ISSN 1018-5593
European Commission
technical steel research
Properties and service performance
Simplified version of Eurocode 3
for usual buildings
STEEL RESEARCH
European Commission
technical steel research
Properties and service performance
Simplified version of Eurocode 3
for usual buildings
P. Chantrain, J.-B. Schleich
ARBED recherches
BP 141
L-4009 Esch-sur-Alzette
Contract No 7210-SA/513
1 July 1991 to 30 June 1994
Final report
Directorate-General
Science, Research and Development
1997
EUR 16839 EN
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!
Cataloguing data can be found at the end of this publication.
Luxembourg: Office for Official Publications of the European Communities, 1997
ISBN 92-828-1485-8
© European Communities, 1997
Reproduction is authorised provided the source is acknowledged.
Printed in Luxembourg
[
I
SIMPLIFIED VERSION OF EIIROCODE 3 FOR USUAL BUILDINGS.
ECSC Agreement 7210-SA/513
Summary
The aim of the following E.C.S.C. research is to elaborate a simple but complete document to
design commonly used buildings in steel construction. This document is completely based on
Eurocode 3 and each paragraph is totally conform to Eurocode 3. Only the design formulas
necessary to design braced or non-sway buildings are taken into account in this document.
Tall buildings (skyscrapers) and halls are not treated. The designers and steel constructors
are able to calculate and erect a commonly used steel building with this design handbook.
Therefore also the important load cases from Eurocode 1 will be included in this document.
The working group of the research project was constituted of 10 European engineering
offices. Firstly that working group has carried out different examples of calculation of braced
or non-sway buildings according to Eurocode 3 Part 1.1: check of existing steel structures and
design of new steel buildings. Afterwards thanks to those examples of calculation the needed
design formulas of Eurocode 3 was highlighted and general procedure of design was
determined. The design handbook "Simplified version of Eurocode 3" is based on that
experience.
The link of the working group to the drafting panel of Eurocode 3 was guaranteed by the
Professor Sedlacek of Aachen University.
Liaison has been ensured with both other E.C.S.C. research projects nr SA/312 and nr S A/419
also dealing with Eurocode 3: respectively, "Application software of Eurocode 3: EC3-tools"
(CTICM, France) and "Design handbook for sway buildings" (CSM-Italy).
VERSION SIMPLIFIEE DE L'EUROCODE 3 POUR LES BATIMENTS COURANTS
Agrément CECA 7210-SA/513
Sommaire
Le but de cette recherche est d'élaborer un document simple mais complet pour calculer des
bâtiments courants en construction métallique. Ce document est entièrement basé sur
l'Eurocode 3 et chaque paragraphe est totalement conforme à VEurocode 3. Il n'a été pris en
compte que les formules nécessaires au calcul de bâtiments contreventés et rigides. Les
bâtiments très élancés (gratte-ciel) et les halls industriels n'y sont pas traités. Les bureaux
d'études et constructeurs métalliques devront être capables de calculer et d'ériger un
bâtiment courant en acier avec ce manuel de dimensionnement. Les cas de charges le plus
importants issus de l'Eurocode 1 seront également inclus dans ce document.
Le groupe de travail du projet de recherche était constitué de 10 bureaux d'études européens.
En première partie ce groupe de travail a effectué différents exemples de calculs de bâtiments
contreventés et rigides conformément à l'Eurocode 3 Partie 1.1: vérification de structures en
acier déjà existantes et dimensionnement de nouveaux bâtiments en acier. Grâce à ces
exemples concrets de calcul, les formules de l'Eurocode 3 utiles au dimensionnement ont été
mises en évidence et une procédure générale de dimensionnement a été déterminée. Le
manuel de dimensionnement "Version simplifiée de l'Eurocode 3" se base sur cette
expérience.
La jonction entre le groupe de travail et le groupe de rédaction de l'Eurocode 3 a été faite par
le professeur Sedlacek de l'Université d'Aix-La-Chapelle.
Une collaboration a été assurée avec deux autres projets de recherche CECA N° SA/312 et N°
SA/419 qui concernent aussi l'Eurocode 3: respectivement, "Logiciel d'application de
l'Eurocode 3: EC3-Tools" (CTICM, France) et "Manuel de dimensionnement de bâtiments
souples (à nœuds déplaçables)" (CSM, Italie)
VEREINFACHTE VERSION DES EUROCODE 3 FÜR ÜBLICHE GEBÄUDE.
EGKS Zulassung7210-SA/513
Zusammenfassung
Dieses EGKS Forschungsprojekt hat zum Ziel, ein einfaches aber vollständiges Dokument
für allgemeine (übliche) Stahlbaubemessung auszuarbeiten. Dieses Dokument ist völlig auf
Eurocode 3 basiert und jeder Paragraph paßt genau zu Eurocode 3. Nur die
Bemessungsformeln, die notwendig sind für ausgesteifte oder unverschiebliche Tragwerke ,
werden berücksichtigt. Hochhäuser (Wolkenkratzen) oder Hallen werden nicht behandelt.
Die Ingenieurbüros und Stahlkonstrukteuren haben die Möglichkeit mit diesem DesignHandbuch einen einfachen Stahlbau zu berechnen und zu bauen. Dafür sind die wichtigsten
Lastfälle von Eurocode 1 in diesem Dokument beinhaltet.
Die Arbeitsgruppe des Forschungssprojekt bestand aus 10 europäischen Ingenieurbüros. Die
Arbeitsgruppe hat, im ersten Teil dieses Forschungsvorhabens, verschiedene
Berechnungsbeispiele mit ausgesteiften oder unverschieblichen Tragwerken nach Eurocode
3 Teil 1.1 durchgefühlt : Berechnungs-Nachweis einer existierenden Stahlstruktur und
Dimensionierung eines neuen Stahlbaus. Anschließend an diese konkreten Beispiele, wurden
die benutzten Bemessungsformeln nach Eurocode 3 hervorgehoben und ein allgemeines
Bemessungsverfahren wurde festgelegt. Das Design-Handbuch "Vereinfachte Version des
Eurocode 3" basiert auf dieser Erfahrung.
Die Verbindung zwischen der Arbeitsgruppe und dem technischen Komitee wurde von
Professor Sedlacek der Aachener Universität hergestellt.
Eine Zusammenarbeit bestand mit zwei anderen EGKS Forschungesprojekten N° SA/312
und N° SA/419, die auch Eurocode 3 behandeln : "Application software of Eurocode 3:
EC3-tools" (CTICM, France) und "Design handbook for sway buildings" (CSM-Italy).
Contents
Summary
3
Sommaire
4
Zusammenfassung
5
Contents
7
1. Introduction
9
2. Working group
10
3. Part 1 : Worked examples
3.1. Exercise 1 : Verification of an existing braced or non-sway structure
3.2. Exercise 2: Verification of a non-sway wind bracing in a building
3.3. Exercise 3: Design of a braced or non-sway structure
11
4. Part 2 : Design handbook ·
12
11
12
12
FIGURES (Ito 8 )
APPENDICES
List of symbols
List of tables
List of
flow-charts
(6 pages)
(3 pages)
(1 page)
"Design handbook according to Eurocode 3 for braced or non-sway steel buildings"
(short title : "EC3 for non-sway buildings")
(196 pages)
15
23
29
32
33
1. Introduction
The research was divided into different parts:
- in the first part worked examples of braced or non-sway structures has been carried out by
European engineering offices according to Eurocode 3 and Eurocode 1.
Different contacts have been taken with different engineering offices in Europe and
professional organisations (E.C.C.S. and C.T.I.C.M.). The working group of this research
project has been constituted with 10 engineering offices.
- in the second part the needed formulae for simple design of braced or non-sway structures
have been selected thanks to the exercises about check and design of steel buildings. The
design handbook has been elaborated on the basis of that experience.
The present final report of this research project presents the design handbook called
"Design handbook according to Eurocode 3 for braced or non-sway steel buildings"
(short title : "EC3for non-sway buildings").
2. Working group
The research project was fully managed and carried out by ProfilARBED-Research (RPS
Department), with the active support of the following working group which is particularly
thanked for the fruitful collaboration :
- the following 10 engineering offices which were involved to perform 3 worked examples :
Reference
Number
Engineering office
City
Country
2
Adem
Mons
Belgium
3
Bureau Delta
Liège
Belgium
4
Varendonck Groep / Steelrrack
Gent
Belgium
6
Ramboll & Hanneman
Copenhagen
Denmark
7
Bureau Veritas
Courbevoie
France
9
Socotec
Saint-Quentin-Yvelines
France
10
Sofresid
Montreuil
France
13
Danieli Ingegneria
Livorno
Italy
14
Schroeder & Associés
Luxembourg
Luxemburg
16
D3BN
Nieuwegein
The Netherlands
- Professor Sedlacek and assistant from Aachen University (Germany) which guaranteed the
link of this working group to the drafting panel of Eurocode 3 and Eurocode 1,
- some other engineering offices which participated to the meetings of the full working group :
Reference
number
City
Country
5
Engineering
office
Verdeyen & Moenart
Associate Partner
Bruxelles
Belgium
12
18
Ingenieur gruppe Bauen
Ove Arup & Partners
Karlsruhe
London
19
ECCS-TCll
Kiel
Germany
United
Kingdom
Germany
10
-some members of CTICM (France) and SIDERCAD (Italy) involved in complementary
research projects about simplified approaches of Eurocode 3 (respectively, "Application
software of Eurocode 3 : EC3-tools" and "Design handbook for sway buildings") :
. which participated to the meetings of the full working group,
. and with which a general flow-chart (FC1) about elastic global analysis of steel frame
according to EC3 has been established.
3. Part 1 : Worked examples
In order to find the needed formulae and to familiarise the engineering offices to the
Eurocodes, it has been decided to perform 3 different exercises (check and design of a steel
structure),
- exercise 1: verification of an existing braced or non-sway steel structure,
- exercise 2: verification of a non-sway steel wind bracing in a building,
- exercise 3: design of a braced or non-sway steel structure,
Different drawings issued from the exercises of the offices are enclosed in the technical report
n° 4 (TR4) showing the type of the calculated buildings and some details :
- office building with bracing system (engineering offices n° 2, 9 and 16),
(Annex 1 of TR4); - car park (engineering office n° 3), (Annex 2 of TR4);
- residential building with bracing system (engineering office n° 7), (Annex 3 of TR4);
- office building with bracing system (engineering office n° 10), (Annex 4 of TR4);
- industrial building with catalytic reactors (engineering office n° 13), (Annex 5 of TR4);
- office building with concrete core (engineering office n° 14), (Annex 6 of TR4);
- office building with concrete core (engineering office n° 4), (Annex 7 of TR4);
- office building with bracing system (engineering office n° 6), (Annex 8 of TR4).
3.1. Exercise 1 : Verification of an existing braced or non-sway structure
The flow-chart of figure 1 shows the procedure followed for the verification of an existing
building with the Eurocodes 1 and 3. This first exercise aimed to find the needed formulae
given by the Eurocodes in order to check the safety of the different limit states.
This exercise was not an iterative processes, but was only a verification procedure of an
existing braced or non-sway building.
The flow-chart of figure 1 is divided into 3 subjects:
a. The "Keywords" representing the different steps of a check procedure.
1. conceptional type of structure.
2. occupancies.
3. shape.
4 structural concept.
5 action effects.
6. design and verification.
b. The "Requirements and References" of each step of the verification.
The references are Eurocode 1, Eurocode 3 and the product standards
EN 10025 and EN 10113.
c. The "Object" describing each step of the verification.
11
3.2. Exercise 2: Verification of a non-sway wind bracing in a building
The non-sway wind bracing consisted of a latticed steel structure. The flow-chart of figure 2
gives the procedure of the verification of this wind bracing. This exercise was also not an
iterative process.
The description of the present flow-chart (figure 2) is the same than in the first example
presented in the chapter 3.1 (figure 1).
3.3. Exercise 3: Design of a braced or non-sway structure
After the two first exercises, the engineering offices were familiarised with the Eurocodes 1
and 3. They were able to perform a complete design of a structure by using an iterative
procedure. The aim of this exercise was to analyse the way to find a good solution.
This exercise allowed us to follow step by step the calculation of a structure in practice. The
practical design handbook about the simplified version of the Eurocode 3 follows an
improved way than the one defined in the initial design procedure. The figure 3 shows the
different data for the design and the type of chosen optimisation. The Figure 4 gives the type
of building to be designed.
4. Part 2 : Design handbook
A list of the needed formulae taken from the Eurocode 3 has been established following the
initial procedure defined for the exercises (see figures 5 to 8).
This initial design procedure nearly corresponds to the sequence of the chapters of Eurocode
3. It had to be adapted to common practice.
The solved exercises E3 (design of a building) and the experience of each engineering office
allowed to determine a more suitable design procedure which constitutes the frame of the
design handbook.
About that practical design procedure reference may be made to the enclosed design
handbook which is called "Design handbook according to Eurocode 3 for braced or nonsway steel buildings" (short title : "EC3for non-sway buildings") :
- table of contents
- general flow-chart FC1 about elastic global analysis of steel frames according to
Eurocode 3 (see chapter I of the design handbook); this flow-chart FC1 constitutes the
link with the 2 other researches about simplified approaches of EC3 : from CTICM and
SIDERCAD (see chapter 2 of the present report),
- flow-chart FC3.1 and FC3.2 about general procedures to study structures submitted to
actions (see chapter ΠΊ of the design handbook), with load cases which are respectively
defined :
. by relevant combinations of characteristic values of load arrangements, (g, q, s, w,
...), in general cases,
. or, by relevant combinations of characteristic values for the effects of actions (N,
V, Μ; δ, f,...), in case of first order elastic global analysis.
- flow-chart FC4 about elastic global analysis of braced or non-sway steel frames
according to Eurocode 3 (see chapter IV of the design handbook),
- flow-chart FC 12 about elastic global analysis of bracing system according to Eurocode 3
(see chapter ΧΠ of the design handbook)
12
In general, for the design of buildings we need to :
- define the analysis model of frames (assumptions of plane frames, bracing systems,
connections, members,...)
- characterise the load arrangements and load cases,
- carry out the elastic global analysis of frames in order to determine the effects of actions :
. deformations (δ), vibrations (f) for Serviceability Limit States (SLS) and,
. internal forces and moments (N, V, M) for Ultimate Limit States (ULS).
- check the members at SLS (vertical and horizontal displacements, eigenfrequencies) and
at ULS (resistance of cross-sections, stability of members and stability of webs) for :
. members in tens on (braces,...)
. members in compression (columns,...)
. members in bending (beams,...)
. members with combined axial load force and bending moment (beam-columns,...)
- check the local effects of transverse forces on webs at ULS (resistance and stability of
webs),
- check the connections at SLS and at ULS.
Especially for members to be checked at ULS specific tables are given in the concerned
chapters of the handbook, with list of checks according to different types of loading (separate
or combined internal forces and moments : N, V, M).
The design handbook which is enclosed to this final report of the research project, intends to
be a design aid in supplement to the complete document Eurocode 3 - Part 1.1 in order to
facilitate the use of Eurocode 3 for the design of such steel structures which are usual in
common practice : braced or non-sway steel structures.
Although the present design handbook has been carefully established and intends to be selfsufficient it does not substitute in any case for the complete document Eurocode 3 - Part 1.1,
which should be consulted in conjunction with the NAD, in case of doubt or need for
clarification.
All references to Eurocode 3 - Part 1.1 which appear systematically, are made in [...].
Any other text, tables or figures not quoted from Eurocode 3 are considered to satisfy the
rules specified in Eurocode 3 - Part 1.1.
The lists of all symbols, tables and flow-charts included in the "Design Handbook" are
enclosed to the present appendices.
13
1. conceptional rype of structure
different braced non sway structures <, 20 storeys
: Classification
non­sway: Vsd / VCT <, 0.1
braced: φ h £ 0.2 φ ^
I
2. occupancies
types of occupancy
­ ware house
­ office building
­ industrial hall
*
3.shape
shape of the building
(
Basis of design. Imposed
loads on floor and roofs
^
■cEC 1: Wind loads, Snow loads
Τ
4. structural concept
structural model
Geometric dimensions
Non­structural elements
Load bearing structure
Joints
Profiles
3
}
EC 3: Non­sway
Product standards:
EN 10025, EN 10113
EC 3 ­> b /1 classification
Floor structure
Material properties
5. action effects
determination of the action effects
(global and local)
EC 1: Load cases
EC 3: Load combinations
elastic or plastic model
SLS
ULS
I
6. dimensioning and verification
SLS limits
ULS limits
Frame stability
deformations
vibrations
Static equilibrium
Resistance of cross section
EC 3: Imperfections
EC 3: Modelling depending on
b /1 classification
1 s t order analysis
V_
• tension
• comprei»ion
• bending moment
- bending montent vid »hear
· bending momtia and axial force
- bending moment, sbear and axial force
- abear
· transverse farces c a webs
- ine ar boe kl mg
Resistance of members (stability)
• compression members : bocfcling
- lateral torsional buckling of beams
- bending and axial ami ion
• bending and axial compresaseli
Legend
Keywords
Γ
I
Requirement & References
)
C
Object
Connection
I
- joints
• base of colorons
Exercise 1. Verification of an existing braced non­sway structure
Figure 1
15
I
1. conceptiqnal type of structure
non­sway wind bracing in a building
(latticed structure )
EC 3: Classification
non­sway :VSd / VCT <, 0.1
1 s t order theory
. Τ .„
2. occupancies
part of an office building
EC 1: Basis of design, vertical loading
m
Horizontal loading
3. shape
­ position in the building
­ locations from load introduction and con­
nections from floors, roofs, claddings etc.
J
4. structural concept
structural model
Geometric dimensions
Non­structural elements
Joints
Profiles
Vertical f orces from gravity loads,
imposed loads, snow and wind loads
Horizontal forces from wind,
imperfections
EC 3: Non­sway
Product standards:
EN 10025, EN 10113
EC 3 ­> b / 1 classification
Material properties
J
5. action effects
EC 1: Load cases
EC 3: Load combinations
determination of the action effects
(global and local)
EC 3: Imperfections
EC 3: Modelling depending on
b / 1 classification
elastic or plastic model
SLS
ULS
1 s t order analysis
6. dirnehsioning and verification
SLS limits
ULS limits
deformations
Frame stability
vibrations
Static equilibrium
Resistance of cross section
- tension
- compression
• bending moment
- shear
- bending moment and shear
- bending moment and axial force
- bending moment, shear and axial force
- transverse forces on webs
■ shear buckling
Resistance of members (stability)
- compression members : buckling
- lateral torsional budding of beams
- bending and axial tension
- bending and axial compress ion
Legend
Keywords
WÊÉBÊÊ
Requirement & References
C
Object
Connection
-joints
• baae of columns
Exercise 2.Verification of a non­sway wind bracing in a building
Figure 2
16
1. conceptional type of structure
Braced non sway structure (defined)
I
2. occupancies
types of occupancy (defined)
- office building
3. shape
shape of the building (defined)
1
J
4. structural concept
structural model
Geometric dimensions (defined)
Non-structural elements (not defined)
Load bearing structure (not defined)
Type of joints (defined)
Profiles ( not defined)
7. optimisation of the weight
Floor structure (not defined)
Material properties (not defined)
Profiles:
- max 3 different profiles
for the columns
Type of joints:
- hinged or rigid
connections
Steel: FeE 235
or FeE 355
or FeE 460 grades
I
5. action effects
determination of the action effects
(global and local)
elastic or plastic model
SLS
t
ULS
6. dimensioning and
SLS limits
deformations
vibrations
t
verification
ULS limits
Frame stability
Static equilibrium
Resistance of cross section
and axial force
abear and axial force
■ compression
- sbear bedding
Resistance of members (stability)
- cflfupyraHin membera ι bockung
- lateral torsional bedding of beams
• ^"Hiraj and axial tension
* bending and axial compresiion
Connection
•joints
- base of columns
Exercise 3.Design of a braced non-sway structure
Figure 3
17
plane view
lift
Χ
front view
Ν
ζ
!
II
ce
I
CA
C
Reference
number
2
6
7
9
10
13
Engineering office
n°
X Y storeys Joints
n=
(m) (m)
Adem
1 30 10
5
Rigid
Rambøll, Hannemann & Højlund
2 30 10
15
Rigid
Veritas
3 50 14
10
Hinged
Socotec
4 50 14
15
Hinged
Sofresid
5 50 18
20
Rigid
Danieli
6 50 18
15
Hinged
Exercise 3 : Type of building to be designed
Figure 4
18
Eurocode 3 Formulae References
···· · i
Λ. C« · ■ ·
1. Conceptional type of structure.
1.1. non-sway
­> Chapter 5.2.5.2
1.2. braced
­> Chapter 5.2.5.3
1.3
storeys
2. Occupancies.
2.1. Type of building, (category,...)
2.2. Imposed loads on floors and roof (p and P) ­> Chapter EC 1, part 2.4: Imposed load
3, Shape,
3.1. Wind loads fw) ­> Chapter EC1 Part 2.7: Wind loads.
3.2. Snow loads (s) ­> Chapter EC1 Part 2.5: Snow loads.
4. Structural concept.
4.1. Structural model.
4.2. Geometric dimensions.
4.3. Non structural elements.
4.4. Load bearing structure.
4.5. Joints.
4.6. Profiles.
4.7. Floor structure.
4.8. Material properties.
5. Action effects.
5.1. Load cases. ­> EC1.
­ permanent loads: g and G
­ variable loads: q and Q:
­ imposed loads: ρ and Ρ (presentparagraph 2.2.)
- wind loads: w (presentparagraph 3.1.)
- snow loads: s (presentparagraph 32.)
5.2. Load combinations. ­> EC3.
SLS:
-> Chapter 2.3.4 clause (5), formulae (2.17) and (2.18)
ULS:
­> Chapter 2.3.3.1 clause (5), formulae (2.11) and (2.12)
5.3. Imperfections. -> EC3.
Frame :
­> Chapter 5.2.4.3
clause (1) formula (5.2)
Bracing system: ­> Chapter 5.2.4.4
clause (1) formulae (5.3) and (5.4)
[Members :
­> Chapter 5.2.4.2.
clause (4) formula (5.1)7
5.4. Elastic or plastic model -> EC3: Chapter 5.3: classification of cross­sections (b/t ratios).
Flange:
­> table 5.3.1 (sheet 3)
Web:
­> table 5.3.1 (sheet 1)
­> Chapter 5.4.6
clause (7) shear buckling
=> (presentparagraph 72.9 )
Section:
­> Chapter 5.3.4 for elastic global analysis
­> Chapter 5.3.3 for plastic global analysis
Figure 5
19
Eurocode 3 Formulae References
6. Verification SLS. -> Chapter 4
6.1. Global analysis. -> beams, portal f rames,structural frames Calculation for
- > bracing system
δ vertical
and δ horizontal
6.2. Deformations.
6.3. Vibrations.
-> Chapter 4.2.2 clause (1) δ vertical table 4.1, figure 4.1
clause (4) δ horizontal
-> Chapter 4.3. (ECCSpublication n°65: table4.4;... ;
7. Verification ULS.
7.1. Global analysis.
= > internal forces: Μ, Ν and V
- Elastic analysis -> Chapter 5.2.1.3
- Plastic analysis -> Chapter 5.2.1.4
- 1st or 2nd order analysis (present paragraph 1.1 )
7.2. Resistance of cross-sections. -> Chapter 5.4
7.2.1. tension.
-> Chapter 5.4.3 clause (1) formula (5.13)
7.2.2. compression.^ Chapter 5.4.4 clause (1) formula (5.16)
7.2.3. bending moment.-> Chapter 5.4.5
-> Chapter 5.4.5.1
clause (1) formula (5.17)
clause (2) formula (5.18)
f Ύ
-> Chapter 5.4.5.3
clause(l) formula(5.19) => A v n e t > -2- ,­iÖ­
U
' Mo
0.9
(remark: y m factors should be ignored)
7.2.4. shear.
­> Chapter 5.4.6 clause (1) formula (5.20)
clause (2): Ayz
Avy: ECCS publication n°65: table 5.14
clause (8) formula (5.21)
clause (9)
7.2.5. bending and shear.
-> Chapter 5.4.7 clause (2)
clause (3) a), b) formula (5.22)
for cross­sections with unequal flanges:
M S d <L M V j R d = Mf>Rd + (Mp 1>Rd ­ Mf >Rd )1 ­
VSd ­ 1
'pl,Rd
^Mc,Rd
7.2.6. bending and axial force.
Class 1 and 2 cross-sections:
-> Chapter 5.4.8.1
clause (3)
clause (4) formulae (5.25) and (5.26)
clause (11) formula (5.35)
Class 3 cross-sections:
-> Chapter 5.4.8.2
clause (1) formula (5.37)
7.2.7. bending, shear and axial force.
-> Chapter 5.4,9
clause (2)
clause (3)
­> biaxial bending: (ECCS publication η °65: tables 5.15 and 5.16)
Figure 6
20
Eurocode 3 Formulae References
7.2.8. transverse forces on webs.
-> Chapter 5.4.10
clause (3) -> clause (1) formula (5.41)
clause (2) formula (5.42) figure 5.4.3
or
-> clause (4) formula (5.43)
clause (5) formula (5.44)
-> Chapter 5.7.1
clause (3) figure 5.7.1 (a)
clause (4) figure 5.7.1 (b)
clause (5)
-> Chapter 5.7.2
clause (3) figure 5.7.2
-> Chapter 5.7.3 Crushing clause (1) formulae (5.71) and (5.72)
ƒ clause (4) formula (5.74) J
-> Chapter 5.7.4 Crippling clause (1) formula (5.77)
clause (2) formula (5.78)
-> Chapter 5.7.5 Buckling clause (1) formula (5.79)
clause (3) figure 5.7.3
7.2.9. shear buckling. -> Chapter 5.6.1 clause (1) limit condition (present paragraph 5.4 )
7.2.10 flange-induced buckling.
-> Chapter 5.7.7
ECCS publication n °65: table 520
7.3. Resistance of members. (->for 1 st order analysis)
7.3.1. compression members: buckling.
-for 1 st order elastic analysis:
-> Chapter 5.5.1.1
clause (1) formula (5.45)
-> Chapter 5.5.1.2
clause (1) formula (5.46) with table 5.5.1, or table 5.5.2
-> Chapter 5.5.1.4
clause (1) table 5.5.3
clause (3) formula (5.47)
-> Chapter 5.5.1.5
clause (2) Annex E
- for 2 "d order elastic analysis:
-> Chapter 5.2.6.2
clause (2)
7.3.2. lateral-torsional buckling of beams.
-> Chapter 5.5.2
clause (1) formula (5.48)
clause (2) formula (5.49)
clause (3)
clause (5)
clause (6) Annex F
clause (7) limit condition
clause (8)
7.3.3. bending and axial tension.
-> Chapter 5.5.3
7.3.4. bending and axial compression.
