Compendium of en 1993-1-1

Eurobuild in Steel

Compendium of EN1993-1-1

COMPENDIUM OF EN 1993-1-1

ECSC EURO BUILD IN STEEL PAGE 1 OF 84

Forschungsvereinigung Stahlanwendung e.V. (Contractor)

Peiner Träger GmbH (Contractor)

University of Dortmund – Institute for Steel Construction (Sub-contractor)

This document is part of the ECSC EuroBuild project

Compendium of

EN 1993-1-1

Summary, design aids and flow charts



Table of contents

1 BASICS 5

1.1 Symbols 5

1.2 Concept of design 8

1.3 Partial factors γ 9 1.3.1 Partial factor γF for loads 9 1.3.2 Partial factor γM for resistances 9

1.4 Materials 9 1.4.1 Design values of material coefficients 9 1.4.2 Material properties 10

2 BASIS OF DESIGN 11

2.1 Classification of cross-sections 11

2.2 Internal forces and moments 14 2.2.1 Influence of second-order analysis 14 2.2.2 Influence of second-order analysis: portal frames structures 14

2.3 Structural stability of frames 15

2.4 Imperfection 16 2.4.1 Global initial sway imperfection φ 16 2.4.2 Initial local bow imperfection 17 2.4.3 Imperfection for analysis of bracing systems 19

2.5 Structural analysis 21

3 ULTIMATE LIMIT STATE 23

3.1 General 23 3.1.1 Von-Mises yield criterion 23 3.1.2 Section properties and resistances 23

3.2 Structural analysis of cross-section 24 3.2.1 Tension 24 3.2.2 Compression 27 3.2.3 Bending moment about one axis 28



3.2.4 Bending moment about both axes 29 3.2.5 Shear 29 3.2.6 Torsion 31 3.2.7 Interaction of torsion and shear 32 3.2.8 Interaction of bending and shear 33 3.2.9 Interaction of uniaxial bending and axial force 34 3.2.10 Interaction of bi-axial bending and axial force 36 3.2.11 Interaction of bending, shear and axial force 36

3.3 Structural analysis of members 39 3.3.1 Buckling length Lcr 39 3.3.2 Uniform members in compression 40 3.3.3 Uniform members in bending – Lateral torsional buckling 44 3.3.4 Uniform members in bending and axial compression – I-, H- and

hollow sections 48 3.3.5 Interaction factor kij according to Annex B 51

4 DESIGN AIDS 53

4.1 Initial sway imperfection 53

4.2 Classification of cross-sections 54

4.3 Effective shear area AV 56

4.4 Interaction of bending and shear 57

4.5 Interaction of uniaxial bending and axial force 58

4.6 Reduction factor χ and χLT 61

5 FLOW CHARTS 63

5.1 Design of steel structures 64 Flow chart 5.1: General procedure of the design of steel structures 64 Flow chart 5.1 (1): Continuation of General procedure of the design of steel structures 65

5.2 Basis of design 66 Flow chart 5.2: Initial sway imperfection φ 66

5.2.1 Classification of cross-sections 67 Flow chart 5.3: Classification of one side supported compression parts 67 Flow chart 5.4: Classification of both-side supported compression parts 68



5.3 Structural analysis of cross-sections 69 Flow chart 5.5: Tension 69 Flow chart 5.6: Compression 70 Flow chart 5.7: Bending 71 Flow chart 5.8: shear 72

5.3.1 Interaction 73 Flow chart 5.9: Interaction bending and shear of I-sections V + M 73 Flow chart 5.10: Interaction bending and axial force N + My 74 Flow chart 5.11: Interaction bending about z-z axis and axial force N + Mz 75 Flow chart 5.12: Interaction of uniaxial bending, shear and axial force N + V + My 76 Flow chart 5.12 (1): Continuation of interaction N + V + My 77 Flow chart 5.12 (2): Continuation of interaction N + V + My 78

5.4 Structural analysis of members 79 Flow chart 5.13: Centrical compression – flexural buckling 79 Flow chart 5.14: Lateral torsional buckling 80 Flow chart 5.14 (1): Continuation of lateral torsional buckling 81 Flow chart 5.15: Bending and compression 82 Flow chart 5.15 (1): Continuation of Bending and compression 83

6 LITERATURE 84



1 Basics

1.1 Symbols

This chapter notes and describes the basic symbols. The extra list of indices al-lows by a combination with the major symbols, that all notations could be men-tioned in a short way.

Material propertiesfy yield strength

fu ultimate tensile strength

fyb yield strength of a bolt

fub ultimate strength of a bolt

fyp yield strength of a pin

fup ultimate strength of a pin

fur ultimate strength of a rivet

Variables of load, resistance and cross sectionE force and load; modulus of

elasticity

R resistance

G dead load

Q variable load

F load; force

N axial force

V shear force

M bending moment

T torsional moment

∆M additional moment from shift of the centre of the effective area Aeff relative to the cen-tre of gravity of the gross cross section

eN,i shift of the centre of the area Aeff relative to the centre of gravity of the gross cross section

γMi partial factor for resistance

γFi partial factor for material

ψ combination factor

A gross area of cross-section

S first moment of inertia

I second moment of inertia

W section modulus

i radius of gyration

b width of a cross-section

h depth of a cross-section

c width or depth of a part of a cross section

t thickness of a part of a cross section

σ axial stress

τ shear stress

a effective throat thickness



leff effective length

d nominal bolt diameter

d0 hole diameter for a bolt, a pin or a rivet

Variables of the systemFcr elastic critical buckling load

for global instability mode based on initial elastic stiff-nesses

Ncr elastic critical force for the relevant buckling mode based on the gross cross sectional properties

Mcr elastic critical moment for lateral torsional buckling

αcr amplification factor by which the design loads have to be increased to reach elastic critical loads

λ non-dimensional slender-ness

λ1 slenderness value to deter-mine the relative slender-ness

Indicesi; j general: variable, replace-

ment character

x; y; z symbol of cross-section axes

k nominal value

d design value

E stress

R resistance

A exceeding

ser serviceability

c cross section

pl plastic

el elastic

eff effectiv

net net

LT lateral torsion, torsional buckling

u; t ultimate, tension

w welding

b bolt, bearing, buckling

v shearing

s slip

f flange

w web

V reduced by the presence of shear force

N reduced by the presence of normal force

⊥ vertical

II parallel



Figure 1.1: Dimensions and axes of cross-sections



EN 1990, 6.4.1(b)

1.2 Concept of design

The safety concept considers temporal and spatial variations as well as insecurities of the mechanical and stochastic models by using partial fac-tor to reduce the resistance and to increase the forces, compare Figure 1.2. By this differentiated concept a more realistic design of steel struc-tures is possible in comparison to the concept of the global factors of older standards.

dt∫∞−

=x

f(x)F(X) ; f: density function

Figure 1.2: Density function f(x) of load E and resistance R

From this facts follow the concept of design for all checks:

General concept of design

M

kFkdd

REREγ

γ ≤⋅⇔≤

with

Ed: design value of loads in ultimate limit state

Rd: design value of resistance

Ek: characteristic value of loads

Rk: characteristic value of resistance

γFi: partial factor of loads

γMi: partial factor of resistances



EN 1993-1-1, 3.2.6

EN 1990, Table A.1.2 (B)

EN 1993-1-1, 6.1 γM1 = 1,0

1.3 Partial factors γ

1.3.1 Partial factor γF for loads

Table 1.1: Partial factor γF for loads for the design of ultimate limit state

effects permanent load variable load

unfavourable 35,1sup,jG =γ 50,1

supQ =γ

favourable 00,1inf,jG =γ 0

inf,Q =γ

1.3.2 Partial factor γM for resistances

Table 1.2: partial factor γM for resistances

partial factor limit states

γM0 = 1,00 resistance of cross-sections

γM1 = 1,00 resistance of members due to instability

γM2 = 1,25 resistance of cross-sections in tension to fracture

The partial factors γMi may be defined in the National Annex. In Germany for example γM1 = 1,10 is recommend to DIN-FB 103 for the design of bridges.

1.4 Materials

1.4.1 Design values of material coefficients

• modulus of elasticity: ]mm/N[000.210E 2=

• shear modulus: ]mm/N[000.81)1(2

EG 2≈−

=ν

• Poisson’s ratio in elastic stage: 3,0=ν

• coefficient of linear thermal expansion: ]K[10x12 16 −−=α )C100Tfor( °≤



1.4.2 Material properties

Table 1.3: Nominal value of yield strength and ultimate tensile strength for hot rolled structural steel according to EN 1993-1-1, Table 3.1

nominal thickness t fy fu Steel grade EN 10025 [mm] [N/mm²] [N/mm²]

mm40t ≤ 235 360 S 235

mm80tmm40 ≤< 215 360

mm40t ≤ 275 430 S 275

mm80tmm40 ≤< 255 410

mm40t ≤ 355 510 S 355

mm80tmm40 ≤< 335 470

mm40t ≤ 420 520 S 420 N/NL

mm80tmm40 ≤< 390 520

mm40t ≤ 440 550 S 450

mm80tmm40 ≤< 410 550

mm40t ≤ 460 540 S 460 N/NL

mm80tmm40 ≤< 430 540

Table 1.4: Nominal value of yield strength and ultimate tensile strength for hot rolled structural hollow sections according to EN 1993-1-1, Table 3.1

nominal thickness t fy fu Steel grade EN 10210-1 [mm] [N/mm²] [N/mm²]

mm40t ≤ 235 360 S 235 H

mm65tmm40 ≤< 215 340

mm40t ≤ 275 430 S 275 H

mm65tmm40 ≤< 255 410

mm40t ≤ 355 510 S 355 H

mm65tmm40 ≤< 335 490

EN 1993-1-1, Table 3.1 proper thickness of material: rolled

welded

EN 1993-1-1, Table 3.1



EN 1993-1-1, 5.5.2

2 Basis of design

2.1 Classification of cross-sections

Table 2.1: Classification of the analysis on the basis of the class of cross-section

Class Criterion Structural analysis

1 Cross-sections with rotation capacity to form plastic hinges and -zones plastic-plastic

2 Cross-sections with limited rotation capac-ity, but able to develop plastic moment resistance

elastic-plastic

3 Cross-sections which achieve the yield strength in the outer compression fibre, without plastic moment resistance

elastic-elastic

4 Cross-sections which fail of local buckling before the yield strength will achieve.

elastic-elastic in consideration of local buckling on EN 1993-1-5

The following four tables include the criterion of the width-and-thickness for the cross-section classes 1-3. It differentiates compressed plates that are supported at one or both sides respectively and angels and hollow sections. If the criterion of a class 3 cross-section fails, the cross-section has to classify to class 4. Then an analysis on the basis of EN 1993-1-5 is necessary.



Table 2.2: Maximum width-to-thickness ratio for both-side supported compression parts

Class Part subject to

bending Part subject to compression

Part subject to bending and compression

:5,0>α113

396−αε

1 ε72 ε33 :5,0≤α

αε36

:5,0>α113

456−αε

2 ε83 ε38 :5,0≤α

αε5,41

:1−>ψψ

ε33,067,0

42+

3 ε124 ε42

:1−≤ψ ( ) ( )ψψε −−162

S235 S275 S355 S420 S460

yf235

=ε 1,0 0,92 0,81 0,75 0,71

Table 2.3: Maximum width-to-thickness ratio for one side supported compression parts

part subject to bending and compression Class

part subject to compression Tip in compression Tip in tension

1 ε9 α

ε9 ααε9

2 ε10 α

ε10 ααε10

3 ε14 σε k21 kσ see Table 2.6

1

2

σσψ =

with σ1 maximum compres-sive stress The compressive stress is defined positive.

EN 1993-1-1, Tab. 5.2, sheet 2

EN 1993-1-1, Tab. 5.2, sheet 1



Table 2.4: Maximum width-to-thickness ratio of angles

Class Section in compression

3 εε 5,11t2hb:15t/h ≤

+≤

Table 2.5: Maximum width-to-thickness ratio of tubular sections

Class Section in bending and/or compres-

sion 1 50 ε² 2 70 ε² 3 90 ε²

fy S235 S275 S355 S420 S460ε² 1,0 0,85 0,66 0,56 0,51

Table 2.6: Buckling factor kσ for internal and outstand compression elements

Internal compression elements ψ 1 1 > ψ > 0 0 0 > ψ > -1 -1 -1 > ψ > -3

kσ 4,0 )05,1(2,8

ψ+ 7,81 7,81 - 6,29ψ + 9,78ψ² 23,9 5,98 (1 - ψ)²

Outstand compression elements – Tip under compression

ψ 1 0 -1 1 > ψ > -3 kσ 0,43 0,57 0,85 0,57 – 0,21ψ + 0,07ψ²

Outstand compression elements – Tip under tension

ψ 1 1 > ψ > 0 0 0 > ψ > -1 -1

kσ 0,43 )34,0(578,0

+ψ 1,70 1,7 - 5ψ + 17,1ψ² 23,8

Buckling factor kσ on EN 1993-1-5, Table 4.1 and 4.2

EN 1993-1-1, Tab. 5.2, sheet 3

1

2

σσ

ψ =

with σ1 maximum compres-sive stress



EN 1993-1-1, 5.2.1, (4)B

2.2 Internal forces and moments

First-order analysis

The internal forces and moments have to be determined in consideration of the initial geometry of the structure with initial sway imperfections re-placed by equivalent horizontal forces.

Second-order analysis

The internal forces and moments have to be determined in consideration of the deformation of the structure and the imperfections, which are the cause of an increasing moment.

2.2.1 Influence of second-order analysis

Structures, which fulfil the following conditions, are classified to stiff struc-tures. Therefore the effects of the horizontal deformation does not have to be considered. Accordingly a second-order analysis is not necessary.

For Structures, which don’t achieve this criterions, the second-order ef-fects have to be considered.

• Elastic analysis

10FF

Ed

crcr ≥=α (2.1)

• Plastic analysis

15FF

Ed

crcr ≥=α (2.2)

with FEd design loading on the structure

Fcr elastic critical buckling load for global instability mode based on initial elastic stiffnesses

2.2.2 Influence of second-order analysis: portal frames structures

Especially for portal frames constructions with a slope steeper of 1:2 or 26° respectively, the mentioned criterions also apply. Additionally, the non-dimensional slenderness of the beams or rafters has to restrain the following condition and it has to be provided, that the ends of the system length are hinged.

2

2

cr LEIF ⋅

=π

EN 1993-1-1, equation (5.1)



EN 1993-1-1, equation (5.4)

EN 1993-1-1, 5.2.2, (5)B

Criterion: Ed

y

NfA

3,0⋅

≥λ (2.3)

The factor αcr is defined as follows:

15.bzw10hVH

Ed,HEd

Edcr ≥⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=

δα (2.4)

Figure 2.1: Definition of HEd, VEd und δH,Ed

2.3 Structural stability of frames

The influences of a second order analysis and imperfections and the fol-lowing analysis of stability can occur in three ways.

