bottom flanges respectively.
5.5 Verification
 The verification should be performed as follows:
where V_{Ed} is the design shear force including shear from torque.
6 Resistance to transverse forces
6.1 Basis
 The design resistance of the webs of rolled beams and welded girders should be determined in accordance with 6.2, provided that the compression flange is adequately restrained in the lateral direction.
 The load is applied as follows:
 through the flange and resisted by shear forces in the web, see Figure 6.1 (a);
 through one flange and transferred through the web directly to the other flange, see Figure 6.1 (b).
 through one flange adjacent to an unstiffened end, see Figure 6.1 (c)
25
 For box girders with inclined webs the resistance of both the web and flange should be checked. The internal forces to be taken into account are the components of the external load in the plane of the web and flange respectively.
 The interaction of the transverse force, bending moment and axial force should be verified using 7.2.
Figure 6.1: Buckling coefficients for different types of load application
6.2 Design resistance
 For unstiffened or stiffened webs the design resistance to local buckling under transverse forces should be taken as
where
t_{w} 
is the thickness of the web; 
f_{yw} 
is the yield strength of the web; 
L_{eff} 
is the effective length for resistance to transverse forces, which should be determined from 
L_{eff} = χ_{F} ℓ_{y} 
where
ℓ_{y} 
is the effective loaded length, see 6.5, appropriate to the length of stiff bearing s_{s}, see 6.3; 
χ_{F} 
is the reduction factor due to local buckling, see 6.4(1). 
6.3 Length of stiff bearing
 The length of stiff bearing s_{s} on the flange should be taken as the distance over which the applied load is effectively distributed at a slope of 1:1, see Figure 6.2. However, s_{s} should not be taken as larger than h_{w}.
 If several concentrated forces are closely spaced, the resistance should be checked for each individual force as well as for the total load with s_{s} as the centretocentre distance between the outer loads.
Figure 6.2: Length of stiff bearing
26
 If the bearing surface of the applied load rests at an angle to the flange surface, see Figure 6.2, s_{s} should be taken as zero.
6.4 Reduction factor χ_{F} for effective length for resistance
 The reduction factor χ_{F} should be obtained from:
where
 For webs without longitudinal stiffeners k_{F} should be obtained from Figure 6.
NOTE: For webs with longitudinal stiffeners information may be given in the National Annex. The following rules are recommended:
For webs with longitudinal stiffeners k_{F} may be taken as
where 
b_{1} 
is the depth of the loaded subpanel taken as the clear distance between the loaded flange and the stiffener 
where 
I_{sℓ,1} 
is the second moment of area of the stiffener closest to the loaded flange including contributing parts of the web according to Figure 9.1. 
Equation (6.6) is valid for and and loading according to type a) in Figure 6.1.
 ℓ_{y} should be obtained from 6.5.
6.5 Effective loaded length
 The effective loaded length ℓ_{y} should be calculated as follows:
For box girders, b_{f} in equation (6.8) should be limited to 15εt_{f} on each side of the web.
 For types a) and b) in Figure 6.1, ℓ_{y} should be obtained using:
but ℓ_{y} ≤ distance between adjacent transverse stiffeners (6.10)
27
 For type c) ℓ_{y} should be taken as the smallest value obtained from the equations (6.11) and (6.12).
Where
6.6 Verification
 The verification should be performed as follows:
where
F_{Ed} 
is the design transverse force; 
L_{eff} 
is the effective length for resistance to transverse forces, see 6.2(1); 
t_{w} 
is the thickness of the plate. 
7 Interaction
7.1 Interaction between shear force, bending moment and axial force
 Provided that (see below) does not exceed 0,5, the design resistance to bending moment and axial force need not be reduced to allow for the shear force. If is more than 0,5 the combined effects of bending and shear in the web of an I or box girder should satisfy:
where
M_{f,Rd} 
is the design plastic moment of resistance of the section consisting of the effective area of the flanges; 
M_{pl,Rd} 
is the design plastic resistance of the cross section consisting of the effective area of the flanges and the fully effective web irrespective of its section class. 

for V_{bw,Rd} see expression (5.2). 
In addition the requirements in sections 4.6 and 5.5 should be met.
Action effects should include global second order effects of members where relevant.
 The criterion given in (1) should be verified at all sections other than those located at a distance less than h_{w}/2 from a support with vertical stiffeners. 28
 The plastic moment of resistance M_{f,Rd} may be taken as the product of the yield strength, the effective area of the flange with the smallest value of A_{f}f_{y}/γ_{M0} and the distance between the centroids of the flanges.
 If an axial force N_{Ed} is present, M_{pl,Rd} and M_{f,Rd} should be reduced in accordance with 6.2.9 of EN 199311 and 5.4(2) respectively. When the axial force is so large that the whole web is in compression 7.1(5) should be applied.