-> Chapter 5.5.4 -without lateral-torsional buckling:
clause (1) formula (5.51) class 1 and 2 cross-sections
clause (3) formula (5.53) class 3 cross-sections
- with lateral-torsional buckling:
clause (2) formula (5.52) class 1 and 2 cross-sections
clause (4) formula (5.54) class 3 cross-sections
clause (7) figure 5.5.3
7.4. Resistance of connections.
7.4.1. boltedjoints. -> Chapter 6.5
7.4.1.1. Positioning of holes.
-> Chapter 6.5.1 figures 6.5.1 to 6.5.4
(ECCSpublication n°65: table 62 )
7.4.1.2. Design shear rupture resistance.
-> Chapter 6.5.2.2
clause (2) formula (6.1)
clause (3) figure 6.5.5
Figure 7
21
Eurocode 3 Formulae References
7.4.1.3. Angles.
-> Chapter 6.5.2.3
clause (2) formulae (6.2) to (6.4)
clause (3) figure 6.5.6
7.4.1.4. Categories of bolted connections.
-> Chapter 6.5.3 and table 6.5.2
7.4.1.5. Distribution offorces between fasteners.
-> Chapter 6.5.4
figure
6.5.7
7.4.1.6. Design resistance of bolts.
-> Chapter 6.5.5
clause (2) table 6.5.3
clause (3)
clause (4) formula (6.5)
clause (5) formula (6.6)
clause (9)
clause (10)
(ECCSpublication n°65: tables 6.6, 6.7and6.8)
7.4.1.7. High strength bolts in slip-resistant connections
-> Chapter 6.5.8
-> Chapter 6.5.9
Annex J
-> Chapter 6.5.10
clause (1) formula (6.11) and figure 6.5.10
[-> Chapter 6.5.11
clause (2) formula (6.12)7
[-> Chapter 6.5.12
clause (1) formula (6.13)7
-> Chapter 6.5.13.
tables 6.5.6 and 6.5.7, figure 6.5.12
[7.4.2 Joints with rivets.
-> Chapter 6.5.67
-> Chapter 6.6
7.4.3 Welded connections.
clause (3)7
[-> Chapter 6.6.3
clause (1)
-> Chapter 6.6.4
clause (4)
clause (7)
-> Chapter 6.6.5.1
clause (2)
-> Chapter 6.6.5.2
clause (2)
-> Chapter 6.6.5.3
clause (1) Annex M
clause (3) formula (6.14)
clause (4) formula (6.15)
clause (5)
-> Chapter 6.6.8
clause (2) formula (6.16)
clause (3)
[-> Chapter 6.6.9
clause (1)7
[
clause (3) formula (6.18)7
-> Chapter 6.6.10
clause (2)
clause (3)
7.4.4 Beam-to-column connections. -> Chapter 6.9 and Annex J
7.4.5. Column bases.
-> Chapter 6.11 and Annex L
7.5. Frame stability.
-> Chapter 5.2.6.1
clause (1)
clause (3)
clause (4)
7.6. Static equilibrium.
-> Chapter 2.3.2.4
clauses (1) to (12)
Figure 8
22
1. List of symbols in the "Design Handbook"
1.
List of symbols (1/6)
Latin symbols
a
a
a<i
ay
aup
ao
a i , &2
A
ci )
eo,d
designation of a buckling curve
throat thickness of filllet weld
geometrical data of the effects of actions
geometrical data for the resistance
design throat thickness for submerged arc welding
designation of a buckling curve
distance between fastener holes and edge
accidental action; area of building loaded by external pressure of wind;
area of gross cross-section
effective area of class 4 cross-section
effective area of class 4 cross-section subject to uniform compression
(single N x .sd)
effective area of class 4 cross-section subject to uniaxial bending
(single My.sd or single M z .sd)
net area of cross-section
reference area for C f (wind force)
tensile stress area of bolt
shear area of cross-section
effective shear area for resistance to block shear
shear area of cross-section according to yy axis
shear area óf cross-section according to zz axis
designation of a buckling curve; flange width; building width
effective breadth
design punching shear resistance of the bolt head and the nut
designation of a buckling curve; out stand distance
altitude factor for reference wind velocity
dynamic factor for wind force
direction factor for reference wind velocity
exposure coefficient for wind pressure and wind force
wind force coefficient
external pressure coefficient for wind pressure
roughness coefficient for determination of c e
topography coefficient for determination of c e
temporary (seasonal) factor for reference wind velocity
nominal value related to the design effect of actions
factors for determination of F v jyc
factors for determination of MC T
designation of a buckling curve; web depth
bolt diameter
mean diameter of inscribed and circumscribed circles of bolt head or nut
hole diameter
shift of relevant centroidal axis of the class 4 effective cross-section subject to
uniform compression (single N x .sd)
shift of the y centroidal axis of the class 4 effective cross-section subject to
uniform compression
shift of the ζ centroidal axis of the class 4 effective cross-section subject to
uniform compression
shift of relevant centroidal axis of the class 4 effective cross-section subject to
uniaxial bending (single My.Sd or single Mz.Sd)
equivalent initial bow imperfection
design value of equivalent initial bow imperfection
ei, e2
E
ECCS
ECSC
distance between hole fastener and edge
modulus of elasticity or Young Modulus; effect of actions at SLS
European Convention for Constructional Steelwork
European Community of Steel and Coal
Aeff
Atff.N
Aeff.M
Anet
A re f
Ag
Av
Av.net
A v .y
A v .z
b
beff
Bp.Rd
c
CALT
ca
DIR
ce
Cf
Cpe
cr
ct
C
CTEM
Cd
C i , C2
C1.C2.C3
d
d
d,,,
do
βΝ
eNy
βΝζ
eM
23
EC 1
EC 3
EC 8
Ed
Ek
fd
fe
fmin
fu
fub
f,
¿b
fyb
fyw
F, Fi, F2
FC
FbUd
FbJUc
Fd
Ffr
Fh.sd
Fk
Fp.Rd
Fsd
Fsk
FsHd
Fs.Rd.ser
Fs.Rk
Ft.Rd
Ft .Rk
Ft.sd
FvRd
FvRk
Fv.sd
Fv.sd.ser
Fw
Fw.Rk
Fw.sd
g
G
Gd
Gk
h
ho
H
i
I
Ieff
It
Iw
Iz
k
k
kur
k<y
kw
ky, kz
List of symbols (2/6)
Eurocode 1
(/l/)
Eurocode 3
(/2/)
Eurocode 8
(/3/)
design value of the effect of action
characteristic value of effects of actions at SLS
design natural frequency
natural frequency
recommended limit of natural frequency
ultimate tensile strength
nominal value of ultimate tensile strength for bolt
yield strength
basic yield strength of the flat steel material before cold forming
nominal value of yield strength for bolt
yield strength of the web
action (load, transverse force, imposed deformations,...)
flow-chart
design bearing resistance per bolt
characteristic value of bearing resistance per bolt
design value of action
friction force
force on bolt calculted from Msd and/or Fbjtd
characteristic value of action
design punching shear resistance per bolt
design transverse force applied on web through the flange
characteristic value of transverse force
design slip resistance per bolt at the ultimate limit state
design slip resistance per bolt at the serviceability limit state
caracteristic slip-resistance per bolt and per friction interface
design tension resistance per bolt
characteristic value of tension resistance per bolt
design tensile force per bolt for the ultimate limit state
design shear resistance per bolt
characteristic value of shear resistance per bolt and per shear plane
design shear force per bolt for the ultimate limit state
design shear force per bolt for the serviceability limit state
resultant wind force
characteristic value of resistance force of fillet weld
design force of fillet weld
distributed permanent action; dead load
permanent action
design permanent action
characteristic value of permanent action
overall depth of cross-section; storey height; building height
overall height of structure
total horizontal load
radius of gyration about relevant axis using the properties of gross cross-section
second moment of area A
second moment of effective area Aeff (class 4 cross-section)
torsional constant
warping constant
second moment of area about zz axis
subscript meaning characteristic (unfactored) value
effective length factor
factor for lateral-torsional buckling with N-M interaction
buckling factor for outstand flanges
effective length factor for warping end condition
factors for N-M interaction
24
ί
L
Lb
LTB
Ly
m
max
min
M
M b .Rd
MCT
M
cRd
M€
Mf.Rd
MN.Rd
MN.VJld
MN.V.yJRd
MN.V.z.Rd
MN.y.Rd
MN.z.Rd
Mpf
Mp£Rd
Mp/iw.Rd
Mp£y.Rd
MptzJRd
MRd
Msd
Mv.Rd
Mw.sd
My
My.Sd
Mz
Mz.sd
η
rie
nr
n8
Ν
NAD
NbÄd
Nb.yJld
NbiJld
Ν compression
Ner
NcRd
Nxsd
NpCRd
List of symbols (3/6)
roughness factor of the terrain
portion of a member
effective length for out­of­plane bending
system length; span length; weld length
buckling length of member
lateral­torsional buckling
distance between extreme fastener holes
mass per unit length
maximum
minimum
bending moment
design resistance moment for lateral­torsional buckling
elastic critical moment for lateral­torsional buckling
design resistance moment of the cross­section
torsional moment
elastic moment capacity
design plastic resistance moment of the cross­section consisting of the flanges
only
reduced design plastic resistance moment allowing for axial force Ν
reduced design plastic resistance moment allowing for axial force Ν and by
shear
force V
reduced design plastic resistance moment about yy axis allowing for axial force
Ν and shear force V
reduced design plastic resistance moment about zz axis allowing for axial force
Ν and shear force V
reduced design plastic resistance moment about yy axis allowing for axial
force Ν
reduced design plastic resistance moment about zz axis axial force Ν
plastic moment capacity
design plastic resistance moment of the cross­section
design plastic resistance moment of the web
design plastic resistance moment of the cross­section about yy axis
design plastic resistance moment of the cross­section about zz axis
design bending moment resistance of the member
design bending moment applied to the member
design plastic resistance moment reduced by shear force
design value of moment applied to the web
bending moment about yy axis
design bending moment about yy axis applied to the member
bending moment about zz axis
design bending moment about zz axis applied to the member
number of fastener holes on the block shear failure path
number of columns in plane
number of members to be restrained by the bracing system
number of storeys
normal force; axial load
National Application Document
design buckling resistance of the member
design buckling resistance of the member according to yy axis
design buckling resistance of the member according to zz axis
normal force in compression
elastic critical axial force
design compression resistance of the cross­section
design value of tensile force applied perpendicular to the fillet weld
design plastic resistance of the gross cross­section
25
NRd
Nsd
NLRd
rN tension
Nu.Rd
N x .sd
Pl»P2
Ρ
q
qk
qref
Q
Qd
Qk
Vkmax
r
R
Ra,Rd
Rb,Rd
Rd
Rk
Ry,Rd
S
S
Sd
Sk
Ss
S
Sd
Sk
SLS
t
tf
tp
tp
tw
U
ULS
v
Vref
Vref.O
V
VbaJld
Ver
V//Sd
Vj.sd
Vp£Rd
VpiyJld
Vp£ z Jld
vRd
Vsd
Vy
Vy.Sd
vz
VZ.Sd
w
List of symbols (4/6)
design resistance for tension or compression member
design value of tensile force or compressive force
design tension resistance of the cross-section
normal force in tension
design ultimate resistance of the net cross-section at holes for fasteners
design internal axial force applied to member according to xx axis
distances between bolt holes
Point load
imposed variable distributed load
characteristic value of imposed variable distributed load
reference mean wind pressure
imposed variable point load
design variable action
characteristic value of imposed variable point load
variable action which causes the largest effect
radius of root fillet
rolled sections
design crippling resistance of the web
design buckling resistance of the web
design resistance of the member subject to internal forces or moment
characteristic value of Rd
design crushing resistance of the web
snow load
thickness of fillet weld
design snow load
characteristic value of the snow load on the ground
length of stiff bearing
effects of actions at ULS
design value of an internal force or moment applied to the member
characteristic value of effects of actions at ULS
Serviceability Limit states
design thickness, nominal thickness of element, material thickness
flange thickness
thickness of the plate under the bolt head or the nut
thickness of a plate welded to an unstiffened flange
web thickness
major axis
Ultimate Limit States
minor axis
reference wind velocity
basic value of the reference wind velocity
shear force; total vertical load
design shear buckling resistance
elastic critical value of the total vertical load
design value of shear force applied parallel to the fillet weld
design value of shear force applied perpendicular to the fillet weld
design shear plastic resistance of cross-section
design shear plastic resistance of cross-section according to yy axis (// to web)
design shear plastic resistance of cross-section according to zz axis (_L to flange)
design shear resistance of the member
design shear force applied to the member; design value of the total vertical load
shear forces applied parallel to yy axis
design shear force applied to the member parallel to yy axis
shear force parallel to zz axis
design internal shear forces applied to the member parallel to zz axis
wind pressure on a surface
26
List of symbols (5/6)
design wind load
wind pressure on external surface
welded sections
elastic section modulus of effective class 4 cross­section
elastic section modulus of effective class 4 cross­section according to yy axis
elastic section modulus of effective class 4 cross­section according to zz axis
elastic section modulus of class 3 cross­section
elastic section modulus of class 3 cross­section according to yy axis
elastic section modulus of class 3 cross­section according to zz axis
plastic section modulus of class 1 or 2 cross­section
plastic section modulus of class 1 or 2 cross­section according to yy axis
plastic section modulus of class 1 or 2 cross­section according to zz axis
axis along the member
characteristic value of the material properties
principal axis of cross section (parallel to flanges, in general)
principal axis of cross section (parallel to the web, in general)
reference height for evaluation of c e
Wd
we
W
Weff
Weff.y
Weff.z
Wef
We£y
We£z
Wpi
WpÉy
Wp£Z
x, xx
Xk
y, yy
z, zz
Ze
2u
Oreek symbols
α
α
α
aa
PA
βM
ßMl/r
ßMy
βκίζ
βw
ßw
YF
YG
YM
YMb
7Ms.ser
TMW
YMO
coefficient of frequency of the basis mode vibration
coefficient of linear thermal expansion
factor to determine the position of the neutral axis
coefficient of critical amplification or coefficient of remoteness of critical state
of the frame
non­dimensional coefficient for buckling
equivalent uniform moment factor for flexural buckling
equivalent uniform moment factor for lateral­torsional buckling
equivalent uniform moment factor for flexural buckling about yy axis
equivalent uniform moment factor for flexural buckling about zz axis
non-dimensional coefficient for lateral-torsional buckling
correlation factor (for a fillet weld)
partial safety factor for force or for action
partial safety factor for permanent action
partial safety factor for the resistance at ULS
ΎΜΙ
YM2
YQ
δ
Ob
5d
6dv
partial safety factor for the resistance of bolted connections
partial safety factor for the slip resistance of preloaded bolts
partial safety factor for the resistance of welded connections
partial safety factor for resistance at ULS of class 1,2 or 3 cross-sections
(plasticity or yielding)
partial safety factor for resistance of class 4 cross-sections
(local buckling resistance)
partial safety factor for the resistance of member to buckling
partial safety factor for the resistance of net section at bolt holes
partial safety factor for variable action
relative horizontal displacement of top and bottom of a storey
horizontal displacement of the braced frame
design deflection
design vertical deflection of floors, b e a m s , . . .
¿¿d
OHmax
δς
design horizontal deflection of frames
recommended limit of horizontal deflection
in plane deflection of the bracing system due to q plus any external loads
YMI
27
δς
δ\,
Ôvd
°Vmax
δο
δι
θ2
Δ
θ
λ
λι
λ
λβο.ν
λβΚ.y
λβιϊ.ζ
XLT
λρ
λν
λy
λζ
μ
Hi
μι,τ
μγ
μζ
ρ
ρ
py
pz
σ
List of symbols (6/6)
deflection due to variable load (q)
horizontal displacement of the unbraced frame
design vertical deflection of floors, beams,...
recommended limit of vertical deflection
pre­camber (hogging) of the beam in the unloaded state (state 0)
svariation of the deflection of the beam due to permanent loads (G) immediatly
after loading (state 1)
variation of the deflection of the beam due to the variable loading (Q) (state 2)
displacement
235
(with fy in N/mm2)
J
rotation
slenderness of the member for the relevant buckling mode
Euler slenderness for buckling
non­dimensional slenderness ratio of the member for buckling
effective non­dimensional slenderness of the member for buckling about w axis
effective non­dimensional slenderness of the member for buckling about yy axis
effective non-dimensional slenderness of the member for buckling about zz axis
non-dimensional slenderness ratio of the member for lateral-torsional buckling
plate slenderness ratio for class 4 effective cross-sections
non-dimensional slenderness of the member for buckling about vv axis
non dimensional slenderness ratio of the member for buckling about yy axis
non dimensional slenderness ratio of the member for buckling about zz axis
factor for FsjRk depending on surface class
snow load shape coefficient
factor for N-M interaction with lateral-torsional buckling
factor for N-M interaction
factor for N-M interaction
density
reduction factor due to shear force Vsa
reduction factor due to shear force Vy.sd
reduction factor due to shear force Vz.sd
normal stress
Gq
numerical values for the stabilizing forces of a bracing system
GxEd, <*xm.Ed»
design values of normal stresses for web check with Von Mises criteria
^z£d
τ
υ
φ
χ
%Ul
Xmin
Xy
Xz
shear stresss
Poisson's ratio
initial sway imperfection of the frame
reduction factor for the relevant buckling mode
reduction factor for lateral-torsional buckling
minimum of %y and χ ζ
reduction factor for the relevant buckling mode about yy axis
reduction factor for the relevant buckling mode about zz axis
28
2. List of tables in the "Design Handbook"
O.c
Table 0.1
I
Table 1.1
Table 1.2
Table 1.3
Table 1.4
Table 1.5
Table 1.6
Table 1.7
Table 1.8
Π
Table Π.1
Table Π.2
Table Π.3
Table Π.4
Table Π.5
Table Π.6
Table Π.7
Table Π.8
ΙΠ
Table ΠΙ.1
Table ΠΙ.2
Table ΠΙ.3
Table ΠΙ.4
Table ΠΙ.5
Table ΠΙ.6
Table ΠΙ.7
Table ΙΠ.8
Table ΙΠ.9
IV
Table IV. 1
Table IV.2
Table IV.3
Table IV.4
Table IV.5
Table IV.6
Table V.l
Table V.2
Table V.3
List of tables (1/3)
Pages of the Handbook
SYMBOLS AND NOTATIONS
Dimensions and axes of rolled steel sections
INTRODUCTION
Summary of design requirements
Partial safety factor γΜ for the resistance
Definition of framing for horizontal loads
Checks at Serviceability Limit States
Member submitted to internal forces, moments and transverse forces
Planes within internal forces, moments (Nsd> V$d, Msd) and transverses
forces Fsd are acting
Internal forces, moments and transverse forces to be checked at ULS for
different types of loading
List of references to chapters of the design handbook related to all check
formulas at ULS
STRUCTURAL CONCEPT OF THE BUILDING
Typical types of joints
Modelling of joints
Comparison table of different steel grades designation
Nominal values of yield strength f« and ultimate tensile strength fu for
structural steels to EN 10025 and EN 10113
Maximum thickness for statically loaded structural elements
Maximum thickness for statically loaded structural elements
Nominal values of yield strength fyb and ultimate
tensile strength fub for bolts
Material coefficient
LOAD ARRANGEMENTS AND LOAD CASES
Load arrangements (Ffc) for building design according to EC1
Imposed load (qk, Qk) on floors in buildings
Pressures on surfaces
Exposure coefficient c e as a function of height ζ above ground
External pressure Cpe for buildings depending on the size of
the effected area A
Reference height ZQ depending on h and b
Combinations of actions for serviceability limit states
Combinations of actions for ultimate limit states
Examples for the application of the combinations rules in Table III.8.
All actions (g, q, P, s, w) are considered to originate from different sources
DESIGN OF BRACED OR NON-SWAY FRAME
Modelling of frame for analysis
Modelling of connections
Global imperfections of the frame
Values for the initial sway imperfections φ
Specific actions for braced or non-sway frames
Recommended limits for horizontal deflections
CLASSIFICATION OF CROSS-SECTIONS
Definition of the classification of cross-section
Determinant dimensions of cross-sections for classification
Classification of cross-section : limiting width-to thickness ratios for
class 1 & class 2 I cross-sections submitted to different types of loading
29
46
51
52
63
64
65
66
67
68
70
71
72
73
74
75
75
76
80
81
82
83
83
84
85
86
86
88
87
97
98
99
100
105
112
113
Table V.4
Table V.5
Table V.6
Table V.7
Table V.8
Table V.9
Table V.10
VI
Table VI. 1
Table VI.2
Table VI.3
Table VI.4
vn
Table Vn.l
Table Vfl.2
Table Vfl.3
Table VH.4
Table Vfl.5
Table VH.6
vm
Table Vm.l
Table vm.2
Table Vni.3
Table Vm.4
Table vm.5
Table vm.6
Table vm.7
Table Vffl.8
Table
Vin.9
Table Vm. 10
Table Vm. 11
Table Vm. 12
Table VIfl.13
Table Vm. 14
EX
Table DC. 1
Table DC.2
Table K . 3
List of tables (2/3)
Pages of the Handbook
Classification of cross­section : limiting width­to thickness ratios for
114
class 3 I cross­sections submitted to different types of loading
Buckling factor k<y for outs tan d flanges
115
Classification of cross­section : limiting width­to­thickness ratios for
internal flange elements submitted to different types of loading
116
Classification of cross­section : limiting width­to­thickness ratios for
angles tubular sections submitted to different types of loading
117
Effective cross­sectional data for symmetrical profiles
(class 4 cross­sections)
118
Limiting values of axial load Nsd for web classification of I cross­sections
subject to axial load Nsd and to bending according to major axis My.sd
119
120
Examples of shift of centroidal axis of effective cross­section
MEMBERS IN TENSION (Ntension)
List of checks to be performed at ULS for the member in tension (Ntension) 124
125
Gross and net cross­sections
126
Reduction factors ß2 and ß3
127
Connection of angles
MEMBERS IN COMPRESSION (NCOmpression)
List of checks to be performed at ULS for the member
in compression (NCOmp.)
Imperfection factor α
Value of Euler slenderness λι
Selection of buckling curve for a cross­section
Buckling length of column : LD
Reduction factors χ = ί(λ)
MEMBERS IN BENDING (V ; M ;( V,M))
List of checks to be performed at ULS for the member in bending according
to the applied internal forces and/or moments(V ; M ;( V,M))
Recommended limiting values for vertical deflections
Vertical deflections to be considered
Recommended limiting values for floor vibrations
Shear area A v for cross-sections
Determination of Av.net for block shear resistance
Limiting width-to-thickness ratio related to the shear buckling in web
Simple post-critical shear strength xDa
Buckling factor for shear k t
Reduction factor %LT = f (λυτ) for lateral-torsional buckling
Effective length factors : k, kw
Numerical values for Ci and definition of ψ
Reduced design plastic resistance moment My Rd allowing for shear force
Interaction of shear buckling resistance and moment resistance
with the simple post-critical method
MEMBERS WITH COMBINED AXIAL FORCE AND
BENDING MOMENT ((N, M) ;(N, V, M))
List of checks to be performed at ULS for the member submitted to
combined axial force and bending moment (Ν, M)
Principle of interaction formulas between axial force Nsd
and bending moment Msd
Reduced design plastic resistance moment MN.Rd allowing for axial load
for Class 1 or 2 cross-sections
30
131
134
134
135
136
137
143
147
147
148
150
151
151
152
152
156
156
157
159
160
164
170
171
Table K.4
Table DÍ.5
Table DC.6
Table DC.7
Table Di. 8
Table DÍ.9
Table X.l
Table X.2
Table X.3
Table X.4
Table X.5
Table X.6
Table X.7
Table X.8
Table X.9
XI
Table XI. 1
Table XI.2
Table XI.3
Table XI.4
Table XI.5
Table XI.6
Table XI.7
Table XI.8
Table XI.9
Table XI. 10
Table XI. 11
Table XI. 12
Table XL 13
Table XL 14
Table XL 15
Table XL 16
XII
Table ΧΠ.1
Table ΧΠ.2
Table ΧΠ.3
Table ΧΠ.4
Table D.l
List of tables (3/3)
Pages of the Handb '
Interaction formulas for the (N,M) stability check of members
of Class 1 or 2
175
Interaction formulas for the (NM) stability check of members of Class 3 176
General interaction formulas for the (N,M) stability check of
members of Class 4
177
Supplementary interaction formulas for the (N,M) stability check
of members of Class 4
178
Reduced design resistance Ny.Rd allowing for shear force
179
Reduced design plastic resistance moment MN.VJM allowing for axial load
and shear force for Class 1 or 2 cross­sections
181
TRANSVERSE FORCES ON WEBS (F ; (F,N,V,M))
Failure modes due to load introduction
187
Stresses in web panel due to bending moment, axial force
and transverse force
188
Yield criteria to be satisfied by the web
189
Load introduction
190
Length of stiff bearing, s s
190
Interaction formula of crippling resistance and moment resistance
191
Effective breadth beff for web buckling resistance
192
Compression flange buckling in plane of the web
1*93
Maximum width­to­thickness ratio d/tw
193
CONNECTIONS
Designation of distances between bolts
196
Linear distribution of loads between fasteners
196
Possible plastic distribution of loads between fasteners. Any realistic
combination could be used, e.g.
197
Prying forces
197
Categories of bolted connections
198
Bearing resistance per bolt for recommended detailing
for t = 10 mm in [kN]
199
Shear resistance per bolt and shear plane in [kN]
200
200
Long joints
201
Tension resistance per bolt in [kN]
201
Interaction formula of shear resistance and tension resistance of bolts
Characteristic sup resistance per bolt and friction interface for 8.8 and 10.
bolts, where the holes in all the plies have standard nominal clearances
202
Common types of welded joints
203
204
Throat thickness
204
Action effects in fillet welds
205
Resistance of a fillet weld
206
Effective breadth of an unstiffened tee joint
DESIGN OF BRACING SYSTEM
Load arrangements of the bracing system
216
Bracing system imperfections
218
Values for the equivalent stabilizing force Zq
218
Bracing system imperfections (examples)
219
APPENDK D
List of references to Eurocode 3 Part 1.1 related
to all check formulas at ULS
221
31
3. List offlow-chartsin the "Design Handbook"
©
Chapter Pages
Elastic global analysis of steel frames according to Eurocode 3
General
Details
Comments (6 pages)
I
I
14
15
16 to 21
[FC 3.1] Load arrangements & load cases for general global analysis of
m
38
ni
39
(FCM)Elastic global analysis of braced or non­sway steel frames
according to EC 3
General
Details
Comments (4 pages)
IV
IV
IV
50
51
52 to 55
ÍFC 5. lì Classification of I cross­section
V
62
(FC 5.2J Calculation of effective cross­section properties of Class 4
rv
63
VI
82
VI
83
vn
90
vm
102
ΧΠ
ΧΠ
ΧΠ
168
169
170 to 175
the structure
[FC 3.2J Load arrangements & load cases for first order elastic global
analysis of the structure
cross­section
(FC6.Ì)
(FC 6.11 Members in tension (Ntension)
(FC6.2)
(FC7)
Angles connected by one leg and submitted to tension
Members in compression (Ncompression)
(FC 8Î Design of I members in uniaxial bending (Vz;My;(Vz,My)) or
(Vy;Mz;(VyJvIz);
ÍFC 12ÌElastic global analysis of bracing system according to
Eurocode 3
General
Details
Comments (6 pages)
32
Design handbook according
to Eurocode 3 for
braced or non-sway steel buildings
Short title: EC 3 for non-sway buildings
Profil ARBED-Recherches
Chantrain Ph.
Conan Y.
Mauer Th.