Table 2.7: Summary of the analysis of stability

Analysis Proper internal forces and mo-ments Stability

1 second order analysis in considera-tion of all imperfections Resistance of cross-sections

2 second order analysis in considera-tion of the deformation released by initial sway imperfection

Buckling resistance with end moments of the members and buckling length = system length

3 first order analysis

Buckling resistance on the basis of the appropriate buckling length (= equivalent column method)

Second order analysis by using the factor q for increase Criterion: 3cr ≥α (2.5)

If αcr complies with the requirement, the factor q can be determined:

cr

11

1q

α−

= (2.6)



EN 1993-1-1, 5.3.2, equation (5.5)

The analysis of the initial forces and moments is carried out by increasing the horizontal forces Fh,Ed’.

IEd,h

IIEd,h FqF ⋅= (2.7)

with

Fh,Ed’ design value of the horizontal forces including the equivalent horizontal forces of the initial sway imperfection

2.4 Imperfection

2.4.1 Global initial sway imperfection φ

If the following criterion is fulfilled, the initial sway imperfection does not have to be considered.

Criterion: EdEd V15,0H ≥ (2.8)

with

HEd sum of the horizontal forces at the bottom of the system

VEd sum of the vertical forces at the bottom of the system

Initial sway imperfection mh0 ααφφ ⋅⋅= (2.9)

with

φ0 basic value: φ0 = 1/200

αh reduction factor for height h applicable to columns

h

2h =α , but 0,1

32

h ≤≤ α

h height of the structure in meters

αm reduction factor for the number of columns in one row

⎟⎠⎞

⎜⎝⎛ +=

m115,0mα

m the number of columns in one row including only those columns which carry a vertical load NEd not less than 50% of the average value of the column in the vertical plane considered

EN 1993-1-1, 5.3.2, (4)B



EN 1993-1-1, Table 5.1

Figure 2.2: Equivalent initial sway imperfection

2.4.2 Initial local bow imperfection

A local bow imperfection is considered by a second-order analysis, if the condition (1) or (2) is fulfilled. (1)

Ed

y

NfA

5,0>λ (2.10)

with

λ the in-plane non-dimensional slenderness calculated for the member considered as hinged at its ends (β =1)

(2) at least one moment resistant joint at one member end

Figure 2.3: Initial bow imperfection

Table 2.8: Value of the initial bow imperfection e0 e0 Buckling curves elastic plastic

a0 l/350 l/300 a l/300 l/250 b l/250 l/200 c l/200 l/150 d l/150 l/100

Class 1up to 3:

cr

y

NfA

=λ

EN 1993-1-1, 5.3.2, (6)



EN 1993-1-1, 5.3.4, (3)

EN 1993-1-1, 5.3.2, (7) and Figure 5.4

Equivalent horizontal forces

The imperfections can also be considered by equivalent horizontal forces. For multi-storey frame structures with the same geometry and stiffness in each storey, the equivalent horizontal forces can be deter-mined as shown in Figure 2.5.

Figure 2.4: Replacement of initial imperfection by equivalent horizontal forces

Figure 2.5: Equivalent horizontal forces for multi-storey frame constructions

Initial bow imperfections for lateral-torsional buckling

For members, that should be checked for lateral-torsional buckling on the basis of a second-order analysis and an analysis according to chapter 0, initial bow imperfections about the minor axis have to be considered, see (2.11). The resulting and additional moments have to apply by determin-ing the buckling resistances.

00 eke~ ⋅= (2.11)

with

e0 value of the initial bow imperfection about the minor axis

k reduction factor; k = 0,5 is recommended



2.4.3 Imperfection for analysis of bracing systems

Initial bow imperfection of the members to be restrained

500Le m0 α= (2.12)

with

αm reduction factor for the number of members to be restrained

⎟⎠⎞

⎜⎝⎛ +=

m115,0mα

m number of members to be restrained

Simplified, the initial bow imperfection can convert in an equivalent stabi-lising forces as shown in Figure 2.6.

Figure 2.6: Equivalent stabilising force q

Equivalent stabilising force q

∑+

= 2q0

Ed Le

8Nqδ

(2.13)

with

δq deformation of the bracing system

L length of the bracing system

NEd design value of the compression load in the flange or chord of the members to be restrained, it can be accepted:

h/MN EdEd =

MEd maximum bending moment in the beam

h overall height of the beam

EN 1993-1-1, 5.3.3

equation (5.12)

EN 1993-1-1, equation (5.13)



Bracing forces at splices

Figure 2.7: Local forces to bracing system

Sway imperfections of vertical members to be restrained

/2

/2

NEd

NEd

h

h

Hi = NEd

nordnung der Anfangsschiefst

Hi = NEd

NEd

NEd

h

fstellung für Horizontalkräfte

Hi: additional load on horizontal bracing members Figure 2.8: Configuration of sway imperfections φ for horizontal forces on floor dia-

phragms

EN 1993-1-1, 5.3.2, (5)B

for example: floor

0m φαφ =

200/10 =φ

αm see equation (2.12)

force at the splice

100/NN2 EdmEd αφ =

EN 1993-1-1,5.3.3, (4) and Figure 5.7

splice

brac

ing

syst

em



EN 1993-1-1, 5.4.1 and 5.4.2 5.4.1, (4)B

EN 1993-1-1, 5.4.1 and 5.4.3

2.5 Structural analysis

Figure 2.9: Stress-strain relationship for elastic and plastic structural analysis

Elastic global analysis

The elastic global analysis admits redistribution of the support bending moment in continuous beams, when they exceed the plastic bending re-sistance of 15%. The requirements of a redistribution are:

1. after the redistribution, the internal forces an moments remain in equilibrium with the applied loads

2. all members in which the moment are reduced have Class 1 or 2 cross-sections

3. lateral torsional buckling of the members is prevented.

Figure 2.10: Criterion on redistribution of the support moment

Plastic global analysis

Requirements:

1. The members have class 1 cross-sections with a sufficient rotation capacity.

2. Futhermore the stability of the members at plastic hinges has to be assured.



The analysis of the internal forces and moments can be done by three methods:

1. elastic-plastic analysis with plastified sections and/or joints as plastic hinges

2. non-linear plastic analysis considering the partial plastification of members in plastic zones

3. rigid plastic analysis neglecting the elastic behavior between hinges



3 Ultimate limit states

3.1 General

3.1.1 Von-Mises yield criterion

1f

3ffff

2

0My

Ed

0My

Ed,z

0My

Ed,x

2

0My

Ed,z

2

0My

Ed,x ≤⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

γτ

γσ

γσ

γσ

γσ

(3.1)

The von-Mises yield criterion applies if no other criterion of interaction or analysis will be mentioned.

3.1.2 Section properties and resistances

Table 3.1: Resistance against classes of section

Class Resistance Section properties compression

Section properties bending

1 plastic A Wpl,y, Wpl,z

2 plastic A Wpl,y, Wpl,z

3 elastic A Wel,y, Wel,z

4

resistance on the basis of the effective cross-

section, see EN 1993-1-5 and Figure 3.1

Aeff Weff,y, Weff,z

The effective section properties Aeff and Weff have to be calculated on the basis of a reduced cross-section due to local buckling, compare Figure 3.1, according to EN 1993-1-5.

Figure 3.1: Effective area Aeff of Class 4 cross-sections under bending and compression

From this, additional moments ∆My,Ed or/and ∆Mz,Ed , that depend on the shift of the major axes of the effective section area as regards to the axes of the gross cross-section area result. They must be calculated by multi-plying the axial force with the distance to the new balance point.

NyEdEd,y eNM ⋅=∆ and NzEdEd,z eNM ⋅=∆

EN 1993-1-1, equation (6.1)

EN 1993-1-1, Table 6.7

6.3.3, NOTE 3



EN 1993-1-1, equation (6.6) and (6.7)

EN 1993-1-1,6.2.2.2,(3)

3.2 Structural analysis of cross-section

The assimilation of the analysis (el.-el., el.-pl., pl.-pl.) results from the different classes in consideration of the cross-section properties and the elastic or plastic analysis of the internal forces and moments. Respec-tively, cross-sections with local buckling can be verified on the basis of the following analysis, also, compare 3.1.2: Section properties and resis-tances.

3.2.1 Tension

Check: 0,1NN

Rd,t

Ed ≤ (3.2)

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⋅⋅=

⋅==

=

2M

unetRd,u

0M

yRd,elRd,pl

Rd,t

fA9,0N

fANN

minN

γ

γ (3.3)

with

A gross cross-section area

Anet net area along the critical fracture line

Nt,Rd for cross-sections with bolted connections of category C ac-cording to EN 1993-1-8 The cross-section with a bolted connection of category C has to be veri-fied in the critical fracture line, in addition to the above named analysis.

0M

ynetRd,netRd,t

fANN

γ⋅

== (3.4)

Net area Anet For symmetric bolt connections, the critical fracture line is defined by the line which runs rectangular to the axis of the member and through the maximum numbers of holes, compare Figure 3.2.

EN 1993-1-1, equation (6.5)



Figure 3.2: Critical fracture line in symmetrical bolted connections

If the bolt connections are staggered, compare Figure 3.3. The area ∆A, that has to deducted from the gross area, is the maximum of the follow-ing two values (1) and (2).

Figure 3.3: Critical fracture line in staggered bolted connections

(1) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅=∆ ∑ p4

sdntA2

line 1, compare Figure 3.3

with

s 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

t thickness

n number of holes extending over the fracture line

d diameter of hole

(2) ∆A is like the deduction of non-staggered holes, line 2

Angels connected by one leg in tension

If the following criterions of bolted and welded connections are satisfied, the eccentricity and the additional moments in the joints do not have to be considered. Bolted connections

Figure 3.4: Configuration und pitches of holes

EN 1993-1-1,6.2.2.2,(4)

EN 1993-1-8, 3.10.3



EN 1993-1-8, 3.10.2

EN 1993-1-8, 4.13

• connection with 1 bolt ( )

2M

u02Rd,u

ftd5,0e0,2Nγ

⋅−= (3.5)

• connection with 2 bolts

2M

unet2Rd,u

fANγ

β ⋅⋅= (3.6)

• connection with 3 or more bolts

2M

unet3Rd,u

fANγ

β ⋅⋅= (3.7)

Table 3.2: Reduction factors β2 and β3

pitch p1 ≤ 2,5 d0 ≥ 5,0 d0

2 bolts β2 0,4 0,7

3 or more bolts β3 0,5 0,7

This factors may be interpolated linear, if the pitches p1 are different from the defined values in the table.

Welded connections

Figure 3.5: Definition of the area A for welded connection of angels by one leg

Block tearing Tension members, which are connected with a connection plate by bolts, have to achieve the analysis of block tearing, equation (3.8), additionally.

Check: 0,1VN

Rd,1,eff

Ed ≤ (3.8)



0M

nvy

2M

ntuRd,1,eff 3

AfAfVγγ ⋅

⋅+

⋅= (3.9)

with

Ant net area subjected to tension

Anv net area subjected to shear

Figure 3.6: Tearing of a symmetrical bolted connections under centrical force

3.2.2 Compression

For members in compression instability failure has to be analysed, com-pare chapter 3.3.2: Uniform members in compression. Furthermore the cross-sections at the end of the members have to satisfy equation (3.10).

Check: 0,1NN

Rd,c

Ed ≤ (3.10)

Class 1, 2 and 3 cross-sections

0M

yRd,elRd,plRd,c

fANNN

γ⋅

=== (3.11)

Class 4 cross-sections

0M

yeffRd,c

fAN

γ⋅

= (3.12)

The area of the cross-section or the effective area should be determined without a deduction of holes due to fasteners. But all other holes have to be considered.

EN 1993-1-1, equation (6.10)

EN 1993-1-1, equation (6.9)

EN 1993-1-1, equation (6.11)

EN 1993-1-8, equation (3.9)



EN 1993-1-1, 6.2.5

EN 1993-1-1, 6.2.5, (4)-(6)

3.2.3 Bending moment about one axis

Check: 0,1MM

Rd,c

Ed ≤ (3.13)

Class 1 and 2 cross-sections

0M

yplRd,plRd,c

fWMM

γ⋅

== (3.14)


0M

yelRd,elRd,c

fWMM

γ⋅

== (3.15)


0M

yeffRd,c

fWM

γ⋅

= (3.16)

The section modulus Wpl, Wel und Weff must be calculated for the respec-tive axis.

Reduction in the tensile zone of the section

Reduction of the cross-section as a result of holes in the tension zone, compare Figure 3.7, have to be considered by the analysis of the bend-ing moment. If the requirement (3.17) for the flange in tension is not at-tained, the section modulus Wy and Wz will be calculated by the consid-eration of the new cross-section. Webs in tension have to be observed as well. The mentioned criterion above must be applied to the whole tension zone, accordingly the flange and part of web in tension.

Figure 3.7: Deduction of the flange

considering the holes in the tension zone



equation (6.18)

EN 1993-1-1, 6.2.1(7) or 6.2.9.1(6), α = β = 1, respectively

EN 1993-1-1, 6.2.6, (7) or EN 1993-1-8, 3.10.2 respectively EN 1993-1-1, equation (6.17)

Criterion: 0M

yf

2M

unet,f fAf9,0Aγγ

⋅≥

⋅⋅ (3.17)

with

Af,net net area of the tension flange

Af gross area of the tension flange

3.2.4 Bending moment about both axes To combine the bending moment about the y-y and z-z axes in a conser-vative way, the utilisations have to be added, compare the von-Mises yield criterion (α = β = 1). This analysis can be applied to all classes of cross-section.

Check: 0,1MM

MM

Rd,z,c

Ed,z

Rd,y,c

Ed,y ≤⎥⎦

⎤⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡βα

(3.18)

For a differentiated analysis in consideration of the form of the cross-section, the exponents α and β are defined in chapter 3.2.10: Interaction of bi-axial bending and axial force.