 A flange in a box girder should be verified using 7.1(1) taking M_{f,Rd} = 0 and τ_{Ed} taken as the average shear stress in the flange which should not be less than half the maximum shear stress in the flange and is taken as η_{1} according to 4.6(1). In addition the subpanels should be checked using the average shear stress within the subpanel and χ_{w} determined for shear buckling of the subpanel according to 5.3, assuming the longitudinal stiffeners to be rigid.
7.2 Interaction between transverse force, bending moment and axial force
 If the girder is subjected to a concentrated transverse force acting on the compression flange in conjunction with bending and axial force, the resistance should be verified using 4.6, 6.6 and the following interaction expression:
η_{2} + 0,8 η_{1} ≥ 1,4 (7.2)
 If the concentrated load is acting on the tension flange the resistance should be verified according to section 6. Additionally 6.2.1(5) of EN ^99311 should be met.
8 Flange induced buckling
 To prevent the compression flange buckling in the plane of the web, the following criterion should be met:
A_{w} 
is the cross section area of the web; 
A_{fc} 
is the effective cross section area of the compression flange; 
h_{w} 
is the depth of the web; 
t_{w} 
is the thickness of the web. 
The value of the factor k should be taken as follows:
– 
plastic rotation utilized 
k = 0,3 
– 
plastic moment resistance utilized 
k = 0,4 
– 
elastic moment resistance utilized 
k = 0,55 
 When the girder is curved in elevation, with the compression flange on the concave face, the following criterion should be met:
r is the radius of curvature of the compression flange.
NOTE: The National Annex may give further information on flange induced buckling.
29
9 Stiffeners and detailing
9.1 General
 This section gives design rules for stiffeners in plated structures which supplement the plate buckling rules specified in sections 4 to 7.
NOTE: The National Annex may give further requirements on stiffeners for specific applications.
 When checking the buckling resistance, the section of a stiffener may be taken as the gross area comprising the stiffener plus a width of plate equal to 15εt but not more than the actual dimension available, on each side of the stiffener avoiding any overlap of contributing parts to adjacent stiffeners, see Figure 9.1.
 The axial force in a transverse stiffener should be taken as the sum of the force resulting from shear (see 9.3.3(3)) and any external loads.
Figure 9.1: Effective crosssection of stiffener
9.2 Direct stresses
9.2.1 Minimum requirements for transverse stiffeners
 In order to provide a rigid support for a plate with or without longitudinal stiffeners, intermediate transverse stiffeners should satisfy the criteria given below.
 The transverse stiffener should be treated as a simply supported member subject to lateral loading with an initial sinusoidal imperfection w_{0} equal to s/300, where s is the smallest of a_{l}, a_{2} or b, see Figure 9.2, where a_{1} and a_{2} are the lengths of the panels adjacent to the transverse stiffener under consideration and b is the height between the centroids of the flanges or span of the transverse stiffener. Eccentricities should be accounted for.
Figure 9.2: Transverse stiffener
 The transverse stiffener should carry the deviation forces from the adjacent compressed panels under the assumption that both adjacent transverse stiffeners are rigid and straight together with any external load
30
and axial force according to the NOTE to 9.3.3(3). The compressed panels and the longitudinal stiffeners are considered to be simply supported at the transverse stiffeners.
 It should be verified that using a second order elastic method analysis both the following criteria are satisfied at the ultimate limit state:
 – that the maximum stress in the stiffener should not exceed f_{y}/γ_{:M1}
 – that the additional deflection should not exceed b/300.
 In the absence of an axial force in the transverse stiffener both the criteria in (4) above may be assumed to be satisfied provided that the second moment of area I_{st} of the transverse stiffeners is not less than:
e_{max} 
is the maximum distance from the extreme fibre of the stiffener to the centroid of the stiffener; 
N_{Ed} 
is the maximum compressive force of the adjacent panels but not less than the maximum compressive stress times half the effective^{P} compression area of the panel including stiffeners; 
σ_{cr,c}, σ_{cr,p} are defined in 4.5.3 and Annex A. 
NOTE: Where out of plane loading is applied to the transverse stiffeners reference should be made to EN 19932 and EN 199317.
 If the stiffener carries axial compression this should be increased by ΔN_{st} = σ_{m}b^{2} / π^{2} in order to account for deviation forces. The criteria in (4) apply but ΔN_{st} need not be considered when calculating the uniform stresses from axial load in the stiffener.