TABLE OF CONTENTS
0
PRELIMINARIES
0.a
O.u.l
0.a.2
0.a.3
O.a.4
0.a.5
O.b
Foreword
Generalities
Objective of this design handbook
Warning
How to read this design handbook
Acknowledgements
References
41
41
41
41
41
42
42
43
O.c
Symbols and notations
O.c.l
Symbols
O.c.2
Convention for member axes
0.C.3
Dimensions and axes of rolled steel sections
0.C.4
Notations in flow-charts
O.d
Definitions and units
O.d.1
Definition of special terms
0.d.2
Units
44
44
44
44
45
47
47
' 47
INTRODUCTION
La
Basis of design
I.a.1
Fundamental requirements
I.a.2
Definitions
I.a.2.1
Limit states
I.a.2.2
Actions
I.a.2.3
Material properties
I.a.3
Design requirements
I.a.3.1
General
I.a.3.2
Serviceability Limit States
I.a.3.3
Ultimate Limit States
Lb
General flow-charts about elastic global analysis
I.b.l
Flow-chart FC LElastic global analysis of steel frames according to EC 3
Lb. 1.1
Flow-chart FC 1: general
I.b.1.2
Flow-chart FC 1: details
I. b. 1.3
Comments on flow-chart FC 1
Le
Content of the design handbook
I.c.1
Scope of the handbook
I.C.2
Definition of the braced frames and non-sway frames
I.C.3
Summary of the table of contents
I.C.4
Checks at Serviceability Limit States
I.C.5
Checks of members at Ultimate Limit States
35
f
._
.R
7°
~™
49
49
50
50
50
50
53
53
53
53
56
61
61
62
64
64
65
TABLE OF CONTENTS
Π
STRUCTURAL CONCEPT OF THE BUILDING
69
n.a
Structural model
69
n.b
Geometric dimensions
69
n.c
Non structural elements
69
n.d
Load bearing structure
69
n.e
Joints ? 0
n.f
Profiles
'Jl
U.g
Floor structure
7­
U.h
Material properties
72
n.h. 1
Nominal values for hot rolled steel
7~
n.h.2
Fracture toughness
n.h.3
Connecting devices
\?
75
n.h.3.1
Bolts
U.h.3.2
Welding consumables
76
n.h.4
Design values of material coefficients
76
m
IV
LOAD ARRANGEMENTS AND LOAD CASES
77
ULa
Generalities
ni.b
Load arrangements
m.b.l
Permanent loads (g and G)
m.b.2
Variable loads (q, Q, w and s)
III.b.2.1
Imposed loads on floors and roof (q and Q)
m.b.2.2
Wind loads (we,i, F w )
m.b.2.2.1 Wind pressure (we¿)
m.b.2.2.2 Wind force (Fw)
m.b.2.3
Snow loads (s)
ULc
Load cases
m.c.l
Load cases for serviceability limit states
m.c.2
Load cases for ultimate limit states
77
80
80
80
80
81
81
84
84
85
85
86
DESIGN OF BRACED OR NON­SWAY FRAME
87
rV.a
Generalities
IV.a.l
Analysis models for frames
IV.a.2
Flow­chart FC 4:Elastic global analysis of braced or non­sway steel frames
according to Eurocode 3
IV.a.2.1
Flow­chart FC 4 general
rV.a.2.2
How­chart FC 4 details
rv.a.2.3
Comments on flow­chart FC 4
rv.b
Static equilibrium
rv.c
Load arrangements and load cases
IV.c.l
Generalities
IV.C.2
Frame imperfections
rV.d
Frame stability
rv.e
First order elastic global analysis
rV.e.l
Methods of analysis
IV.e.2
Effects of deformations
IV.e.3
Elastic global analysis
rv.f
Verifications at SLS
rv.f.l
Deflections of frames
rv.g
Verifications at ULS
rv.g.l
Classification of the frame
rv.g. 1.1
Hypothesis for braced frame
rv.g. 1.2
Hypothesis for non­sway frame
rv.g.2
ULS checks
87
87
36
89
89
89
92
96
96
96
96
97
98
98
98
99
99
100
10
°
*00
*00
100
100
TABLE OF CONTENTS
V
VI
VII
CLASSIFICATION OF CROSS-SECTIONS
ΙΟΙ
V.a
Generalities
V.b
Definition of the cross-sections classification
V.c
Criteria of the cross-sections classification
V.c. 1
Classification of compression elements of cross-sections
V.C.2
Classification of cross-sections
V.c.3
Properties of class 4 effective cross-sections
V.d
Procedures of cross-sections classification for different loadings
V.d. 1
Classification of cross-sections in compression
V.d.2
Classification of cross-section in bending
V.d. 3
Classification of cross-sections in combined (N,M)
101
104
106
106
106
106
109
109
109
110
M E M B E R S IN TENSION (Ntension)
Vl.a
Generalities
VLb
General verifications at ULS
Vl.b. 1
Resistance of gross cross-section to Ntension
VI.b.2
Resistance of net cross-section to Ntension
VLc
Particular verifications at ULS for angles connected by one leg
VI.c. 1
Connection with a single row of bolts
VI.C.2
Connection by welding
M E M B E R S IN COMPRESSION (NCOnipression)
Vn.a Generalities
VILb Classification of cross-sections
VII.c General verifications at ULS
Vn.c.l
Resistance of cross-section to Ncompression
VII.C.2 Stability of member to Ncompression
Vn.c.2.1 Resistance to flexural buckling
Vn.c.2.2 Resistance to torsionnal buckting and to flexural-torsional buckling
VILd Particular verifications at ULS for class 4 monosymmetrical cross-section
VILd. 1 Resistance of cross-section to Ncompression
VII.d.2 Stability of member to Ncompression
VILe Particular verifications at ULS for angle connected by one leg
VILe. 1 Connection with a single row of bolts
Vn.e. 1.1 Resistance of cross-section to NCOmpression
Vn.e. 1.2 Stability of member to Ncompression
VII.e.2 Connection by welding
Vn.e.2.1 Resistance of cross-section to Ncompression
Vn.e.2.2 Stability of member to NCOmpression
VIII MEMBERS IN BENDING (V ; M ; (V, M))
VHLa Generalities
VHLb Verifications at SLS
Vm.b.l Deflections
Vin.b.2 Dynamic effects - vibrations
VIII.c Classification of cross-section
VIILd Verifications at ULS to shear force Vsd
Vin.d. 1 Resistance of cross-section to Vsd
VIII.d.2 Stability of web to Vz.sd
Vin.e Verifications at ULS to bending moment Msd
Vin.e. 1 Resistance of cross-section to Msd
Vm.e.2 Stability of member to My.sd
Vin.f Verifications at ULS to biaxial bending moment (My.sd> Mz.sd)
Vin.f. 1 Resistance of cross-section to (My.sd, Mz.sd)
Vm.f.2 Stability of member to (My.sd, Mz.sd)
37
121
121
124
124
125
126
126
· 128
129
129
132
J 33
^
133
137
J38
^ °
^8
139
139
^9
139
139
139
139
140
140
145
145
147
147
148
148
150
152
152
I53
156
156
157
TABLE OF CONTENTS
LX
X
XI
Vm.g Verifications at ULS to combined (Vsd, Msd)
Vrn.g.1 Resistance of cross-section to (Vsd. Msd)
Vin.g. 1.1 Shear force Vsd and uniaxial bending Msd
Vin.g. 1.2 Shear force Vsd and biaxial bending moment Msd
Vm.g.2 Stability of web to (Vz.Sd,My.Sd)
MEMBERS WITH COMBINED AXIAL FORCE AND
BENDING MOMENT ((N, M) ; (N, V; M))
157
157
157
158
159
161
Di.a
Generalities
K.b
Verifications at SLS
LX.b.l
Deflections
LX.b.2
Vibrations
DC.c
Classification of cross-section
DCd
Verifications at ULS to (N,M)
LX.d. 1
Resistance of cross-section to (Nsd, Msd)
LX.d. 1.1
Uniaxial bending of class 1 or 2 cross-sections
LX.d. 1.2
Biaxial bending of class 1 or 2 cross-sections
LX.d. 1.3
Bending of class 3 cross-sections
LX.d. 1.4
Bending of class 4 cross-sections
LX.d.2 Stability of member to (Nsd,Msd)
K.d.2.1
Stability of member to (Ntension» My.sd)
LX.d.2.2
Stability of member to (Ncompression* Msd)
DC.e
Verifications at ULS for (N$d ,Vsd)
LX.e.l
Resistance of cross-section to (Nsd.Vsd)
JX.f
Verifications at ULS to (Nsd ,Vsd,Msd)
LX.f.l
Resistance of cross-section to (Nsd>Vsd>Msd)
LX.f. 1.1
Uniaxial bending of class 1 or 2 cross-section
LX.f. 1.2
Biaxial bending of class 1 or 2 cross-section
LX.f. 1.3
Bending of class 3 cross-section
LX.f. 1.4
Bending of class 4 cross-section
LX.f.2 Stability of web to (Nx.Sd, Vz.Sd, My.Sd)
161
167
167
167
167
167
1 ^7
167
170
170
171
171
171
172
X.a
Generalities
X.b
Classification of cross-section
X.c
Resistance of webs to (F,N,V,M)
X.C.1
Yield criterion to (F,N,V,M)
X.c.2
Crushing resistance to F
X.d
Stability of webs to (F ; (F, M))
X.d.l
Crippling resistance to (F;(F, M))
X.d. 1.1
Crippling resistance to F
X.d. 1.2
Crippling resistance to (F,M)
X.d.2
Buckling resistance to F
X.e
Stability of webs to compression flange buckling
184
185
I8 5
185
187
188
188
188
188
189
190
TRANSVERSE FORCES ON WEBS (F ; (F, N, V, M))
CONNECTIONS
XLa
Generalities
XI.b
Bolted connections
XLb. 1
Positioning of holes
XI.b.2
Distribution of forces between bolts
XI.b.3
Prying forces
XI.b.4
Categories of bolted connections
XI.b.5
Design ULS resistance of bolts
XI.b.5.1
Bearing resistance
XI.b.5.2
Shear resistance
XI.b.5.2.1 General case
38
176
I77
177
^7^
178
180
180
181
182
184
191
191
191
191
191
193
193
194
194
196
196
TABLE OF CONTENTS
XI.b.5.2.2 Long joints
XI.b.5.3
Tension resistance
XI.b.5.4
Punching shear resistance
XI.b.5.5
Shear and tension interaction
XI.b.6
ULS resistance of element with bolt holes
XI.b.6.1
Net section ULS resistance
XI.b.6.2
ULS resistance of angle with a single row of bolt
XI.b.6.3
Block shear ULS resistance
XI.b.7
High strength bolts in slip-resistant connections at SLS
XI.c
Welded connections
XI.c. 1
Type of weld
XI.C.2
Fillet weld
XI.C.3
Design resistance of fillet weld
XI.C.3.1
Throat thickness
XI.c.3.2
Design resistance
XI.C.4
Design resistance of butt weld
XI.c. 5
Joints to unstiffened flanges
Xl.d
Pin connections
XI.e
Beam-to-column connections
Xl.f
Design of column bases
196
197
197
197
198
198
198
198
198
199
199
199
200
200
201
201
202
202
202
202
XII DESIGN OF BRACING SYSTEM
203
203
203
203
203
206
212
212
212
213
216
216
216
216
216
216
216
APPENDIX A :
List of symbols
217
APPENDIX Β
List of tables
List of flow-charts
223
List of references to Eurocode 3 Part 1.1 related to
all check formulas at ULS
227
XILa Generalities
XILa. 1 Flow-chart FC 12:Elastic global analysis of bracing system according to EC 3
XILa. 1.1 Flow-chart FC 12: general
XD.a.1.2 Flow-chart FC 12: details
XILa. 1.3 Comments on flow-chart FC 12
Xll.b Static equilibrium
XII.c Load arrangements and load cases
XII.c.l
Generalities
XII.C.2 Global imperfections of the bracing system
XILd Bracing system stability
XILe First order elastic global analysis
XILf Verifications at SLS
Xll.g Verifications at ULS
Xll.g. 1 Classification of the bracing system
Xll.g. 1.1 Non-sway bracing system
XII.g.2 ULS checks
APPENDIX C
APPENDIX D
226
39
PRELIMINARIES
O.a
Foreword
0-a.l
Generalities
(1) The Eurocodes are being prepared to harmonize design procedures between countries
which are members of CEN (European Committee for Standardization).
(2) Eurocode 3 - Part 1.1 "Design of Steel Structures ¡General Rules and Rules for Buildings'
has been published initially as an ENV document (European pre-standard - a prospective
European Standard for provisional application).
(3) The national authorities of the members states have issued National Application
Documents (NAD) to make Eurocode 3 - Part 1.1 operative whilst it has ENV-status
(ENV 1993-1-1).
0.a.2
Objective of this design handbook
(1) The present publication is intended to be a design aid in supplement to the complete
document Eurocode 3 - Part 1.1 in order to facilitate the use of Eurocode 3 for the design
of such steel structures which are usual in common practice : braced or non-sway steel
structures.
(2) Therefore, the "Design handbook according to Eurocode 3 for braced or non-sway steel
buildings" presents the main design formulas and rules extracted from Eurocode 3 - Part
1.1, which are needed to deal with :
- elastic global analysis of buildings and similar structures in steel,
- checks of structural members and connections at limit states,
- in case of braced or non-sway structures,
- according to the european standard Eurocode 3 - Part 1.1 (ENV 1993-1-1).
0-a.3
Warning
(1) Although the present design handbook has been carefully established and intends to be
self-sufficient it does not substitute in any case for the complete document Eurocode 3 Part 1.1, which should be consulted in conjunction with the NAD, in case of doubt or need
for clarification.
(2) All references to Eurocode 3 - Part 1.1 are made in [...].
(3) Any other text, tables or figures not quoted from Eurocode 3 are considered to satisfy the
rules specified in Eurocode 3 - Part 1.1.
41
O.a.4
How to read this design handbook
(1) Example of numbering of chapters and paragraphs : VIE . a . 1 . 2
(2) Layout of pages :
EC 3 for non-sway buildings - VI Members in tension
| Ref.
f
\
left column
short title
for references of the handbook
k
References
t
Page 68
t
concerned chapter
number of the page
Main text with a following example about layout of chapters:
(...)
Π
STRUCTURAL CONCEPT OF THE BUILDING
(...)
ILh
Material properties
(...)
n.h.3
(...)
Connecting devices
II.h.3.2
Welding consumables
(...)
(3) In the left column of each page (Ref.): references to Eurocode 3 are always included
between brackets [...]; the other references are specified without brackets; the word
"form." means "formula"
(4) References to Eurocode 3 are also given in the text between brackets [...]
O.a. 5
Acknowledgements
(1) Particular thanks for fruitful collaboration are addressed to:
. 15 engineering offices : Adem (Belgium), Bureau Delta (Belgium), Varendonck
Groep/Steeltrak (Belgium), VM Associate Partner (Belgium), Rambøll,
Hannemann & Højlund (Denmark), Bureau Veritas (France), Socotec (France),
Sofresid (France), CPU Ingenieurbüro (Germany), IGB-Ingenieurgrappe Bauen
(Germany), Danieli Ingegneria (Italy), Schroeder & Associés (Luxemburg),
D3BN (the Netherlands), Ove Arup & Partners (United Kingdom), ECCS / TC 11
(Germany),
. RWTH : Steel Construction Department from Aachen University with Professor
SEDLACEK G. and GROTMANN D.,
. SIDERCAD (Italy) with MM. BANDINIM. and CATTANEO F.,
. CnCM (France) with MM. CHABROLIN B., GALEA Y. and BUREAU A.
(2) Grateful thanks are also expressed to :
. the ECSC which supported this work in the scope of the european research
n° P2724(contract n° 7210 - SA/513),
. the F6 executive committee which has followed and advised the working group
of the research,
. anyone who has contributed to the work: MM. CONAN Yves, MAUER Thierry,
GERARDY LC.
42
Ή
O.b
References
- in the left column of each page (Ref.): references to Eurocode 3 are always included
between brackets [...]; the other references are specified without brackets.
- references to Eurocode 3 are also given in the text between brackets [...]
- the reference "i" given in this chapter is designated in the text by IM.
/ l / Eurocode 1, draft version, Basis of Design and Actions on Structures (Parts 1, 2.2,2.4,
2.5,2.7, 10)
C
(E 1)
HI Eurocode 3, ENV 1993-1-1, Design of steel structures Part 1.1 General rules and rules
for Buildings (EC 3)
131 Eurocode 8, draft version, Design of structures for earthquake resistance (EC8)
141 EC C S technical publication n°65, Essentials of Eurocode 3 Design Manual for Steel
Structures in Building, 1991, First Edition
151 Practical exercises showing applications of design formulas of Eurocode 3 :
ECCS technical publication n°71, Examples to Eurocode 3,1993, First Edition
161 "Design handbook for sway buildings", from Sidercad (Italy)
ΠI Software for the check of main formulas in Eurocode 3:"EC 3 tools"
(available for PC computer, Windows 3.1), from CT1CM (France)
/8/ Eurocode 3 Background Document 5.03 : "Evaluation of test results on columns, beams
and beam-columns with cross-sectional classes 1 - 3 in order to obtain strength functions
and suitable model factors", April 1989.
191 Paper "Application de l'Eurocode 3 : classement des sections transversales en I", by
Bureau A. and Galea Y., (CTICM), Construction métallique, n° 1-1991.
43
[1.6]
O.c
Symbols and notations
O.e.!
Symbols
(1) See Appendix A for a list of symbols used in this design handbook. Those symbols
are conform to Eurocode 3.
Q.C.2
[1.6.7]
Convention for member axes
(1) For steel members, the conventions used for cross-section axes are:
xx along the member
. generally:
yy cross-section axis parallel to the flanges
zz cross-section axis perpendicular to the flanges or parallel to the web
. for angle sections:
yy
axis parallel to the smaller leg
zz
axis perpendicular to the smaller leg
. where necessary:
uu
major axis (where this does not coincide with the yy axis)
vv
minor axis (where this does not coincide with the zz axis)
(2) The convention used for subscripts which indicate axes for moments is:
"Use the axis about which the moment acts."
(3) For example, for an I-section a moment acting in the plane of the web is denoted M y
because it acts about the cross-section axis parallel to the flanges.
0.C.3
Dimensions and axes of rolled steel sections
(1) "asymmetrical" (I and D ) and "monosymmetrical" ( [, Τ and L) rolled steel sections
are shown in table 0.1.
44
0-C.4
Notations in flow-charts
(1) AU the flow­charts appearing in the present design handbook should be read according
to the following rules :
­ reading from the top to the bottom, in general
­ the references to Eurocode 3 are given in [...]
­ "n.f" means that the checks are not fulfilled and that stronger sections or joints have to
be selected.
­ convention for flow­charts:
(FC χ)
Flow­chart number (x)
Title
,___L__,
[ Assumption j
—ï—
Action: determination, calculation,
I
<
^ Z 7 o ^ o , y ^ ì
otherflow­chax,number(y)
yes 1
ι
τ
» the dotted line (
) means that path
has to be followed through the box
(^
Results
J
45
Table 0.1
Dimensions and axes of rolled steel sections
"?'
ΓΪ7
y —
1 =£tf
+
*·
I
ζ
ttf
y
Htw
46
■7e
— y
[14.2 (i)]
O.d
Definitions and units
iLdJ
Definition of special terms
(1) The following terms are used in Part 1.1 of Eurocode 3 with the following meanings:
Frame: Portion of a structure, comprising an assembly of directly connected
structural elements, designed to act together to resist load. This term refers to both
rigid-jointed frames and triangulated frames. It covers both plane frames and threedimensional frames.
Sub-frame: A frame which forms part of a larger frame, but is treated as an isolated
frame in a structural analysis.
Type of framing: Terms used to distinguish between frames which are either:
Semi-continuous, in which the structural properties of the connections need
explicit consideration in the global analysis.
Continuous, in which only the structural properties of the members need explicit
consideration in the global analysis.
Simple, in which the joints are not required to resist moments.
Global analysis: The determination of a consistent set of internal forces and moments
(N, V, M) in a structure, which are in equilibrium with a particular set of actions on
the structure.
First order global analysis: Global analysis using the initial geometry of the structure
and neglecting the deformation of the structure which influences the effects of
actions (no Ρ-Δ effects).
Second order global analysis: Global analysis taking into account the deformation of
the structure which influences the effects of actions (Ρ-Δ effects).
Elastic global analysis: First-order or second-order global analysis based on the
assumption that the stress-strain behaviour of the material is linear, whatever the
stress level; this assumption may be maintained even where the resistance of a crosssection is based on its plastic resistance (see chapter V about classification of crosssections).
System length: Distance between two adjacent points at which a member is braced
against lateral displacement in a given plane, or between one such point and the end
of the member.
Buckling length: System length of an otherwise similar member with pinned ends,
which has the same buckling resistance as a given member.
Designer: Appropriately qualified and experienced person responsible for the
structural design.
Q&2
[1.5 (2)]
ilniiS
(1) For calculations the following units are recommended in accordance with ISO 1000:
Forces and loads
Unit mass
Unit weight
Stresses and strengths
Moments (bending....)
kN, kN/ m , kN/ m 2
:
kg/m3
:
kN/ m 3
:
N/mm2 (=MN/ m 2 or MPa)
kNm.
47
I INTRODUCTION
La
Basis of design
(1) The table 1.1 summarizes this chapter La providing the practical principles of design
requirements. Details and explanations are given in the following sub-chapters I.a.l to
I.a.3.
I.a.l
Fundamental requirements
[2.1 (l)]
(1) A structure shall be designed and constructed in such a way that:
. with acceptable probability, it will remain fit for the use for which it is required,
having due to regard to its intended live and its cost, and
. with appropriate degrees of reliability, it will sustain all actions and influences
likely to occur during execution (i.e. the construction period) and use (i.e. the
service period) and have adequate durability in relation to maintenance costs.
[2.1 (2)]
(2) A structure shall also be designed in such a way that it will not be damaged by events like
explosions, impact or consequences of human errors, to an extent disproportionate to the
original cause.
[2.1 (4)]
(3) The above requirements shall be met by the choice of suitable materials, by appropriate
design and detailing and by specifying control procedures for production, construction and
use as relevant for the particular project.
I.a.2
Definitions
I.a.2.1
Limit states
(1) Eurocode 3 is a limit state design code in which principles and rules are given for the
verification of:
. Serviceability Limit States (SLS) and,
. Ultimate Limit States (ULS).
[2.2.1.1 (l)] (2) The limit states are states beyond which the structure no longer satisfies the design
performance requirements.
(3) These limit states are referred to physical phenomena as for instance:
[2.2.1.1 (6)]
a) for SLS, problems which may limit the serviceability because of:
. deformations or deflections which adversely affect the appearance of effective use
of the structure (including the proper functioning of machines or services) or cause
damage to finishes or non-structural elements,
. vibration which causes discomfort to people, damage to the building or its
contents, or which limits its functional effectiveness.
[2.2.1.1 (4)]
b) for ULS, problems which may endanger the safety of people and thus be regarded as
ultimate limit because of:
. loss of equilibrium of structure or any part of it, considered as a rigid body,
. failure by excessive deformation, rupture, or loss of stability of the structure or any
part of it, including supports and foundations.
48
ΙΛ.22
Actions
(1) Details about actions are provided in Eurocode 1
[2.2.2.1 (i)] (2) An action (F) is:
. a force (load) applied to the structure (direct action), or
. an imposed deformation (indirect action); for example, temperatures effects or
differential settlement.
[2.2.2.1 (2)] (3) Actions (F) are classified as:
. permanent actions (G), e.g. self-weight of structures, fittings, ancillaries and fixed
equipment
. variable actions (Q), e.g. imposed loads (q), wind loads (w) or snow loads (s)
. accidental actions (A), e.g. explosions or impact from vehicles.
[2.2.2.2 (l)] (4) Characteristic values F^ of actions are specified:
. in Eurocode 1 or other relevant loading codes, or
. by client, or the designer in consultation with the client, provided that the
minimum provisions specified in the relevant loading codes or by the competent
authority are observed.
[2.2.2.4(1)] (5) The design (factored) values Fd of an action (for instance Gd, Q<u Wd, Sd) is expressed in
general terms as:
[form. (2.1)]
F d =YpFk
where F k is the characteristic (unfactored) value of action.
YF is the partial safety factor for the action considered - taking into
account of, for example, the possibility of unfavourable deviations of
the actions, the possibility of inaccurate modelling of the actions,
uncertainties in the assessment of effects of actions and uncertainties in
the assessment of the limit state considered (the values of γρ are given
in chapter ΠΙ: YG (permanent actions), YQ (variable actions),...).
[2.2.2.5]
[form. (2.2)]
[2.2.3.1 (3)]
[2.2.3.2 (2)]
[form. (2.3)]
(6) The combinations of actions respectively for ULS and for SLS are given in chapter ΙΠ.
(7) Design values of the effects of actions:
The effects of actions (E) are responses (for example, internal forces and moments
(Nsd. Vsd» Msd), stresses, strains, deflections, rotations) of the structure to the
actions. Design values of the effects of actions (Ed) are determined from the design
values of the actions, geometrical data (ad) and material properties when relevant:
Ed = E(Fd>ad,...)
I.a.2 3 Material properties
(1) characteristic values of material properties:
Material properties for steel structures are generally represented by nominal values
used as characteristic values (unfactored) (Xk)·
(2) design values of material properties:
For steel structures, the design (factored) resistance Rd (for example, design
resistance for tension (NRd), buckling (NRd), shear (VRd) , bending (MRd)) is
generally determined directly from the characteristic (unfactored) values of the
material properties (Xk) and geometrical data (a^):
R d =R(X k a k ,. ) / γ Μ
where YM is the partial safety factor for the resistance(the different YM factors are
explicitly introduced in the design formulas and their values are given in
table 1.2)
49
I.a.3
[2.3.1 (l)]
[2.3.1 (2)]
[2.3.1 (3)]
[2.3.1 (4)]
Design requirements
r.a.3.1 General
(1) It shall be verified that no relevant limit state is exceeded
(2) All relevant design situations and load cases shall be considered.
(3) Possible deviations from the assumed directions or positions of actions shall be considered.
(4) Calculations shall be performed using appropriate design models (supplemented, if
necessary, by tests) involving all relevant variables. The models shall be sufficiently
precise to predict the structural behaviour, commensurate with the standard of
workmanship likely to be achieved, and with the reliability of the information on which
the design is based.
I.a.3 2 Serviceability Limit States
[2.3.4 (l )] (1 ) It shall be verified that:
[form. (2.13)]
Ed^Cd or E d < R d
where
Ed
is the design effect of actions, determined on the basis of one
of the combinations defined below,
Cd
is a nominal value or a function of certain properties of
materials related to the design effect of actions considered.
(2) Practical checks of SLS (see chapter I.b.3) in floors and frames for instance:
( g Vd> S Hd) ^ (5vma*> S Hmax)
f d ^ f min
is the design vertical deflection of floors
(recommended limits oVmax = L/250» —)
is the design horizontal deflection of frames
ÖHd
(recommended limits δππ,^ = h/300» —)
is the design natural frequency of floors
fd
(recommended limits fmin = 3 Hz,...)
I,a,3,3 Ultimate Limit States
[2.3.2.1 (2)] (1) When considering a limit state of rupture or excessive deformation of a section, member
or connection (fatigue excluded) it shall be verified that:
where
[form. (2.7)]
övd
Sd^Rd
Sd
is the design value of an internal force or moment (or of a
respective vector of several internal forces or moments)
Rd
is the corresponding design resistance,
associating all structural properties with the respective design values.
(2) Practical checks of ULS (see chapter I.b.4) in members for instance:
where
where
(N S d ,V S d ,M S d )<(N R d ,V R d ,M R d )
condition concerning separate internal forces or moments or, interaction
between them ((V,M), (Ν, M),...)
(Nsd. Vsd, Msd) are design internal forces and moments applied to
the members,
(NRd, VRd, MRd) are design resistance of the members.
50
Table LI
Summary of design requirements
11 frame submitted to SLS and ULS combinations of design actions Fd (G d , Qd, w d , s d ,...):
[form. (2.1)]
|Fd=Y F Fk
where
F^
YF
is the characteristic value of actions
is the partial safety factor for the considered action
(see chapter ΠΓ)
2) after global analysis of the frame:
­ design effects of actions (e.g. deflections, frequencies) (for SLS):
Ed H 5 d , fd) )
­ design values of internal forces and moments (for ULS):
S d (=(Nsd, Vsd, M S d ) )
3) verification conditions at limit states:
­ for SLS checks (see chapter I.b.4):
[2.3.4(1)]
. in general:
where
EH<C
Cd
is the nominal value related to the design effect of
considered actions (design capacity).