3.2.5 Shear


Check: 0,1VV

Rd,c

Ed ≤ (3.19)

( )0M

yvRd,plRd,c

3fAVV

γ== (3.20)

with

Av effective shear area, Table 3.3

Figure 3.8: Effective shear area of parallel to the web loaded cross-sections



EN 1993-1-1, 6.2.6, (3) η = 1,0, compare DIN-FB 103 tf flange thickness tw web thickness b overall breadth h overall depth hw depth of the web r root radius

Table 3.3: Effective shear area Av

cross-section direction of load effective shear area Av

rolled I- and H-sections parallel to web wwfwf tht)r2t(bt2A η≥++−

rolled U-sections parallel to web fwf t)rt(bt2A ++−

rolled T-sections parallel to web )btA(9,0 f−

welded I- ,H- and box sec-tions parallel to web ∑ )th( wwη

rolled I- and H-sections parallel to flange ftb2 welded I- ,H- , U- and box sections parallel to flange )th(A ww∑−

rolled rectangular hollow sections

parallel to depth

parallel to width

)hb/(Ah +

)hb/(Ab +

tubular hollow sections and tubes of uniform thickness - π/A2


Check: 0,1)3(f 0My

Ed ≤γτ (3.21)

tISVEd

Ed ⋅⋅

=τ (3.22)

For I- and H-sections with a distinctive flange, which means that the ratio area of flange to area of web. Af/Aw ≥ 0,6 applies:

w

EdEd A

V=τ (3.23)

with

Aw area of the web; www thA ⋅=

Shear buckling for web without stiffeners

Ckeck: ηε72

th

t

w ≤ (3.24)

with

EN 1993-1-1, equation (6.19), (6.20)

EN 1993-1-1, equation (6.21)

EN 1993-1-1, 6.2.6, (6)

yf235

=ε



EN 1993-1-8, 3.10.2

EN 1993-1-8, equation (3.10)

η = 1,0 (conservative value)

Block tearing at ends of members

Check: 0,1V

V

Rd,2,eff

Ed ≤ (3.25)

0M

nvy

2M

ntuRd,2,eff 3

AfAf5,0Vγγ ⋅

⋅+

⋅⋅= (3.26)

with

Ant net area subjected to tension

Anv net area subjected to shear

Figure 3.9: Tearing under shear force

3.2.6 Torsion

For the verification of cross-sections loaded by torsion, the elastic yield-criterion must be satisfied for all classes of cross-section.

For I-sections and other open cross-sections shear stresses subjected to torsion result from St. Venant torsion moment and warping torsion mo-ment, which arise from applicable bearing and furthermore from a chang-ing torsional moment.

The analysis of I-sections evaluated by the shear stress and the geome-try of the cross-section is defined as follows:

Check: 0M

yEd,wEd,tEd 3

f

γτττ

⋅≤+= (3.27)

with

Ed,tτ shear stress due to St. Venant torsion (maximum in the flange)

EN 1993-1-1, 6.2.7

compare EN 1993-1-1, equation (6.23) and (6.24)

For a calculation of stress in I-section, compare literature



EN 1993-1-1, 6.2.7, (9)

EN 1993-1-1, equation (6.25)

EN 1993-1-1, equation (6.26) up to (6.28)

t

fEd,tEd,t I

tT ⋅=τ

Ed,wτ shear stress due to warping torsion

ω

ωτ

I4/bT MEd,w

Ed,w

⋅⋅=

Tt,Ed design value of the St. Venant torsional moment

Tw,Ed design value of the warping torsional moment

For I-sections with warping normal stress the following analysis has to be satisfied:

Check: 0M

yEdB

fI

BEd γ

σω

≤= (3.28)

with

σBEd normal stress to the bimoment

BEd bimoment

For an analysis of other cross-sections, design aids in literature may be used.

3.2.7 Interaction of torsion and shear


Check: 0,1V

V

Rd,T,pl

Ed ≤ (3.29)

The shear stresses must be determined in the shear loaded parts of the section.

• I- and H-sections

( ) Rd,pl0My

Ed,tRd,T,pl V

/3/f25,11V ⋅−=

γ

τ (3.30)

• U-sections

( ) ( ) Rd,pl0My

Ed,w

0My

Ed,tRd,T,pl V

/3/f/3/f25,11V ⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−=

γ

τ

γ

τ (3.31)



EN 1993-1-1, 6.2.8, (2)

EN 1993-1-1, 6.2.8(3) or 6.2.10 NOTE respectively

EN 1993-1-1, 6.2.7, (5)

• Hollow sections

( ) Rd,pl0My

Ed,tRd,T,pl V

/3/f25,11V ⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡−=

γ

τ (3.32)


The interaction of Class 3 and 4 sections results from the analysis of the yield criterion equation (3.1).

3.2.8 Interaction of bending and shear

Criterion:

5,0VV

Rd,pl

Ed ≤ (3.33)

If the condition is not satisfied, the interaction will take place by reducing the loadbearing capacity of the shear loaded part of cross-section either by

1. reducing the yield strength ( ) yred,y f1f ⋅−= ρ or

2. reducing the thickness ( ) wred,w t1t ⋅−= ρ

2

Rd,pl

Ed 1V

V2⎟⎟⎠

⎞⎜⎜⎝

⎛−=ρ (3.34)

General Check: Rd,cRd,VEd MMM ≤≤ (3.35)

with

MV,Rd reduced design plastic resistance because of shear force

Double-symmetric I-cross-section with bending about the major axis

Rd,y,c0M

yw

2w

y,pl

Rd,y,V Mf

t4AW

M ≤⎥⎥⎦

⎤

⎢⎢⎣

⎡−

=γ

ρ

(3.36)

This definition is valid for all classes of section. Therefore the limit value Mc,y,Rd must be determined depending on the class of section, compare chapter 3.2.3.

EN 1993-1-1, equation (6.30) compare DIN FB 103, 5.4.7, (103)



EN 1993-1-1, equation (6.33) and (6.34)

3.2.9 Interaction of uniaxial bending and axial force

Class 1 and 2 cross-sections Check: Rd,NEd MM ≤ (3.37)

• Rectangular sections

( )[ ]2Rd,plEdRd,plRd,N N/N1MM −= (3.38)

• I- and H-sections and sections with flanges

Bending about y-y axis If the design value of the axial force does not apply to both conditions given in (3.39), the bending moment resistance will have to be reduced.

Criterion: ⎪⎩

⎪⎨⎧

<ydw

Rd,pl

EdfA5,0

N25,0N with γM0 (3.39)

The reduced moment resistance must calculated in consideration of the form of the cross-section. It must be differentiated between I, H sections and hollow and welded box sections.

I- and H-sections

Rd,y,plRd,y,plRd,y,N Ma5,01

n1MM ≤−

−= (3.40)

with

Rd,pl

Ed

NNn = and 5,0

Atb2Aa f ≤

−=

Hollow and welded box sections

Rd,y,plw

Rd,y,plRd,y,N Ma5,01

n1MM ≤−

−= (3.41)

with

- hollow section: ( ) 5,0A/tb2Aaw ≤−=

- box section: ( ) 5,0A/tb2Aa fw ≤−=

EN 1993-1-1, equation (6.39) and (6.40)

EN 1993-1-1, equation (6.31)



Bending about z-z axis Criterion: ydwEd fAN < with γM0 (3.42)

I- and H-sections

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

−−

−=>2

Rd,z,plRd,z,N a1an1MM:an (3.43)

Rd,z,plRd,z,N MM:an =≤ (3.44)

with

Rd,pl

Ed

NNn = and 5,0

Atb2Aa f ≤

−=

Hollow and welded box sections

Rd,z,plf

Rd,z,plRd,z,N Ma5,01

n1MM ≤−

−= (3.45)

with

- hollow sections: ( ) 5,0A/th2Aaf ≤−=

- box sections: ( ) 5,0A/th2Aa wf ≤−=

Class 3 cross-section

0M

yEd,x

fγ

σ ≤ (3.46)

The insertion of all variables follows:

Check: 0,1fW

MfA

N

d,yel

Ed

d,y

Ed ≤⋅

+⋅

(3.47)


Check: 0,1fWMM

fAN

d,yeff

EdEd

d,yeff

Ed ≤⋅∆+

+⋅

(3.48)

For the determination of ∆MEd see the descriptions in chapter 3.1.2: Sec-tion properties and resistances.

EN 1993-1-1, equation (6.35)

EN 1993-1-1, equation (6.39) and (6.40)

EN 1993-1-1, 6.2.9.2 or EN 1993-1-1, 6.2.1, (7) respectively

EN 1993-1-1, equation (6.31)



3.2.10 Interaction of bi-axial bending and axial force Class 1 and 2 cross-sections

Check: 0,1MM

MM

Rd,z,N

Ed,z

Rd,y,N

Ed,y ≤⎥⎦

⎤⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡βα

(3.49)

with - I -and H-sections: 0,1n52 ≥== βα

- circular hollow sections: 22 == βα

- rectangular hollow sections: 0,6n13,11

66,12 ≤

−== βα


Check: 0,1fW

MfW

MfA

N

d,yz,el

Ed,z

d,yy,el

Ed,y

d,y

Ed ≤⋅

+⋅

+⋅

(3.50)


Check: 0,1fWMM

fWMM

fAN

d,yz,eff

Ed,zEd,z

d,yy,eff

Ed,yEd,y

d,yeff

Ed ≤⋅∆+

+⋅

∆++

⋅ (3.51)

3.2.11 Interaction of bending, shear and axial force

Cross-sections, that are subjected to bending, shear and axial force have to be checked, whether an interaction of the loads are necessary. In con-sideration of the criterion in chapter 3.2.8, 3.2.9 and 3.2.10, the plastic moment resistance Mpl,Rd has to be determined.

First of all, it must be checked, whether an interaction due to shear load is required. If this is the case, the yield strength or the thickness of the shear loaded section part has to be reduced by the factor (1-ρ), see chapter 3.2.8. The reduced resistance has to be used also for the plastic axial resistance. Finally the plastic moment resistance has to be reduced by the shear and/or axial force.

For double-symmetric I- and H-sections, the interaction criteria are evalu-ated for all classes of cross-section and forces, see Table 3.5 and 3.6.

EN 1993-1-1, equation (6.41)

EN 1993-1-1, equation (6.44)



Table 3.4: Interaction of double symmetric I- and H-sections with the internal forces NEd, Vz,Ed and My,Ed

Cla

ssC

riter

ion

Rd

,pl

Ed

,zV5,0

VC

riter

ion

Rd

,pl

Ed

,zV5,0

V

1 2

Rd

,pl

Ed

N5,0N

or

d,yw

Ed

fA

25,0N

0M

yy,

plR

d,y,

plE

d,y

fW

MM

Rd

,V

Ed

N5,0

N

or

d,yw,

VE

df

A25,0

NR

d,y,c

0M

yw

2w

y,pl

Rd

,y,V

Ed

,yM

ft

4AW

MM

30,1

MMNN

Rd

,y,c

Ed

,y

Rd

,cEd

0,1MM

NN

Rd

,y,V

Ed

,y

Rd

,cEd

4

alw

ays

inte

ract

ion

N

+ M

0,1M

MM

NN

Rd

,y,c

Ed

,yE

d,y

Rd

,cEd

alw

ays

inte

ract

ion

N +

M

0,1M

MM

NN

Rd

,y,V

Ed

,yE

d,y

Rd

,cEd

1 2

Rd

,pl

Ed

N5,0N

and

d,yw

Ed

fA

25,0N

Rd

,y,pl

Rd

,y,pl

Rd

,y,N

Ed

,yM

a5,01

n1

MM

M

Rd

,V

Ed

N5,0N

and

d,yw,

VE

df

A25,0

N

Rd

,y,pl

V

VR

d,y,

VR

d,y,

V,N

Ed

,yM

a5,01

n1

MM

M

30,1

MMNN

Rd

,y,c

Ed

,y

Rd

,cEd

0,1MM

NN

Rd

,y,V,

N

Ed

,y

Rd

,cEd

4

alw

ays

inte

ract

ion

N +

M0,1

MM

MNN

Rd

,y,c

Ed

,yE

d,y

Rd

,cEd

alw

ays

inte

ract

ion

N +

M0,1

MM

MNN

Rd

,y,V,

N

Ed

,yE

d,y

Rd

,cEd

Rd,y,c0M

yw

2w

y,pl

Rd,y,V M

ft4

AW

M ≤⎥⎥⎦

⎤

⎢⎢⎣

⎡−

=γ

ρ

Mc,y,Rd must be consid-ered on the basis of the class of cross-section. Reduction factor ρ

2

Rd,i,pl

Ed,i 1V

V2⎟⎟⎠

⎞⎜⎜⎝

⎛−=ρ

Reducing the area of web

( ) wred t1t ⋅−= ρ Aw area of web

www thA ⋅= AV,w reduced area of web

( ) ww,V A1A ⋅−= ρ Ared,V reduced area of

section wV,red AAA ⋅−= ρ

Additional moment ∆MEd depending on the shift of the major axes

NiEdEd,i eNM ⋅=∆ n = NEd/Npl,Rd nV = NEd/NV,Rd NV,Rd reduced resis-

tance of axial force as a result of a reduced web thickness

0M

yV,redRd,V

fAN

γ⋅

=

( ) 5,0A/tb2Aa f ≤−= ( ) a1aV ⋅−= ρ



Table 3.5: Interaction of double symmetric I- and H-sections with internal forces NEd, Vy,Ed und Mz,Ed

Cla

ssC

riter

ion

Rd

,pl

Ed,yV5,0

VC

riter

ion

Rd

,pl

Ed,yV5,0

V

1 20

M

yw

Ed

fA

N0

M

yz,

plR

d,z,

plEd,z

fW

MM

0M

yw,

VEd

fA

NR

d,z,

plR

d,z,

VEd,z

MM

M

30,1

MMNN

Rd

,z,c

Ed,z

Rd

,cEd0,1

MMNN

Rd

,z,V

Ed,z

Rd

,cEd

4

alw

ays

inte

ract

ion

N

+ M

0,1M

MM

NN

Rd

,z,c

Ed,zEd,z

Rd

,cEd

alw

ays

inte

ract

ion

N +

M

0,1M

MM

NN

Rd

,z,V

Ed,zEd,z

Rd

,cEd

1 20

M

yw

Ed

fA

Nn

> a:

2

Rd

,z,pl

Rd

,z,N

Ed,za

1a

n1

MM

M

n a

:R

d,z,

plR

d,z,

NEd,z

MM

M0

M

yw,

VEd

fA

Nn V

> a

V:2

V

VV

Rd

,z,V

Rd

,z,V,

NEd,z

a1

an

1M

MM

n V a

V:R

d,z,

VR

d,z,

V,N

Ed,zM

MM

30,1

MMNN

Rd

,z,c

Ed,z

Rd

,cEd0,1

MMNN

Rd

,z,V

Ed,z

Rd

,cEd

4

alw

ays

inte

ract

ion

N

+ M

0,1M

MM

NN

Rd

,z,c

Ed,zEd,z

Rd

,cEd

alw

ays

inte

ract

ion

N +

M

0,1M

MM

NN

Rd

,z,V

Ed,zEd,z

Rd

,cEd

Mc,z,Rd must be consid-ered on the basis of the class of cross-section. Reduction factor ρ

2

Rd,i,pl

Ed,i 1V

V2⎟⎟⎠

⎞⎜⎜⎝

⎛−=ρ

Reducing the area of web

( ) wred t1t ⋅−= ρ Following to the reduc-tion of the web thick-ness, the section modulus Wel,i,red, Wpl,i,red, Weff,i,red must calculated new.