 As a simplification the requirement of (4) may, in the absence of axial forces, be verified using a first order elastic analysis taking account of the following additional equivalent uniformly distributed lateral load q acting on the length b:
where
σ_{m} 
is defined in (5) above; 
w_{0} 
is defined in Figure 9.2; 
w_{el} 
is the elastic deformation, that may be either determined iteratively or be taken as the maximum additional deflection b/300. 
 Unless a more advanced method of analysis is carried out in order to prevent torsional buckling of stiffeners with open crosssections, the following criterion should be satisfied:
where
I_{p} 
is the polar second moment of area of the stiffener alone around the edge fixed to the plate; 
I_{T} 
is the St. Venant torsional constant for the stiffener alone. 
31
 Where warping stiffness is considered stiffeners should either fulfil (8) or the criterion
σ_{cr} ≥ θ f_{y} (9.4)
where
σ_{cr} 
is the elastic critical stress for torsional buckling not considering rotational restraint from the plate; 
θ 
is a parameter to ensure class 3 behaviour. 
NOTE: The parameter θ may be given in the National Annex. The value θ = 6 is recommended.
9.2.2 Minimum requirements for longitudinal stiffeners
 The requirements concerning torsional buckling in 9.2.1(8) and (9) also apply to longitudinal stiffeners.
 Discontinuous longitudinal stiffeners that do not pass through openings made in the transverse stiffeners or are not connected to either side of the transverse stiffeners should be:
 – used only for webs (i.e. not allowed in flanges);
 – neglected in global analysis;
 – neglected in the calculation of stresses;
 – considered in the calculation of effective^{P} widths of web subpanels;
 – considered in the calculation of the elastic critical stresses.
 Strength assessments for stiffeners should be performed according to 4.5.3 and 4.6.
9.2.3 Welded plates
 Plates with changes in plate thickness should be welded adjacent to the transverse stiffener, see Figure 9.3. The effects of eccentricity need not be taken into account unless the distance to the stiffener from the welded junction exceeds b_{0}/2 or 200 mm whichever is the smallest, where b_{0} is the width of the plate between longitudinal stiffeners.
Figure 9.3: Welded plates
32
9.2.4 Cut outs in stiffeners
 The dimensions of cut outs in longitudinal stiffeners should be as shown in Figure 9.4.
Figure 9.4: Cut outs in longitudinal stiffeners
 The length ℓ should not exceed:
ℓ ≤ 6 t_{min} 
for flat stiffeners in compression 
ℓ ≤ 8 t_{min} 
for other stiffeners in compression 
ℓ ≤ 15 t_{min} 
for stiffeners without compression 
where t_{min} is the lesser of the plate thicknesses 
 The limiting values ℓ in (2) for stiffeners in compression may be increased by when
σ_{x,Ed} ≤ σ _{x,Rd} and ℓ ≤ 15t_{min}.
σ_{x,Ed} is the compression stress at the location of the cutout
 The dimensions of cut outs in transverse stiffeners should be as shown in Figure 9.5.
Figure 9.5: Cut outs in transverse stiffeners
 The gross web adjacent to the cut out should resist a shear force V_{Ed}, where
I_{net} 
is the second moment of area for the net section of the transverse stiffener; 
e 
is the maximum distance from the underside of the flange plate to the neutral axis of net section, see Figure 9.5; 
b_{G} 
is the length of the transverse stiffener between the flanges. 
33
9.3 Shear
9.3.1 Rigid end post
 The rigid end post (see Figure 5.1) should act as a bearing stiffener resisting the reaction from the support (see 9.4), and should be designed as a short beam resisting the longitudinal membrane stresses in the plane of the web.
NOTE: For the effects of eccentricity due to movements of bearings, see EN 19932.
 A rigid end post should comprise of two doublesided transverse stiffeners that form the flanges of a short beam of length h_{w}, see Figure 5.1 (b). The strip of web plate between the stiffeners forms the web of the short beam. Alternatively, a rigid end post may be in the form of a rolled section, connected to the end of the web plate as shown in Figure 9.6.
Figure 9.6: Rolled section forming an endpost
 Each double sided stiffener consisting of flats should have a cross sectional area of at least 4h_{w}t^{2} / e, where e is the centre to centre distance between the stiffeners and e > 0,1 h_{w}, see Figure 5.1 (b). Where a rolled section other than flats is used for the endpost its section modulus should be not less than 4h_{w}t^{2} for bending around a horizontal axis perpendicular to the web.
 As an alternative the girder end may be provided with a single doublesided stiffener and a vertical stiffener adjacent to the support so that the subpanel resists the maximum shear when designed with a nonrigid end post.
9.3.2 Stiffeners acting as nonrigid end post
 A nonrigid end post may be a single double sided stiffener as shown in Figure 5.1 (c). It may act as a bearing stiffener resisting the reaction at the girder support (see 9.4).