(ô V d>5 Hd ) < (ôvmax'a H m a x
for instance:
f d ^ f min
where
is the design vertical deflection of floors
OVd
is the design horizontal deflection of frames
ÖHd
is the design natural frequency of floors
fd
°Vmax .8Hmax»fmin are recommended limits (for instance:
L/250, h/300, 3
Hz
>
for ULS checks (see chapter I.b.5):
[2.3.2.1 (2)]
. in general:
where
Sd < R
Rd
is the design resistance (=(NRd, V R d , M R d )):
Rd = Rk/YM
[form. (2.3)]
where
for instance:
Rk
YM
is the characteristic value of the used material
is the partial safety factor for the resistance
(see table 1.2)
(NSd,VSd>Msd)<;(NRd,VRd,MRd)
condition concerning separate internal forces or
moments or, interaction between them ((V,M), (N,M),..)
51
Table 1.2
Partial safety factor YM for the resistance
The design values of
- the resistances of cross-sections
- the buckling resistance of members
- and, the resistances of connections
shall be determined with the following partial safety factors YM:
- at Ultimate Limit States;
=
M
=
W
- resistance of class 1,2 or 3 cross-section*) :
(plasticity or yielding)
ΎΜΟ
- resistance of class 4 cross-section*)
(local buckling)
:
YMI
- resistance of members to buckling
(global and local buckling)
:
ΎΜΙ = Μ
- resistance of net section at bolt holes
:
YM2 = 1 . 2 5
- resistance of bolted connections
:
ÏMb=1.25
- resistance of welded connections
:
YMW = 1'25
:
ÏMs.ser ~~*>*
- at Serviceability Limit States:
- slip resistance of preloaded bolts
Note 1 :
The different JM factors are explicitly introduced in 1he design formulas.
Note 2:
The yui factors are provided according to the official version of Eurocode 3.
Those "boxed" values are only indicative. The value s of YM to be used in
practice are fixed by the national authorities in each country and published in
the relevant National Application Document (NAD)
*) The classification of cross-sections is defined in chapter V
52
Lb
General flow-charts about elastic global analysis
(1) Chapter Lb. 1 presents flow-chart FC 1 about elastic global analysis of steel frames (in
general) according to Eurocode 3.
(2) Chapter IV.a.2 presents flow-chart FC 4 about elastic global analysis of braced or nonsway steel frames according to Eurocode 3.
(3) Chapter XILa. 1 presents flow-chart FC 12 about elastic global analysis of bracing system
according to Eurocode 3.
Lb. 1
Flow-chart FC 1 : Elastic global analysis of steel frames according to Eurocode 3
(1) The flow-chart FC 1 aims to provide a general presentation of elastic global analysis of
steel frames according to Eurocode 3.
(2) The present design handbook only deals with the path φ of FC 1 elastic global analysis
of braced or non-sway frames (presented in FC 4 in chapter IV). All the details are given
in chapters Π to XI of the handbook.
(3) The elastic global analysis of sway frames is out of the scope of the present design
handbook; the assumptions of the elastic global analysis of sway frames are briefly
presented - just for information - in the paths (D to (D of FC 1.
(4) The flow-chart FC 1 refers to flow-chart FC 12 about elastic global analysis of bracing
system according to Eurocode 3. The flow-chart FC 12 and all the details about bracing
system design are given in chapter ΧΠ.
(5) The flow-chart FC 1 is divided in 3 parts:
Lb. 1.1 general part (1 page)
Lb. 1.2 details (1 page)
Lb. 1.3 comments (6 pages)
Ib.1.1
Flow-chart FC 1: cenerai
see the following page
LJLLZ
Flow-chart FC 1: details
see the second following page
53
Flow-chart ΓFC 1) : Elastic global analysis of steel frames according to Eurocode 3 (General)
row:
Actions
Predesign
SLS checks
Choice of the type of global analysis
for ULS
10
ULS global analysis of the frame
to determine the internal forces and moments (N, V, M)
13
14
ULS checks of members
16
submitted to internal forces and moments (N, V, M)
II
19
ULS checks of local effects
ULS checks of connections
54
20
Flow­chart
( FC l) : Elastic global analysis oí Steel frames according to Eurocode 3
(Details)
row:
Determination of load arrangements (EC1 and EC 8)
1
Load cases
for SLS [2.3.4.]
Load cases
for ULS [2.3.3.]
C~
^
Predesign of members^beams & columns => Sections^
with pinned and/or rigid connections
ι
Frame with bracing system /
~^l
JT
not fulfilled
notfulfilled
SLS checks
[Chap. 4]
ULS checks
[Chap. 5]
Design of the
bracing system
ι
Frame without bracing system
Classification
of the frame
, Braced framed
yes
S.
\no
Global imperfections
of the frame
[5.2.4.3.]
6 b £ 0 , 2 5u
[5.2.5.3. (2)]
1
Non-swayframeyes /Non­sway frame [5.2.5.2.Λ
Vsd £ 0 , 1
£.
«δ,
ε:
Sway frame
m
1
λ > 0,5 [A.fy / NSd] 0 · 5 V T
no
[5.2.4.2.(4)]
v
yes / 0,1 < ^ . < 0,25 \nov
Ver
[5.2.6.2. (4)]
FIRST
'ORDER ANALYSIS
±
Non­sway mode buckling
length approach
Sway mode buckling
length approach
[5.2.6.2(1) a)][5.2.6.2. (7)]:
[5.2.6.2(1) b)][5.2.6.2. (8)1:
with sway moments
amplified by factor
l/(l-VSd/Vcr) [5.2.6.2.(3)]
Ό
ι SECOND
©-
Mjembers imperfectiojis
l eo,d
with sway moments
amplified by factor 1,2 in
beams & connections
©
i
eo,d where
necessary
152.62.(2)]
[5.2.4.5.(3)]
--0
I Sway mode L b )
ÍNon­sway mode
ï
' Classification of cross­section [Chap. 5.3] '
1
±
±
Checks of the in­plane stability: members buckling [Chap. 5 J ]
φ-:
yes
<5
16
not fulfilled (n.f.)
I
Checks of local effects (buckling and resistance of webs) [Chap. 5.6 and 5.7]
55
[5.2.4.5.(2)]
«a,
Checks of resistance of cross­sections [Chap. 5.4]
Checks of connections [Chap. 6 and Annex J]
ï
eo,d in all
members
Lb
Checks of the out­of­plane stability: members and/or frame buckling [Chap. 5.5]
f
[5.2.4.5.]
—fá—/members \_
\ with eo.d /
Non­sway mode iL b
r-
ORDER ANALYSIS'
nf.
yi
yi
->n£
I.b.l .3
Comments on flow-chart FC 1
comments (1/6) on flow-chart FC 1:
* Generalities about Eurocode 3:
- AU checks of (ULS) Ultimate Limit States and all checks of (SLS) Serviceability Limit
States are necessary to be fulfilled.
- According to the classification of cross-sections at ULS (row 14; chapter V of the design
handbook) Eurocode 3 allows to perform:
[5.2.1.2(1)]
. plastic global analysis of a structure only composed of class 1 cross-sections when
required rotations are not calculated [5.3.3 (4)] or,
. elastic global analysis of a structure composed of class 1. 2. 3 or 4 cross-sections
assuming for ULS checks, either a plastic resistance of cross-sections (class 1 and 2)
or, an elastic resistance of the cross-sections, without local buckling (class 3) or, with
local buckling (class 4 with effective cross-section).
- In order to determine the internal forces and moments (N. V. M) in a structure Eurocode 3
allows the use of different types of elastic global analysis either:
a) first order analysis using the initial geometry of the structure or,
[5.2.1.2 (2)]
b) second order analysis taking into account the influence of the deformation of the
structure
- First order analysis (row 11) may be used for the elastic global analysis in the following
cases (types of frames):
The first order elastic global analysis of the frame should
take into account
actions
types of
frames
1) braced frames &)
the vertical
loads
the horizontal
loads
[5.2.5.3 (5)]
the member
imperfections
(row 12)
X(b)
(path®)
[5.2.5.3 (3)]
the global
imperfections of
the frame
(row 7)
2) non-sway frames
(path φ )
X
X
3) sway frames (c)
(paths © and © )
X
X
Notes : (a) braced frames are frames which may be treated as fully supported laterally by a
bracing system.
(b) only the part of horizontal loads which are applied to the frame but not assumed
to be transmitteii to the bracing system through the floors.
(c) use of design methods which make indirect allowance for second-order effects.
56
comments (2/6) onflow-chartFC 1:
[5.2.1.2(3)]
- Second order analysis (row 11) may be used in all cases (types of frames) :
^v.
actions
types of ^«x.
frames
^ \
1) for sway frames
The second order elastic global analysis of the frame should
take into account
the member
the horizontal
the global
the vertical
loads
imperfections of the imperfections
loads
frame
(row 7)
(row 12)
(path ® )
(path <§»
2) for frames
in general
(path ©)(*>)
[5.24.5(3)]
X
X
X
X
X
X
X(a)
X
X
X
X
Notes : fa) members imperfections are introduced where necessarv.
(b) the more complex possibility of second order global analysis of the frame
(path © ) could be conservative because it allows the bypass of the
"sway or non-sway frame" classification and consequently :
- either the first order analysis might be sufficient,
- or, the introduction of member imperfections would not be necessary in all
members.
On the other hand, particular care shall be brought to the introduction of
member imperfections ( eo,d) which would be imposed for the global analysis
in the realistic directions corresponding to the deformations of the members
for the failure mode of the frame; that failure mode of the frame is related to
the combination of applied external loads; otherwise, with more favourable
direction of member imperfections, the second order global analysis might
overestimate the bearing capacity of the frame.
- in the flow-chart FC 1 from path φ to path (D (from left to right) the proposed
methods for global analysis become more and more sophisticated.
* row 1:
EC 1: Draft
EC 3: ENV 1993-1-1
EC 8: Draft
Eurocode 1 Basis of design and actions on structures
Eurocode 3 Design of steel structures, Part 1.1:
general rules and rules for buildings.
Eurocode 8 Design of structures for earthquake resistance
57
comments (3/6) on flow-chart FC 1:
[Chap. 5]
[Chap. 4]
* rows 2.4:
-ULS
-SLS
means Ultimate Limit States
means Serviceability Limit States
* row 3:
This flow­chart concerns structures using pinned and/or rigid joints.
In the case of semi-rigid joints whose behaviour is between pinned and rigid joints,
the designer shall take into account the moment­rotation characteristics of the joints
(moment resistance, rotational stiffness and rotation capacity) at each step of the
design (predesign, global analysis, SLS and ULS checks). The semi­rigid joints
should be designed according to chapter 6.9 and the Annex J of Eurocode 3.
[4.2.1 (5)]
[5.2.5.3 (2)]
row 4:
For SLS checks, the deflections should be calculated making due allowance for any
second order effects, the rotational stiffness of any semi­rigid joints and the possible
occurrence of any plastic deformations.
* row 6:
braced frame
unbraced frame
5 b <0,2ô u
The frame is braced if:
where δι>:
horizontal displacement of the frame with the bracing system
ôu:
horizontal displacement of the unbraced frame,
according to first order elastic global analysis of the frame either with hypothetic
horizontal loads or with each ULS load case.
Note : in the case of simple frames with all beam­column nodes nominally pinned, the
frame without bracing would be hypostatic, hence c\j is infinite and thus the
condition 6b ^ 0,2 δ\ι is always fulfilled.
[5.2.4.3]
* row 7:
global imperfections of the frame
initial sway imperfections of the frame
F2
equivalent horizontal forces
F2
JMii
Σ
could be applied
in the form of
φ Fi
»
φ (Fi + F2)
2
58
φ (Fi + F2)
2
comments (4/6) on flow-chart FC 1:
* row 8:
[5.2.5.2]
classification of swav or non­swav frame:
A frame may be classified as non-sway if according to first order elastic global analysis
of the frame for each ULS load case, one of the following criteria (see row 9) is satisfied:
either, ai in general
[5.2.5.2 (3)]
Sd _
cr
where Vsd:
VCT:
α cr
, condition which is equivalent to
aCT > 10
design value of the total vertical load
elastic critical value of the total vertical load for failure in a sway mode
( = π2ΕΙ / L2 with L, buckling length for a column in a sway mode; V cr of
a column does not correspond necessarily to V cr of the frame including
that column)
coefficient of critical amplification or coefficient of remoteness of critical
state of the frame
aCT :
or,
< 0,1
b) in case of building structures with beams connecting each columns at each storey level:
[5.2.5.2(4)]
δ.χν_δ.(ν 1 -Γ·ν 2 )
h.^H
where H, V:
δ:
h:
H, V, δ
h.(H 1 + H 2 )
< 0,1
total horizontal and vertical reactions at the bottom of the storey.
relative horizontal displacement of top and bottom of the storey,
height of the storey.
are deduced from a first order analysis of the frame submitted to both
horizontal and vertical design loads and to the global imperfections of the
frame applied in the form of equivalent horizontal forces
(see comments on row 7).
Notes:
­ A same frame could be classified as sway according to a load case (V$dl for
instance) and as non­sway according to another load case ( Vsd2 for instance).
For multi­storeys buildings the relevant condition is
condition which is equivalent to
' ν * Λ or åen are related to the storey i.
where
v
V cny
59
Sd
cr
= maximum
'2*i
a w = minimum (Ocri)»
comments (5/6) on flow-chart FC 1:
[5.2.4.2 (4)]
* row 9:
Members imperfections may be neglected except in sway frames in the cases
of members which are subject to axial compression and which have moment
resisting connections, if :
λ>0,5
"Afyl
.condition which is equivalent to
.NSd.
where λ :
fv,:
A:
Nsd·'
Ncr:
N
Ν
« >
4"
π
or equivalent to ε > —
2
non-dimensional slenderness ratio calculated with a buckling length equal
to the system length
yield strength
area of the cross-section
design value of the compressive force
elastic critical axial force ( = π2ΕΙ/ L2, with L = system length)
factor (= Li I——, with L = system length)
EI
ε:
[5.2.6.2(4)]
0,5
* row 10: According to the definition of occr introduced in comment on row 8
0 , l < - ^ - < 0,25 .condition which is equivalent to
4 < a c r < 10
* row 11:
The actions to be considered in first order elastic global analysis and in second order
elastic global analysis are listed in the "generalities about Eurocode 3" (see the first
comments on flow-chart 1) in function of the type of frame.
[5.2.4.5]
* rows 12.13.14 :
- path @ : Sway moments amplified by factor 1,2 in beams and beam-to-column
connections and not in the columns. The definition of "sway moments"
is provided in [5.2.6.2 (5)].
- paths (5) and (6) :
the introduction of member imperfections eo,d should be
considered equivalent to the introduction of distributed loads
along the members :
eo,d
Nsd
Nsd
equivalent to
q
Nsd
i
,;
i
,
Nsd
'
,
wimiq
L
= 8.N Sd .e 0 , d / L 2
|Q = 4.N S d .e 0 < d /L
Q
0
Note : the equivalence of eo,d and (q, Q) loading is proposed here for a practical point of
view but it is not included in Eurocode 3.
60
comments (6/6) on flow­chart FC 1:
* row 13:
Vsd
For the meaning of the ratio ——, refer to comment on row 8.
"cr
* row 15:
[Annex E]
L¡,, buckling length of members for sway or non­sway mode
***
Nsd
^ .
»O
Nsd
CH
Lb
* row 16:
The classification of cross-sections have to be determined before all the
ULS checks of members, cross­sections and webs (rows 17 to 20).
*rows 17,18,19.20,21;
The sequence of the Ultimate Limit States checks is not imposed and it is up
to the designer to choose the order of the ULS checks which are anyhow all
necessary to be fulfilled. On the contrary, the sequence of steps to select the
type of analysis is well fixed and defined in rows 5 to 10.
* row 19: When the member imperfections eo,d are used in a second order analysis
(paths (D and © ) , the resistance of the cross-sections shall be verified as
specified in chapter [5.4] but using the partial safety factor γηΙ in place of v mo
[5J. 1.3 (6)]
Lc
Content of the design handbook
LSLl
Scope of the handbook
(1) Actions (loadarrangements) on buildings to be taken into account in the design are
presented as described in Eurocode 1 111,
(2) The load cases for SLS and for ULS to be considered in the design are defined as
prescribed in Eurocode 3 Part 1.1 /2/,
(3) The elastic global analysis of steel structures in braced or non-sway buildings according
to Eurocode 3 Part 1.1 HI is assumed to be carried out :
a) by elastic global analysis of the structure to determine:
. the vertical deflections of beams, the horizontal displacements of frames
and vibrations of floors and,
. the internal forces and moments (N, V, M) in the members and,
b) by check of requirements for the Serviceability Limit States and,
61
c) by check of requirements for the Ultimate Limit States :
c.l) by check of the resistance of cross-sections and,
C.2) by check of the buckling resistance of members and,
C.3) by check of local effects (buckling and resistance of webs) and,
C.4) by check of joints and connections,
for all members characterised by a class of cross-sections at ULS:
. classes 1 and 2, which assume a full plastic distribution of stresses
over the cross- section at the level of yield strength or,
. class 3, which is based on an elastic distribution of stresses across
the cross-section with the yield strength reached at the extreme
fibres or,
. class 4, which makes explicit allowances for the effects of local
buckling appearing in the cross-section.
(4) The elastic global analysis of steel bracing system according to Eurocode 3 Part 1.1/2/ is
assumed to be carried out with the same hypothesis than for steel structures but with
specific actions:
loads and effects of global imperfections:
. from the bracing system itself and,
. from all the frames which it braces.
(5) This design handbook deals with the analysis of braced or non-sway steel structures
subject to static loading. Eurocode 3 (121) and Eurocode 8 (131) should be consulted for the
following problems which are not considered here: fatigue, resistance to fire, dynamic
analysis or seismic analysis.
[9.1.4 (i)]
(6) No fatigue assessment is normally required for building structures except in the following
cases:
a) members supporting lifting appliances or rolling loads,
b) members subject to repeated stress cycles from vibrating machinery,
c) members subject to wind-induced oscillations,
d) members subject to crowd-induced oscillations.
For those fatigue problems the chapter 9 of Eurocode 3 Part 1.1 (¡If) should be consulted.
I.C.2
Definition of the braced frames and non-sway frames
[5.2.5.1 (l)] (1) All structures shall have sufficient stiffness to resist to the horizontal forces and to limit
lateral sway. This may be supplied by:
a) the sway stiffness of the bracing systems, which may be:
. triangulated frames
. rigid-jointed frames
. shear walls, cores and the like
b) the sway stiffness of the frames, which may be supplied by one or more of the
following:
. triangulation
. stiffness of the connections
. cantilever columns
62
[Annex J]
Semi-rigid connections may be used, provided that they can be demonstrated to provide
sufficient reliable rotational stiffness (see [6.9.4]) to satisfy the requirements for sway-mode
frame stability (see [5.2.6]).
(2) Framing for resistance to the horizontal loads and to sway. Two examples are given in
table 1.3:
[5.2.5.3 (i)]
a) typical example of a frame with "bracing system", which could be sufficiently
stiff:
. for the frame to be classified as a "bracedframe"
. and, to assume that all in-plane horizontal loads are resisted by the bracing
system.
[5.2.5.3 (2)]
[5.2.5.2 (l)]
The criterion of classification as braced or unbraced frames is explained in
chapter IV.g. 1.1.
b) example of an unbraced frame which could have sufficiently stiff momentresisting joints between the beams and the columns:
. for the frame to be classified as a "non-sway frame"
. and, to neglect any additional internal forces or moments arising from
in-plane horizontal displacements of the nodes of the frame.
The criteria of classification as sway or non-sway frames are detailed in
chapter IV.g. 1.2.
[5.2.5.2 (3). (4)]
[Annex H]
Definition of framing for horizontal loads
Table L3
1) With bracing system :
Γ
μ
'
AL
w, ir
m
wWT
=
Γ Γ r
y
ν
r
ν
r
Γ ν ι
fl
il
Η
mw
mw
iiflv
BRACED FRAME
2) Non-sway frames :
*
"
Àf\
FRAME WITH BRACING
Μ
i'
AL
w
mm
i'
n
63
wftrr
+
BRACING
SYSTEM
T.c.3
Summarv of the table of contents
- chapter I : . Limit States (SLS, ULS), design requirements;
. flow-chart about elastic global analysis of steel frames according to EC 3.
. scope, definitions;
. tables of SLS and ULS checks;
- chapter Π : complete set of data of the structure
- chapter III : determination of load arrangements and load cases for
. Ultimate Limit States and,
. Serviceability Limit States
- chapter IV : . frame design and,
. SLS checks for frames (see chapter I.c.4).
. ULS classifications of frames
. braced frame condition and,
. non-sway frame condition
- chapter V : classification of cross-sections at Ultimate Limit States
- chapter VI to LX :
. SLS checks for beams (see chapter I.c.4).
. ULS checks of members (beams and columns,...) submitted to internal
forces and moments (N, V, M) considering the resistance of crosssections, the overall buckling of members (buckling, lateral-torsional
buckling) and local effects (shear buckling of webs (V)): see chapter I.c.5
- chapter X : . ULS checks of local effects: resistance of webs to transverse forces F
(yield criterion, crushing, crippling, local buckling, flange induced
buckling): see chapter I.c.5
- chapter XI : ULS and SLS checks of connections.
- chapter ΧΠ: design of steel bracing system
I.c.4
Checks at Serviceability Limit States
(1) The table 1.4 presents the different checks which shall be fulfilled by beams and frames
at Serviceability Limit States with references to the design handbook:
|| Table 1.4
Checks at Serviceability Limit States
Type of checks Vertical deflections
of beams
Beams
Frames
Chapter Vm.b.l
Chapter Vin.b.l
Horizontal
deflections of frames
Vibration of floors
_
Chapter VlII.b.2
Chapter VIII.b.2
Chapter IV.f.l
64
Lía
Checks of members at Ultimate Limit States
(1) The following tables define the different checks which shall be fulfilled at Ultimate Limit
States:
- by all the members of frames submitted to internal forces and moments (N, V,M),
­ by all webs of cross­sections submitted to transverse forces F.
Table 1.5
Member submitted to internal forces, moments and transverse forces
F F v
λ/f*)
m
torsion
Ncompression
C
_r
- ï » 3£„
XX
._
y¡ Π
^
fl
x?* . lvlbendin^\ V
intension
"
Λ
*U*
Al>
fi..
^.Ncompression Μ , * ^ η 1
±fi «w-r™
U 0 M bending x f
-^tension
IF
Note:
[5.4]
[5.4]
[5.4]
[5.4]
[5.7]
[5.7]
[5.3]
[5.5]
[5.5]
[5.6]
[5.7]
[5.7]
*) the effects of torsion are not considered in the handbook because the
Annex G of Eurocode 3 is not officially available yet.
taMe 1,6
Definition of the planes of cross-sections within internal forces, moments
(Nsd, Vsd, Msd) and transverses forces Fsd are acting.
table 1.7:
For different types of loading on the members and on the Webs (tension,
compression, bending, combined (N,M), transverse forces) the table 1.7
provides the internal forces, moments (N (Ntension» Ncompression).
V (Vy,Vz), M(My,Mz)), transverse forces (F) and interactions
between them ((V,M),(N,M),(N,V),(N,V,M),...)
to be checked at Ultimate Limit States.
table 1.8:
List of references to the design handbook related to all the check formulas
at Ultimate Limit States, for different types of loading.
The different types of loading on the members and on the webs includes
internal forces, moments, transverse forces and interactions between them
(see also the more detailed table 1.7).
Two types of ULS checks are defined (resistance of cross­sections and
stability of members or webs) and refer to the following physical
phenomena:
. (R) resistance of cross-sections:
. tension
. compression
.shear
. bending
. resistance on webs to transverse forces
. crushing of webs to transverse forces
. (5) stability of members or webs (global and local buckling):
. local buckling for class 4 cross­sections
. Ν buckling and N­M buckling of members
. lateral­torsional buckling of members
. shear buckling of webs
. stability of webs to transverse forces: crippling, buckling
. web buckling induced by compression flange
65
The formulas of ULS checks include different parameters depending on the
class of the cross-section (see chapter V); they may consider the
following cross-section properties:
. plastic properties for class 1 or 2 cross-section (Wpf, ...)
. elastic properties for class 3 cross-section (We/ ,...)
. effective properties for class 4 cross-section (Weff,...) taking into
account the occurrence of local buckling.
The table 1.8 is related to the classes of cross-section and shows if there are
differences between check formulas in function of those classes of crosssection.
In Appendix D of the design handbook a similar table (table D.l) is
provided (for information) presenting a list of references to Eurocode 3
Part 1.1 (J2f) also related to all check formulas at Ultimate Limit States for
different types of loading.
(2) In respective following chapters tables present complete lists of the checks to be
performed at Ultimate Limit States for members or webs submitted to different loading:
in chapter VI, table VI. 1 for members in tension,
in chapter VH, table VILI for members in compression,
in chapter VIH, table VIILI for members in bending,
in chapter LX, table IX. 1 for members with combined axial force and bending
moment.
Planes within
Table 1.6
H
internal forces, moments (N$d, Vsd, Msd)
and transverses forces Fsd are acting
Fsd
Í tP
<£
-rl
&
Vz.sd My.sd
FsdT
Nsd
{
xy
(xaxis)
xz
Mx.sd
·
y.Sd
'z.Sd
xy
xz
My.Sd
Mz.Sd
Fsd
xz
xy
xz
moment of torsion (is not considered in the handbook because the Annex G
of Eurocode 3 is not officially available yet).
66
Table L7
Internal forces, moments and transverse forces to be checked at ULS
for different types of loading
Internal forces, moments and transverse forces
and, interactions between them
Type of loading on the members
and on the webs
*
x—
N x.Sd
.y
·— X
y"
Members in tension
(braces,...) : chapter VI
e£
I
I
ζ
Ν tension
Nx.sd
x —
Λ—X
Members in c ompression
(columns,...) : chapter Vu
I
I
ζ
Ν,compression
ζ
I
e.
X
M z.Sd
-'■' 1,3 .Xp
—
Vz;sdMy.Sd
Members in bending
(beams,...) : chapter VIH
Μ
(Μν,Μ,)
(V,M)
(V,My,Mz)
Μ z.Sd
y.Sd .
Nx.sd
Κ lo XP
χ -
Members with c ombined (Ν, M)
(beams-columns,...) : chapter LX
Vz:sdMy.sd
ι
ι
ζ
(N,M)
(N,MV,MZ)
u
(N,V)
(N,V,M)
(N,V,MV,MZ)
Fsd I |Fsd ?