0M

yred,iRd,z,V

fWM

γ⋅

=

Aw area of a web

www thA ⋅= AV,w reduced area of web

( ) ww,V A1A ⋅−= ρ Ared,V reduced area of

section wV,red AAA ⋅−= ρ

Additional moment ∆MEd depending on the shift of the major axes

NiEdEd,i eNM ⋅=∆ n = NEd/Npl,Rd nV = NEd/NV,Rd NV,Rd reduced resis-

tance of axial force as a result of a reduced web thickness

0M

yV,redRd,V

fAN

γ⋅

=

( ) 5,0A/tb2Aa f ≤−= ( ) a1aV ⋅−= ρ



3.3 Structural analysis of members

In the following explanations the analyses are presented in general. Re-spectively, they have a validity for all classes of cross-section. Therefore the resistances and sections properties have to be adapted to the appli-cable class of cross-section, compare 3.1.2: Section properties and resis-tances. But all limit states can be proved simplified by the elastic resis-tances.

The buckling resistance check of members will be carried out at a single span member regarded cut out of the system and in consideration of

1. the global deformation and the first-order analysis or

2. the internal forces and moments by a second-order analysis. Then the buckling length is equal to the system length.

3.3.1 Buckling length Lcr

General

The buckling length Lcr will be equal to the length of the global deforma-tion, if second-order effects are neglected (first-order analysis).

Generally, the condition (3.52) applies.

LLcr ⋅= β (3.52)

with

β coefficient of buckling length

L system length

If the internal forces and moments are determined on the basis oft a sec-ond-order analysis, the buckling length will be equal to the system length.

Triangulated and lattice structures

Table 3.6: Buckling length Lcr of triangulated and lattic structures Buckling Members in-plane out-of-plane

General Lcr = L Lcr = L I- and H-sections Lcr = 0,9 L Lcr = L Chord mem-

bers Hollow section Lcr = 0,9 L Lcr = 0,9 L General Lcr = 0,9 L Lcr = L Triangulated

structures Angels, connected by one bolt Lcr = L Lcr = L

The stiffness of triangulated structures of angles, that are connected with one bolt, must be considered by an effective non-dimensional slender-

Annex BB.1

L = system length



EN 1993-1-1, 6.3.1.2, (4) Critical buckling force

2cr

2

cr LEIN π

=

EN 1993-1-1, equation (6.46) and (6.47) or (6.48) respec-tively

ness. The slenderness for buckling about each axis should be taken as follows:

- vv,eff 7,035,0 λλ += ; for buckling about v-v axis

- yy,eff 7,050,0 λλ += ; for buckling about y-y axis

- zz,eff 7,050,0 λλ += ; for buckling about z-z axis

This specifications of the buckling length will be applicable, if no exact length is assessed, see for example [13].

3.3.2 Uniform members in compression

Simplified assessment method for flexural buckling

If the following condition is satisfied, i.e. the non-dimensional slenderness is 2,0≤λ , so flexural buckling does not occur.

Criterion: 04,0NN

cr

Ed ≤ (3.53)

with


Flexural buckling

Check: 0,1NN

Rd,b

Ed ≤ (3.54)

1M

yRd,b

fAN

γχ ⋅⋅

= (3.55)

Class 4 sections with additional moments as a result of eccentricity of the axial force must comply with the interaction, that is given in 3.2.9.



EN 1993-1-1, equation (6.50) or (6.51)

2cr

2

cr LEIN π

=

y1 f

E⋅= πλ

Reduction factor χ

• Non-dimensional slenderness λ I-, H- and rectangular cross-sections

1

cr

cr

y 1i

LN

fAλ

λ ⋅=⋅

= (3.56)

1

eff

cr

cr

yeff AA

iL

NfA

λλ ⋅=

⋅= (3.57)

with

A gross cross-section

Aeff effective cross-section


Lcr buckling length for the plane considered; chapter 3.3.1: Buckling length Lcr

i radius of gyration about the relevant axis, determined the proper-ties of the gross cross-section

λ1 slenderness value

λ1= 93,9ε with yf

235=ε

For compression members, which are symmetrical to one axis or have a restrained axis, for example T- and U- sections, it has to be verified, which of the two failures, flexural or torsional buckling is the proper fail-ure. Therefore both slenderness ratios λ and Tλ have to be assessed. The maximum value must be used to analyses buckling.

Non-dimensional slenderness Tλ for torsional buckling

cr

yT N

fA ⋅=λ (3.58)

fy 235 275 355 420 460 λ1 93,9 86,4 76,0 70,4 66,7

EN 1993-1-1, equation (6.52)

Class 1-3 cross-sections




with

Ncr = Ncr,TF < Ncr,T Ncr,TF elastic torsional-flexural buckling force

Ncr,T elastic torsional buckling force

For the resistance against torsional buckling, the buckling curve, relates to the z-z axis is valid.

• Imperfection factor α

First of all, the buckling curve must be selected on the basis of the ge-ometry of the cross section and the loaded axis, therefore see Table 3.7.

Red

uctio

n fa

ctor

χ

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1,1

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 3,0_

χχ χχ

a0

bcd

a

Non-dimensional slenderness λ Figure 3.10: Buckling curves

T,crN see [13]

EN 1993-1-1, Figure 6.4 In addition to the known buckling curve, a buck-ling curve a0 for S 460 is mentioned.



Table 3.7: Selection of buckling curve for a cross-section Buckling curve

Cross section Limits Buckling

aboutaxis

S 235 S 275 S 355 S 420

S 460

tf ≤ 40 mm y – y z – z

ab

a0

a0

h/b

> 1

,2

40 mm < tf ≤ 100 y – y z – z

bc

aa

tf ≤ 100 mm y – y z – z

bc

aa

Rol

led

sect

ions

b

h y y

z

z

t f

h/b

≤ 1,

2

tf > 100 mm y – y z – z

dd

cc

tf ≤ 40 mm y – y z – z

bc

bc

Wel

ded

I-

sect

ions

tt ff

y yy y

z ztf > 40 mm

y – y z – z

cd

cd

hot finished any a a0

Hol

low

se

ctio

ns

cold formed any c c

generally (except as below)

any b b

Wel

ded

box

sect

ions

t

t

f

b

h yy

z

z

wthick welds: a > 0,5tf

b/tf < 30 h/tw <30

any c c

U-,

T-

and

solid

sec

tions

any c c

L-s

ectio

ns

any b b

Table 3.8: Imperfection factor α for buckling curves

Buckling curve a0 a b c d Imperfection factor α or αLT 0,13 0,21 0,34 0,49 0,76

• Reduction factor χ

0,1122

≤−+

=λφφ

χ (3.59)

The value φ is defined as follows:

( )[ ]²2,015,0 λλαφ +−+= (3.60)

with

λ maximum of non-dimensional slenderness for buckling or tor-sional buckling

In range of 2,0≤λ , χ = 1,0.

EN 1993-1-1, equation (6.49) and 6.3.1.2 (1)

EN 1993-1-1, Tab. 6.1, in addition: a0 for S 460

EN 1993-1-1, Table 6.2



EN 1993-1-1, Annex B, BB.2.1

EN 1993-1-1, Annex B, BB.2.2

3.3.3 Uniform members in bending: Lateral torsional buckling

Continuous lateral restraints

This simplified check of lateral-torsional buckling for structures with trapezoidal sheeting profiles according to EN 1993-1-3 can be used, if the profiles are connected with the beam at each rib. For profile, that are connected only at each second rip, the shear stiffness has to be reduced as 0,2 S.

2

22

2

zt2

2

w h70h25,0

LEIGI

LEIS ⎟⎟

⎠

⎞⎜⎜⎝

⎛++≥

ππ (3.61)

with

S shear stiffness provided by the sheeting regarding its deforma-tion in the plane and connected to the beam at each rib, see EN 1993-1-3

Iw warping constant

It torsion constant

Iz second moment of inertia of the cross-section about the minor axis of the cross-section

L beam length

h depth of the beam

Continuous torsional restraints

Lateral-torsional buckling can also be avoided, if a rotation is restrained by abutting members.

υϑϑ KKEI

MC

z

2k,pl

k, > (3.62)

with

Cϑ,k rational stiffness provided to the beam by the stabilising contin-uum and the connections

Kυ 0,35 for elastic analysis

Kυ 1,0 for plastic analysis

Kϑ factor for considering the moment distribution and the type of restraint

Mpl,Rk characteristic value of the plastic moment of the beam



EN 1993-1-1, 6.3.2.4 (1)

k,Dk,Ck,Rk, C1

C1

C1

C1

ϑϑϑϑ

++= (3.63)

with

CϑR,k rotational stiffness provided by the stabilising continuum to the beam assuming a stiff connection to the member

CϑC,k rotational stiffness of the connection between the beam and the stabilising continuum

CϑD,k rotational stiffness deduced from an analysis of the distorsional deformations of the beam cross sections, where the flange in compression not restrained; where the compression flange is the restrained or where distorsional deformations of the cross sec-tions may be neglected (e.g. for usual rolled profiles) CϑD,k = ∞.

For more information see EN 1993-1-3 or [13].

Table 3.9: Factor Kϑ for considering the moment distribution and the type of restraint

Case Moment distribution without

translational restraint

with translational

restraint

1 M 4,0 0

2aM

M0,12

2bM

M

M

3,5

0,23

3 M 2,8 0

4 M 1,6 1,0

5M

M-0,3

1,0 0,7

Simplified calculation method for lateral-torsional buckling

Figure 3.11: Definition of the regarded part of the cross-section in compression to use in

the simplified calculation method

Ed,y

Rd,c0c

1z,f

ccf M

Mi

Lk λλ

λ ≤= (3.64)

compression



EN 1993-1-1, 6.3.2.4 (2)

EN 1993-1-1, 6.3.2.4 NOTE 2B

with

My,Ed maximum design value of the bending moment within the re-straint spacing

1M

yyRd,c

fWM

γ⋅

=

Wy

appropriate section modulus corresponding to the compression flange

kc slenderness correction factor for moment distribution between restraints; see Table 3.10

Lc length between restraints

if,z radius of gyration of the compression flange including 1/3 of the compressed part of the web area, about the minor axis of the section

0cλ slenderness parameter of the above compression element; 5,01,04,01,00,LT0c =+=+≤ λλ

Table 3.10: Correction factors kc

Moment distribution kc

ψ = 1 1,0

-1 ≤ ψ ≤ 1 ψ− 33,033,1

1

0,94

0,90

0,91

0,86

0,77

0,82

If the formula (3.64) is not satisfied, the following check has to be carried out:

Check: 0,1MM

Rd,b

Ed ≤ (3.65)

Rd,cRd,cflRd,b MMkM ≤= χ (3.66)



with

kfl modification factor accounting to the conservatism of the equiva-lent compression flange method

kfl = 1,10

Mc,Rd design resistance for bending

1M

yyRd,c

fWM

γ⋅

=

Reduction factor χ for the simplified calculation method

The reduction factor χ must be determined on the basis of equation (3.59), that is comparable to the resistance of flexural buckling. But the non-dimensional slenderness λ must be simply substituted by the non-

dimensional slenderness fλ only. Moreover the imperfection factor α is defined in Table 3.11.

Table 3.11: Buckling curve for the simplified calculation method of lateral-torsional buck-ling

Cross section Limits Buckling curve

welded sections h/tf ≤ 44ε h/tf > 44ε

d c

other sections - c

Lateral torsional buckling

Check: 0,1MM

Rd,b

Ed ≤ (3.67)

1M

yyLTRd,b

fWM

γχ ⋅⋅= (3.68)

Reduction factor χLT

• Non-dimensional slenderness LTλ

cr

yyLT M

fW ⋅=λ (3.69)

with

EN 1993-1-1, 6.3.2.1

derived from EN 1993-1-1, 6.3.2.4, (3)B h overall depth of the

cross-section tf thickness of the

compression flange

For the determination of Mcr see information in literature, for exam-ple [13] or EN 1993 (1993), Annex F.



Mcr elastic critical moment for lateral-torsional buckling

Lateral-torsional buckling does not have to be considered, if the non-dimensional slenderness will be 4,0LT ≤λ .

• Reduction factor χLT for torsional buckling (general case)

The verification of the factor χLT is like the one for flexural buckling. But the buckling curves are different, compare Table 3.12.

Table 3.12: Selection of buckling curves to determine χLT (general case)

Cross sections Limits Buckling curves

rolled I-sections h/b ≤ 2 h/b > 2

a b

welded I-sections h/b ≤ 2 h/b > 2

c d

other cross sections - d

• Reduction factor χLT for torsional buckling of rolled sections and equivalent welded sections

⎪⎩

⎪⎨⎧

≤−+

=²

10,1

²²1

LTLTLTLT

LT

λλβφφχ (3.70)

The value φLT is defined as follows:

( )[ ]²15,0 LT0,LTLTLTLT λβλλαφ +−+= (3.71)

with

4,00,LT ≤λ and 75,0≥β .

For a simple determination, the reduction factor χLT is summarised tabu-larly for LTλ in chapter 4.6.

Table 3.13: Selection of buckling curves to determine χLT according to equation (3.70)

Cross section Limits Buckling curve

rolled I-sections h/b ≤ 2 h/b > 2

b c

welded I-sections h/b ≤ 2 h/b > 2

c d

Cross sections, which are not classified for this method, for example cross sections with larger dimensions then rolled sections have to be proved in consideration of the general case.

EN 1993-1-1, 6.3.2.3 equation (6.57)

hyperbola of Euler

EN 1993-1-1, Table 6.4

EN 1993-1-1, Table 6.5



3.3.4 Uniform members in bending and axial compression: I-, H- and hollow sections

The interaction formulae are based on the modelling of simply supported single span members with end fork conditions, which are subjected to compression forces, end moments or transverse loads.

(1) Uniaxial bending and axial force

Members without torsional deformations - Deflection normal to y-y axis

0,1MMM

kNN

1M

Rk,y

Ed,yEd,yyy

1M

Rky

Ed ≤∆+

+

γγχ

(3.72)

- Deflection normal to z-z axis Simplifying, it should be verified (kzy = 0):

0,1NN

1M

Rkz

Ed ≤

γχ

(3.73)

Members with torsional deformations

- Deflection normal to y-y axis

0,1M

MMk

NN

1M

Rk,yLT

Ed,yEd,yyy

1M

Rky

Ed ≤∆+

+

γχ

γχ

(3.74)

- Deflection normal to z-z axis

0,1M

MMk

NN

1M

Rk,yLT

Ed,yEd,yzy

1M

Rkz

Ed ≤∆+

+

γχ

γχ

(3.75)

EN 1993-1-1, equation (6.61) interpreted for cross-sections without susceptible torsional deformations

EN 1993-1-1, equation (6.62) interpreted compare note in table B.1 last line; kzy = 0

EN 1993-1-1, equation (6.61) and (6.62) inter-preted



(2) Bi-axial bending and axial force

For bi-axial bending and axial force, the maxima bending moments My,Ed and Mz,Ed must be used independently from their point of occurrence.