9.3.3 Intermediate transverse stiffeners
 Intermediate stiffeners that act as rigid supports to interior panels of the web should be designed for strength and stiffness.
 When flexible intermediate transverse stiffeners are used, their stiffness should be considered in the calculation of k_{τ} in 5.3(5). 34
 The effective section of intermediate stiffeners acting as rigid supports for web panels should have a minimum second moment of area I_{st}:
NOTE: Intermediate rigid stiffeners may be designed for an axial force equal to according to 9.2.1(3). In the case of variable shear forces the check is performed for the shear force at the distance 0,5h_{w} from the edge of the panel with the largest shear force.
9.3.4 Longitudinal stiffeners
 If longitudinal stiffeners are taken into account in the stress analysis they should be checked for direct stresses for the cross sectional resistance.
9.3.5 Welds
 The web to flange welds may be designed for the nominal shear flow V_{Ed} / h_{w} if V_{Ed} does not exceed For larger values V_{Ed} the weld between flanges and webs should be designed for the
 In all other cases welds should be designed to transfer forces along and across welds making up sections taking into account analysis method (elastic/plastic) and second order effects.
9.4 Transverse loads
 If the design resistance of an unstiffened web is insufficient, transverse stiffeners should be provided.
 The outofplane buckling resistance of the transverse stiffener under transverse loads and shear force (see 9.3.3(3)) should be determined from 6.3.3 or 6.3.4 of EN 199311, using buckling curve c. When both ends are assumed to be fixed laterally a buckling length ℓ of not less than 0,75h_{w} should be used. A larger value of ℓ should be used for conditions that provide less end restraint. If the stiffeners have cut outs at the loaded end, the cross sectional resistance should be checked at this end.
 Where single sided or other asymmetric stiffeners are used, the resulting eccentricity should be allowed for using 6.3.3 or 6.3.4 of EN 199311. If the stiffeners are assumed to provide lateral restraint to the compression flange they should comply with the stiffness and strength criteria in the design for lateral torsional buckling.
35
10 Reduced stress method
 The reduced stress method may be used to determine the stress limits for stiffened or unstiffened plates.
NOTE 1: This method is an alternative to the effective width method specified in section 4 to 7 in respect of the following:
 – σ_{x,Ed}, σ_{z,Ed} and τ_{Ed} are considered as acting together
 – the stress limits of the weakest part of the cross section may govern the resistance of the full cross section.
NOTE 2: The stress limits may also be used to determine equivalent effective areas. The National Annex may give limits of application for the methods.
 For unstiffened or stiffened panels subjected to combined stresses σ_{x,Ed}, σ_{z,Ed} and τ_{Ed} class 3 section properties may be assumed, where
where
α_{ult,k} 
is the minimum load amplifier for the design loads to reach the characteristic value of resistance of the most critical point of the plate, see (4); 
ρ 
is the reduction factor depending on the plate slenderness to take account of plate buckling, see (5); 
γ_{M1} 
is the partial factor applied to this method. 
 The modified plate slenderness should be taken from
where
α_{cr} 
is the minimum load amplifier for the design loads to reach the elastic critical load of the plate under the complete stress field, see (6) 
NOTE 1: For calculating σ_{cr} for the complete stress field, the stiffened plate may be modelled using the rules in section 4 and 5 without reduction of the second moment of area of longitudinal stiffeners as specified in 5.3(4).
NOTE 2: When α_{cr} cannot be determined for the panel and its subpanels as a whole, separate checks for the subpanel and the full panel may be applied.
 In determining α_{ult,k} the yield criterion may be used for resistance:
where σ_{Ed}, σ_{z,Ed} and τ_{Ed} are the components of the stress field in the ultimate limit state.
NOTE: By using the equation (10.3) it is assumed that the resistance is reached when yielding occurs without plate buckling.
 The reduction factor ρ may be determined using either of the following methods:
 the minimum value of the following reduction factors:
ρ_{x} 
for longitudinal stresses from 4.5.4(1) taking into account columnlike behaviour where relevant; 
ρ_{z} 
for transverse stresses from 4.5.4(1) taking into account columnlike behaviour where relevant; 
χ_{w} 
for shear stresses from 5.3(1) ; 
36
each calculated for the modified plate slenderness according to equation (10.2).
NOTE: This method leads to the verification formula:
NOTE: For determining ρ_{z} for transverse stresses the rules in section 4 for direct stresses σ_{x} should be applied to σ_{z} in the zdirection. For consistency section 6 should not be applied.
 a value interpolated between the values of ρ_{x}, ρ_{z} and χ_{w} as determined in a) by using the formula for α_{ult,k} as interpolation function
NOTE: This method leads to the verification format:
NOTE 1: Since verification formulae (10.3), (10.4) and (10.5) include an interaction between shear force, bending moment, axial force and transverse force, section 7 should not be applied.
