compendium of en 1993-1-1
TRANSCRIPT
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
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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
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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
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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
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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
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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
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Figure 1.1: Dimensions and axes of cross-sections
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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
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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( °≤
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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
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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.
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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
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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
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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)
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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)
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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
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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)
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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
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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)
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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
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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.
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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
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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
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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)
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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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 26 OF 84
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)
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 27 OF 84
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)
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 28 OF 84
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)
Class 3 cross-sections
0M
yelRd,elRd,c
fWMM
γ⋅
== (3.15)
Class 4 cross-sections
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 29 OF 84
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
Class 1 and 2 cross-sections
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 30 OF 84
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
Class 3 and 4 cross-sections
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 31 OF 84
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 32 OF 84
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
Class 1 and 2 cross-sections
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)
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 33 OF 84
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)
Class 3 and 4 cross-sections
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)
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 34 OF 84
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)
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 35 OF 84
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)
Class 4 cross-section
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)
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 36 OF 84
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
Class 3 cross-sections
Check: 0,1fW
MfW
MfA
N
d,yz,el
Ed,z
d,yy,el
Ed,y
d,y
Ed ≤⋅
+⋅
+⋅
(3.50)
Class 4 cross-sections
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)
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 37 OF 84
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 38 OF 84
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 39 OF 84
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 40 OF 84
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
Ncr elastic critical force for the relevant buckling mode based on the gross cross sectional properties
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.
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 41 OF 84
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
Ncr elastic critical force for the relevant buckling mode based on the gross cross sectional properties
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
Class 4 cross-sections
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 42 OF 84
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.
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 43 OF 84
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 44 OF 84
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
COMPENDIUM OF EN 1993-1-1
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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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 46 OF 84
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)
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 47 OF 84
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.
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 48 OF 84
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
COMPENDIUM OF EN 1993-1-1
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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
COMPENDIUM OF EN 1993-1-1
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(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)
- Deflection normal to z-z axis
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)
- Deflection normal to z-z axis
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)
COMPENDIUM OF EN 1993-1-1
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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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 52 OF 84
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
⋅==
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 53 OF 84
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
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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
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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
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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
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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
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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 ≤−
=
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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 =
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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 =
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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
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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
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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.
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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
(2) at least one moment resistant joint at one member end
Ed
y
NfA
5,0⋅
⋅>λ
Check, whether an additional bow imperfection has to be permitted.Condition:(1) or
(2) at least one moment resistant joint at one member end
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 ⋅=⋅
=
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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!
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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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 67 OF 84
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
Class 1 cross-section
Class 2 cross-section
Class 3 cross-section
Class 4 cross-section
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 68 OF 84
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 69 OF 84
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 70 OF 84
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
⇒
Class 4 cross-section
0M
yeffRd,c
fAN
γ⋅
=
Design value of the axial force NEd
Check
0,1NN
Rd,c
Ed ≤
γM0 = 1,00 γM2 = 1,25
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 71 OF 84
γM0 = 1,00 γM2 = 1,25
Flow chart 5.7: Bending
The cross-section is ananlysed!
yes
Classification of the cross-section according to Flow chart 5.3 and 5.4
Class 4 cross-section
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
⋅==
Bending resistance Mc,Rd
0M
yelRd,elRd,c
fWMM
γ
⋅==
Bending resistance Mc,Rd
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 72 OF 84
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
Classification of the cross-section according to Flow chart 5.3 and 5.4
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 73 OF 84
5.3.1 Interaction
Flow chart 5.9: Interaction bending and shear of I-sections V + M
The cross-section is analysed!
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
Class 3 and 4 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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 74 OF 84
Aw area of web www htA ⋅=
fy,d = fy/γM0
Flow chart 5.10: Interaction bending and axial force N + My
The cross-section is analysed!
Classification of the cross-section according to Flow chart 5.3 and 5.4
Determination of the normal stress of the cross -section
Class 1 or 2 cross-section
Class 3 or 4 cross-section
Interaction is necessary!
Interaction isn’t 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 ≤+
Class 4 cross-section
Class 3 cross-section
no
yes
Criterion of interaction
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 75 OF 84
Flow chart 5.11: Interaction bending about z-z axis and axial force N + Mz
The cross-section is analysed!
Classification of the cross-section according to Flow chart 5.3 and 5.4
Determination of the normal stress
Class 1or 2 cross-section
Class 3 or 4 cross-section
Interaction is necessary!
Interaction isn’t 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
one
d ne
w!
Plastic resistance to normal forces Npl,Rd
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 76 OF 84
Flow chart 5.12: Interaction of uniaxial bending, shear and axial force N + V + My
Classification of the cross-section according to Flow chart 5.3 and 5.4
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
!
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 77 OF 84
Flow chart 5.12 (1): Continuation of interaction N + V + My
The cross-section is analysed!
Interaction isn’t necessary!
yes
Interaction is necessary!
no no
yes yes yes
no
The
cros
s-se
ctio
n ha
s to
be
dim
ensi
oned
new
!
13
Criterion of interaction
5,0VV
Rd,z,pl
Ed,z ≤
0,1VV
Rd,z,pl
Ed,z ≤
Interaction isn’t necessary!
Interaction is necessary!
Interaction isn’t necessary!
Plastic shear resistance Vpl,Rd
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
Plastic resistance to normal forces Npl,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 ≤
−=
Interaction is necessary!
Criterionof interaction
and5,0
NN
Rd,pl
Ed <
d,ywEd fA25,0N <
Criterion of interaction
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⋅
=
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 78 OF 84
Flow chart 5.12 (2): Continuation of interaction N + V + My
The cross-section is analysed!
Interaction due to shear force is necessary!
Interaction due to shear force isn’t necessary!
ja no
Class 3 cross-section
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 ≤∆+
+
Plastic shear resistance Vpl,Rd
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
⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡−=
Criterion of interaction
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 79 OF 84
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
Classification of the cross-section according to Flow chart 5.3 and 5.4
Class 4 cross-section
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 =
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 80 OF 84
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
Classification of the cross-section according to Flow chart 5.3 and 5.4
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*
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 81 OF 84
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
The member is analysed!
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 82 OF 84
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 83 OF 84
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
The member is analysed!
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
COMPENDIUM OF EN 1993-1-1
ECSC EURO BUILD IN STEEL PAGE 84 OF 84
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