Members without torsional deformations - Deflection normal to y-y axis

0,1M

MMk

MMM

kN

N

1M

Rk,z

Ed,zEd,zyz

1M

Rk,y

Ed,yEd,yyy

1M

Rky

Ed ≤∆+

+∆+

+

γγγχ

(3.76)


0,1M

MMk

MMM

kN

N

1M

Rk,z

Ed,zEd,zzz

1M

Rk,y

Ed,yEd,yzy

1M

Rkz

Ed ≤∆+

+∆+

+

γγγχ

(3.77)

Members with torsional deformations - Deflection normal to y-y axis

0,1M

MMk

MMM

kN

N

1M

Rk,z

Ed,zEd,zyz

1M

Rk,yLT

Ed,yEd,yyy

1M

Rky

Ed ≤∆+

+∆+

+

γγχ

γχ

(3.78)


0,1M

MMk

MMM

kN

N

1M

Rk,z

Ed,zEd,zzz

1M

Rk,yLT

Ed,yEd,yzy

1M

Rkz

Ed ≤∆+

+∆+

+

γγχ

γχ

(3.79)

with

χy; χz reduction factors due to flexural buckling according to equation (3.59)

χLT reduction factors due to torsional buckling according to equation (3.70)

kij interaction factors

EN 1993-1-1, equation (6.61) and (6.62) inter-preted for cross-sections without sus-ceptible torsional de-formations

EN 1993-1-1, equation (6.61) and (6.62)



3.3.5 Interaction factor kij according to Annex B

Table 3.14: Equivalent uniform moment factors Cm

Cmy, Cmz, CmLT Moment diagram range Uniform loading Concentrated load

11 ≤≤− ψ 4,04,06,0 ≥+ ψ

10 s ≤≤ α 11 ≤≤− ψ 4,08,02,0 s ≥+ α 4,08,02,0 s ≥+ α

10 ≤≤ ψ 4,08,01,0 s ≥− α 4,08,0 s ≥− α

hss M/M=α 01 s <≤− α 01 <≤− ψ ( ) 4,08,011,0 s ≥−− αψ ( ) 4,08,02,0 s ≥−− αψ

10 h ≤≤ α 11 ≤≤− ψ h05,095,0 α+ h10,09,0 α+

10 ≤≤ ψ h05,095,0 α+ h10,09,0 α+

shh M/M=α 01 h <≤− α 01 <≤− ψ ( )ψα 2105,095,0 h ++ ( )ψα 2110,090,0 h +−

The part of the beam between the restraints and the bending moment, that produces the corresponding failure are decisive for the assessment of Cm.

- Cmy: My with restraints in z-z plane

- CmLT: My with restraints in y-y plane

- Cmz: Mz with restraints in y-y plane

Table 3.15: Interaction factor kij’for members not susceptible to torsional deformations Interaction

factor Type of section Class 1 and 2 Class 3 and 4

kyy I-sections RHS-sections

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+≤

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

Rdy

Edmy

Rdy

Edymy

NN8,01C

NN2,01C

χ

χλ

⎟⎟⎠

⎞⎜⎜⎝

⎛+≤

⎟⎟⎠

⎞⎜⎜⎝

⎛+

Rdy

Edmy

Rdy

Edymy

NN6,01C

NN6,01C

χ

χλ

kyz I-sections RHS-sections 0,6 kzz kzz

kzy I-sections RHS-sections 0,6 kyy 0,8 kyy

I-sections

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+≤

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

Rdz

Edmz

Rdz

Edzmz

NN8,01C

NN6,021C

χ

χλ

kzz

RHS-sections

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+≤

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

Rdz

Edmz

Rdz

Edzmz

NN8,01C

NN2,01C

χ

χλ

⎟⎟⎠

⎞⎜⎜⎝

⎛+≤

⎟⎟⎠

⎞⎜⎜⎝

⎛+

Rdz

Edmz

Rdz

Edzmz

NN6,01C

NN6,01C

χ

χλ

with 1M

y

1M

RkRd

fANNγγ

⋅==

EN 1993-1-1, Annex B, Table B.3 L distance between

the applicable re-straint

Mh bending moment at the restraints

Ms bending moment at the half length between the re-straints

EN 1993-1-1, Annex B, Table B.1



EN 1993-1-1, Annex B, Table B.2 The factors kyy, kyz and kz are identical with the factor of members without torsional deforma-tions.

Table 3.16: Interaction factor kij’for members susceptible to torsional deformations Interaction

factor Type of sec-

tions Class 1 and 2 Class 3 and 4

kyy I-sections

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+≤

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

Rdy

Edmy

Rdy

Edymy

NN8,01C

NN2,01C

χ

χλ

⎟⎟⎠

⎞⎜⎜⎝

⎛+≤

⎟⎟⎠

⎞⎜⎜⎝

⎛+

Rdy

Edmy

Rdy

Edymy

NN6,01C

NN6,01C

χ

χλ

kyz I-sections 0,6 kzz kzz

kzy I-sections

( )

( ) ⎥⎦

⎤⎢⎣

⎡−

−≥

⎥⎦

⎤⎢⎣

⎡

−−

Rdz

Ed

mlT

Rdz

Ed

mLT

z

NN

25,0C1,01

NN

25,0C1,01

χ

χλ

for 4,0z <λ

( ) Rdz

Ed

mLT

z

z

NN

25,0C1,01

6,0

χλ

λ

−−≤

+

( )

( ) ⎥⎦

⎤⎢⎣

⎡−

−≥

⎥⎦

⎤⎢⎣

⎡

−−

Rdz

Ed

mLT

Rdz

Ed

mLT

z

NN

25,0C05,01

NN

25,0C05,01

χ

χλ

kzz I-sections

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+≤

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

Rdz

Edmz

Rdz

Edzmz

NN8,01C

NN6,021C

χ

χλ

⎟⎟⎠

⎞⎜⎜⎝

⎛+≤

⎟⎟⎠

⎞⎜⎜⎝

⎛+

Rdz

Edmz

Rdz

Edzmz

NN6,01C

NN6,01C

χ

χλ

with 1M

y

1M

RkRd

fANNγγ

⋅==



4 Design aids

4.1 Initial sway imperfection

Table 4.1: Initial sway imperfection φ in· 10-3 (%)

m h 2 3 4 5 6 7 8 9 10

≤ 4 4,330 4,082 3,953 3,873 3,819 3,780 3,750 3,727 3,708 4,2 4,226 3,984 3,858 3,780 3,727 3,689 3,660 3,637 3,619 4,4 4,129 3,892 3,769 3,693 3,641 3,604 3,575 3,553 3,536 4,6 4,038 3,807 3,686 3,612 3,561 3,525 3,497 3,475 3,458 4,8 3,953 3,727 3,608 3,536 3,486 3,450 3,423 3,402 3,385 5 3,873 3,651 3,536 3,464 3,416 3,381 3,354 3,333 3,317

5,2 3,798 3,581 3,467 3,397 3,349 3,315 3,289 3,269 3,252 5,4 3,727 3,514 3,402 3,333 3,287 3,253 3,227 3,208 3,191 5,6 3,660 3,450 3,341 3,273 3,227 3,194 3,169 3,150 3,134 5,8 3,596 3,390 3,283 3,216 3,171 3,139 3,114 3,095 3,079 6 3,536 3,333 3,227 3,162 3,118 3,086 3,062 3,043 3,028

6,2 3,478 3,279 3,175 3,111 3,067 3,036 3,012 2,993 2,978 6,4 3,423 3,227 3,125 3,062 3,019 2,988 2,965 2,946 2,932 6,6 3,371 3,178 3,077 3,015 2,973 2,942 2,919 2,901 2,887 6,8 3,321 3,131 3,032 2,970 2,929 2,899 2,876 2,858 2,844 7 3,273 3,086 2,988 2,928 2,887 2,857 2,835 2,817 2,803

7,2 3,227 3,043 2,946 2,887 2,846 2,817 2,795 2,778 2,764 7,4 3,184 3,002 2,906 2,847 2,808 2,779 2,757 2,740 2,726 7,6 3,141 2,962 2,868 2,810 2,770 2,742 2,721 2,704 2,690 7,8 3,101 2,924 2,831 2,774 2,735 2,707 2,685 2,669 2,655 8 3,062 2,887 2,795 2,739 2,700 2,673 2,652 2,635 2,622

8,2 3,024 2,851 2,761 2,705 2,667 2,640 2,619 2,603 2,590 8,4 2,988 2,817 2,728 2,673 2,635 2,608 2,588 2,572 2,559 8,6 2,953 2,784 2,696 2,641 2,604 2,578 2,557 2,542 2,529 8,8 2,919 2,752 2,665 2,611 2,575 2,548 2,528 2,513 2,500 9 ≤ 2,887 2,722 2,635 2,582 2,546 2,520 2,500 2,485 2,472

Initial sway imperfection: mh0 ααφφ ⋅⋅= φ0 = 1/200

h2

h =α but 0,132

h ≤≤ α

h: height of the structure in meters

⎟⎠

⎞⎜⎝

⎛ +=m115,0mα

m: the number of columns in one row including only those columns which carry a verti-cal load NEd not less than 50% of the average value of the column in the vertical plane considered



4.2 Classification of cross-sections Table 4.2: Classification of IPE-sections

S 235 S 275 S 355 S 460 S 235 S 275 S 355 S 46080 1 1 1 1 1 1 1 1100 1 1 1 1 1 1 1 1120 1 1 1 1 1 1 1 1140 1 1 1 2 1 1 1 1160 1 1 1 2 1 1 1 1180 1 1 2 3 1 1 1 1200 1 1 2 3 1 1 1 1220 1 1 2 4 1 1 1 1240 1 2 2 4 1 1 1 1260 - - - - - - - -270 2 2 3 4 1 1 1 1280 - - - - - - - -300 2 2 4 4 1 1 1 1320 - - - - - - - -330 2 3 4 4 1 1 1 1340 - - - - - - - -360 2 3 4 4 1 1 1 1400 3 3 4 4 1 1 1 1450 3 4 4 4 1 1 1 1500 3 4 4 4 1 1 1 1550 4 4 4 4 1 1 1 1600 4 4 4 4 1 1 1 1650 - - - - - - - -700 - - - - - - - -800 - - - - - - - -900 - - - - - - - -

1000 - - - - - - - -

Compression BendingIPE

Table 4.3: Classification of HEA-sections

S 235 S 275 S 355 S 460 S 235 S 275 S 355 S 46080 - - - - - - - -100 1 1 1 1 1 1 1 1120 1 1 1 1 1 1 1 1140 1 1 1 2 1 1 1 2160 1 1 1 2 1 1 1 2180 1 1 2 3 1 1 2 3200 1 1 2 3 1 1 2 3220 1 1 2 3 1 1 2 3240 1 1 2 3 1 1 2 3260 1 1 3 3 1 1 3 3270 - - - - - - - -280 1 2 3 3 1 2 3 3300 1 2 3 3 1 2 3 3320 1 1 2 3 1 1 2 3330 - - - - - - - -340 1 1 1 3 1 1 1 3360 1 1 1 2 1 1 1 2400 1 1 2 3 1 1 1 1450 1 1 2 4 1 1 1 1500 1 2 3 4 1 1 1 1550 2 3 4 4 1 1 1 1600 2 3 4 4 1 1 1 1650 3 4 4 4 1 1 1 1700 3 4 4 4 1 1 1 1800 4 4 4 4 1 1 1 1900 4 4 4 4 1 1 1 1

1000 4 4 4 4 1 1 1 2

Compression BendingHEA



Table 4.4: Classification of HEB-sections

S 235 S 275 S 355 S 460 S 235 S 275 S 355 S 46080 - - - - - - - -100 1 1 1 1 1 1 1 1120 1 1 1 1 1 1 1 1140 1 1 1 1 1 1 1 1160 1 1 1 1 1 1 1 1180 1 1 1 1 1 1 1 1200 1 1 1 1 1 1 1 1220 1 1 1 1 1 1 1 1240 1 1 1 1 1 1 1 1260 1 1 1 1 1 1 1 1270 - - - - - - - -280 1 1 1 1 1 1 1 1300 1 1 1 1 1 1 1 1320 1 1 1 1 1 1 1 1330 - - - - - - - -340 1 1 1 1 1 1 1 1360 1 1 1 1 1 1 1 1400 1 1 1 1 1 1 1 1450 1 1 1 2 1 1 1 1500 1 1 2 2 1 1 1 1550 1 1 2 3 1 1 1 1600 1 2 3 4 1 1 1 1650 2 2 3 4 1 1 1 1700 2 2 4 4 1 1 1 1800 3 3 4 4 1 1 1 1900 3 4 4 4 1 1 1 11000 4 4 4 4 1 1 1 1

Compression BendingHEB

Table 4.5: Classification of HEM-sections

S 235 S 275 S 355 S 460 S 235 S 275 S 355 S 46080 - - - - - - - -100 1 1 1 1 1 1 1 1120 1 1 1 1 1 1 1 1140 1 1 1 1 1 1 1 1160 1 1 1 1 1 1 1 1180 1 1 1 1 1 1 1 1200 1 1 1 1 1 1 1 1220 1 1 1 1 1 1 1 1240 1 1 1 1 1 1 1 1260 1 1 1 1 1 1 1 1270 - - - - - - - -280 1 1 1 1 1 1 1 1300 1 1 1 1 1 1 1 1320 1 1 1 1 1 1 1 1330 - - - - - - - -340 1 1 1 1 1 1 1 1360 1 1 1 1 1 1 1 1400 1 1 1 1 1 1 1 1450 1 1 1 1 1 1 1 1500 1 1 1 1 1 1 1 1550 1 1 1 1 1 1 1 1600 1 1 1 1 1 1 1 1650 1 1 1 2 1 1 1 1700 1 1 2 3 1 1 1 1800 1 2 3 4 1 1 1 1900 2 3 4 4 1 1 1 11000 3 4 4 4 1 1 1 1