NOTE 2: The National Annex may give further information on the use of equations (10.4) and (10.5). In case of panels with tension and compression it is recommended to apply equations (10.4) and (10.5) only for the compressive parts.
 Where α_{cr} values for the complete stress field are not available and only α_{cr,i} values for the various components of the stress field σ_{x,Ed}, σ_{Z,Ed} and τ_{Ed} can be used, the α_{cr} value may be determined from:
and α_{cr,x}, α_{cr,z}, τ_{cr}, ψ_{x} and ψ_{z} are determined from sections 4 to 6.
 Stiffeners and detailing of plate panels should be designed according to section 9.
37
Annex A – Calculation of critical stresses for stiffened plates
[informative]
A.1 Equivalent orthotropic plate
 Plates with at least three longitudinal stiffeners may be treated as equivalent orthotropic plates.
 The elastic critical plate buckling stress of the equivalent orthotropic plate may be taken as:
α_{cr,p} = k_{σ,p} σ_{E} (A.1)
where

k_{σ,p} 
is the buckling coefficient according to orthotropic plate theory with the stiffeners smeared over the plate; 
b 
is defined in Figure A. 1; 
t 
is the thickness of the plate. 
NOTE 1: The buckling coefficient k_{αp} is obtained either from appropriate charts for smeared stiffeners or relevant computer simulations; alternatively charts for discretely located stiffeners may be used provided local buckling in the subpanels can be ignored and treated separately.
NOTE 2: σ_{cr,p} is the elastic critical plate buckling stress at the edge of the panel where the maximum compression stress occurs, see Figure A. 1.
NOTE 3: Where a web is of concern, the width h in equations (A.1) and (A.2) should be replaced by h_{w}
NOTE 4: For stiffened plates with at least three equally spaced longitudinal stiffeners the plate buckling coefficient k_{σ,p} (global buckling of the stiffened panel) may be approximated by:
with
where:
I_{st} 
is the second moment of area of the whole stiffened plate; 
I_{p} 
is the second moment of area for bending of the plate 
A^{st} 
is the sum of the gross areas of the individual longitudinal stiffeners; 
A_{p} 
is the gross area of the plate = bt; 
σ_{1} 
is the larger edge stress; 
σ_{2} 
is the smaller edge stress; 
38
Figure A.1 : Notations for longitudinally stiffened plates
39
A.2 Critical plate buckling stress for plates with one or two stiffeners in the compression zone
A.2.1 General procedure
 If the stiffened plate has only one longitudinal stiffener in the compression zone the procedure in A.1 may be simplified by a fictitious isolated strut supported on an elastic foundation reflecting the plate effect in the direction perpendicular to this strut. The elastic critical stress of the strut may be obtained from A.2.2.
 For calculation of A_{st,1}, and I_{st,1} the gross crosssection of the column should be taken as the gross area of the stiffener and adjacent parts of the plate described as follows. If the subpanel is fully in compression, a portion (3 – ψ)/(5 – ψ) of its width b_{1} should be taken at the edge of the panel and 2/(5 – ψ) at the edge with the highest stress. If the stress changes from compression to tension within the subpanel, a portion 0,4 of the width b_{c} of the compressed part of this subpanel should be taken as part of the column, see Figure A.2 and also Table 4.1. ψ is the stress ratio relative to the subpanel in consideration.
 The effective^{P} crosssectional area A_{st,eff} of the column should be taken as the effective^{P} crosssection of the stiffener and the adjacent effective^{P} parts of the plate, see Figure A.1. The slenderness of the plate elements in the column may be determined according to 4.4(4), with σ_{com,Ed} calculated for the gross crosssection of the plate.
 If ρ_{c}f_{y}/γ_{M1}, with ρ_{c} determined according to 4.5.4(1), is greater than the average stress in the column σcom.Ed no further reduction of the effective^{P} area of the column should be made. Otherwise the effective area in (4.6) should be modified as follows:
 The reduction mentioned in A.2.1(4) should be applied only to the area of the column. No reduction need be applied to other compressed parts of the plate, except for checking buckling of subpanels.
 As an alternative to using an effective^{p} area according to A.2.1 (4), the resistance of the column may be determined from A.2.1 (5) to (7) and checked to ensure that it exceeds the average stress σ_{com,Ed}.
NOTE: The method outlined in (6) may be used in the case of multiple stiffeners in which the restraining effect from the plate is neglected, that is the fictitious column is considered free to buckle out of the plane of the web.