V
τ—Γ
r
ι
Nx.sd
'
N
x.sd
My.Sd
ι
ζ
Fsd
(F,N)
(F,VZ)
z.Sd
ι
.-y
x—
Transverse forc es
on webs : chapter X
N x . Sd
(F,N,VZ)
67
(F,MV)
(F,N,MV)
(F,V z ,My)(F,N,V z ,M v )
Table 1.8 List of references to chapters of the design handbook related to all check formulas at ULS
Typ«; References to the design handbook for ULS checks
Internal forces
moments, and
Physical phenomena
of in function of classes of cross­sections (chapter V) :
class 3
class 4
transverse forces check s classes 1 or 2 |
tension resistance (gross & net section)
VI.b.1 (1) + VI.b.2 (1) + VI.c.1 (l) + VI.c.2 (1)
R
1. Ntension
compression resistance
vn.c.l (1)
vn.c.i (1)
2. Ncompression R
Ν buckling of members
VII.c.2.1(2) + VII.c.2.2
VHc.2.1 (2)
S
shear and block shear resistances
Vin.d.l (1)
R
3. V
shear buckling
VIII.d.2 (5)
S
uniaxial bending resistance
VIII.e.l (1)
VHI.e.1 (1)
R
VIII.e.1 Í1)
4. M
lateral­torsional buckling (My) (LTB)
Vin.e.2
(4)
Vm.e.2
(4)
Vin.e.2 (4)
S
5. (My,Mz)
biaxial bending resistance
Vm.f.l (1)
vm.f.i (1)
R
Vffl.f.1 (1)
6. (V,M)
(V z ,M y )
S vm.f.2 (1) + (2) Vm.f.2 (1) + (2) VIII.f.2 (1) + (2) biaxial flexural buckling
uniaxial bending & shear resistance
vm.g.i.i(i)
R
uniaxial bending & shear buckling
VIH.g.2(3)
S
7. ( ν , Μ ^ Μ ζ )
R
(V z ,M y ,M z )
8. (N,M)
(Ntension,My)
(NComp..My)
(Ncomp^Mz)
9. (N,M y ,Mz)
S
R
S
S
s
R
11. (N,V,M)
S
R
S
R
(N,VZ,My)
12. (N,VJvlyJVlz)
S
R
10. (N, V)
(N,V z ,M y ,M z ) S
13. F,(F,N),(F,My), R
biaxial bending & shear resistance
uniaxial bending & shear buckling
vm.g.2 (3)
uniaxial bending & axial force resistance
IX.d.1.4 (2)
DCd.l.l(l)
DC.d.l.3(2)
lateral­torsional buckling (LTB)
IX.d.2.1 (1)
K.d.2.1 (1)
IX.d.2.2 (2), (3) IX.d.2.2 (2), (3) IX.d.2.2 (2), (3) N­M buckling + LTB
IX.d.2.2 (3), (4) IX.d.2.2 (3), (4) IX.d.2.2 (3), (4) N­M buckling
IX.d.1.4 (1)
biaxial bending & axial force resistance
IX.d.1.2 (1)
IX.d. 1.3(1)
IX.d.2.2 (1), (2) IX.d.2.2 (1), (2) rX.d.2.2(l),(2) (N­biaxial M) buckling + LTB
rx.e.l (1)
rx.e.l (1)
shear and axial load resistance
shear buckling
Vm.d.2 (5)
uniaxial bending &
IX.f.1.4 (2)
IX.f.1.1 (1)
IXI.1.3 (2)
shear and axial force resistance
IX.f.2 (3)
(N­uniaxial M) resistance & shear buckling
Vin.g.l.2(2)
Vm.g.l.2(3)
Vm.g.l.2(3)
rx.f.i.2(i)
IX.f.1.3 (1)
rX.f.1.4 (1)
X.c.1 (1)
IX.f.2 (3)
X.c.1 (1)
X.c.1 (1)
biaxial bending & shear and axial force
resistance
(N­uniaxial M) resistance & shear buckling
transverse force (+N, +M y ) resistance
j
(FFMy)
F
R
S
X.c.2 (1)
X.d.1.1 ((1), (2)) + X.d.2 ((1), (2), (3))
(F%)
S
X.d. 1.2(1)
S
14. (F.V^.CF^.Vz),
R
(F,V z ,M y ),
(F,N,Vz,My)
S
tvpe of loading
tvpe of checks:
crushing
crippling + buckling
crippling
X.e(l)
X.e(l)
X.e(l)
X.c.1 (1)
X.c.1 (1)
X.c.1 (1)
IX.f.2 (3)
compression flange induced buckling
transverse forces + shear V z (+N, +M y )
resistance
(N­uniaxial M) resistance & shear buckling
1.
= tension members
2.
= compression memb•ers
3. to 7.
= members in bendin g
8. to 12.
= members with combined N­M
13. to 14.
» transverse forces oi
ι webs
R = resistance of cross­sections ([5.4]
5 = Stability of memt>ers ([5.5]) or we)bs ([5.6Π5.7])
68
II
STRUCTURAL CONCEPT OF THE BUILDING
(1) This chapter intends to list the data of the analysed building concerning the types of
structure, members and joints, the geometry and the material properties. The load
arrangements applied to the building are defined in chapter m .
II.a
Structural model
(1) The type of structure, the type of the bracing system and all the different prescriptions of
the project (office building, housing, sport or exhibition hall, parking areas,....) should be
defined.
Ill)
Geometric dimensions
(1) The geometry of the building should be defined:
- the height, the width and the length of the structure, the number of storeys of the
building and the dimensions of the architectural element.
- definition of storeys: plane frame with 3 storeys:
II.C
Non structural elements
(1) All the elements of the building which do not bear any loads have to be considered in the
evaluation of the self-weight loads: walls, claddings, ceilings, coverings,...
I I.d
Load bearing structure
(1) All the elements which bear the loads should be defined : frames, beams, columns, bracing
system, concrete core,....
69
Joints
Il.e
(1) The design handbook assumes the use of pinned or rigid joints (see chapter XI).
Semi-rigid joints are not considered in the design handbook. In the case of semi-rigid
joints whose behaviour is between pinned and rigid joints, the designer shall take into
account the moment-rotation characteristics of the joints (moment resistance, rotational
stiffness and rotation capacity) at each step of the design (predesign, global analysis, SLS
and ULS checks). The semi-rigid joints should be designed according to chapter 6.9 and
the Annex J of Eurocode 3. Table ILI presents typical types of joints.
Typical types of joints
Table 11.1
Pi
r^
0
0
0
o
0
0
0
<"
0
o
0 t^^^^m
0
3^"
-vRigid joints
■■u
■r
Hr
Φ*
£=
* \ -
Semi-rigid joints
Pinned joints
70
(2) The table II.2 presents the modelling of joints. The joints may be modelled by nodes offset
from the member centrelines to reflect the actual locations of the connections.
ΓTable IL2
Type of joint
Modelling of joints
Modelling
Behaviour
M
11
Mu
Φ
RIGID Joint
SEMI­RIGID Joint
M
K>
i S S v - S l * ν î-
PINNED Joint
71
Φ
n.f
Profiles
(1) The selected steel profiles used as beams and columns in the structure and as elements in
the bracing system should be listed and precisely referred.
n.g
Floor structure
(1) Composition of the floor system (in situ concrete slab, precast concrete slab, steel sheet
deckings, slim floor,...) is needed to determine the self-weight loads. Composite effect
between the floor and the beams is not considered in this design handbook.
Il.h
Material properties
[3]
(1) The material properties given in this chapter are nominal values to be adopted as
characteristic (unfactored) values in design calculations.
[3.2.2.1]
n.h. 1
Nominal values for hot rolled steel
(1) The nominal values of the yield strength fy and the ultimate strength fu for hot rolled steel
are given in table II.4 for steel grades S 235, S 275 and S 355 in accordance with EN
10025 and for steel grades S 275 and S355 in accordance with EN 10113.
(2) The european standard EN 10025 specifies the requirements for long and flat products of
hot rolled weldable non-alloy structural steels (steel grades: S 235, S 275, S 355).
The european standard EN 10113 specifies the requirements for long and flat products of
hot rolled weldable fine grain structural steels (steel grades: S 275, S 355, S 420, S 460).
(3) Similar values as defined in table Π.4 may be adopted for hot finished structural hollow
sections.
(4) For a larger range of thicknesses the values specified in EN 10025 and EN 10113 may be
used.
(5) For high strength steels (S 420 and S 460) specific rules are given in the normative
Annex D of Eurocode 3. Their material properties are introduced in table Π.4.
(6) The table Π.3 compares the symbolic designations of steel grades according to various
standards. The design handbook always uses the single designation of structural steels
defined by the european
standard EN 10027-1: "S" followed by the value of yield strength
expressed in N/mm2 (=MPa).
Comparison table of different steel grades designation
Table II.3
EN 10027-1
S 235
\ S275
S 355
S 420
S 460
EN 10113
FeE 275
FeE 355
FeE 420
FeE 460
EN 10025 NF A 35-504/ NF A 35-501 DIN 17102 DIN 17100 BS 4360 ASTM
NF A 36-201
Fe 360
Fe 430
Fe 510
E 24
E 28
E 36
E 355
E 420
E 460
72
StE285
StE355
StE420
StE460
St 37-3
St44-3
St 52-3
40 D
43 D
50D
55 C
gr. 50
gr. 60
gr. 65
Table H.4
Nominal values of yield strength fy and ultimate tensile strength fu
for structural steels according to EN 10025 and EN 10113
Thickness t (mm)*)
Nominal steel grade
EN 10027-1
Designation
S 235
S 275
S 355
EN 10025
Standard
Fe 360
Fe 430
Fe 510
EN 10113
Standard
FeE 275
FeE 355
S 420 M
S 460 M
t<40mm
fy (N/mm2)
fu (N/mm2)
235
275
355
360
430
510
40 mm < t < 100 mm**)
fy (N/mm2)
fu (N/mm2)
215
255
335
340
410
490
255
390
S 275
275
370
335
355
490
S 355
470
420
390
500
S 420
500
460
430
530
S 460
530
Notes:
i
J—
") t is the nominal thickness of the element
(Γ
ir
■-t
- of the flange of rolled sections (t = tf)
- of the particular elements of the welded sections
**) the condition 40 mm < t < 63 mm should be taken for plates and other flat products
in steels of delivery condition TM to EN 10113-3.
II.h.2
Fracture toughness
[3.2.2.3]
(1) The material shall have sufficient fracture toughness to avoid brittle fracture at the lowest
service temperature expected to occur within the intended life of the structure.
(2) In normal cases of welded or non-welded members in building structures subject to static
loading or fatigue loading (but not impact loading), no further check against brittle
fracture is necessary if the conditions given in tables Π.5 and Π.6 are satisfied.
Tables Π.5 and Π.6 provide the maximum thicknesses of the structural elements which are
allowable for certain lowest service temperatures and for different steel grades in
accordance to the EN 10025 and EN 10113 standards.
(3) For all other cases reference should be made to informative Annex C.
73
[table 3.2]
Table Π.5
Maximum thickness for statically loaded structural elements
Steel grade and quality
Maximum thickness (mm)
for lowest service temperature of
-10°C
0°C
Service condition 5)
-20°C
SI
S2
SI
S2
SI
S2
108
250
250
30
75
212
74
187
250
22
53
150
63
150
250
19
45
127
45
123
250
14
33
84
EN 10027
EN 10025 D
S235JR
S 235 JO
S 235 J2
Fe 360 Β
Fe 360 C
Fe 360 D
150
250
250
S 275 JR
S 275 JO
S 275 J2
Fe 430 Β
Fe 430 C
Fe 430 D
90
250
250
41
110
250
26
63
150
S 355 JR
S 355 JO
S 355 J2
S355K2
Fe 510 Β
Fe 510 C
Fe 510 D
Fe 510 DD 2)
40
106
250
250
12
29
73
128
29
73
177
250
9
21
52
85
21
52
150
250
6
16
38
59
EN 10113 3)
S 275 M
S 275 ML
Fe E 275 KG 4 )
Fe E 275 KT
250
250
250
250
250
250
192
250
250
250
150
250
S 355 M
S 355 ML
Fe E 355 KG 4 )
Fe E 355 KT
250
250
128
250
250
250
85
250
250
250
59
150
Service conditions 5 ):
SI
either:
. non-welded, or
. in compression
S2
as welded, in tension
In both cases of service conditions this table assumes loading rate RI (normal static or slow
loading; no impact loading) and consequences of failure condition C2 (fracture critical
members or joints with potential complete structural collapse).5)
Notes:
1)
For rolled sections over 100 mm thick, the minimum Charpy V-notch energy specified in EN 10025 is
subject to agreement. For thicknesses up to 150 mm, a minimum value of 27 J at the relevant specified
tests temperature is required and 23 J for thicknesses over 150 mm up to 250 mm.
2)
For steel grade Fe 510 DD to EN 10025, the specified minimum Charpy V-notch energy value is 40 J
at -20°C. The entries in this row assume an equivalent value of 27 J at -30°C.
3)
For steels of delivery condition N to EN 10113-2 over 150 mm thick and for steels of delivery
condition TM to EN 10113-3 over 150 mm thickforlong products and over 63 mm thickforflat
products, the minimum Charpy V-notch energy specified in EN 10113 is subject to agreement. For
thicknesses up to 150 mm, a minimum value of 27 J is required and 23 J for thicknesses over 150 mm
up to 250 mm. The test temperature should be -30°C for KG quality steel and -50°C for KT quality
steel.
4)
For steel of quality KG to EN 10113, the specified minimum values of Charpy V-notch energy go
down to 40 J at -20°C. The entries in this row assume an equivalent value of 27 J at -30°C.
5)
For full details, refer to informative Annex C of Eurocode 3.
74
[Üble D.2]
Maximum thickness for statically loaded structural elements
Table Π.6
Maximum thickness (mm)
for lowest service temperature of
Steel grade and quality
0°C
Service condition 4 )
SI
S2
SI
S2
SI
S2
50
145
38
101
140
250
36
94
99
250
28
69
EN 10027
EN 10113 D
S 420 M
S 420 ML
S 420 KG 2)
S 420 KT 3)
250
250
70
172
162
250
S 460 M
S 460 ML
S 460 KG 2)
S 460 KT 3)
179
250
53
150
150
250
Service conditions 4 ):
51
52
-20°C
­10°C
either:
. non­welded, or
. in compression
as welded, in tension
In both cases of service conditions this table assumes loading rate Rl (normal static or slow
loading; no impact loading) and consequences of failure condition C2 (fracture critical
members or joints with potential complete structural collapse).4)
Notes:
For steels of delivery condition N to EN 10113­2 over 100 mm thick for steel grade S 460 and over
D
150 mm thick for steel grade S 420, and for steels of delivery condition M to EN 10113­3 over 150
mm thick for long products and over 63 mm thick for flat products, the minimum Charpy V­notch
energy specified in EN 10113 is subject to agreement. Up to 150 mm thick, a minimum value of 27 J
is required and 23 J over 150 mm thick up to 250 mm. The test temperature should be ­30°C for steel
qualities S 460 KG and S 420 KG and ­50°C for steel qualities S 460 KT and S 420 KT.
2)
For steel qualities S 460 KG and S 420 KG the specified minimum Charpy V­notch energy in EN
10113 only go down as far as 40 J at ­20°C. The entries in this row assume an equivalent value of 27 J
at­30°C.
3)
For steel qualities S 460 KT and S 420 KT the specified minimum Charpy V­notch energy in EN
10113 is 27Jat­50°C.
4)
For full details, refer to informative Annex C of Eurocode 3.
II.h.3
Connecting devices
[3.3]
II.h.3.1 B olts
[3.3.2]
(1) The nominal values of the yield strength fyb and the ultimate strength fUb ( to be adopted
as characteristic values in calculations) are given in table II.7.
Table IL7
Nominal values of yield strength .fø and ultimate tensile strength/„ft for bolts
4.6
4.8
5.6
5.8
6.8
8.8
10.9
1 fyb (N/mm2)
240
320
300
400
480
640
900
| fub (N/mm2)
400
400
500
500
600
800
1000
Bolt grade
75
II.h.3 2
[3.3.5 (2)]
(1) The specified yield strength, ultimate tensile strength, elongation at failure and minimum
Charpy V-notch energy value of the filler metal, shall all be either equal to, or better than,
the corresponding values specified for the steel grade being welded.
n.h.4
[3.2.5]
Welding consumables
Design values of material coefficients
(1) The material coefficients to be adopted in calculations for the steels covered by Eurocode
3 shall be taken as follows:
Table II.8
Material coefficient
:
E=
210 000
N/mm 2
.
G=
80700
N/mm 2
2(1+ v)
. coefficient of linear thermal expansion
:
a =
12. IO"6
1/°C
. density
:
P=
7850
kg/m^
. Poisson's ratio
:
V =
0,3
. modulus of elasticity
. MICdl IIIUUIUUS
VJ —
"
E
76
III
LOAD ARRANGEMENTS AND LOAD CASES
III.a G eneralities
[2.2J5 (l)]
[2.25 (2)]
(1) A load arrangement identifies the position, magnitude and direction of a free action.
(2) A load case identifies compatible load arrangements, set of deformations and
imperfections considered for a particular verification.
(3) For the definitions of actions Goad arrangements: F= G, Q,...) and effects of actions (E, S)
and for the design requirements it should be referred to chapter La (Basis of design).
(4) Flow-chart FC 3.1 presents the general procedure to study structures submitted to actions :
all load cases are defined by relevant combinations of characteristic (unfactored)
values of load arrangements (Fø,
for each load case the global analysis of the structure determines the design values
for the effects of actions (Ed = oV,6h, f,... ; Sd = N, V, Μ, σ,...) which shall be
checked at SLS (Cd limits) and at ULS (Rd resistances).
This general procedure is used in the flow-charts about elastic design of:
steel frames (in general) (flow-chart FC 1; see chapter I),
braced or non-sway frames (flow-chart FC 4; see chapter IV),
bracing system (flow-chart FC 12; see chapter ΧΠ),
according to Eurocode 3. Moreover references to those general flow-charts FC 1, FC 4
and FC 12 are specified at the different steps of the general procedure presented in the
flow-chart FC 3.1.
(5) For braced or non-sway buildings it is explained in chapter I.b.l (flow-chart FC 1) and in
chapter IV.a.2 (flow-chart FC 4) that the elastic global analysis of the structure could be
based on first order theory. In that case of first order elastic global analysis the principle of
superposition is applicable because the effects of actions (E, S) are linear functions of the
applied actions (F = G, Q,...) (no Ρ-Δ effects and used material with an elastic linear
behaviour).
The principle of superposition allows to consider a particular procedure to study structures
submitted to actions. This procedure illustrated in flow-chart FC 3.2 could be more
practical because it should simplify the decision of which load case gives the worst effect.
For each single characteristic (unfactored) value of load arrangement (Fk) the global
analysis of the structure determines characteristic (unfactored) values for the effects of
actions : Ek = (Ov,6h, f,..)k ; Sk = (N, V, M, a,...)k.
All load cases are defined by relevant combinations of the characteristic (unfactored)
values for the effects of actions (E^Sk). All these load cases directly furnish the design
values for the effects of actions (Ed = δγ,δι,, f,... ; Sd = N, V, M, σ,...) which shall be
checked at SLS (Cd limits) and at ULS (Rd resistances).
77
Flow­chart LFC3.1J : Load arrangements and load cases for cenerai global analysis of the structure
rows:
\s>^>/
C
rows:
'.
"N
Determine all load arrangements with characteristic (unfactored) values of actions Fk (Gk, Qk,...)
- row 1 of flow-chart FC 1
- row 2 of flow-chart FC 4
Determine all ULS load cases with relevant
ULS combinations of load arrangements (Fk)
(with partial safety factors 7F = γσ, 7Q,...):
Determine all SLS load cases with relevant
SLS combinations of load arrangements (Fk):
- row 2 of flow-chart FC 1 and FC 12
- row 4 of flow-chart FC 4
- row 2 of flow-chart FC 1 and FC 12
- row 4 of flow-chart FC 4
yes
3 r*
All ULS load cases
analysed ?
All SLS load cases
analysed ?
ULS checks
SLS checks
I
yes
Classification of the frame:
braced or non­sway frame
- rows 5 to 8 of flow-chart FC 1
- rows 8 to 14 of flow-chart FC 4
- rows 5 and 6 of flow-chart FC 12
Global analysis of the structure submitted
to the considered load case in order to
determine the design values for the ULS
effects of actions:
Sd = N, V, Μ, σ,...
Global analysis of the structure submitted
to the considered load case in order to
determine the design values for the SLS
effects of actions:
Ed = oh, δν, f,...
- rows 11 to 13 of flow-chart FC 1
- row 15 of flow-chart FC 4
- rows 9 to 11 of flow-chart FC 12
- row 4 of flow-chart FC 1 and FC 12
- row 6 of flow-chart FC 4
Determine ULS resistances (Rd):
Determine SLS limits (Cd):
- rows 14 to 21 of flow-chart FC 1
- rows 16 to 22 of flow-chart FC 4
- rows 12 to 19 of flow-chart FC 12
- row 4 of flow-chart FC 1 and FC 12
- row 7 of flow-chart FC 4
»­Í Select stronger section(s) or joint(s)
10
j
Adopt the structure if both ULS and SLS checks are fulfilled V
Note: . for the definition of Fk (Gk, Qk), γρ (JG, γο), Sd. Rd, Ed, Cd : see chapter La (Basis of design).
. references are done to flow­chart FC 1 (Elastic design of steel frames according to Eurocode 3),
flow­chart FC 4 (Elastic design of braced or non­sway steel frames according to Eurocode 3),
and flow­chart FC 12 (Elastic design of bracing system according to Eurocode 3).
78
10
Flow-chart (FC 3.2) .Load arrangements and load cases for tirsi order elastic elobal analysis of the structure
rows:
i
rows:
f Determine all load arrangements with characteristic (unfactored) values of actions Fk (Gk, Qk, ...)J
y"/AH load arrangements
analysed?
Global analysis of the structure submitted to the considered single
load arrangement (Fk) in order to determine characteristic
(unfactored) values for the effects of actions :
- for ULS checks: Sk = (N, V, Μ, σ, ...)k
- for SLS checks: Ek = (δη, δν, f,^)k
1
1
ULS checks
SLS checks
I
i
Determine all ULS load cases with relevant
ULS combinations of effects of actions (Sk)
(with partial safety factors 7F = 7G, γο,...)
Determine all SLS load cases with relevant
SLS combinations of effects of actions (Ek)
(Design values for the effects of actionsPN
V.
Ed ■ oh. δν, f,...
J
Design values for the effects of actions:
Sd = N, V, Μ, σ,...
All ULS load cases \ v*
analysed?
/
yes
All SLS load cases
analysed?
Classification of frame:
braced or non-sway frame
CDetermine ULS resistances (ftp)
10
11
12
Γ Determine SLS limits (Cd) )
yes
yes
Adopt the structure if both
ULS and SLS checks are fulfilled
Τ
Γ Select stronger section(s) or joint(s) J-
Note: for the definition of Fk (Gk, Qk), ^F (yc, *fQ), Sd. Rd. Ed, Cd : see chapter La (Basis of design)
79
1
IIl.b
Load arrangements
(1) The following load arrangements are characteristic (unfactored) values of actions (Fk) to
be applied to the structure. The characteristic values of load arrangements given hereafter
are issued from Eurocode 1 (/l/).
(2) The table ΠΙ.1 provides a list of
all the load arrangements (Fk) to be taken into account in building design and,
the references to the chapters of the handbook where details are given about those
load arrangements.
Load arrangements Fk for building design according to EC 1
Table ΙΠ.1
Load arrangements (Fk)
distributed, g
Permanent loads :
concentrated, G
1)
2)
Variable loads:
Imposed loads on floors and roof:
- Wind loads:
Snow loads:
ECl
1.5.1 (4)
ECl
1.5.1 (4)
ΠΙ. b. 1
distributed, q
concentrated, Q
wind pressure, we4
wind force, F w
distributed, s
Reference to the handbook
ni.b.l
m.b.2.1
ffl.b.2.1
m.b.2.2
m.b.2.2
m.b.2.3
Permanent loads (g and G)
(1) Action which is likely to act throughout a given design situation and for which the
variation in magnitude with time is negligible in relation to the mean value, or for which
the variation is always in the same direction until the action attains a certain limit value.
m.b.2
Variable loads (q, Q, w and s)
(1) Action which is unlikely to act throughout a given design situation or for which the
variation in magnitude with time is not negligible in relation to the mean value nor
monotonie.
m.b.2.1 Imposed loads on floors and roof (q and Q)
ECl
2.1.5.1. (1)
ECl
2.1.6.1. (1)
(1) Categories of areas: areas in offices, housing, warehouses, parkings, dwellings, etc. are
divided into six categories according to their specific use:
- Category A:
areas for domestic and residential activities.
- Category B:
areas where people may congregate.
- Category C:
areas susceptible to overcrowding, including access areas.
- Category D:
areas susceptible to accumulation of goods, including access areas.
- Category E:
- Category F:
traffic and parking areas for light vehicles,
traffic and parking areas for medium vehicles.
(2) The values of imposed loads on floors and roof are given in table ΠΊ.2 according to the
category of areas and the loaded areas.
80
Table II1.2
Imposed load (qk, Qk) on floors in buildings
Categories of areas
Loaded areas
qk (kN/m2)
Qk(kN)
Category A;
general
stairs
balconies
2,0
3,0
4,0
2,0
2,0
2,0
Category B;
general
stairs, balconies
3,0
4,0
2,0
2,0
Category C;
with fixed seats
other
4,0
5,0
4,0
4,0
Category P;
general
5,0
7,0
Category E:
vehicles weight: Ú 30 kN
2,0
10
Category F;
vehicles weight: 30 ­160 kN
5,0
45
III.h.2.2 Wind loads (weJ, Fw)
EC1,6.4P(1)
(1) The wind load is presented either as a wind pressure or a wind force. The action on the
structure caused by the wind pressure is assumed to act normal to the surface except where
otherwise specified; e.g. for tangential friction forces.
ECl, 6.4 (3) (2) The wind action is given by:
w
wind pressure on a surface (see m.b.2.2.1).
Fw
resulting wind force: see m.b.2.2.2 or obtained by integrating the wind
pressure.
Me
torsional moment, refer to ECI, part 2.3
Ffr
friction force, refer to ECl, part 23
III.h.2.2.1 Wind pressure (wCii)
ECl, 6.5.4
(1) The net wind pressure across a wall or an element is the difference of the pressures on
each surface taking due account of their signs (Pressure is taken as positive, when
directed towards the surface and is negative when represents a suction) (see table ΠΙ.3).
81
Table ΙΠ.3
Pressures on surfaces
Θ
0
Θ:
®:
®
<2C
©r , Θ
^
—*- s
·"·
"*" ^
**
— r=r
0
r©;
—► s —»-
¡^ s
;
, 0
EÉ
:©
ECl, 6.5.2 (2) The wind pressure acting on:
. the external surfaces of a structure, w e , shall be obtained from:
We=qref-Ce(Ze)-Cpe
. the internal surfaces of a structure, WJ, shall be obtained from:
w
ECl, 6.7.1
i=qref-ce(zi)-cpi
where qref is the reference mean wind pressure determined from:
Ρ 2
ECl, form. (6.7.1)
where
Ρ
Vref
ECl, form. (6.7.2)
qref = Tpref
is the air density (generally = 1,25 kg/m3)
is the reference wind velocity taken as follows
v
ref - c DIR- c TEM- c ALT- v ref,0
where vref,o
ECl, 6.8.1
ECl, form. (6.8.1)
basic value of the reference wind velocity at sea level given by
the wind maps of the countries (Annex 6.A of ECl).
CDIR
direction factor to be taken as 1,0 unless otherwise specified in
the wind maps.
C
TEM temporary (seasonal) factor to be taken as 1,0 unless otherwise
specified in the wind maps.
c ALT altitude factor to be taken as 1,0 unless otherwise specified in the
wind maps.
where ce(ze) is the exposure coefficient for ζ = ze is defined by:
Ce(ze) = C?.C?+7Kr.Cr.Ct
where Kr, cr (z), c t (z) are given for more details in [ECl, 6.8.1]
For flat terrain (i.e. upwind slope < 5% in the wind direction), c t =1,0. For such
conditions the exposure coefficient c e is given in the table III.4.
82
ECl,
Fig. 6.8.1
ECl
Table 6.8.1
1 Table III.4
Exposure coefficient ce as a function of height ζ above ground
Terrain C ategory:
I
Rough open sea, lake shores with at least 2 km fetch upwind and
smooth flat country without obstacles.