Compression BendingHEM



4.3 Effective shear area AV

I-sections: Loaded parallel to the web

Table 4.6: Effective shear area AV [cm²]

nominal height

I IPE IPE a IPE o HEA A HEA HEB HEM

80 3,30 3,57 - - - - - - 100 4,72 5,06 - - 6,15 7,52 9,00 18,00 120 6,45 6,15 5,38 - 6,95 8,42 10,96 21,15 140 8,32 7,62 6,22 - 7,90 10,11 13,12 24,50 160 10,54 9,67 7,82 - 10,42 13,24 17,64 30,86 180 13,00 11,20 9,22 12,70 12,13 14,52 20,29 34,40 200 15,60 12,18 11,50 15,49 15,42 18,05 24,85 40,75 220 18,55 13,92 13,59 17,67 17,67 20,63 27,88 44,87 240 21,75 16,48 16,30 21,34 21,57 25,14 33,24 60,48 260 25,41 - - - 24,78 28,74 37,15 67,25 270 - 43,24 18,70 25,19 - - - - 280 29,43 - - - 27,50 31,78 40,73 71,87 300 33,75 50,70 22,22 29,02 32,36 37,75 47,35 90,45 320 38,34 - - - 35,42 40,77 51,43 94,80 330 - 59,05 26,95 34,87 - - - - 340 43,26 - - - 39,19 44,48 56,19 98,80 360 48,84 68,53 29,80 40,18 42,56 49,20 60,96 102,60 400 60,37 79,81 35,78 47,99 48,26 57,35 70,20 110,40 450 76,19 93,45 42,22 59,74 54,64 65,76 79,68 119,40 500 93,68 109,68 50,31 70,50 62,03 75,18 90,18 129,20 550 109,10 126,88 60,01 82,61 72,83 83,96 100,01 139,20 600 135,68 147,75 70,12 104,60 81,23 92,75 110,85 150,00 650 - - - - 90,64 103,55 121,70 160,00 700 - - - - 100,39 116,50 136,72 169,80 800 - - - - 123,32 139,00 161,58 194,00 900 - - - - 147,00 163,80 188,48 214,80

1000 - - - - 171,96 184,72 212,44 234,80



4.4 Interaction of bending and shear

Table 4.7: Reduction factor ρ dependent on the utilisation ratio n

n ρ (1-ρ) n ρ (1-ρ) 0,51 0,0004 0,9996 0,76 0,2704 0,7296 0,52 0,0016 0,9984 0,77 0,2916 0,7084 0,53 0,0036 0,9964 0,78 0,3136 0,6864 0,54 0,0064 0,9936 0,79 0,3364 0,6636 0,55 0,0100 0,9900 0,80 0,3600 0,6400 0,56 0,0144 0,9856 0,81 0,3844 0,6156 0,57 0,0196 0,9804 0,82 0,4096 0,5904 0,58 0,0256 0,9744 0,83 0,4356 0,5644 0,59 0,0324 0,9676 0,84 0,4624 0,5376 0,60 0,0400 0,9600 0,85 0,4900 0,5100 0,61 0,0484 0,9516 0,86 0,5184 0,4816 0,62 0,0576 0,9424 0,87 0,5476 0,4524 0,63 0,0676 0,9324 0,88 0,5776 0,4224 0,64 0,0784 0,9216 0,89 0,6084 0,3916 0,65 0,0900 0,9100 0,90 0,6400 0,3600 0,66 0,1024 0,8976 0,91 0,6724 0,3276 0,67 0,1156 0,8844 0,92 0,7056 0,2944 0,68 0,1296 0,8704 0,93 0,7396 0,2604 0,69 0,1444 0,8556 0,94 0,7744 0,2256 0,70 0,1600 0,8400 0,95 0,8100 0,1900 0,71 0,1764 0,8236 0,96 0,8464 0,1536 0,72 0,1936 0,8064 0,97 0,8836 0,1164 0,73 0,2116 0,7884 0,98 0,9216 0,0784 0,74 0,2304 0,7696 0,99 0,9604 0,0396 0,75 0,2500 0,7500 1,00 1,0000 -

Utilisation ratio: Rd,pl

Ed

VVn =

Reduction factor: 2

Rd,pl

Ed 1V

V2⎟⎟⎠

⎞⎜⎜⎝

⎛−=ρ

reducing the yield strength ( ) yred,y f1f ⋅−= ρ or

reducing the thickness ( ) wred,w t1t ⋅−= ρ



4.5 Interaction of uniaxial bending and axial force

Table 4.8: Ratio a according to EN 1993-1-1, 6.2.9.1 (5)

nominal height I IPE IPE a IPE o HEA HEA A HEB HEM

80 0,35 0,37 - - - - - - 100 0,36 0,39 - - 0,25 0,29 0,23 0,20 120 0,37 0,39 0,41 - 0,24 0,29 0,22 0,20 140 0,38 0,39 0,39 - 0,24 0,27 0,22 0,20 160 0,38 0,40 0,40 - 0,26 0,26 0,23 0,21 180 0,39 0,39 0,40 0,39 0,25 0,26 0,23 0,21 200 0,39 0,40 0,40 0,39 0,25 0,27 0,23 0,21 220 0,39 0,39 0,40 0,39 0,25 0,27 0,23 0,21 240 0,40 0,40 0,40 0,40 0,25 0,28 0,23 0,21 260 0,40 - - - 0,25 0,28 0,23 0,21 270 - 0,40 0,40 0,38 - - - - 280 0,41 - - - 0,25 0,28 0,23 0,21 300 0,41 0,40 0,41 0,39 0,26 0,29 0,23 0,20 320 0,42 - - - 0,25 0,30 0,24 0,21 330 - 0,41 0,41 0,40 - - - - 340 0,42 - - - 0,26 0,32 0,25 0,22 360 0,43 0,41 0,39 0,40 0,27 0,33 0,25 0,23 400 0,43 0,42 0,41 0,41 0,28 0,34 0,27 0,25 450 0,44 0,44 0,42 0,43 0,29 0,36 0,28 0,27 500 0,44 0,45 0,43 0,44 0,30 0,39 0,30 0,29 550 0,43 0,46 0,44 0,45 0,32 0,41 0,31 0,31 600 0,45 0,46 0,44 0,45 0,34 0,43 0,33 0,33 650 - - - - 0,36 0,45 0,35 0,35 700 - - - - 0,38 0,47 0,37 0,37 800 - - - - 0,41 0,50 0,41 0,40 900 - - - - 0,44 0,50 0,43 0,43

1000 - - - - 0,46 0,50 0,46 0,46

Ratio of the web-area to the area of cross-section: 5,0A

tb2Aa f ≤−

=



Table 4.9: Factor f in [kNm] to determine MN,y,Rd of I-sections, S 235

nominal height I IPE IPE a IPE o HEA A HEA HEB HEM

80 6,48 6,71 - - - - - - 100 11,40 11,51 - - 16,10 22,23 27,68 61,73 120 18,35 17,74 14,69 - 23,15 31,91 43,71 91,54 140 27,61 25,74 20,90 - 33,63 46,36 64,90 129,20 160 39,54 36,28 29,13 - 51,52 66,36 94,21 177,33 180 54,49 48,60 39,69 55,20 69,70 86,77 127,86 232,10 200 73,11 64,77 53,49 73,18 94,78 115,45 170,64 298,90 220 94,86 83,70 70,56 93,95 121,43 152,31 219,41 373,11 240 120,84 107,41 91,75 120,21 156,19 199,82 279,91 555,53 260 151,19 - - - 195,57 247,25 340,19 661,00 270 - 142,18 120,97 166,86 - - - - 280 186,46 - - - 239,11 298,96 407,46 776,34 300 225,68 184,86 159,86 216,55 293,22 373,12 497,40 1.066,51320 271,31 - - - 331,11 437,23 570,19 1.164,31330 - 237,98 208,16 276,28 - - - - 340 321,60 - - - 374,18 498,46 642,96 1.244,70360 380,79 300,76 264,33 348,11 420,30 563,70 721,48 1.320,57400 513,94 390,29 337,96 445,31 516,12 700,76 881,62 1.495,74450 722,13 512,28 443,79 609,63 625,60 886,14 1.090,35 1.719,31500 977,34 666,36 580,99 789,21 751,70 1.091,24 1.330,30 1.949,61550 1.272,51 848,94 745,46 989,14 923,29 1.293,08 1.561,98 2.206,17600 1.657,21 1.077,16 944,79 1.362,15 1.085,72 1.514,20 1.810,44 2.470,53650 - - - - 1.265,13 1.754,68 2.084,65 2.747,63700 - - - - 1.482,89 2.038,60 2.402,78 3.029,87800 - - - - 1.948,93 2.575,89 3.015,67 3.666,00900 - - - - 2.506,67 3.258,31 3.775,52 4.323,32

1000 - - - - 3.064,40 3.922,73 4.535,19 5.040,46

To determine MN,y,Rd use the following equation:

( ) fn1M Rd,y,N ⋅−= with utilisation ratio Rd,pl

Ed

NNn =



Table 4.10: Factor f in [kNm] to determine MN,y,Rd of I-sections, S 355

nominal height I IPE IPE a IPE o HEA A HEA HEB HEM

80 9,78 10,13 - - - - - - 100 17,21 17,39 - - 24,32 33,58 41,82 93,24 120 27,72 26,80 22,19 - 34,97 48,20 66,03 138,29 140 41,71 38,88 31,57 - 50,80 70,03 98,05 195,18 160 59,73 54,81 44,01 - 77,83 100,25 142,31 267,88 180 82,31 73,42 59,95 83,38 105,29 131,08 193,15 350,62 200 110,44 97,84 80,80 110,55 143,18 174,41 257,78 451,54 220 143,29 126,44 106,59 141,92 183,44 230,09 331,46 563,64 240 182,55 162,26 138,61 181,59 235,94 301,85 422,84 839,21 260 228,39 - - - 295,43 373,50 513,90 998,54 270 - 214,78 182,74 252,07 - - - - 280 281,67 - - - 361,21 451,62 615,52 1.172,77300 340,92 279,26 241,49 327,13 442,95 563,65 751,39 1.611,11320 409,85 - - - 500,19 660,50 861,35 1.758,85330 - 359,50 314,46 417,37 - - - - 340 485,83 - - - 565,24 753,00 971,28 1.880,29360 575,23 454,34 399,31 525,87 634,92 851,54 1.089,89 1.994,91400 776,38 589,58 510,53 672,71 779,67 1.058,60 1331,81 2.259,53450 1.090,88 773,87 670,41 920,94 945,05 1.338,63 1.647,12 2.597,26500 1.476,41 1.006,62 877,66 1.192,21 1.135,55 1.648,47 2.009,60 2.945,15550 1.922,30 1.282,43 1.126,13 1.494,23 1.394,76 1.953,38 2.359,59 3.332,73600 2.503,45 1.627,19 1.427,23 2.057,72 1.640,13 2.287,41 2.734,92 3.732,08650 - - - - 1.911,15 2.650,69 3.149,15 4.150,67700 - - - - 2.240,11 3.079,58 3.629,73 4.577,04800 - - - - 2.944,13 3.891,24 4.555,58 5.538,00900 - - - - 3.786,67 4.922,13 5.703,44 6.530,98

1000 - - - - 4.629,20 5.925,82 6.851,04 7.614,32

To determine MN,y,Rd use the following equation:

( ) fn1M Rd,y,N ⋅−= with utilisation ratio Rd,pl

Ed

NNn =



4.6 Reduction factor χ and χLT

Table 4.11: Reduction factor χ for flexural buckling

a0 a b c d0,2 1,0000 1,0000 1,0000 1,0000 1,00000,3 0,9859 0,9775 0,9641 0,9491 0,92350,4 0,9701 0,9528 0,9261 0,8973 0,85040,5 0,9513 0,9243 0,8842 0,8430 0,77930,6 0,9276 0,8900 0,8371 0,7854 0,71000,7 0,8961 0,8477 0,7837 0,7247 0,64310,8 0,8533 0,7957 0,7245 0,6622 0,57970,9 0,7961 0,7339 0,6612 0,5998 0,52081,0 0,7253 0,6656 0,5970 0,5399 0,46711,1 0,6482 0,5960 0,5352 0,4842 0,41891,2 0,5732 0,5300 0,4781 0,4338 0,37621,3 0,5053 0,4703 0,4269 0,3888 0,33851,4 0,4461 0,4179 0,3817 0,3492 0,30551,5 0,3953 0,3724 0,3422 0,3145 0,27661,6 0,3520 0,3332 0,3079 0,2842 0,25121,7 0,3150 0,2994 0,2781 0,2577 0,22891,8 0,2833 0,2702 0,2521 0,2345 0,20931,9 0,2559 0,2449 0,2294 0,2141 0,19202,0 0,2323 0,2229 0,2095 0,1962 0,17662,1 0,2117 0,2036 0,1920 0,1803 0,16302,2 0,1937 0,1867 0,1765 0,1662 0,15082,3 0,1779 0,1717 0,1628 0,1537 0,13992,4 0,1639 0,1585 0,1506 0,1425 0,13022,5 0,1515 0,1467 0,1397 0,1325 0,12142,6 0,1404 0,1362 0,1299 0,1234 0,11342,7 0,1305 0,1267 0,1211 0,1153 0,10622,8 0,1216 0,1182 0,1132 0,1079 0,09972,9 0,1136 0,1105 0,1060 0,1012 0,09373,0 0,1063 0,1036 0,0994 0,0951 0,0882

Buckling curveλ



Table 4.12: Reduction factor χLT for lateral torsional buckling, equation (3.70)

b c d0,4 1,0000 1,0000 1,00000,5 0,9602 0,8126 0,68920,6 0,9171 0,7989 0,67830,7 0,8696 0,7798 0,66210,8 0,8171 0,7529 0,63820,9 0,7600 0,7158 0,60461,0 0,6997 0,6676 0,56051,1 0,6386 0,6101 0,50761,2 0,5792 0,5476 0,44941,3 0,5236 0,4851 0,39051,4 0,4728 0,4262 0,33431,5 0,4273 0,3728 0,28311,6 0,3868 0,3256 0,23781,7 0,3460 0,2845 0,19861,8 0,3086 0,2490 0,16511,9 0,2770 0,2183 0,13682,0 0,2500 0,1918 0,11312,1 0,2268 0,1690 0,09342,2 0,2066 0,1493 0,07692,3 0,1890 0,1322 0,06342,4 0,1736 0,1174 0,05222,5 0,1600 0,1045 0,04302,6 0,1479 0,0933 0,03542,7 0,1372 0,0835 0,02922,8 0,1276 0,0749 0,02412,9 0,1189 0,0673 0,01993,0 0,1111 0,0607 0,0165

Buckling curveLTλ



5 Flow charts

The following flow charts should clarify the workflow for a structure analysis that is defined in the chapters above. The first flow chart: General structural analysis shows the complete workflow for the design of steel structures. The following diagrams apply to the first, but they can be regarded separately also.

The sheets should be used as follows:

yes

operation: It determines the basic values for the structural analysis.

junction: It seperats the workflows on the basis of criterions.

beginning or ending: It points the beginning and ending of the flow chart.

change: It connects flow charts.

subprogram: It chracterizes values, that have to be calculated by another flow chart.

flow line: It connects the sheets and shows the direction of reading.

junction: It separats workflows due to checks.



Definition of the hori-zontal deflection

5.1 Design of steel structures

Flow chart 5.1: General procedure of the design of steel structures

10cr ≥α A first-order analysis is permitted!