Figure A.2: Notations for a web plate with single stiffener in the compression zone
 If the stiffened plate has two longitudinal stiffeners in the compression zone, the one stiffener procedure described in A.2.1(1) may be applied, see Figure A.3. First, it is assumed that one of the stiffeners 40 buckles while the other one acts as a rigid support. Buckling of both the stiffeners simultaneously is accounted for by considering a single lumped stiffener that is substituted for both individual ones such that:
 its crosssectional area and its second moment of area I_{sℓ} are respectively the sum of that for the individual stiffeners
 it is positioned at the location of the resultant of the respective forces in the individual stiffeners
For each of these situations illustrated in Figure A.3 a relevant value of σ_{cr,p} is computed, see A.2.2(1), with and and , see Figure A.3.
Figure A.3: Notations for plate with two stiffeners in the compression zone
A.2.2 Simplified model using a column restrained by the plate
 In the case of a stiffened plate with one longitudinal stiffener located in the compression zone, the elastic critical buckling stress of the stiffener can be calculated as follows ignoring stiffeners in the tension zone:
with
where
A_{sℓ,1} 
is the gross area of the column obtained from A.2.1 (2) 
I_{sℓ,1} 
is the second moment of area of the gross crosssection of the column defined in A.2.1(2) about an axis through its centroid and parallel to the plane of the plate; 
b_{1}, b_{2} 
are the distances from the longitudinal edges of the web to the stiffener (b_{1}+b_{2} = b). 
note deleted 
 In the case of a stiffened plate with two longitudinal stiffeners located in the compression zone the elastic critical plate buckling stress should be taken as the lowest of those computed for the three cases using 41 equation (A.4) with , and b = B*. The stiffeners in the tension zone should be ignored in the calculation.
A.3 Shear buckling coefficients
 For plates with rigid transverse stiffeners and without longitudinal stiffeners or with more than two longitudinal stiffeners, the shear buckling coefficient k_{τ} can be obtained as follows:
where
but not less than
a 
is the distance between transverse stiffeners (see Figure 5.3); 
I_{sℓ} 
is the second moment of area of the longitudinal stiffener about the zz axis, see Figure 5.3 (b). 
For webs with longitudinal stiffeners, not necessarily equally spaced, I_{sℓ} is the sum of the stiffness of the individual stiffeners. 
NOTE: No intermediate nonrigid transverse stiffeners are allowed for in equation (A.5).
 The equation (A.5) also applies to plates with one or two longitudinal stiffeners, if the aspect ratio satisfies α ≥ 3. For plates with one or two longitudinal stiffeners and an aspect ratio α < 3 the shear buckling coefficient should be taken from:
42
Annex B – Nonuniform members
[informative]
B.1 General
 The rules in section 10 are applicable to webs of members with non parallel flanges as in haunched beams and to webs with regular or irregular openings and non orthogonal stiffeners.
 α_{ult} and α_{crit} may be obtained from FEmethods, see Annex C.
 The reduction factors ρ_{x}, ρ_{z} and χ_{w} for may be obtained from the appropriate plate buckling curves, see sections 4 and 5.
NOTE: The reduction factor ρ may be obtained as follows:
This procedure applies to ρ_{x}, ρ_{z} and χ_{w}. The values of and α_{p} are given in Table B.1. These values have been calibrated against the plate buckling curves in sections 4 and 5 and give a direct correlation to the equivalent geometric imperfection, by :
Table B.1: Values for and α_{p}
Product 
predominant buckling mode 
α_{P} 

hot rolled 
direct stress for ψ ≥ 0 
0,13 
0,70 
direct stress for ψ < 0 shear transverse stress 
0,80 
welded or cold formed 
direct stress for ψ ≥ 0 
0,34 
0,70 
direct stress for ψ < 0 shear transverse stress 
0,80 
43
B.2 Interaction of plate buckling and lateral torsional buckling
 The method given in B.1 may be extended to the verification of combined plate buckling and lateral torsional buckling of members by calculating α_{ult} and α_{cr} as follows:
α_{ult} 
is the minimum load amplifier for the design loads to reach the characteristic value of resistance of the most critical cross section, neglecting any plate buckling and lateral torsional buckling; 
α_{cr} 
is the minimum load amplifier for the design loads to reach the elastic critical loading of the member including plate buckling and lateral torsional buckling modes. 
 When α_{cr} contains lateral torsional buckling modes, the reduction factor ρ used should be the minimum of the reduction factor according to B.1(3) and the χ_{LT} – value for lateral torsional buckling according to 6.3.3 of EN 199311.
44
Annex C – Finite Element Methods of analysis (FEM)
[informative]
C.1 General
 Annex C gives guidance on the use of FEmethods for ultimate limit state, serviceability limit state or fatigue verifications of plated structures.