Π Farmland with boundary hedges, occasional small farm structures,
houses or trees.
ΠΙ Suburban or industrial areas and permanent forests.
FV Urban areas in which at least 15% of the surface is covered with
buildings and their average height exceeds 15 m.
z(m)
1000 ■ , ,
IV TTT π Τ
Ψ
inn
1UVJ
—
10 -
Ce( ζ )
1
_
'
0,1X)
where c
ECl,table6.10.2.1
Pe
1,130
2,(X)
3,1X)
4,1Χ)
I
5,00
is the external pressure coefficient given in ECl, 6.9, which depend on the
size of the effected area A and the shape of the building (see table ΠΙ.5).
Table ΙΠ.5 External pressure Cpe for buildings depending on the size
of the effected area A
Cpe = Cpei ι
A<lm2
Cpe = Cpe, 1 + (Cpe, 10" Cpe, i) l o g i o A
1 m2 < A < 10 m 2
Cpe = Cpe, 10
A > 10 m2
The values of are given in the chapters 6.9.2.2 to 6.9.2.8 of ECI for the different
83
shapes of the buildings.
where Ze is the reference height appropriate to the relevant pressure coefficient (see table
ΙΠ.6).
ECl,
Fig. 6.9.2.1
Reference height ze depending on h and b
Table II 1.6
h>2b
b < h < 2b
| Ze = h
h<b
Ze = b
ECl, 6.9.2.2 (4)
Buildings whose height h is greater than 2b shall be considered to be in multiple
parts, comprising: a lower part extending upwards from the ground by a height
equal to b for which Ze = b; and a middle region, between the upper and lower
parts, divided into as many horizontal strips as desired and for which Ze is the
height of the top of each strip.
and where Cpi is the internal pressure coefficient. For a homogeneous distribution of openings
ECl, 6.10.2.9
the value Cpi = - 0,25 shall be used
III.b2.2.2
Wind force (F..)
(1) The global force, F w , shall be obtained form the following expression:
ECl, form. (6.6.1)
F
w
-°tref-Ce(ze)-Cf-Aref-cd
. qref is the reference mean wind pressure (see m.b.2.2.1)
- ce(ze) is the exposure coefficient for ζ = Ze (see m.b.2.2.1)
. Ze is the reference height appropriate to the relevant pressure coefficient (see
m.b.2.2.1)
. Cf is the force coefficient derived from ECl, part 2.3, chapter 10, if available
- Aref is the reference area for Cf
- ca is the dynamic factor
HI,b,2,3 Snow loads (s)
(1) The snow loads are given by:
s = ^.ce.ct.sk
where μι
Sk
ce
ct
is the snow load shape coefficient
is the characteristic value of the snow load on the ground (kN/mm 2 )
is the exposure coefficient, which usually has the value 1,0
is the thermal coefficient, which usually has the value 1,0
84
111 χ
Load cases
(1) The following load cases are related to the general procedure to study structures submitted
to actions (see flow­chart FC 3.1 and comment (4) in chapter IILa): all load cases are
defined by relevant combinations of characteristic (unfactored) values of load
arrangements (Fk).
[2.3.2.2 (l)]
For each load case, design values (Ed, Sd) for the effects of actions shall be detenrtined
from global analysis of the structure submitted to the design values of actions
(Fd = 7 F · Fk) involved by combination rules as given:
­ in table ΠΙ.7, for SLS
­ in table ΠΙ.8 and table m.9, for ULS
(2) In the case of the particular procedure defined in flow­chart FC 3.2 (see also comment (5)
in chapter IILa), the characteristic (unfactored) values for the effects of actions (Ek, Sk)
are obtained from global analysis of the structure submitted to each single characteristic
(unfactored) value of load arrangement (Fk).
For each load case, design values (Ed, Sd) for the effects of actions shall be determined
from combination rules defined in tables ΠΙ.7 to m.9 where values of load arrangements
(Fk = Gk, Qk, g, q, s, w, P) are replaced by the characteristic values for the effects of
actions (Ek = (ov,Ôh, f,..)k ; S k = (N, V, M, o,...)k).
For instance, in the case of the third example in table m.9, the general load case 1. ,
(l,35.gk + 1,50 wk) should be replaced by the following particular load case 1. considering
the elements or the cross­sections with
. their worst effects of actions (for columns: axial force (N)k; for beams: shear force (V)k
and bending moment (M)k) and,
. their worst combined effects of actions (for beam­columns: (N)k + (M)k ; ...):
­ max N = 1,35 (N)k(due to gk) + l,50.(N)k.max(due to Wk),
­ max V = 1,35 (V)k(due to gk) + l,50.(V)k.max(due to Wk),
­ max M = 1,35 (M)k(due to gk) + l,50.(M)k.max(due to Wk),
­ max N + associated M,
­ max M + associated N,...
(3) In the following chapters UI.c.l and ffl.c.2, the proposed combinations of actions are
simplifications adapted to building structures (for SLS, [2.3.4 (5)] and for ULS, [2.3.3.1 (5)]).
(4) If the limitations imposed at SLS and at ULS are difficult to be respected, more favourable
combinations of actions could be used instead of the respective simplified proposals of
table m.7 (then see [2.3.4 (2)] of EC 3) or tables ΙΠ.8 and ΠΙ.9 (then see [23.2.2(2)] of EC3).
in.c. 1
ECCS n°65
table 2.3
Load cases for serviceability limit states
Table 111.7
Combinations of actions for serviceability limit states
Load combinations to be considered:
with only the most unfavourable variable
actions (Qicmax):
1.
£ G k +Qk.max
with all unfavourable variable actions (Qk):
2.
£ G k +0,9^Qk
Gk ­
Qk ­
permanent actions, e.g. self weight
variable actions, e.g. imposed loads
on floors, snow loads, wind loads
Qk.max ­the variable action which causes the
largest effect
The load combination which gives the largest effect (i.e. deformations, deflections) is
decisive
85
m.c.2
ECCSn°65
table 2.1
Load cases for ultimate limit states
Table ΙΠ.8
Combinations of actions for ultimate limit state
Load combinations to be considered:
with only the most unfavourable variable
actions (Qk.max):
**
YG­XGk+YQ­Qk.max
l,35*.EG k +l,50**.Q k.max
1.
with all unfavourable variable actions (Qk):
YGEGk+0,9YQ.XQk
2.
permanent actions, e.g. self weight
Qk­
variable actions, e.g. imposed loads
on floors, snow load, wind loads
Qk.max ­the variable action which causes the
largest effect
partial safety factor for permanent
YGactions
partial safety factor for variable
YQactions
Gk­
l,35*.]TG k +l,35**.£Q k
* If the dead load G counteracts the variable
action Q(meaning a favourable effect of G):
YG = LOO
■
' '' t " M " ♦ windload Q
ν , Ι ν ι „ „ ι deadload G
-κ
t πι titt
"if the variable load Q counteracts the dominant
loading (meaning a favourable effect of Q):
YQ = 0
The load combination which gives the largest effect (i.e.internal forces or moment ) is decisive
ECCSn0 65
table 2.2
Examples for the application of the combinations rules in table ΙΠ.8.
All actions (g, q, P, s, w) are considered to originate from different sources
load cases combinations of actions
ŒEUm s
1.
l,35.g+l,50.q
J
I
q
2.
l,35.g+l,50.s
l,35.(g + q + s)
3.
A
A
Table ΙΠ.9
f
α ιρg
LXLTEDq
* O i i i Q M D s
ο π *i π
-A
w
g­
q-
P­
s­
w­
ΓΤΤΤ
ΊΊ
ΕΡΖΕΕΠ q
ŒEEEED g
1.
2.
3.
4.
l,35.g+l,50.q
LSS.g+LSO.P*)
1,35. g+1,50. s
L35.(g+q+s + P*))
1.
2.
3.
4.
l,35.g+l,50.w
l,35.g+l,50.q
1,35. g+1,50. s
l,35.(g + q + w + s)
) assuming Ρ is independent of g, q, s and w
dead load
imposed load
Point load
snow load
wind load
86
IV
[5.1.2 (l)]
[5.1.2(2)]
ECCS n° 65
table 5.2
DESIGN OF BRACED OR NON-SWAY FRAME
I V.a
Generalities
(1) Frames shall be checked :
. at Serviceability Limit States:
- for horizontal deflections (see chapter IV.f.l) ,
. at Ultimate Limit States:
- for static equilibrium (see chapter IV.b),
- for frame stability (see chapter IV.d),
- for resistance of cross-sections, members and connections (see chapter IV.g) .
(2) When checking the resistance of cross-sections and members of a frame, each member
may be treated as isolated from the frame, with forces and moments applied to each end as
determined from the frame analysis. The conditions of restraint at each end should be
determined by considering the member as part of the frame and should be consistent with
the type of analysis and mode failure.
IV-a.l Analysis models for frames
(1) In general spacial frame structures may be separated into several plane frames that may be
considered as laterally supported at the spacial nodes (see table IV. 1, part 1.).
In the first step for the inplane loading of these plane frames out-of-plane deflections
between the lateral supports are neglected and only the inplane monoaxial action effects
are determined.
In the second step the individual members of the plane frame between the lateral supports,
i.e. the beams and the columns, are separated from the plane frame, to consider lateral
buckling and lateral-torsional buckling, under monoaxial bending and compression.
Members which are common to two different frames, e.g. columns, may be verified for
biaxial bending and compression (see table IV. 1, part 2.).
(2) Table IV.2 shows the modelling of connections in the global analysis depending on their
rotational stiffness.
Table IV.2
Modelling of connections
Type of connection Symbols in the analysis Designed for
Pinned connection
tension,
compression
or shear only
O
moment, shear,
tension or
compression
from an elastic
or plastic global
analysis
Rigid connection
Design or detail criteria
Small restraint to
sufficient rotations
Small rotations,
sufficient elastic moment
and shear strength
ì E
For semi-rigid connections see Eurocode 3, Part 1.1 (J2f)
(3) Guidance on assumptions for reliable simplified modelling of buildings is provided in the
Annex H of Eurocode 3 Part 1.1 (J2f) which is in preparation:
"Modelling of building structures for static analysis".
87
Table IV.l
Modelling of frame for analysis
1. Separation of plane frames from the spacial frame :
FRAME 2
Tflrr
2. Separation of individual members from plane frame:
FRAME 1
ÌK t
t
Ml
cHnW
7Û7T
Isolated beam
Isolated column
N1 + N2
88
IV.a.2
Flow-chart FC 4:Elastic global analysis of braced or non-s wax steel frames
according to Eurocode 3
(1) The flow-chart FC 4 aims to provide a general presentation of the subject dealt in the
present design handbook:
elastic global analysis of braced or non-sway steel frames according to Eurocode 3.
All the details are given in chapters II to XI of the handbook.
(2) The flow-chart FC 4 refers to other flow-charts:
- flow-chart FC 1 about elastic global analysis of steel frames in general according to
EC 3 (the flow-chart FC 1 is provided in chapter I).
- flow-chart FC 12 about elastic global analysis of bracing system according to EC 3
(the flow-chart FC 12 and all the details about bracing system design are given in
chapter XII) and,
(3) The flow-chart FC 4 is a part of flow-chart FC 1 which gives a general presentation of:
- elastic global analysis of braced or non-sway frames (= flow chart FC 4 = path Φ
of flow-chart FC 1) and,
- elastic global analysis of sway frames which are out of the scope of the present
design handbook (= paths (D to (D of flow-chart FC 1).
(4) The flow-chart FC 4 is divided in 3 parts:
rv.a.2.1 general part (1 page)
rv.a.2.2 detail (1 page)
IV.a.2.3 comments (4 pages)
IV.a.2.1 Flow-chart FC 4 general
see the following page
IV.a.2.2 Flow-chart FC 4 details
see the second following page
89
Flow-chart 4ÍFC 4J: Elastic global analysis of braced or non-swav steel frames according to EC 3 rowi
(General)
Actions
Predesign
SLS checks
and
Classification of the frame
for ULS
(braced or non-sway frame)
10
11
12
13
ULS global analysis of the frame
to determine the internal forces and moments (N, V, M)
IS
16
ULS checks of members
submitted to internal forces and moments (N, V, M)
17
18
19
20
ULS checks of local effects
ÜLS checks of connections
90
Flow­chart 4 \FC 4 j : Elastic global analysis of braced
steel frames according to EC 3
or non-swav
Í Assumptions of the frame modelling J
Τ
[Determination of load arrangements (ECl and EC 8)J
(Details)
row:
ι
2
SLS checks [Chap. 4]
ULS checks [Chap. 5]
Τ
JL.
:
Load cases
Load cases
for ULS [2.3.3.]
for SLS
[2.3.4.]
notfulfilled
Predësign of members: beams & columns => Sections
with pinned and/or rigid connections
First order elastic global
analysis of the frame
=> 6v, Oh, f,...
t
SLS checks
Frame with bracing system
Classification
of the frame
notfulfilled
[Chap. 4]
Frame without bracing system
1
First order elastic global analysis of the frame
submitted to hypothetic horizontal loads:
1) with bracing system
=> 5b
and
/ C h o i c e of criterion
**—(of sway / non­sway
\
frame
δΣν
ηΣΗ
Vsd
fcr
±
2) without bracing system => 5u
Vertical
loads
Horizontal &
vertical loads
Design of the
bracing system
, Braced frame\
yes/
o
Global imperfections of the frame
=> equivalent horizontal loads [5.2.4.3]
\no
Ob £ 0,2 5u
[5.2.53. (2)]
First order elastic global analysis of the frame
for each ULS load case
Non-sway frame
First order elastic global analysis of the frame
for all concerned ULS load cases:
either, laterally supported if braced frame
or, without special lateral boundary
conditions if non-sway frame
yes
Non­sway frame [5.2.52.]
δΣν
<ο,ι Ο Γ ™ so.i
ηΣΗ
F C 1 J­
Classification of the cross­sections [Chap. 5.3]
±
L b, buckling length of members for non­sway mode [Annex E]
J
J
IB
notfulfilled
Checks of the out­of­plane stability: members buckling [Chap. 5.5.]
Checks of resistance of cross­sections [Chap. 5.4.]
C
1
Checks of local effects (buckling and resistance of webs) [Chap. 5.6 and 5.7]
Checks of connections [Chap. 6 and Annex J]
91
17
notfulfilled
Checks of the in­plane stability: members buckling [Chap. 5 J.]
(
14
Vcr
■ L - -
C
Design of
sway frames
19
notfulfilled
>
J­
>
notfulfilled
notfulfilled
rV.a.2.3 Comments onflow-chart FC 4
comments (1/4) on flow-chart FC 4:
* Generalities about Eurocode 3:
- All checks of (ULS) Ultimate Limit States and all checks of (SLS) Serviceability Limit
States are necessary to be fulfilled.
- According to the classification of cross-sections at ULS (row 16; chapter V of the design
handbook) Eurocode 3 allows to perform:
. plastic global analysis of a structure only composed of class 1 cross-sections when
required rotations are not calculated [5.3.3 (4)] or,
. elastic global analysis of a structure composed of class 1. 2. 3 or 4 cross-sections
assurning for ULS checks, either a plastic resistance of cross-sections (class 1 and 2)
or, an elastic resistance of the cross-sections, without local buckling (class 3) or, with
local buckling (class 4 with effective cross-section).
[5.2.1.2(1)]
- In order to determine the internal forces and moments (N. V. M) in a structure Eurocode 3
allows the use of different types of elastic global analysis either:
a) first order global analysis using the initial geometry of the structure or,
b) second order global analysis taking into account the influence of the deformation of
the structure
[5.2.1.2 (2)]
[Annex H]
- First order global analysis may be used for the elastic global analysis in the cases of braced
or non-sway frames (row 15).
* row 1:
Assumptions of the frame modelling: examples are provided in the present chapter
rv.a. 1 and more details are presented in the [Annex H] of Eurocode 3
("Modelling of building structures for analysis").
* row 2:
[Chap. 5]
[Chap. 4]
EC 1: Draft
EC 3: ENV 1993-1-1
Eurocode 1
Eurocode 3
Basis of design and actions on structures
Design of steel structures, Part 1.1:
general rules and rules for buildings.
Design of structures for earthquake resistance
EC 8: Draft
Eurocode 8
* rows 3.4:
-ULS
-SLS
means Ultimate Limit States
means Serviceability Limit States
* row 5:
This flow-chart concerns structures using pinned and/or rigid joints.
In the case of semi-rigid joints whose behaviour is between pinned and rigid joints,
the designer shall take into account the moment-rotation characteristics of the joints
(moment resistance, rotational stiffness and rotation capacity) at each step of the
design (predesigri, global analysis, SLS and ULS checks). The semi-rigid joints
should be designed according to chapter 6.9 and the Annex J of Eurocode 3.
92
comments (2/4) on flow-chart FC 4:
[4.2.1 (5)]
* row 6:
For SLS checks, the deflections should be calculated making due allowance for any
second order effects, the rotational stiffness of any semi­rigid joints and the possible
occurrence of any plastic deformations.
[5.2.5.3 (2)]
»nbraçed frame
The frame is braced if:
δ 0 <0,2δ„
where
δ^,:
horizontal displacement of the frame with the bracing system
oV
horizontal displacement of the unbraced frame,
according to first order elastic global analysis of the frame submitted to hypothetic
horizontal loads.
Note: in the case of simple frames with all beam­column nodes nominally pinned, the
frame without bracing would be hypostatic, hence δα is infinite and thus the
condition Ob £ 0,2 δ„ is always fulfilled.
[5.2.4.3]
* row 12:
global imperfections of the frame
initial sway imperfections of the frame
F2
ζ α *
equivalent horizontal forces
F2
0F2
could be applied
in the form of
fc1
­J
Fi
φ Fl
i i i i ·
fc
f
φ (Fl + F2)
93
i i i ιι
^β
1
r—
φ (Fi + F2)
comments (3/4) on flow-chart FC 4:
* row 14:
[5.2.5.2]
classification of sway or non­swav frame:
A frame may be classified as non-sway if according to first order elastic global analysis
of the frame for each ULS load case, one of the following criteria (see row 91 is satisfied:
either, al in general :
[5.2.5.2 (3)]
^ ­ = — < 0,1 , condition which is equivalent to
ï
a
cr
a C T > 10
"­cr
design value of the total vertical load (see row 10)
elastic critical value of the total vertical load for failure in a sway mode
( = π2ΕΙ / L2 with L, buckling length for a column in a sway mode; VCT of
a column does not correspond necessarily to V cr of the frame including that
column)
ac
coefficient of critical amplification or coefficient of remoteness of critical
state of the frame
b) in case of building structures with beams connecting each columns at each storey level:
where Vsa:
Vcr:
or,
[5.2.5.2(4)]
ο·Σν­δ·(νι­τ­ν2)
< 0,1
h.£H
h.(H 1 + H 2 )
where H, V: total horizontal and vertical reactions at the bottom of the storey,
δ:
relative horizontal displacement of top and bottom of the storey,
h:
height of the storey.
Η,ν,δ are deduced from a first order analysis of the frame submitted to both
horizontal and vertical design loads (see row 10) and to the global
imperfections of the frame applied in the form of equivalent horizontal
forces (see comments on row 12).
Notes:
­ A same frame could be classified as sway according to a load case (Vsdl for
instance) and as non­sway according to another load case ( Vsd2 for instance)(see
row 13).
V ­ = maximum sdi
For multi­storeys buildings the relevant condition is
V
V cri
condition which is equivalent to
where
Λ
sdi
>
V
V vcny
or acrj are related to the storey i.
94
otcr = minimum (oten),
comments (4/4) on flow-chart FC 4:
* row 15:
At this step of the ULS checks procedure the type of frame is defined as
- braced frame and the first order elastic global analysis of the frame should
be carried out for all ULS load cases,
- or, non-sway frame and the first order elastic global analysis of the frame
might have already been done for all concerned ULS load cases when the
syv
criterion —^— has been chosen (rows 9 to 13).
h2)H
The load cases should consider specific actions in case of braced or non-sway frames
as provided in the table below.
The global analysis of the frame determines the internal forces and moments
(N,V,M) in the members.
The first order elastic global analysis of the frame should take into account
the horizontal
the global
the vertical
actions
loads
loads
imperfections of the
types of
frame
(row 12)'
frames
X(b)
1) braced frames ($
2) non-sway frames (c)
15.253 (3)]
Notes : (a) braced frames are frames which may be treated as fully supported laterally by a
bracing system.
(b) only the part of horizontal loads which are applied to the frame but not assumed
to be transmitted to the bracing system through the floors.
[5.2.5.3(5)]
(c) no special lateral boundary conditions are considered in the frame modelling.
* row 16;
The classification of cross-sections have to be determined before all the ULS checks
of members, cross-sections and webs (rows 18 to 21).
[Annex E]
* row 17:
Nsd
Lh, buckling length of members for non-sway mode
►c^
Lb
* rows 18Λ19,2Q> 2h 22;
The sequence of the Ultimate Limit States checks is not imposed and it is up to the
designer to choose the order of the ULS checks which are anyhow all necessary to
be fulfilled. On the contrary, the sequence of steps to define the assumptions for
the global analysis (row IS) is well fixed and defined in rows 8 to 14.
95
[2.3.2.4]
I V.b
Static equilibrium
(1) For the verification of static equilibrium, destabilizing (unfavourable) actions shall be
represented by upper design values and stabilizing (favourable) actions by lower design
values.
(2) For stabilizing effects, only those actions which can reliably be assumed to be present in
the situation considered shall be included in the relevant combination.
(3) Variable actions should be applied where they increase the destabilizing effects but
omitted where they would increase the stabilizing effects (γς> = 0, in table III.8).
(4) Account should be taken of the possibility that non-structural elements might be omitted
or removed.
(5) For building structures, the normal partial safety factor given in table ΓΠ.8 of chapter ΙΠ
apply to permanent actions (YG = 1,0 if favourable actions).
(6) Where uncertainty of the value of a geometrical dimension significantly affects the
verification of static equilibrium, this dimension shall be represented in this verification
by the most unfavourable value that it is reasonably possible for it to reach.
I V.c
Load arrangements and load cases
r v . c l Generalities
(1) Load arrangements which may be applied to buildings are provided in chapter ULb.
(2) Load cases (see chapter ni.c) may be established according to two procedures to study
structures submitted to actions:
a general procedure presented in flow-chart FC 3.1 (chapter ΠΙ) or,
a particular procedure presented in flow-chart FC 3.2 (chapter ΠΓ) which is
applicable for braced or non-sway buildings because such structure may be studied
by first order elastic global analysis.
(3) Two types of load cases shall be considered:
load cases for Serviceability Limit States and,
load cases for Ultimate Limit States,
where differences are related to combination rules:
see table ΓΠ.7 for SLS combinations of actions
see table ΙΠ.8 for ULS combinations of actions
rv.c.2 Frame imperfections
[5.2.5.3 (4)] (1) In case of braced frame global imperfections are not necessary for the design of the braced
frame itself but they shall be taken into account in the design of the bracing system (see
chapter XII).
(2) In case of non-sway frame global imperfections are needed for the design of the frame.
[5.2.4.1 (l)] (3) Appropriate allowances shall be incorporated to cover the effects of practical
imperfections, including residual stresses and geometrical imperfections such as lack of
vertically, lack of straightness due to welding or lack of fit and the unavoidable minor
eccentricities present in practical connections.
[5.2.4.3 (l)] (4) The effects of imperfections shall be allowed for in frame analysis by means of :
- an equivalent geometric imperfection in the form of an initial sway imperfection φ or,
- equivalent horizontal forces according to table IV.3, either method is permissible.
(5) As shown in table IV.3 the initial sway imperfections of a frame are directly proportionate
to the relevant applied vertical loads of each load case.
Therefore global imperfections of a frame should be calculated for each load case.
96
Global imperfections of the frame
Table IV.3
Initial sway imperfections φ of the frame
equivalent horizontal forces
ECCS ηβ65
table 5 J
F2
1 i i i ·
tel
φΡ2
Fi
'
φ Fi
•
l i l i -
■ νy
φ (Fi + F2)
^
' •
φ (Fi + F2)
[5.2/4.3 (4)] (6) The initial sway imperfections φ apply in all horizontal directions but need only be.
considered in one direction at a time. The table IV.4 gives the numerical values for φ:
φ = k c ks φ 0
[form. (5.2)]
where
Φο=
Ξ55'
k c =Jo,5 + — < 1,0 and
V
nr
k g = j 0 , 2 + — <Ξ1,0
V
nc
is the number of columns per plane
where
n„
is the number of storeys
nc
[5.2.4.3 (2)] (7) Only those columns which carry a vertical load Nsd of at least 50% of mean value of the
vertical load per column in the considered plane, shall be included in nc.
[5.2.4.3 (3)] (8) Only those columns which extend through all the storeys included in n s shall be included
in nc . Only those floor or roof levels which are connected to all the columns included in
nc shall be included when determining n*
[5.2.6.1 (l)]
[5.2.6.1 (2)]
[5.2.6.1 (3)]
[5.2.6.1 (4)]
IV.d
Frame stability
(1) All frames shall have adequate resistance to failure in a sway mode. However, where the
frame is shown to be non­sway, no further sway mode verification is required.
(2) All frames including sway frames, shall also be checked for adequate resistance to failure
in non­sway modes.
(3) A check should be included for the possibility of local storey­height failure mode.
(4) Frames with non­triangulated pitched roofs shall also be checked for snap­through
buckling.
97
Table IV.4
ECCS n°65
table 5.6
Values for the initial sway imperfections φ
number of
columns
inplane
nc = 2
η<;=3
nc = 4
nç=5
Τ
number
of
storeys
I
ι
I
n.= 1
1/200
1/220
1/230
1/240
1/280
ns = 2
1/240
1/260
1/275
1/285
1/335
ns=3
1/275
1/300
1/315
1/325
1/385
ns = 4
1/300
1/325
1/345
1/355
1/420
nSs = oo
1/445
1/490
1/515
1/535
1/630
—
ï
[5.2.1.1]
[5.2.1.2]
!!<;=,
i
I V.e
First order elastic global analysis
IV.e.l
Methods of analysis
(1) The internal forces and moments in a statically determinate structure shall be obtained
using statics.
(2) The internal forces and moments in a statically indeterminate structure may generally be
determined using either:
elastic global analysis
plastic global analysis
(3) Elastic global analysis may be used in all cases.
rv.e.2
Effects of deformations
(1) The internal forces and moments may generally be determined using either:
first order theory, using initial geometry of the structure.
second order theory, taking into account the influence of the deformation of the
structure.
(2) First order theory may be used for the global analysis in the following cases:
braced frames,
non­sway frames,
design methods which make indirect allowances for second­order effects.
(3) Second order theory may be used for the global analysis in all cases.
98
IV.C.3
[5.2.1.3]
Elastic global analysis
(1) Elastic global analysis shall be based on the assumption that the stress-strain behaviour of
the material is linear, whatever the stress level.
(2) This assumption may be maintained for both first-order and second-order elastic analysis,
even where the resistance of a cross-section is based on its plastic resistance (see chapter
V about classification of cross-section).
(3) In order to determine the internal forces and moments (N, V, M) in braced or non-sway
frames, first order elastic global analysis may be used.
(4) Following a first order elastic global analysis, the calculated bending moments may be
modified by redistributing up to 15% of the peak calculated moment in any member,
provided that:
- the internal forces and moments in the frame remain in equilibrium with the applied
loads and,
- all the members in which the moments are reduced have class 1 or 2 cross-sections
(see chapter V).
(5) The load cases should consider specific actions in case of braced or non-sway frames as
provided in table IV. 5 (issued from comments on row 15 in flow-chart FC 4).