Determination of the initial sway imperfection of the

structure and replacement by equivalent horizontal forces

Determination of the internal forces and moments and the horizontal

deformations by a first-order analysis in consideration of the equivalent forces

Check, whether an first-order analysis is

permitted

Internal forces and moments by second-order analysis considering

the initial sway imperfections

Structural analysis according to the iternal forces and moments that are determined above (first-

order analysis)

1. Determination of the elastic decisive internal forces and moments

Definition of the load case combination for the ulitmate limit state and the serviceability limit state

no

yes

yes

no

Determination of the local bow imperfection e0/l according to

Table 2.12 and of the additional iternal forces and moments

Determination of the local bow imperfection e0/l according to

Table 2.12 and of the additional iternal forces and moments

yes

Internal forces and moments by second-order analysis considering

the initial sway imperfections

Three way to consider the additional initial forces and moments due to effects of imperfections and second-order analysis

1. consideration of all imperfections and second-order effects in the global structural analysis

2. Consideration of the initial sway imperfections in a global structural analysis. The influences of the local imperfections and the second-order analysis are comprehend in the analysis of members. Futhermore the applicable buckling length is equal to the system length.

3. The iternal forces and moments are determined on the basis of an first -order analysis. The design of the structure will be achieved by a member design according to EC 3, chapter 6. The buckling length corresponds to the global deformations.

1

Check, whether an additional bow imperfection has to be permitted.Condition:(1) or


Ed

y

NfA

5,0⋅

⋅>λ

Check, whether an additional bow imperfection has to be permitted.Condition:(1) or


Ed

y

NfA

5,0⋅

⋅>λ

Determination of

(1) bracing structure

(2) other structures (for example columns)

crα

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛=

Ed,HEd

Edcr

hVH

δα

Ed

crcr F

F=α

Non-dimensional slender-ness

1

cr

cr

y 1i

LN

fAλ

λ ⋅=⋅

=



Flow chart 5.1 (1): Continuation of General procedure of the design of steel structures

Analysis of cross-sections according to Flow chart 5.5 up

to 5.12 (2)

Analysis of members according to Flow chart 5.13 up to 5.15with Lcr = L (System length)

Analysis of members according to Flow chart 5.13 up to 5.15

with Lcr = distance between the turningpoints of the buckling

deformation

2. Ultimate limit state

3. Serviceability limit state

Determination of the maximum horizontal and vertical deformations

1

Check

Vertical deflection

Horizontal deflection

maxww ≤

maxuu ≤

The definition of limits for deflection should be specified for each project and agreed with the clients or taken from ENV 1993-1-1 (1993), Table 4.1 and 4.2.2

3. Design of joints

Determination of design loads of the connections and fasteners

Bolted connection welded connection

Check of the bolted connections

Check of the welded connections

The design of the structure is totally done!



5.2 Basis of design

Flow chart 5.2: Initial sway imperfection φ

4h ≤

0,1h =αh

2h =α

Determination of the average value of all columns in the vertical plane Nm for every storey

m: number of columns that achieves the

condition: NEd > 0,5 Nm

yes

yes

no

Determitation of height of the structure in meters

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

m11

21

mα

9h ≥

Initial sway imperfection

mh2001

ααφ ⋅⋅=

32

h =α

no



Therefore it is possible to classify the cross-sections in a more favourable class of cross-section. The criterions are given in EN 1993-1-1, 5.5.2, (9)-(12).

fy ε 235 1,0 275 0,92355 0,81420 0,75460 0,71

yf235

=ε

α stress distribution about a section part

σk buckling value, Table 2.6

5.2.1 Classification of cross-sections

Through the following two flow charts, the classification of one and both side supported compression parts of cross-section is given. The proce-dure has to be apply to all total or partial compression parts of cross-section. Finally, the cross-section is classified like the highest class of its compression part.

Flow chart 5.3: Classification of one side supported compression parts

yes

Compression Compression and bendingnein

no

nein

yes

yes

no

yes

yes

yes

no

yes

Tip in compression Tip in tensionno

yes





Determination of c/tc

tc

tc

t

tc

c/t max c/t = 9

≤ε

c/t max c/t = 10

≤ε

c/t max c/t = 14

≤ε

c/t max c/t =

≤

σε k21

c/t max c/t =

≤

αε9

c/t max c/t =

≤

αα

ε9

c/t max c/t =

≤

αε10

c/t max c/t =

≤

αα

ε10

no

no

yes

yes

no

no



fy ε 235 1,0 275 0,92355 0,81420 0,75460 0,71

yf235

=ε

1

2

σσ

ψ =

with σ1 maximum compres-

sion stress α stress distribution

about a section part

Flow chart 5.4: Classification of both-side supported compression parts

yes

Com

pres

sion

Com

pres

sion

and

ben

ding

no no

yes

yes

no

yes

no

no

yes no

Ben

ding no no

yes

yes

yes

no

nono

yes

no

no

yes

yes

yes

yes

nono

yes

yes

yes

Cla

ss 2

cro

ss-s

ectio

n

Cla

ss 3

cro

ss-s

ectio

n

Cla

ss 4

cro

ss-s

ectio

n

Cla

ss 1

cro

ss-s

ectio

n

Cla

ss 2

cro

ss-s

ectio

n

Cla

ss 3

cro

ss-s

ectio

n

Cla

ss 4

cro

ss-s

ectio

n

Det

erm

inat

ion

of v

orh

c/t

c

tt

cc

t

c t

c

t

c/t

m

ax c

/t =

33

≤ε

Cla

ss 1

cro

ss-s

ectio

n

c/t

max

c/t

= 3

8≤

ε

c/t

m

ax c

/t =

42

≤ε

c

/t

max

c/t

= 1

24≤

ε

c/

t

m

ax c

/t =

≤

c/t

m

ax c

/t =

≤

c/

t

max

c/t

= ≤

c/

t

m

ax c

/t =

≤

c

/t

m

ax c

/t =

≤

c/t

m

ax c

/t =

≤

no

5,0>

α5,0

≥α

11339

6 −α

ε 11345

6 −α

ε

1−>

ψ

()

()

ψψ

ε−

−162

ψε 33,0

67,042 +α

ε5,41α

ε36 1−

≤ψ

c/t

m

ax c

/t =

72

≤ε

c

/t

m

ax c

/t =

83

≤ε



5.3 Structural analysis of cross-sections

Flow chart 5.5: Tension

Determination of a net area of a cross-section Anet

yes

Cross-section is analysed!

ja

Nt,Rd = min(Npl,Rd; Nu,Rd)

Cro

ss-s

ectio

n ha

s to

be

dim

ensi

oned

new

!

no

Npl,Rd or Nel,Rd respectively

0M

yRd,elRd,pl

fANN

γ⋅

==

Design value of the axial force NEd

Bolted connection Welded connection

Npl,Rd or Nel,Rd respectively

0M

yRd,elRd,plRd,t

fANNN

γ

⋅===

Determination of Nu,Rd

Angels connected by one leg

- 1 bolt:

- more bolts:

see Table 3.2

2M

ynetRd,u

fA9,0N

γ⋅⋅

=

( )2M

u02Rd,u

ftd5,0e0,2Nγ

⋅−=

2M

unetRd,u

fANγ

β ⋅⋅=

β

no

Check

0,1NN

Rd,t

Ed ≤

effA = AgrossA = A

- the larger leg- the smaller leg

connected by:

effA

Determination of the cross-section area A or Aeff for angels

Anet net area along the critical fracture line

γM0 = 1,00 γM2 = 1,25



Flow chart 5.6: Compression

Class 1, 2, 3 cross-sections

Cross-section is analysed!

ja

0M

yRd,elRd,plRd,c

fANNN

γ

⋅===

Classification of the cross-section according to Flow chart 5.3 and 5.4

Class 4 cross-sectionCheck of local buckling according to EN 1993-1-5

Determination of the effective cross-section properties and the shift of the major axes e N

yes

no

The

cros

s-se

ctio

n ha

s to

be

dim

ensi

oned

new

!

no

⇒


0M

yeffRd,c

fAN

γ⋅

=

Design value of the axial force NEd

Check

0,1NN

Rd,c

Ed ≤

γM0 = 1,00 γM2 = 1,25



γM0 = 1,00 γM2 = 1,25

Flow chart 5.7: Bending

The cross-section is ananlysed!

yes



yes

no

The

Cro

ss-s

ectio

n ha

s to

be

dim

ensi

oned

new

!

Determination of the normal stress of the cross-section

Class 1 or 2Cross-section

yes

Class 3 cross-sectionno

Bending Mz,Ed about thez-z axis

no

yes

no

Design value of the bending moment M Ed

Deduction in the tensile zone of the cross-section?

Determination of the section modulus according to the

deduction

Bending resistance Mc,Rd

0M

yplRd,plRd,c

fWMM

γ

⋅==


0M

yelRd,elRd,c

fWMM

γ

⋅==


0M

yeffRd,c

fWM

γ

⋅=

no

yes

Check of local buckling, see EN 1993-1-5Determination of the effective cross-section properties

and the shift of the major axes eN

⇒

yes

no

No deduction

Check

0,1MM

MM

Rd,z,c

Ed,z

Rd,y,c

Ed,y ≤+

0M

yf

2M

unet,f fAf9,0Aγγ

⋅≥

⋅⋅

Af area of flange

Af,net net area of the tension flange with deduction of the bolts



fy ε 235 1,0 275 0,92355 0,81420 0,75460 0,71

Flow chart 5.8: shear

The cross-section is analysed!

yes


Determination of the normal stress of the cross-section

Class 1 and 2 cross-section

Class 3 and 4 cross-section

no

no

Effective shear area Av according to Table 3.3

I-, H-sections with Af/Aw 0,6 ≥

Plate buckling, see EN 1993-1-5, chapter 5

no

ja

ja

no

Shear stress

w

EdEd A

V=τ

Edτ Shear stress

tISVEd

Ed ⋅⋅

=τ

Edτ

yes

yes

Shear resistance Vc,Rd

0M

yv

Rd,plRd,c

3/fAVV

γ

⎟⎠⎞⎜

⎝⎛

==

Schub- beulgefahr?

ηε72

th

w

w ≤

Design shear force VEd

Check

0,1VV

Rd,c

Ed ≤

Check

0M

yEd

3

f

γτ

⋅≤

no

The

cros

s-se

ctio

n ha

s to

be

dim

ensi

oned

new

!

η = 1,2; for S460 η = 1,0

area of flange

ff tbA ⋅= area of web

www thA ⋅= γM0 = 1,00



5.3.1 Interaction

Flow chart 5.9: Interaction bending and shear of I-sections V + M


Classifiction of the cross-section according to Flow chart 5.3 and 5.4

Determination of the normal stress of the cross -section

Interaction is necessary!no

yes

Check of shear according to Flow chart 5.8

no

Interaction isn’t necessary!

Reduction factor ρ

2

Rd,pl

Ed 1V

V2⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−=ρ

Plastic shear resistance Vpl,Rd

0M

yv

Rd,pl

3/fAV

γ

⎟⎠⎞⎜

⎝⎛

=

Que

rsch

nitt

neu

dim

ensi

onie

ren

Design shear force VEd and bending moment MEd

no

no

Criterion of interaction

5,0VV

Rd,pl

Ed ≤

yes

yes

yes

Check

0,1VV

Rd,pl

Ed ≤

ja

Class 1 or 2 cross-sections


no

y-y-axis

z-z-axisMV,z,Rd: Dertermination by reducing

- the area or - the yield strength

of the shear loaded parts of the cross -section

0M

yw

2w

y,pl

Rd,y,V

ft4

AW

Mγ

ρ⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

=

t)1(tred ⋅−= ρyred f)1(f ⋅−= ρ

Elastic bending resistance

0M

yiRd,c

fWM

γ⋅

=

Check

Rd,i,cRd,i,VEd,i MMM ≤≤

Check

Rd,cEd MM ≤

Av effective shear area see Table 3.3

Aw area of web www thA ⋅=

γM0 = 1,00



Aw area of web www htA ⋅=

fy,d = fy/γM0

Flow chart 5.10: Interaction bending and axial force N + My



Determination of the normal stress of the cross -section

Class 1 or 2 cross-section


Interaction is necessary!


no yes

yes

no

yes

no

Interaction is always necessary!

yes

yes

The

cro

ss-s

ectio

n ha

s to

be

dim

ensi

oned

new

!

yes

no no

Design normal force NEd and bending moment My,Ed

no

Check

0M

yy,plRd,y,cEd,y

fWMM

γ⋅

=≤

Check

0,1MM

NN

Rd,y,c

Ed,y

Rd,c

Ed ≤+



no

yes


and

5,0NN

Rd,pl

Ed <

d,ywEd fA25,0N <

Plastic resistance to normal forces Npl,Rd

0M

yRd,pl

fAN

γ⋅

=

Check

0,1NN

Rd,pl

Ed ≤

Factor r to reduce the plastic moment resistance Mpl,Rd

with

a5,01n1r

−−

=

Rd,pl

Ed

NNn = ( ) 5,0A/bt2Aa f ≤−=

yes

no

Check

Rd,y,plRd,y,plRd,y,NEd,y MMrMM ≤⋅=≤

Check

0,1M

MMNN

Rd,y,c

Ed,yEd,y

Rd,c

Ed ≤∆+

+

r additional definition for an easier use; The coefficient r isn’t defined in EN 1993-1-1

0M

yiRd,y,c

fWM

γ⋅

=

of the corresponding class of cross-section



Flow chart 5.11: Interaction bending about z-z axis and axial force N + Mz



Determination of the normal stress

Class 1or 2 cross-section




no yes

yes

no

yes

no

Interaction is always necessary!

yes

yes

The

cro

ss-s

ectio

n ha

s to

be

dim

ensi

one

d ne

w!


0M

yRd,pl

fAN

γ

⋅=

yes

no no

Design normal force NEd and bending moment Mz,Ed about the z-z axis

no

Interaction criterion

d,ywEd fAN <

Class 4 cross-section Class 3 cross-sectionno

no

yes

Check

0,1MM

NN

Rd,z,c

Ed,z

Rd,c

Ed ≤+

Check

0M

yz,plRd,z,cEd,z

fWMM

γ⋅

=≤

Check

Rd,z,plRd,z,NEd,z MMM ≤≤

Reduced moment resistance MN,Rd

withRd,pl

Ed

NNn = ( ) 5,0A/bt2Aa f ≤−=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=>2

Rd,z,plRd,z,N a1an1MM:zuanfor

Rd,z,plRd,z,N MM:zuanfor =≤

Check

0,1M

MMNN

Rd,z,c

Ed,zEd,z

Rd,c

Ed ≤∆+

+

Check

0,1NN

Rd,pl

Ed ≤

yes

Aw web area www htA ⋅=

fy,d = fy/γM0

γM0 = 1,00



Flow chart 5.12: Interaction of uniaxial bending, shear and axial force N + V + My


Determination of the normal stress

Design normal force NEd, shear force VEd and bending moment My,Ed

2

Class 1 or 2 cross-section Class 3 or 4 cross-section

13

no

yes

The

cros

s-se

ctio

n ha

s to

be

dim

ensi

oned

new

!