NOTE 1: For FEcalculation of shell structures see EN 199316.
NOTE 2: This guidance is intended for engineers who are experienced in the use of Finite Element methods.
 The choice of the FEmethod depends on the problem to be analysed and based on the following assumptions:
Table C.1: Assumptions for FEmethods
No 
Material behaviour 
Geometric behaviour 
Imperfections, see section C.5 
Example of use 
1 
linear 
linear 
no 
elastic shear lag effect, elastic resistance 
2 
non linear 
linear 
no 
plastic resistance in ULS 
3 
linear 
non linear 
no 
critical plate buckling load 
4 
linear 
non linear 
yes 
elastic plate buckling resistance 
5 
non linear 
non linear 
yes 
elasticplastic resistance in ULS 
C.2 Use
 In using FEM for design special care should be taken to
 – the modelling of the structural component and its boundary conditions;
 – the choice of software and documentation;
 – the use of imperfections;
 – the modelling of material properties;
 – the modelling of loads;
 – the modelling of limit state criteria;
 – the partial factors to be applied.
NOTE: The National Annex may define the conditions for the use of FEM analysis in design.
C.3 Modelling
 The choice of FEmodels (shell models or volume models) and the size of mesh determine the accuracy of results. For validation sensitivity checks with successive refinement may be carried out.
 The FEmodelling may be carried out either for:
 – the component as a whole or
 – a substructure as a part of the whole structure.
NOTE: An example for a component could be the web and/or the bottom plate of continuous box girders in the region of an intermediate support where the bottom plate is in compression. An example for a substructure could be a subpanel of a bottom plate subject to biaxial stresses.
 The boundary conditions for supports, interfaces and applied loads should be chosen such that results obtained are conservative. 45
 Geometric properties should be taken as nominal.
 Ail imperfections should be based on the shapes and amplitudes as given in section C.5.
 Material properties should conform to C.6(2).
C.4 Choice of software and documentation
 The software should be suitable for the task and be proven reliable.
NOTE: Reliability can be proven by appropriate bench mark tests.
 The mesh size, loading, boundary conditions and other input data as well as the output should be documented in a way that they can be reproduced by third parties.
C.5 Use of imperfections
 Where imperfections need to be included in the FEmodel these imperfections should include both geometric and structural imperfections.
 Unless a more refined analysis of the geometric imperfections and the structural imperfections is carried out, equivalent geometric imperfections may be used.
NOTE 1: Geometric imperfections may be based on the shape of the critical plate buckling modes with amplitudes given in the National Annex. 80 % of the geometric fabrication tolerances is recommended.
NOTE 2: Structural imperfections in terms of residual stresses may be represented by a stress pattern from the fabrication process with amplitudes equivalent to the mean (expected) values.
 The direction of the applied imperfection should be such that the lowest resistance is obtained.
 For applying equivalent geometric imperfections Table C.2 and Figure C.1 may be used.
Table C.2: Equivalent geometric imperfections
Type of imperfection 
Component 
Shape 
Magnitude 
global 
member with length ℓ 
bow 
see EN 199311, Table 5.1 
global 
longitudinal stiffener with length a 
bow 
min (a/400, b/400) 
local 
panel or subpanel with short span a or b 
buckling shape 
min (a/200, b/200) 
local 
stiffener or flange subject to twist 
bow twist 
1/50 
46
Figure C.1 : Modelling of equivalent geometric imperfections
47
 In combining imperfections a leading imperfection should be chosen and the accompanying imperfections may have their values reduced to 70%.
NOTE 1: Any type of imperfection should be taken as the leading imperfection and the others may be taken as the accompanying imperfections.
NOTE 2: Equivalent geometric imperfections may be substituted by the appropriate fictitious forces acting on the member.
C.6 Material properties
 Material properties should be taken as characteristic values.
 Depending on the accuracy and the allowable strain required for the analysis the following assumptions for the material behaviour may be used, see Figure C.2:
 elasticplastic without strain hardening;
 elasticplastic with a nominal plateau slope;
 elasticplastic with linear strain hardening;
 true stressstrain curve modified from the test results as follows:
Figure C.2: Modelling of material behaviour
48
NOTE: For the elastic modulus E the nominal value is relevant.
C.7 Loads
 The loads applied to the structures should include relevant load factors and load combination factors. For simplicity a single load multiplier α may be used.
C.8 Limit state criteria
 The ultimate limit state criteria should be used as follows:
 for structures susceptible to buckling:
attainment of the maximum load.
 for regions subjected to tensile stresses:
attainment of a limiting value of the principal membrane strain.
NOTE 1: The National Annex may specify the limiting of principal strain. A value of 5% is recommended.