Table IV.5
Specific actions for braced or non-sway frames
The first order elastic global analysis of the frame should take into account
*^^
actions
the horizontal
the vertical
the global imperfections
loads
of the frame
| types of ^ " ^ ^ ^ ^ ^
loads
| frames
^*"""-^^
X(b)
X
1) bracedframes (¿)
2) non-sway frames (ς)
[5.2.5.3 (3)]
[5.2.5.3 (5)]
X
X
X
Notes : (al braced frames are frames which mav be treated as fully supported laterally bv
the bracing system.
(b) only the part of horizontal loads which are applied to the frame but not assumed
to be transmitted to the bracing system through the floors.
(c) no special lateral boundary conditions are considered in the frame modelling.
(6) In case of first order elastic global analysis the principle of superposition is applicable
because the effects of actions (E, S) are linear functions of the applied actions (F = G, Q,
...) (no Ρ-Δ effects and used material with an elastic linear behaviour).
The principle of superposition allows to consider a particular procedure to study structure
submitted to actions. This procedure illustrated in flow-chart FC 3.2 could be more
practical because it should simplify the decision of which load case gives the worst effect
(see chapter ΙΠ).
For each single characteristic (unfactored) value of load arrangement (Fk) the global
analysis of the structure determines characteristic (unfactored) values for the effects of
actions : Ek = (δν,δη, f,..)k ; Sk = (N, V, Μ, σ,..\.
AU load cases are defined by relevant combinations of the characteristic (unfactored)
values for the effects of actions (E/dSk). AH these load cases directly furnish the design
values for the effects of actions (Ed = Oy.ôh, f,.·· ; Sd = Ν, V, Μ, σ,...) which shall be
checked at SLS (Cd limits) and at ULS (Rd resistances).
IV.f
Verifications at SLS
(1) The limiting values for vertical deflections and vibrations of beams are given respectively
in chapters Vin.b. 1 and Vin.b.2 (in chapter Vm about members in bending).
99
[4.2.2 (4)]
ECCS n°65
table 4.3
IV.f. 1 Deflections of frames
(1) The limiting values for horizontal deflections of frames given in table IV.6 are
illustrated by reference to the multi­storey and single­storey frame.
Table IV.6
Recommended limits for horizontal deflections
Multi­storey frame
διδ2
Single storey frame
δι < hi / 300
δ 2 < h2 / 300
ôo<h0/500
Portal frame without
gantry cranes
δ < h / 150
Other buildings
δ < h / 300
IV.g
Verifications at ULS
IV. g. 1 Classification of the frame
IV.g.1.1 Hypothesis for braced frame
[5.2.5.3]
(1) Examples of bracing system are mentioned in chapter I.b.2 and in chapter ΧΠ.
[5.2.5.3 (2)] (2) A steel frame may be classified as braced if the bracing system reduces its horizontal
displacements by at least 80 %.
(3) For practical presentation of the criterion used to classify a frame as braced reference
may be made to comments on row 11 of flow­chart FC 4 (see chapter IV.a.2.3).
[5.2.5.3 (3)] (4) A braced frame may be treated as fully supported laterally.
(5) As the criterion of braced or unbraced frame classification is related to the stiffness of the
frame and on hypothetic horizontal loads, the frame should be classified as braced or not
independently of load cases.
rv.q.1.2 Hypothesis for non-swav frame
(1) Examples of sway frames are mentioned in chapter I.b.2.
[5.2.5.2]
(2) In order to define the criterion used to classify a frame as sway or non­sway reference
may be made to comments on row 14 of flow­chart FC 4 (see chapter IV.a.2.3).
(3) As the criterion of sway or non­sway frame classification depends on the total vertical
load, a same frame could be classified as sway according to a load case and as non­sway
according to another load case. Therefore the criterion of sway or non­sway frame
classification should be checked for each load case.
rv.g.2
ULS checks
[5.1.2(1)]
(1) The frames shall be checked at ultimate limit states for the resistances of cross­sections,
members and connections. For those ULS checks reference may be made to the following
chapters:
­ Classification of cross­sections:
see chapter V
­ Members in tension:
see chapter VI
see chapter VH
­ Members in compression:
see chapter VIII
­ Members in bending:
see chapter LX
­ Members with combined axial force and bending moments:
see chapter X
­ Transverse forces on webs:
see chapter XI
­ Connections:
100
V C LASSIFI
C ATION OF CROSS-SECTIONS
V.a
Generalities
(1) For a designer the usual procedure is to choose a cross­section in such a way that the
maximal capacity is not controlled by local buckling but is associated with the bearing
load of a particular member of the structure (column, beam, beam­column). Therefore the
local buckling plays an important role in the design of structural steel.
The critical level over which local buckling appears, is defined by the classification of
cross­sections.
(2) For the check of cross­sections and members at Ultimate Limit States, the steel cross­
sections shall be classified. The classification of cross­sections allows to evaluate
beforehand their behaviour, their ultimate resistance and their deformation capacity, taking
into account the possible limits on the resistance due to local buckling of compression
elements of cross­sections.
(3) The classification of cross­sections permits (see table V.l):
to guide the selection of global analysis of the structure (elastic or plastic global
analysis),
to determine the criteria to be used for ULS checks of cross­sections and members.
(4) Four classes of cross­section are defined according to (see chapters V.b and V.c):
the slenderness of its compression elements (width­over­thickness ratios of web
or flange),
the yield strength of the steel and,
the applied loading.
(5) It is important to precise that the present classification of cross­sections is only based on
the distribution of normal stresses across the section due to the following separate or
combined axial forces and/or bending moments applied to the cross­section:
é
¿Ρ
-
χ -
y"
ϊΡ
Μ z.Sd
Ν.x.Sd
,y
Νx.Sd
χ
Μy.Sd
(6) The present classification of cross-sections is not affected by shear forces (Vz.sd.Vy.Sd)·
The resistance of webs to shear buckling (induced by V^sd) should be checked in chapter
VUI.d.2.
(7) The flow-chart FC 5.1 presents the general procedure to classify I cross-section (see the
following page) . More details are given in chapters V.b and V.c.
(8) The flow-chart FC 5.2 presents a procedure to calculate the effective cross-section
properties of class 4 cross-section where local buckling occurs (see the second following
page). More details are given in chapter V.c.3.
101
Flow-chart fee 5.lJ : Classification of I cross-section
rows:
rows:
1
1
Determine ε = V 235 / fy
I
Division of the cross-section into elements:
web and flanges
J
Class of cross-section =
highest class of all elements
J
Γ Determine the slenderness of element to classify : d/tw, c/tf,... J
(
Type of loading on element to classify
1
τ
Bending
I Compression ;
1
1
ί
Combined Ν + Μ
ι axial load and bending moment ¡
M
Ncomp.
J
Γ
Determine the position of neutral axis
with plastic distribution of stresses
^ C l a s s 1 or 2 element^
X
(*)
/
­ ^ C l a s s 1 or 2 element \
Determine the position of neutral axis
with elastic distribution of stresses
yes
10 h
11
Class 3 element ?
i lyes s
Class 3 element ?
(*♦)
no
to*)
Class 4 element
.with local buckling
Note :
l>
ς
Class 4 element
ith local buckling
(*) see table V.3
(**) see table V.4
102
J
J
2£
<C Class 3 element ? >
X
(**)
f
10
no
( Class 4 element
Udth local buckling
u
F C 5 . 2 ) : Calculation of effective cross-section properties of Class 4 cross-section
Flow­chart (IFC5.21
Approximate method assuming all elements of the cross­section at Ultimate Limit States:
the maximal compressive stress in each element is equal to yield strength (fy).
rows:
rows:
f
1
r~ ■
2
ι
'
Í
Type of loading on cross­section
-
1
"JL
Combined N + M
axial load and bending moment
A
Bending
M
Compression
Ncomp.
τ
I
Only class 4 elements (web and/or fiantes)
have effective properties
"f
Calculate effective
section area Aeff.N
(from table V.8)
yes/
\
Calculate shift
of centroidal axis e ^
(with Acff.M from table V.8)
Bisymmetrical
cross­section
?
Calculate effective
moment of inertia Ieff
7
Calculate shift
of centroidal axis βΝ
8
Determine additional
bending moment
ΔΜ = Ν . e N
10
11
8
Calculate the lowest effective
section modulus Wefr
9
Combined (N+M)
loading(M Φ 0)?
7
yes
9
10
11
Effective properties:
12
­ for Ν or (N, M) loading: Aeff.N ; e N y ; e N z
­ for M or (N, M) loading: Weff.y ; Weff.z
103
12
[5.3.2]
V.b
Definition of the cross-sections classification
(1) Four classes of cross-sections are defined, as follows:
Class 1 cross-sections are those which can form a plastic hinge with the rotation
capacity required for plastic analysis.
Class 2 cross-sections are those which can develop their plastic moment resistance,
but have limited rotation capacity.
Class 3 cross-sections are those in which the calculated stress in the extreme
compression fibre of the steel member can reach its yield strength, but local buckling
is liable to prevent development of the plastic moment resistance.
Class 4 cross-section s are those in which it is necessary to make explicit allowances
for the effects of local buckling when determining their moment resistance or
compression resistance.
(2) Table V. 1 recapitulates the characteristics of each class of cross-section in case of
simply-supported beam.
(3) The ultimate resistance of cross-sections and of members submitted to bending and/or
compression, depends on class of cross-sections and is based on the following properties
(see table V.l):
Cross-section properties
for ULS check formulas
ULS partial
safety factors
plastic properties (Wp¿)
YMO
Class 3
- elastic distribution
- with yield strength reached
in the extreme fibres
elastic properties (We/)
YMO
Class 4
- elastic distribution across
the effective section taking
into account local buckling
- with yield strength reached
in the extreme fibres.
effective properties
(Aeff, eN, Weff)
YM1
Distribution of stresses
across the section
[5.3.4 (2)]
[5.3.4(3)]
Class 1 or 2 - full plastic distribution
- at the level of yield strength
[5.3.5]
[5.3.4(4)]
[5.3.4(5)]
(4) When elastic global analysis is used, particular exemptions to these rules may be made for
the following specific cases:
when yielding first occurs on the tension side of the neutral axis,
when the cross-section is composed of class 2 compression flange and class 3 web.
Those exemptions are not considered in the handbook and reference may be made to
Eurocode 3 Part 1.1 (HI).
104
Table V.l
Definition of the classification of cross-section
1
^qTTjpjjj^
Class
Behaviour model
M
Mpt--
PLASTIC
across full section
7local^
buckling
M
Mpi-
Wtpt
Mei
M
ƒ
fy
elastic
important
or,
plastic
PLASTIC
across full section
\
local
buckling θ
χ-—
Xocal
buckling θ
Mpi
Met
Design resistance
Available
rotation
capacity
of plastic
hinge
Global
analysis
of
structures
I
limited
elastic
ELASTIC
across full section
ƒ
fy
none
elastic
none
elastic
ELASTIC
across effective section
M
local
buckling θ
ƒ
105
fy
V.c
Criteria of the cross-sections classification
V.c. 1
Classification of compression elements of cross-sections
[5.3.2(3)]
(1) The classification of a cross-section depends on the proportions of each of its compression
elements (width-over-thickness ratios of web or flange), on the yield strength of
the material and on the applied loading.
[5.3.2(4)]
(2) Compression elements include every element of a cross-section which is either totally or
partially in compression, due to axial force or bending moment, under the load
combination considered.
(3) In case of combined actions (Nsd and Msd), the limiting proportions for classification of
elements are related to the position of plastic or elastic neutral axis (parameters α or ψ in
tables V.3 and V.4); that position depends on the stresses distribution across the section in
equilibrium with the applied design values of (NSd, MSd). Therefore the classification of
an element or a cross-section may be different according to the considered combination of
actions (Ν, M).
(4) In case of elements submitted to tension (Ntension) local buckling is not expected and the
concerned elements shall be class 1.
V.C.2
Classification of cross-sections
[5.3.2(5)]
(1) The various compression elements in a cross-section (such as a web or a flange) can, in
general, be in different classes.
[5.3.2(6)]
(2) A cross-section is normally classified by quoting the highest (least favourable) class of
its compression elements.
[5.3.2(7)]
(3) Alternatively the classification of a cross-section may be defined by quoting both the
flange classification and the web classification.
For instance, the compression flange of an I-section may be class 1 and its web may
be class 3. Then this I-section is class 3.
But this I-section may also be defined by quoting its class 1 compression flange and its
class 3 web.
(4) The determinant dimensions of cross-sections for classification are provided in table V.2.
(5) In case oil or Η cross-sections, T-sections and channels ( [ ) , the limiting proportions
for classification of elements (webs and flanges) are given :
- in table V.3, for class 1 and 2
- in table V.4, for class 3
(6) In case of rectangular and square hollow sections the limiting proportions for
classification of internal flanges are given in table V.6 for class 1,2 and 3. For
classification of webs of these sections reference may be made to tables V.3 and V.4.
(7) In case of angles and tubular sections the limiting proportions for classification of
elements are given in table V.7 for class 1,2 and 3.
V.c.3
[5.3.2(8)]
Properties of class 4 effective cross-sections
(1) An element of a cross-section (as such a web or a flange) which fails to satisfy the limits
for class 3 should be taken as class 4.
106
The limiting proportions for class 3 compression elements should be obtained from tables
V.4, V.6 or V.7.
[5.34(6)]
(2) When any of the compression elements of a cross-section is class 4 the cross-section shall
be designed as a class 4 cross-section.
[5.3.2(2)]
(3) Effective widths may be used in class 4 cross-sections to make the necessary allowances
for reductions in resistance due to the effects of local buckling.
[5.3.5]
(4) The effective cross-section properties of class 4 cross-sections (Aeff, en, Weff.y, Weff^)
shall be based on the effective widths of the compression elements. The flow-chart FC 5.2
presents an approximate method to determine the effective cross-section properties
assuming all elements of the cross-section at Ultimate Limit States : the maximal
compressive stress in each element is equal to yield strength, fy.
(5) The effective properties of class 4 cross-sections may be obtained from table V.8 or
from Eurocode 3 (J2f) for other cases.
(6) In general the determination of the effective width of a class 4 element may be carried out
as follows (see [5.3.5(3)] of EC3) :
a) determination of buckling factor k 0 corresponding to the stress ratio ψ (see [table
5.3.2] and [table 5.3.3] of EC3),
b) calculation of the plate slenderness λ ρ ; given by :
in which t
kø
ε=
b
is the relevant thickness of the elements,
is the buckling factor corresponding to the stress ratio ψ,
235 (with f in N/mm2),
y
is the appropriate width as follows :
b= d for webs,
b= b for internal flange elements (except RHS),
b= b - 3t for flanges of RHS,
b= c for outstand flanges,
b=
b = h or
c)
for equal-leg angles,
for unequal-leg angles.
calculation of reduction factor ρ with the following approximation ([formula (5.11)]
ofEC3):
. when λ ρ <, 0,673 : ρ = 1
(λρ-0,22)
.when λ ρ > 0 , 6 7 3 : p=> _ 2 —'-
d)
determination of the effective width beff
107
(7) For cases proposed in table V.8 the effective cross-sectional data may be determined as
follows :
a) calculation of λ ρ according to table V.8,
b) calculation of ρ according to the formula given in V.c.3(6) c),
c) determination of effective zones of class 4 elements according to table V.8.
(8) It is important to mention that only class 4 compression elements (web and/or flange)
shall have effective width. For instance, HEA 500 cross-section in S 460 steel grade
subject to uniform compression, has a class 1 flange and a class 4 web; therefore the
effective area (Aeff) issued from table V.8 is composed of full flanges and an effective
web.
ECCS n°65 (9) Where the stresses Osd from effective cross-sectional data are less than fy, the plate
5.3.5(5)
slenderness λ ρ may be decreased by
, which may cause an increase of the effective width.
[5.3.5(6)]
(10) Generally the centroidal axis of the effective cross-section will shift by a dimension e
compared to the centroidal axis of the gross cross-section. This should be taken into
account when calculating the properties of the effective cross-section. Examples are given
in table V. 10.
[5.3.5(7)]
(11) When the cross-section is subject to an axial force, the method given in chapter IX.d.1.4
should be used to take account of the additional moment ΔΜ given by :
AM = N e N
where eN
Ν
is the shift of the centroidal axis when the effective cross-section is subject
to uniform compression (single N),
is positive for compression.
108
V.d
[5.3.1(2)]
Procedures of cross-sections classification for different loadings
(1) Because elastic global analysis is used for braced or non-sway frames (see chapter IV.e),
any class of cross-section may be used for the members, provided that the design of the
members takes into account the possible limits on the resistance of cross-section due to
local buckling (see table V.l).
(2) The class of a cross-section may specifically be determined according to the applied
loading :
- for cross-sections subject to compression, see chapter V.d.l,
- for cross-sections subject to bending, see chapter V.d.2,
- for cross-sections subject to combined (N, M), see chapter V.d.3.
V-d.l
Classification of cross-sections in compression
(1) For cross-sections submitted to uniform compression (Nx.sd) two steps are required for
classification:
1) if using the plastic compression resistance of the cross-section, the limiting
proportions for class 3 sections shall be met for class 3 flange and web submitted
to single Ncompression: see tables V.4, V.6 or V.7; the cross-sectional area A shall be
used.
2) if an element of the cross-section fails to satisfy the limits for class 3 it should be
taken as class 4. The occurence of local buckling in that element should be
considered in calculating the effective cross-sectional area : Aeff (see table V.8).
In the case of class 4 monosymmetrical cross-section the shift of the relevant
centroidal axis (eN) should also be calculated.
V-d.2
Classification of cross-section in bending
(1) For cross-sections submitted to bending moments (My.sd, Mz.sd) three steps are required
for classification :
1) if using the plastic moment resistance of the cross-section, the limiting proportions
for class 2 sections shall be met for class 2 flange and web submitted to bending
moments (single My.sd and/or single M^sd) '· see tables V.3, V.6 or V.7; the plastic
section modulus Wpi shall be used.
2) if using the elastic moment resistance of the cross-section, the limiting proportions
for class 3 sections shall be met for class 3 flange and web submitted to bending
moments (single My.sd and/or single M^sd) : see tables V.4, V.6 or V.7; the elastic
section modulus Wei shall be used.
3) if an element of the cross-section fails to satisfy the limits for class 3 sections it
should be taken as class 4. The occurence of local buckling in that element should be
considered in calculating the effective section modulus of the cross-section when
subject only to bending moment about the relevant axis (Weff.y from single My.sd;
Weff.z from single Mz.sd) (see table V.8).
109
V­d.3
Classification of cross­sections in combined (NM)
(1) For cross­sections submitted to combined axial load (Nx.sd) and bending moments
(My.sd» Mz.sd) three steps are required for classification:
1)
if using the plastic moment resistance of the cross­section, the limiting proportions
for class 2 sections shall be met for class 2 flange and web submitted to combined
axial load and bending moments ((Ncompression or Ntention) and (My.sd and/or Mz.Sd)) :
see table V.3, V.6 or V.7; the cross­secùonal area A and the plastic section modulus
Wpi shall be used.
2)
if using the elastic moment resistance of the cross­section, the limiting proportions
for class 3 sections shall be met for class 3 flange and web submitted to combined
axial load and bending moments ((Ncompression or Ntention) and (My.Sd and/or Mz.Sd)) :
see table V.4, V.6 or V.7; the cross­secüonal area A and the elastic section modulus
Wei shall be used.
3)
if an element of the cross­section fans to satisfy the limits for class 3 sections it
should be taken as class 4. The occurence of local buckling in that element should be
considered in calculating the effective section properties (see table V.8) :
­ Aeff : the effective area of the cross­section subject to uniform compression
(single Nx.sd);
­ in the case of class 4 monosymmetrical cross­section:
e
N (= eNy> eNz): the shift of the relevant centroidal axis when the cross­section is
subject to uniform compression (single Nx.sd);
­ Weff (=Weff.y, Weff.z) : the effective section modulus of the cross­section when
subject only to bending moment about the relevant axis (single
My.sd, single Mz.sd)·
(2) Difficulties are met to determine immediately the class of an element submitted to
combined (N, M) loading because the classification depends on the design values of the
applied axial load Nsd and the bending moment Msd which are obtained from global
analysis of the structure.
The limiting proportions for classification are related to the position of plastic or elastic
neutral axis (parameters α or ψ); that position depends on stresses distribution across the
section in equilibrium with those design values of (Nsd, M$d)· Therefore the classification
of an element or a cross­section may be different according to the considered combination
of actions (Ν, M).
Then assumptions of class should be tried and verified with the results issued from the
global analysis.
(3) The class of cross­section submitted to combined (Nsd* Msd) loading could simply be
determined in taking into account more severe loading which allows an easier evaluation
of the elements class. If the limiting proportions are met and correspond to a satisfying
class, complex calculations (positioning of neutral axis) should have been avoided. Two
examples illustrate this proposal :
in case of web submitted to Ncompression and My. sd, it is easier to classify firstly the
web submitted to Ncompression>(see tables V.4, V.6 or V.7),
in case of web submitted to Ntension and My.sd, it is easier to classify firstly the web
submitted to single My^sd­(see tables V.3, V.4, V.6 or V.7).
110
(4) In case of I or H-sections submitted to bending about major axis (Mysd) and axial load
(Nx.sd). the classification of the web may be determined with table V.9 by comparison of
the applied design axial load (Nx.sd) with the given limiting axial load (in compression or
in tension).
The table V.9 should be used (/9/) :
firstly by check of the limiting ratios between the applied axial load Nx.sd and the
plastic load of the web (= Aw-fy), to determine if the web is class 1 or class 2 (in this
case the ultimate limit state is based on plastic distribution of stresses across the
section);
and if limits for class 2 are not met, by check of the limiting ratios between the
applied axial load Nx.Sd and the plastic load of the full section (= A.fy), to determine
if the web is class 3 or class 4 (in this case ultimate limit state is based on elastic
distribution of stresses across the section).
111
Determinant dimensions of cross-sections for classification
Table V.2
- Webs (internal elements perpendicular to axis of bending)(see tables V.3 and V.4)
JS
Axis of
bending t -
J
.__d
*)
_h
1
d_.
^W
í Γ
IW
d = h - 3t (t = tf = t j
Rolled sections
Welded sections
- Outstand flanges (see tables V.3, V.4 and V.5) :
4 ί
^Ή
I *Ê j¿ fc 1
i,
c **)
in
—*—
ι
ΓΊ
i
i
1
—*—
-Ί
r
c **)
Welded sections
Rolled sections
- Internal flange elements (internal elements parallel to axis of bending)(see table V.6) :
TT
Axis of
tending
I
.
.
.1
t .
\
Rolled sections
i
%
Welded sections
- Circular tubes and angles (see table V.7) :
Ψ
-Ml—
*)
**)
For a welded section the clear web depth d is measured :
. between welds for section classification
. between flanges for shear calculations (see chapter VIII)
For welded sections the outstand dimension c is measured from the toe of the weld.
112
Table V.3
Types
of
loading
Classification of cross-section : limiting width-to-thickness ratios for
class 1 & class 2 I c ross-sec tions submitted to different types of loading
Class 1
Class 2
Stresses
Web
Flange
distribution for
Web
Flange
class 1 & class 2
oltt<
c/tf£
d/tw*
d/tw*
I
+
I fy
Ncompression
N
1
Μν
33ε
L
ι
+
ι
Ι
+
Ι
fy
-EEf-7'
R
10ε
R
11ε
38ε
W
9ε
W
10ε
R
10ε
R
11ε
83ε
72ε
W
9ε
W
10ε
R
10ε
R
11ε
W
9ε
W
10ε
α > 0,5 : R
10ε
α>0,5
R
11ε
456ε
13α-1
W
10ε
Μ,
ΙΓΓ-ΤΕΓ-Τ
-•N.M.
■Ν comp. - My
Έ
Ι
1
fy
"•àdH±i
Ntcns. - My
N
■5
396ε
13α-1
W
9ε
-«S.My
a<0,5
R
10ε
Ψ
9ε
R
10ε
^
36ε
α
iy
Μ2
Ncomp. " Μ ζ
Ι
­
Ι
α < 0,5 : R
11ε
\
33ε
fy
41,5ε
α
W
10ε
Æ
11ε
38ε
W
9ε
W
10ε
R
10ε/α
R
Ιΐε/α
Ntens. - Μ ζ
Values of d, t w , c, and tf + : stresses in compression
are defined in table V.2 - : stresses in tension
fy (N/mm2)
= ^2357ζ
W 9ε/α
V^ 10ε / α
R = rolled sections ;
W = welded sections
235
460
275
355
420
ε(ιίΐ£40ητπι)
0,92
0,81
0,75
0,71
ε (if 40 mm < t < 100 mm)
0,96
0,84
0,78
0,74
113
Table V.4
Types
of
loading
N,compression
Classification of cross­section : limiting width­to­thickness ratios for
class 3 I cross-sections submitted to différent types of loading
Stresses
Class 3
distribution for
Web
Flange
class 3
d / ui-w^
g/tfjj
I + I fy
R
15ε
N
42ε
W
14ε
R
15ε
W
14ε
R
23ε Λ /057
W
2^057
R
15ε
W
14ε
ψ<­1 :
R
15ε
62ε(1­ψ) Λ /­ψ
W
14ε
ƒ?
23εΛ/ϋ^~(δ)
W
21εΛσ w
/?
23εβζ(α)
i—E
l
l
fv
W
—Γ—
M,
M,
+
~+*sMy
124ε
Η-Ι---Τ
ψ>­1 :
■N comp. " M y
42ε
(l¥fy|<
Ν)
0,67 + 0,33ψ
( lfy/ψΙ < |fy|)
.
d.__
Ntens. ­ M .
i
^
7fy./V
^
■Ν comp. " M z
» ^
- ^
fl-Ff-5"
H 9
fy,
Ntenc - M ,
N
­*>.M y
Values of d, tw , c, and tf
are defined in table V.2
ε = ^2357Τι
^\MZ
+ : stresses m compression
­ : stresses in tension
fv (N/mm2)
ε (if t < 40 mm)
ε(ΐί40πιηι<ΐ<100πιπι)
114
42ε
ψ
2Ì
B ^jk^(a)
R = rolled sections;
W = welded sections
ko is defined in table V.5
235 275 355 420 460
0,92 0,81 0,75 0,71
0,96 0,84 0,78 0,74
Buckling factor k0 for outstand flanges
Table V.5
Ψ
kc
-1,0
-0,9
-0,8
-0,7
-0,6
-0,5
-0,4
-0,3
-0,2
-0,1
-0,0
0,85
0,82
0,78
0,75
0,72
0,69
0,67
0,64
0,61
0,59
0,57
+0,0
+0,1
+0,2
+0,3
+0,4
+0,5
+0,6
+0,7
+0,8
+0,9
+ 1,0
0,57
0,55
0,53
0,51
0,50
0,48
0,47
0,46
0,45
0,44
0,43
Stress distribution
(compression positive)
M
Stress distribution
(compression positive)
kc
23,80
20,05
16,64
Compression
13,58
10,86
8,48
6,44
4,74
3,38
2,37
1,70
Compression
σι
1,70
1,31
1,07
0,90
0,78
0,69
0,61
0,56
0,51
0,47
0,43
(b)
Compression
(c)
Compression
σ
L
J
Η
\
2
, .w
"
\
'l
\
+
ί
\
„ ........................