Flow chart 5.12 (1): Continuation of interaction N + V + My



yes


no no

yes yes yes

no

The

cros

s-se

ctio

n ha

s to

be

dim

ensi

oned

new

!

13


5,0VV

Rd,z,pl

Ed,z ≤

0,1VV

Rd,z,pl

Ed,z ≤





0M

yv

Rd,pl

3/fAV

γ

⎟⎠⎞⎜

⎝⎛

=

yes no

yes

yes

Check

0,1MM

a5,01n1

NN

Rd,y,c

Ed,y

Rd,c

Ed ≤−

−+

0,1VV

Rd,pl

Ed ≤

Check

0,1MM

NN

Rd,y,V

Ed,y

Rd,c

Ed ≤+

0,1VV

Rd,pl

Ed ≤

Reduced resistance to bending due to shear force MV,y,Rd


0M

yRd,pl

fAN

γ⋅

=

Reduced resistance to normal forces due to shear force NV,pl,Rd

0M

yV,redRd,pl,V

fAN

γ⋅

=

Coefficients

and

Rd,pl

Ed

NN

n =

5,0A

tb2Aa f ≤

−=


Criterionof interaction

and5,0

NN

Rd,pl

Ed <

d,ywEd fA25,0N <


and5,0

NN

Rd,pl,V

Ed <

d,yw,VEd fA25,0N <

Check

0,1NN

Rd,pl

Ed ≤

0,1MM

Rd,y,pl

Ed,y ≤0,1VV

Rd,pl

Ed ≤

no no no

Reduction factor

Determination of the reduced cross-section

ρ2

Rd,pl

Ed,i 1VV2

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−=ρ

wV,red AAA ⋅−= ρ

Check

0,1MM

a5,01n1

NN

Rd,y,V

Ed,y

V

V

Rd,c

Ed ≤−

−+

0,1VV

Rd,pl

Ed ≤

Coefficients

and

Rd,pl,V

EdV N

Nn =

5,0A

tb2Aa

V,red

fV,redV ≤

−=

0M

y

w

2w

y,plRd,y,Vf

t4AWM

γρ

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡−=

Av effective shear area, see Table 3.3

γM0 = 1,00 Aw area of web www thA ⋅= tw web thickness hw depth of the web

fy,d = fy/γM0 AV,w reduced web

area

ww,V A)1(A ⋅−= ρ

tf flange thickness

b overall breadth

0M

yy,plRd,y,c

fWM

γ⋅

=

0M

yRd,c

fAN

γ⋅

=



Flow chart 5.12 (2): Continuation of interaction N + V + My


Interaction due to shear force is necessary!

Interaction due to shear force isn’t necessary!

ja no


yes

Class 4 cross-sectionno Class 3 cross-

sectionClass 4 cross-section

yes

no

Structural analysis of shear according to Flow chart 5.8

yes yesyesyes

no

no

no

Reduction factor ρ

2

Rd,z,pl

Ed,zi 1V

V2⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−=ρ

The

cro

ss-s

ectio

n ha

s to

be

dim

ensi

oned

new

!

no

23

Interaction due to normal forces is always necessary!

Check

0,1MM

NN

Rd,y,c

Ed,y

Rd,c

Ed ≤+

Check

0,1M

MMNN

Rd,y,c

Ed,yEd,y

Rd,c

Ed ≤∆+

+

Check Check

0,1MM

NN

Rd,y,V

Ed,y

Rd,c

Ed ≤+0,1M

MMNN

Rd,y,V

Ed,yEd,y

Rd,c

Ed ≤∆+

+


0M

yv

Rd,z,pl

3/fAV

γ

⎟⎠⎞⎜

⎝⎛

=

Reduced resistance to bending moment due to shear force MV,y,Rd

0M

y

w

2w

y,plRd,y,Vf

t4AWM

γρ

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡−=


5,0VV

Rd,z,pl

Ed,z ≤

Av effective shear area, see Table 3.3

γM0 = 1,00

0M

yiRd,y,c

fWM

γ⋅

=

0M

yiRd,c

fAN

γ⋅

=

of the corresponding class of cross-section



γM1 = 1,00 Class Ai

1, 2, 3 A 4 Aeff

5.4 Structural analysis of members

Flow chart 5.13: Centrical compression – flexural buckling

Determination of the buckling curve according to the dimensions of the cross -

section and the direction of flexural buckling, see Table 3.7

no

yes

no

yes

no

The member is analysed!

yes



local buckling, see EN 1993-1-5 Determination of the effective cross-section properties and the shift of the major axes e N

no

yes

Th

e cr

oss-

sect

ion

has

to b

e di

men

sion

ed n

ew

!

Design value to normal forces NEd; Determination of the material properties

⇒

2,0≤λ

0,1=χ

Has buckling to z-z axis

be checked?⊥

Check

0,1NN

Rd,b

Ed ≤

zy;min χχχ =

Design buckling resistance Nb,Rd

1M

yRd,b

fAN

γχ ⋅⋅

=

);max( Tλλλ =

Non-dimesional slenderness

Class 1-3 Class 4

1

cr

cr

y 1i

LN

fAλ

λ ⋅=⋅

=cr

yeff

NfA ⋅

=λ

Imperfection factor

Curve a0 a b c

0,13 0,21 0,34 0,49

d

0,76

α

α

Reduction factor

22

1

λφφχ

−+=

χ

[ ]2)2,0(15,0 λλαφ +−+=

Buckling length Lcr

- Lcr Buckling length is equal to the global deformations , for Euler it is :

Euler 1 2 3 4

β 2,0 1,0 0,7 0,5LLcr ⋅= β

λ1 = 93,3 ε

2cr

2

cr LEIN π

=

radius of gyration:

IAi =



Class Wi,y

1, 2 Wpl,y

3 Wel,y

4 Weff,y

γM1 = 1,00

Flow chart 5.14: Lateral torsional buckling

- Determination of the stress distribution

- Determination of the heigth h* of the compression web area

Compression area of the flange with 1/3 of the compressed web

areabtth

31A fw

*f ⋅+⋅⋅=

Distance Lc between the restraintsCorrection factor kc, see Table 3.10

Imperfection factor

Curve c

0,49

d

0,76

yes

yes

no

no

Corresponding radius of gyration

f

z,fz,f A

Ii =

Design buckling resistance moment Mb,Rd

with kfl = 1,1

Rd,cflRd,b MkM χ=

[ ]2ff )2,0(15,0 λλαφ +−+=

Corresponding moment of inertia If,zAbout the minor axis

3fz,f tb

121I ⋅⋅=

no


The

cros

s-se

ctio

n ha

s to

be

dim

ensi

oned

new

!

Design value of the bending moment about the y -y axis My,Ed; Determination of the material properties

Simplified assessment method for flexural buckling, EN 1993-1-1, 6.3.2.4

α

Check

0,1MM

Rd,b

Ed,y ≤

α

13 2

Analysis

Ed,y

Rd,cf M

M5,0≤λ

Determination of the buckling curve - welded sections

- other sections

ε44t/h f ≤

ε44t/h f >

d

c

c

Reduction factor

2f

2

1

λφφχ

−+=

χ

1z,f

ccf i

Lkλ

λ =

Design resistance for bending moment Mc,Rd

1M

yy,iRd,c

fWM

γ

⋅=

Stress distribution of the cross-section and the definition of h*



Mcr elastic critical mo-ment for lateral tor-sional buckling; For determination see for example [13] or ENV 1993-1-1, Annex F

Flow chart 5.14 (1): Continuation of lateral torsional buckling

Buckling curve, see Table 3.12


Reduction factor

0,1LT =χ

[ ]2LTLTLTLT 75,0)4,0(15,0 λλαφ +−+=

yes

no

yes

LTχ

4,0LT ≤λ

Check

0,1MM

Rd,b

Ed,y ≤

13 2

Design buckling resistance moment Mb,Rd

1M

yy,iLTRd,b

fWM

γχ=

cr

yy,iLT M

fW ⋅=λ

Imperfection factor

Curve a0 a b c

0,13 0,21 0,34 0,49

d

0,76

LTα

LTα

Reduction factor

or see Table 4.6

⎪⎩

⎪⎨

⎧≤

−+=

2LT

2LT

2LTLT

LT 10,1

75,0

1

λλφφχ

LTχ

Structural analysis of lateral torsional buckling, EN 1993-1-1, 6.3.2.1 Determination of ⇒ LTχ



Class Wi,y

1, 2 Wpl,y

3 Wel,y

4 Weff,y

Flow chart 5.15: Bending and compression

Buckling length Lcr

- Lcr : Buckling length according to the global deformations , (first order analysis)- Lcr : Buckling length is equal to the system length L , (second order analysis)

Determination of the buckling curve according to the dimensions of the cross-section and the direction of

buckling, see Table 3.7

0,1=χ

Has buckling to z-z axis

be checked?

no

yes

yes

no

yes

cr

yyLT M

fW ⋅=λ

4,0LT ≤λ

Reduction factor

0,1LT =χ

noDetermination of the buckling curve according to the ratio h/b

see Table 3.12

ja

1 2

no

3

A n

ew c

ross

-sec

tion

mus

t be

elec

ted

or th

e bu

cklin

g le

ngth

has

be

redu

ced.

LTχ

⊥

[ ]2LTLTLTLT 75,0)4,0(15,0 λλαφ +−+=

1M

yiRd

fAN

γ

⋅=

Design forces NEd, My,Ed und Mz,Ed, of first or second order analysis; Determination of the material properties

);max( Tλλλ =

2,0≤λCurve a0 a b c

0,13 0,21 0,34 0,49

d

0,76

Imperfection factor α

α

[ ]2)2,0(15,0 λλαφ +−+=

Imperfection factor

Curve b c

0,34 0,49

d

0,76

LTα

LTα

Non-dimensional slenderness

Class 1-3 Class4

1

cr

cr

y 1i

LN

fAλ

λ ⋅=⋅

=cr

yeff

NfA ⋅

=λ

Determination of LTχ

Reduction factor

or see Table 4.6

⎪⎩

⎪⎨

⎧≤

−+=

2LT

2LT

2LTLT

LT10,1

1

λλφφχ

LTχ

)or( Tλλ

Reduction factor

or see Table 4.5

22

1

λφφχ

−+=

χ

Member with torsional deformations

λ1 = 93,9 ε fy ε 235 1,0 275 0,92355 0,81420 0,75460 0,71

2cr

2

cr LEIN π

=

IAi =

Mcr elastic critical mo-ment for lateral tor-sional buckling; For determination see for example [13] or ENV 1993-1-1, Annex F

γM1 = 1,00 QSK Ai

1, 2, 3 A 4 Aeff



γM1 = 1,00

Class Wi

1, 2 Wpl

3 Wel

4 Weff

Bending moment be-tween the restraints

Flow chart 5.15 (1): Continuation of Bending and compression

1 2

Determination of the equivalent uniform moment factors Cmy and Cmz Mh1= max M

2h

1h

MM

=ψ

Linear bending moment

Determination of Cmy, Cmz and CmLT see Table 3.14

yes

no

Determination of the interaction factors kyy, kzz, kyz and kzy

see Table 3.15 (without torsional deformation) or 3.16 (with torsional deformation)

Determination of the equivalent uniform moment factors

Cmy, Cmz and CmLT


3

no

yes

Design resistance Mi,Rd depending on the class of cross-section and the direction of load

1M

yiRd,i

fWM

γ

⋅=

0,1=ψ

0,1LT =χ

CheckBuckling to y-y axis

Buckling to z-z axis

For members without torsional deformation should follows:

⊥

⊥

0,1M

MMk

MMM

kN

N

Rd,z

Ed,zEd,zyz

Rd,yLT

Ed,yEd,yyy

Rdy

Ed ≤∆+

+∆+

+χχ

0,1M

MMk

MMM

kN

N

Rd,z

Ed,zEd,zzz

Rd,yLT

Ed,yEd,yzy

Rdz

Ed ≤∆+

+∆+

+χχ

0,1LT =χ

Ratio of the support and midspan moment

h

ss M

M=α

s

hh M

M=α

⇒> sh MM

⇒> hs MM



6 Literature

[1] CEN: Eurocode 3: Design of steel structures. Part 1-1: General rules and rules for buildings. English version. prEN 1993-1-1:2003, November 2003

[2] CEN: Eurocode 3: Design of steel structures. Part 1-8: Design of joints. English version. prEN 1993-1-8:2003, November 2003

[3] CEN: Eurocode 3: Bemessung und Konstruktion von Stahlbauten, Teil 1-1: Allgemeine Bemessungsregeln und Regeln für den Hochbau. Deutsche Fassung. prEN 1993-1-1:2003, November 2003

[4] CEN: Eurocode 3: Bemessung und Konstruktion von Stahlbauten, Teil 1-8: Bemessung von Anschlüssen. Deutsche Fassung. prEN 1993-1-8:2003, November 2003

[5] CEN: Eurocode 3: Bemessung und Konstruktion von Stahlbauten, Teil 1-1: Allgemeine Bemessungsregeln und Regeln für den Hochbau. Deutsche Fassung. DIN ENV 1993-1-1:1992, April 1993

[6] Deutsches Institut für Normung e.V.: DIN 18800: Stahlbauten, Teil 1: Bemessung und Konstruktion, November 1990

[7] Deutsches Institut für Normung e.V.: DIN 18800: Stahlbauten, Teil 2: Stabilitätsfälle, Knicken von Stäben und Stabwerken, November 1990

[8] Piechatzek, Erwin: Einführung in den Eurocode 3: Konzept, Bemessung, Beispiele, Tabellen, Vieweg Braunschweig/ Wiesbaden 2002

[9] Vayas, Ioannis; Ermopoulos, John; Ioannidis, George: Bemessungs-beispiele im Stahlbau nach Eurocode 3, Ernst & Sohn Berlin 2001

[10] Fritsch, Reinhold; Pasternak, Hartmut: Stahlbau: Grundlagen und Trag-werke, Vieweg Braunschweig/ Wiesbaden 1999

[11] Falke, Johannes: Ingenieurhochbau: Tragwerke aus Stahl nach Eurocode 3 (DIN V ENV 1993-1-1): Normen, Erläuterungen, Beispiele, Beuth Verlag GmbH, Berlin, Wien, Zürich und Werner-Verlag GmbH Düsseldorf 1996

[12] Schneider, Klaus-Jürgen: Bautabellen für Ingenieure mit Berechnungs-hinweisen und Beispielen, Auflage 13, Werner Verlag Düsseldorf 1998

[13] Petersen, Christian: Statik und Stabilität der Baukonstruktion, Vieweg Braunschweig/ Wiesbaden 1982

[14] Greiner, Richard; Lindner, Joachim: Die neuen Regelungen in der euro-päischen Norm EN 1993-1-1 für Stäbe unter Druck und Biegung, Stahlbau 72 (2003) Heft 3, Ernst & Sohn Berlin 2003

Documents

Compendium of en 1993-1-1