NOTE 2: Other criteria may be used, e.g. attainment of the yielding criterion or limitation of the yielding zone.
C.9 Partial factors
 The load magnification factor α_{u} to the ultimate limit state should be sufficient to achieve the required reliability.
 The magnification factor α_{u} should consist of two factors as follows:
 α_{1} to cover the model uncertainty of the FEmodelling used. It should be obtained from evaluations of test calibrations, see Annex D to EN 1990;
 α_{2} to cover the scatter of the loading and resistance models. It may be taken as γ_{M1} if instability governs and γ_{M2} if fracture governs.
 It should be verified that:
α_{u} > α_{1} α_{2} (C.2)
NOTE: The National Annex may give information on γ_{M1} and γ_{M2}. The use of γ_{M1} and γ_{M2} as specified in the relevant parts of EN 1993 is recommended.
49
Annex D – Plate girders with corrugated webs
[informative]
D.1 General
 Annex D covers design rules for Igirders with trapezoidal or sinusoidal corrugated webs, see Figure D.1.
Figure D.1: Geometric notations
D.2 Ultimate limit state
D.2.1 Moment of resistance
 The moment of resistance My,_{Rd} due to bending should be taken as the minimum of the following:
where
f_{yf, r} 
is the value of yield stress reduced due to transverse moments in the flanges 
f_{yf,r} = f_{yf} f_{T} 

σ_{x}(M_{z}) 
is the stress due to the moment in the flange 
χ 
is the reduction factor for out of plane buckling according to 6.3 of EN 199311 
NOTE 1: The transverse moment M_{z} results from the shear flow in flanges as indicated in Figure D.2.
NOTE 2: For sinusoidally corrugated webs f_{T} is 1,0.
50
Figure D.2: Transverse actions due to shear flow introduction into the flange
 The effective^{p} area of the compression flange should be determined from 4.4(1) using the larger value of the slenderness parameter defined in 4.4(2). The buckling factor k_{σ} should be taken as the larger of a) and b):

where b is the maximum width of the outstand from the toe of the weld to the free edge
a = a_{1} + 2a_{4}
 k_{σ} = 0,60 (D.3)
text deleted
D.2.2 Shear resistance
 The shear resistance V_{bw,Rd} should be taken as:
where
χ_{c} 
is the lesser of the values of reduction factors for local buckling χ_{c,ℓ} global buckling χ_{c,g} obtained from (2) and (3) 
 The reduction factor χ_{c,ℓ} for local buckling should be calculated from:
a_{max} should be taken as the greater of a_{1} and a_{2}.
51
NOTE: For sinusoidally corrugated webs the National Annex may give information on the calculation of τ_{cr,ℓ} and χ_{c,ℓ}.
The use of the following equation is recommended:
where
w is the length of one half wave, see Figure D.1,
s is the unfolded length of one half wave, see Figure D.1
 The reduction factor χ_{c,g} for global buckling should be taken as
I_{z} second moment of area of one corrugation of length w, see Figure D.1
NOTE 1: s and I_{z} are related to the actual shape of the corrugation.
NOTE 2: Equation (D.10) is valid for plates that are assumed to be hinged at the edges.
D.2.3 Requirements for end stiffeners
 Bearing stiffeners should be designed according to section 9. 52
Annex E – Alternative methods for determining effective cross sections
[normative]
E.1 Effective areas for stress levels below the yield strength
 As an alternative to the method given in 4.4(2) the following formulae may be applied to determine effective areas at stress levels lower than the yield strength:
 for internal compression elements:
 for outstand compression elements:
For notations see 4.4(2) and 4.4(4). For calculation of resistance to global buckling 4.4(5) applies.
E.2 Effective areas for stiffness
 For the calculation of effective areas for stiffness the serviceability limit state slenderness may be calculated from:
where
σ_{com,Ed,ser} is defined as the maximum compressive stress (calculated on the basis of the effective cross section) in the relevant element under loads at serviceability limit state.
 The second moment of area may be calculated by an interpolation of the gross cross section and the effective cross section for the relevant load combination using the expression:
where
I_{gr} 
is the second moment of area of the gross cross section 
σ_{gr} 
is the maximum bending stress at serviceability limit states based on the gross cross section 

is the second moment of area of the effective cross section with allowance for local buckling according to E.1 calculated for the maximum stress σ_{com,Ed,ser} ≥ σ_{gr} within the span length considered. 
 The effective second moment of area I_{eff} may be taken as variable along the span according to the most severe locations. Alternatively a uniform value may be used based on the maximum absolute sagging moment under serviceability loading.
 The calculations require iterations, but as a conservative approximation they may be carried out as a single calculation at a stress level equal to or higher than σ_{com,Ed,ser}.
53