(d)
G21 £ Ι σι
Note 1
ψ = σ2/σι
Note 2 :
The diagram shows a rolled section. For welded members the outstand
dimension c is measured from the toe of the weld (see table V.2).
and
115
;
Table V.6 C lassification of cross-section : limiting width-to-thickness ratios for
internal flange elements submitted to different types of loading
Type of loading
Stresses distribution
classes 1,2 and 3
1
h
+
fy
~Ρ
internal flange
ι
Ν
1
1
N compression
R
(b-3tf)/tf
O
b/tf
<42ε
! ι
I
1
- ■ !
r
<42ε
class 1
]fy
class 2
internal flange
r>
R α>-3ΐ£)Λί<33ε R (b-3tf)/tf£38e
O
M
b/tf
<33ε O
b/tf
<38ε
class 3
+
]fy
internal flange
+/ - * ^
-i-— V
£\ J
R
(b-3tf)/tf
O
b/tf
<42ε
<42ε
Values of b and tf are defined in table V.2
+ : stresses m compression
R = rolled hollow sections
- : stresses in tension
O = other sections
fy (N/mm 2 )
ε = Λ/2357Γ3
275
355
420
460
ε (if t{ < 4 0 m m )
0,92
0,81
0,75
0,71
ε (if 40 mm < tf < 100 mm)
0,96
0,84
0,78
0,74
116
235
Table V.7 : Classification of cross-section : limiting width-to-thickness ratios for
angles and tubular sections submitted to different types of loading
Angles
Note : this table does not apply to angles in continuous contact with other components
Type of loading
Stresses distribution
]*
class 1
class 2
class 3
h/t <
h/t <
h / t < 15 ε and
10 ε
11ε
N,compression
b+h
2t
< 11,5 ε
M and,
see table V.3 (classes 1 and 2) and table V.4 (class 3) with limiting
(Ν, M)
width-to-thickness ratios concerning outstand flanges.
Tubular sections
Type of loading
class 1
class 2
class 3
Ν compression
d/t <
M and,
50 ε2
(N,M)
Values of h, b, t and d are defined in table V.2
fy (N/mm2)
ε = ^235/f 3
70 ε2
90 ε2
+ : stresses in compression
275
355
420
460
ε (if t £ 40 m m )
0,92
0,81
0,75
0,71
ε (if 40 m m < t < 100 m m )
0,96
0,84
0,78
0,74
ε2 (if t ^40 m m )
0,85
0,66
0,56
0,51
ε 2 (if 40 m m < t < 100 m m )
0,92
0,70
0,60
0,55
117
235
Table V.8 Effective cross-sectional data for symmetrical profiles (class 4 cross-sections)
Members in compression (N)
gross cross-section
effective cross-section
'Ρ»
V,."®.
Φ
b
1
t. ε 56,8
ft
!
1
b
ί.ε 18,6
μ^
u
s>_
-■ +
Ν
Φ
b
Aeff
tf
1
t. ε 56,8
Il
Il
Il
II
II
II
Aeff
*"
Members in bending (My, Mz)
I
+
I
Φ
p
b
1
t-ε 138,8
i
b
_b£> Φ ^
®-b©
T- ®
6 p
=t°' '
b 1
ί.ε 18,6
kl
φ
b
1
ί.ε 21,4
Φ
b
1
ί.ε 138,8
3D
ε = ^/2357Γ5
118
weff
Weff
T- p ®' b ©
fb©
ft
Weff
°' 6 -P©+ b ©
235
ε (if t < 40 mm)
ε (if 40 mm < t < 100 mm)
®4-Γ b©(
-Ζ£ ρ ® : ω
b
1
ι.ε 56,8
fy (N/mrn^)
°'4-P<D-i-b®
Η
275
355
420
460
0,92
0,96
0,81
0,84
0,75
0,71
0,78
0,74
Limiting values of axial load Nsjfor web classification of I cross-sections 1
subject to axial load NSd and to bending according to major axis Mysd
Coefficient
dl(tw.e)
32
34
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
Nsd/(A w .fy)
Nsd/(A w .fy)
Nsd/(A.fy)
for Classi web
for Class 2 web
for Class 3 web
Ζ
O
HH
CA
CA
S
cu
O
u
~ΊΓ~
74
76
78
80
83
85
90
95
100
105
110
115
120
"Î24
125
130
135
140
145
150
z
o
HH
co
Ζ
ω
|
·)
*)
*)
0,946
0,846
0,757
0,677
0,604
0,538
0,478
0,423
0,372
0,325
0,282
0,242
0,204
0,169
0,136
0,106
0,077
0,050
0,024
0,000
­0,023
­0,045
­0,077
­0,100
­0,133
­0,153
­0,200
­0,242
­0,280
­0,314
­0,345
­0,374
­0,400
­0,419
­0,424
­0,446
­0,467
­0,486
­0,503
­0,520
*)
*)
*)
*)
*)
*)
0,908
0,824
0,748
0,679
0,615
0,557
0,503
0,453
0,407
0,363
0,323
0,285
0,250
0,217
0,186
0,156
0,128
0,102
0,077
0,053
0,031
0,000
­0,024
­0,078
­0,126
­0,170
­0,210
­0,245
­0,278
­0,308
­0,331
­0,336
­0,362
­0,385
­0,407
­0,428
­0,447
*)
COMPRESSION
Table V.9
L_
ζ
o
HH
c«
Ζ
W
Η
Values of d, tw are defined in table V.2
*)
Ζ
O
HH
CA
CC
tí
tu
O
I
ζ
oCA
Ζ
tí
0,931
0,868
0,811
0,758
0,709
0,663
0,621
0,582
0,545
0,511
0,479
0,449
0,421
0,394
0,369
0,345
0,322
0.301
0,280
0,252
0,234
0,192
0,155
0,121
0,091
0,063
0,038
0,015
0,000
­0,004
­0,023
­0,040
­0,057
­0,071
­0,085
Aw = d.tw (web areeι); Α = sectional area
119
Table V. 10 Examples of shift of centroidal axis of effective cross-sections
1. in case of monosvmmetrical class 4 cross-sections submitted to uniform compression
\N compression) ·
TIJ
Γ-1
'N
2. in case of class 4 cross-sections submitted to bending (My.Sd)
T
II
II
II
ι
eMi: =
ι
t: = :=0
My.Sd
"
e
My.Sd
Mf
=*)
1
Notes :
-1-1
centroidal axis of gross cross-section
-2-2
centroidal axis of effective cross-section
- elements :
'
r
non-effective zone of the element, taking into account the
occurence of local buckling.
120
VI
MEMBERS IN TENSION (Ntension)
Vl.a
Generalities
(1) For each load case (see chapter ΙΠ) the global analysis of the frame (see chapter IV)
determines the design values for the following internal force which is applied to
members in tension and which shall be checked at ultimate limit states :
r£
¿?
Nx.Sd
.-y
x· -
&
(2) The flow­chart FC 6.1 presents the general procedure to check members in tension
at ultimate limit states (see the following page).
(3) The flow­chart FC 6.2 presents the particular procedure to check at ultimate limit states
angles connected by one leg and submitted to tension (see the second following page).
(4) The table VI. 1 provides a list of the checks to be performed at Ultimate Limit States for
the member submitted to axial tension (Ntension)· A member shall have sufficient bearing
capacity if all the checks (from φ ( 1 ) to φ ( 4 ) ) are fulfilled. Several checks (from φ ( 2 )
to φ ( 4 ) ) concern particular cases with specific conditions. All the checks have both
references to Eurocode 3 and to the design handbook.
121
Row-chart (FC 6.l) : Members in tension (Ntension)
revs;
C Determine ULS load cases J
ULS checks
1
Select stronger section
Έ
Select beam size (A, Anet) and steel grade (fy, fu)
I
J
Determine the design tensile force from
global analysis of the structure:
Nsd
Calculate the design plastic resistance
of the gross cross-section : Np£Rd
Calculate the design ultimate resistance
of the net cross-section (Anet) at holes for fasteners:
Nu.Rd
Determine the design tension resistance
of the cross-section:
Nt.Rd = min (Np£Rd, Nu.Rd)
no
Select stronger section
J
Adopt section
122
Flow­chart (FC 6.2J : /Inpfes connected bv one lee and submitted to tension
rows:
\^^^y/
rows:
D
Determine ULS load cases
ULS checks
Select stronger section
f—
,
.
Select angle size (A, An«) and steel grade (fy, fu)
J
Determine the design tensile force from
global analysis of the structure:
Nsd
Calculate the design plastic resistance
of the gross cross­section : Np£Rd
yes
Angle connected
by one leg
yes
7
Unequal­leg angle
connected by its
smaller leg ?
Calculate A* as the gross area
of an equivalent equal­leg
angle of leg size equal to that
of the smaller leg
Bolted
connection
10
11
12
ι
' Welded lap joint
end connection
Calculate the net section
A*net from A*
C
I
13
(
Calculate the design ultimate resistance
of the bolted net section or welded gross section:
ï
Determine the design tension resistance
of the cross­section:
NuRd = min (NpCRd, Nu.Rd)
Select stronger section
15
16
UseA*^)
Nu.Rd
Ζ
14
Type of
connection
Consider A* equal to the
gross area of the angle
(A)
C
Adopt section
123
J
»c
16
List of checks to be performed at ULS for the member in tension (Ntension)
Table VI.1
φ
Axial tensile force iV^ Sd ■
** General case:
(1) Resistance of gross cross-section to Nxsd :
[5.4.3 (1)]
Nx.sd — Np£Rd (design plastic resistance of the gross cross­section)
References :
Vl.b.l (1)
** Particular cases:
(2) Resistance of the net cross-section to Nxsd if holes for fasteners :
[5.4.3 (1)]
[5.4.2.2]
Nx.Sd — N u .Rd
(design resistance of the net cross­section considering the
net area of a member or element cross­section, A net )
VI.b.2 (1)
Resistance of net cross-section to Nxsd if angle connected by a
single row of bolts in one leg:
(3)
Nx.Sd
[6.5.2.3 (2)]
— N u .R(j (design ultimate resistance of the net cross­section, A„et)
VI.c.l (1)
considering the following cases for determination of Anet:
­ either, if unequal­leg angle connected by its smaller leg:
A net = the net section area of an equivalent equal­leg
angle of leg size equal to that of the smaller leg,
­ or, in other cases (equal­leg angle or unequal­leg angle
connected by its larger leg) :
A net = the net section area of the angle
Resistance of cross-section to Nxsd if angle connected by
welding in one leg:
(4)
[6.6.10(2)]
[6.6.10(3)]
N x .Sd — N u .Rd (design ultimate resistance of the cross­section, A)
VI.C.2 (1)
considering the following cases for determination of A:
­ either, if unequal­leg angle connected by its smaller leg:
A=
the gross cross­section area of an equivalent
equal­leg angle of leg size equal to that of the
smaller leg,
­ or, in other cases (equal­leg angle or unequal­leg angle
connected by its larger leg) :
A=
the gross cross­section area of the angle
VLb
General verifications at ULS
VI.b. 1
Resistance of gross cross­section to Ntension
(1) For members in axial tension the design value of the tensile force Nx.sd at each
cross­section shall be checked for gross section yielding :
[5.4.3 (1)]
Ν x.Sd < N pf.Rd
where
Af,
ΎΜΟ
NP£Rd is the design plastic resistance of the gross cross-section,
A
fy
is the gross cross-section (see table VI.2),
is the yield strength (see table II.4),
ΎΜΟ
is a partial safety factor (see table 1.2).
124
vi,b.2
Resistance of net cross-section to Nlcn,1(>n
(1) For members in axial tension the design value of the tensile force Nx.sd at each
cross-section shall be checked for net section rupture at holes for fasteners :
I5A3(1)1
[54.2.2]
N
x.Sd^Nu.Rd =
0*9 A net f „
ΎΜ2
where Nu.Rd is the design ultimate resistance of the net cross-section,
A net is the net area of a member or element cross-section with appropriate
deductions for all holes and other openings (see table VI.2),
f„
is the ultimate tensile strength (see table II.4),
7Ki2 is a partial safety factor (see table 1.2).
IF
Table VL2
Note:
Gross and net cross-sections
-A
= gross cross-section
- Anet = net area of cross-section
l) Non staggered, holes ;
Νx.Sd
Nx.Sd
A
= section 1-1
Anet = section 2-2
2Ί Staggered holes :
Ii2
-1—
I
Νx.Sd
■*—&"-rit
l
Νx.Sd
' i
—φ—(f-f1
• i . rÅ.
­A
= section 3­3
­ Anet = smaller of (section 1­1; section 2­2)
3) Angles with holes in both legs :
C
; spacing of the centres of the same two holes measured
perpendicular to the member axis
125
VI.C
[5.4.3 (3)]
Particular verifications at ULS for angles connected by one leg
(1) In these particular cases the effects of eccentricities in the connections may be neglected
with the following considerations of this chapter. Those considerations should also be
given in a similar way to other types of sections connected through outstands such as
T-sections ( Τ ) and channels ( [ ).
(2) The flow-chart FC 6.2 intends to present the particular cases of this chapter.
VI.c.l
Connection with a single row of bolts
[6.5.2.3 (2)] (1) Angles in tension (N x .sd) connected by a single row of bolts in one leg may be treated
as concentrically loaded with the following requirements :
for a 1 bolt connection
Nx.sd £ N u . R d
for a 2 bolts connection
N
for a 3 bolts connection :
Nx.sd ^ Nu.Rd
where
NujRd
&2
do
t
fu
ΎΜ2
ß2, ß3
Anet
ECCS n° 65
table 5.33
2,0(e2-0,5d0)tfu
ΎΜ2
A
f
x.Sd^N u - R d _ ß 2 n e t u
5
YM2
_ ß3 A n e t f u
»
ΎΜ2
is the design ultimate resistance of the net section,
is the edge distance from the center of a fastener hole to the
adjacent edge of the angle (see table VI.4),
is the hole diameter,
is the material thickness,
is the ultimate tensile strength (see table Π.4),
is a partial safety factor (see table 1.2),
are reduction factors dependent on the pitch p i (see table VI.3),
is the net area of the angle (see table VI.4) :
- if unequal-leg angle connected by its smaller leg, then A n e t = net
section area of an equivalent equal-leg angle of leg size equal to
that smaller leg,
- or, in other cases (equal-leg angle or unequal-leg angle connected
by its larger leg) : A n e t = the net section area of the angle.
Table VI.3
Reduction factors & and/k
Pitch
pi
< 2,5 do
3,3 do
3,75 do
4,2 do
>5do
2 bolts
ß2
0,4
0,5
0,55
0,6
0,7
3 bolts and more
ß3
0,5
0,6
0,7
For intermediate values of pi the values of ß 2 and ß3 may be determined by linear interpolation.
126
Table VL4
1)
Connection of angles
Parameters for bolted connections :
t
4
Nx.Sd
N.x.Sd
O-O
** m
» m-
Pi Pi
2)
Sf
e
i
An»t. net area of the bolted angle :
2.1) if unequal­leg angle connected by its smaller leg
-τι
£
o
-e
^
2.2) if unequal­leg angle connected by its larger leg or if equal­leg angle
-II - j j
L
&
"Θ
\ .
3)
A. cross­sectional area of the welded angle :
3.1) if unequal­leg angle connected by its smaller leg :
b
c
((((((((((((((((
~lr
I
ΙΓΓΓΓΓΓΓΓΓΓΓΓΓΤΤΠ
3.2) if unequal­leg angle connected by its larger leg of if equal­leg angle
b^,
h
l i "Tí {
ί
ffrrrrrrtrrrrrrr.
Τ
{πτΤΤΓΓΓΓΓΓΤΤΤΤΤΓ
127
VI.C.2
[6.6.10(2)]
Connection bv welding
(1) Angles in tension (Nx.sd) welded by one leg may be treated as concentrically loaded with
the following assumptions :
N x .sd^N u.Rd where
Af,
YMO
Nu.Rd
is the design ultimate resistance of the cross-section,
A
is the cross-sectional area of the angle (see table VI.4) :
- if unequal-leg angle welded by its smaller leg
then A = the gross cross-section area of an equivalent equal-leg
angle of leg size equal to that of the smaller leg,
- or, in other cases (equal-leg angle or unequal-leg angle welded by
its larger leg) : A = the gross cross-section area of the angle,
fy
is the yield strength (see table Π.4),
YMO
is a partial safety factor (see table 1.2).
128
ΥΠ
MEMBERS IN COMPRESSION (Ncompression)
VJl.a
Generalities
(1) For each !oad case (see chapter ΙΠ) the global analysis of the frame (see chapter IV)
determines the design value for the following internal force which is applied to members
in compression and which shall be checked at ultimate limit states:
i£
3*
-
χ -
Ν x.Sd
..-y
X
&
(2) The flow-chart FC 7 presents the general procedure to check members in compression at
ultimate limit states (see the following page).
(3) The table VILI provides a list of the checks to be performed at Ultimate Limit States for
the member submitted to axial compression (Ncompression)· A member shall have sufficient
bearing capacity if all the checks (from (J)(l) to (J)(9)) are fulfilled. Several checks (from
(T)(3) to φ ( 9 ) ) concern particular cases with specific conditions. All the checks have both
references to Eurocode 3 and to the design handbook.
129
Flow-chart (FC 1) : Members in compression (Ncompression)
rows:
ί Determine ULS load cases J
1
Τ
ULS checks
C
!T~¡
\
Select beam size (A, 1,1) and steel grade (fy)
,
·
χ.
\
­r .^
\
Γ7Τ
Select stronger section
^
ρ
—
ί Determine the design tensile force from global analysis of the structure: Nsd J
i
Classify the cross-section in compression
J
ι
ι
/■
'
\ r
ι Class 1, 2 or 3 cross-section κ·—I Class of cross-section
)
I
'
I
T
V
J
f—
~ι
Calchiate the design compression
resistance of the cross-section: NcRd
yes/
\
1
,
Determine the buckling length Lb
of the member for each axis : Lb.y, Lb.z
Bisymmetrical
cross-section?
Determine additional bending moment
ΔΜ = Ν . e N
to be checked with (N,M) interaction
±Z
ί Class of cross-section J
__J"
I
Calculate the shift
of centroidal axis: e^
ι
Buckling resistance of the member
ι
J
1 of cross-section
Calculate effective area
Aeff and ratio β A = Aeff/ A
Nsd < NcRd
yes j
ι
π Class 4 cross-section ι
L
X
ι
1
1 r--*
τ
ι Class 1,2 or 3 cross-section ι ι Class 4 cross-section ι
_ _ -
s
f
,
1
3
Calculate the non-dimensional slenderness
ratio λιοί the member for each buckling axis: λy, λζ
1
Q Multiply λy and λζ by VßÄ )
C
ι Select appropriate buckling curvei
*
t
J
'
1
I Determinere reduction factor χ for each buckling axis: χ , χ J
j,
ï
1
Is
r
«I
Calculate the design buckling resistance ¡of the
member Nb.Rd for each buckling axis: Nby.F(d, Nbz.Rd
t
ί Multiply Nby.Rd and Nbz.Rd by βΑ J
<^NSd < min(Nby.Rd, Nbz.Rd)^>-
no
23
yes j
Γ Adopt section J
24
130
Table VILI
List of checks to be performed at ULS for the member in compression
IN compression)
(ï) Axia| compressive force N* M :
** General cases:
(1) Resistance of cross-section to Nxsd ■'
[544(1)J
Nx.Sd — Nc.Rd (design compression resistance of the cross-section J
(2)
[Annex G]
[Annex G]
[54.8.3 (2)]
Vn.c.l (1)
Stability of member to Nxsd ■'
Nx.Sd ^ NbJld (designflexuralbuckling resistance of the member)
[5.5.1.1 (1)J
References :
and, Nx.sd ^ design torsional buckling resistance of member
and, Nx.sd ^ design flexural­torsional buckling resistance of member
** Particular cases:
(3) Resistance of cross-section to Nxsd, if class 4 monosymmetrical
cross-section:
interaction (Nx.sd, AMy.sd, AMz.sd) ^ 1
w h e r e A M s d = N X .sd-eN (= additional moment due to the eccentricity of the
centroidal axis of the effective cross-section, eN)
Vn.c.2.1 (2)
Vn.c.2.2
Vn.c.2.2
Vn.d.l (1)
■
(4) Stability of member to Nxsd if class 4 monosymmetrical crosssection:
[5.54(5)]
interaction (Nx.sd , AMy.sd, AMz.sd) ^ 1
Vn.d.2 (1)
where AM$d = N x .Sd-CN (= additional moment due to the eccentricity of the
centroidal axis of the effective cross-section, eN)
(5) Stability of member to Nxsd
if class 4 monosymmetrical cross-section,
i/cN.y
*0and,
if λ w > 0,4 (potential lateral-torsional buckling): Vm.e.2 (3)
[53.2(7)]
[5.5.4 (6)]
interaction (Nx.sd » AM y .sd, AMz.sd) ^ 1
Vn.d.2 (2)
where AMsd = N x .sd-6N (= additional moment due to the eccentricity of the
centroidal axis of the effective cross-section, e^)
(6)
[6523 (2)]
Resistance of net cross-section to Nxsd if angle connected by a
single row of bolts in one leg:
N x .Sd ¡» Nuüd (design ultimate resistance of the net cross-section, Α,,^)
considering the following cases for determination of Anet:
- either, if unequal­leg angle connected by its smaller leg:
Anet = the net section area of an equivalent equal­leg
angle of leg size equal to that of the smaller leg,
­ or, in other cases (equal­leg angle or unequal­leg angle
connected by its larger leg) :
Anet = the net section area of the angle
(checks nr φ to be continued)
131
Vn.e.l.l(l)
Table VILI
List of checks to be performed at ULS for the member in compression
(■N compression)
(T)
References :
Axial compressive force Nr- KA :
** Particular cases:
(continuation)
(7) Stability of member to Nx_sd if angle connected by a single row of
bolts in one leg:
N x .Sd— Nb.Rd (design flexural buckling resistance of the member considering the
gross cross­sectional area of the angle, A)
[6.5.2.3 (3)]
VILe. 1.2 (1)
w i t h Nbjid ^ N u .Rd
(design ultimate resistance of the net cross­section presented in φ(6))
Resistance of cross-section to Nx¿d if angle connected by
welding in one leg:
(8)
[6.6.10(2)]
[6.6.10(3)]
N x .Sd — N u .Rd (design ultimate resistance of the cross­section, A)
Vn.e.2.1 (1)
considering the following cases for determination of A:
­ either, if unequal­leg angle connected by its smaller leg:
A=
the gross cross­section area of an equivalent
equal­leg angle of leg size equal to that of the
smaller leg,
­ or, in other cases (equal­leg angle or unequal­leg angle
connected by its larger leg) :
A=
the gross cross­section area of the angle
(9)
[6.6.10(3)]
Stability of member to Nxsd if angle connected by welding in one
leg:
Nx.Sd — Nbj^d (designflexuralbuckling resistance of the member considering
the gross cross­sectional area of the angle, A)
VILb
Vn.e.2.2 (1)
Classification of cross­sections
(1) At ultimate limit states the resistance of cross­sections may be limited by its local
buckling resistance. In order to take into account that limitation the different elements
(flange, web) of the cross­sections shall be classified according to the rules defined in
chapter V.
(2) For cross­sections submitted to uniform compression (Nx.sd) the classification may
specifically be determined according to the procedure defined in chapter V.d.l.
132
VILe
General verifications at ULS
VII.C. 1 Resistance Of CrOSS-SeCtion tO Ncompression
[544 (1)]
(1) For members in axial compression, the design value for the compressive force Nx.sd at
each cross-section shall satisfy:
N
c.Rd depending on classes of cross-section:
[form. (5.16)]
Nx.Sd ^
N
cRd
where N c R d
Νp£Rd
A
Aeff
fy
ΎΜ0.ΥΜ1
[5.44 (5)]
Class 1,2 or 3
Af v
= Npf.Rd=-^
ΎΜΟ
class 4
_ Aeff f y
YMI
is the design compression resistance of the gross cross-section,
is the design plastic resistance of the cross-section,
is the area of the gross cross-section,
is the effective area of the cross-section (see chapter V),
is the yield strength (see table Π.4),
are partial safety factors (see table 1.2).
(2) Fastener holes need not to be allowed for in compression members, except for oversize
and slotted holes.
VII.C.2 Stability of member to Nœmpiession
(1) The stability of members submitted to concentrical compressive force shall be checked
according to the following buckling modes : flexural buckling, torsional buckling and
flexural-torsional buckling.
VII.c.2.1 Resistance toflexuralbuckling
(1) The compression members shall be checked to flexural buckling mode (buckling by plane
bending) according to both principal axes of the section (major axis: yy; minor axis: zz)
with the appropriate buckling lengths (Lb.y, LD.z).
[53.1.1 (l)] (2) For members in axial compression the design value of the compressive force Nx.sd shall
satisfy:
Nb Rd depending on classes of cross-section:
[form. (5.45)1
N
X.Sd ^ Nby.Rd
Nx.Sd ^ Nbz.Rd
Class 1, 2 or 3
_ Xy A f y
Class 4
_ Xy Aeff fy
ΎΜΙ
_ XZ A f y
ΎΜΙ
_ XZ Aeff f y
ΎΜΙ
ΎΜΙ
where N ^ j ^ , N b ^ R d , Nbjid are the design buckling resistances of compression member
about y and ζ axes, and in general,
Xy, Xz
are the reduction factors for the buckling mode about y and
ζ axes,
A
is the area of the gross cross-section,
Aeff
is the effective area of the cross-section (see chapter V),
is the yield strength (see table Π.4),
ΎΜ1
is a partial safety factor (see table 1.2).
133
[5.5.1.2 (i)] (3) For constant axial compression in members of constant cross-section, the value of χ
(Xy> Xz ) is related to the appropriate non-dimensional slenderness λ ( λ γ , λ ζ ) :
[form. (5.46)]
x = f(A) =
ι2
Φ+νψ -λ?
, buttø <1|
where φ = 0,5[ΐ + α ( λ - 0 , 2 ) + λ 2 ] ,
α
is an imperfection factor (see table VII.2), depending on the appropriate
buckling curve.
The buckling reduction factor χ is given in function of λ and the appropriate buckling
curve in table VII.6. When λ < 0,2 flexural buckling is not a potential failure mode.
Imperfection factors a
Table VH.2
Buckling curve
Imperfection factor α
a
ao
0,13
0,21
b
0,34
c
d
0,49
0,76
(4) The appropriate buckling curve of a member depends on the type of cross-section. For
hot-rolled I-sections the buckling curve also depends on steel grades (see table VQ.4).
(5) The non-dimensional slendernesses (Xy,Xz) shall be taken as:
xy=^VßI and *·*=·?■ VßA
where (λ*νy,λ
) are the slendernesses of the member:
' v zζ
and
[form.(547)l
where L0.y, LD.Z
are the buckling lengths of the member about y and ζ axes,
iy, iz
where
λι
βA
are the radius of gyration about the y and ζ axes determined
using the properties of the gross cross­section (Iy, Iz and A),
is the Euler slenderness for buckling (see table VII. 3),
is a factor considering the effect of local buckling if class 4 cross­
section:
, for class 1, 2 or 3 cross­sections,
ßA=l
D
_ Aeff
, for class 4 cross­sections.
Va ue of Euler slenderness Xj
Table VH.3
Steel grade
S 235
S 275
S 355
S 420
S 460
λ ι = π
93,91
86,81
76,41
70,25
67,12
­ ^
134
[5.5.14(1)]
[table 5.5.3]
Selection of buckling curve for a cross-section
Table VIL4
Cross-section
Limits
Buckling
about axis
with h/b > 1,2 and :
. tf ^ 4 0 mm
Rolled I-sections :
b
. 40<tf < 100 mm
with h/b <, 1,2 and :
tf £ 100 mm
y_-
Buckling curves
for steel grades
S 235
S 420
S 460
to S 355
yy
zz
a
b
b
c
a
a
b
b
ao
ao
a
a
yy
zz
b
c
b
b
a
a
yy
zz
d
d
d
d
c
c
yy
zz
^u
with any h/b and
tf > 100 mm
Welded I-sections
ζ
ζ
E
y-
y
if tf <, 40 mm
yy
zz
b
c
if tf > 40 mm
yy
zz
c
d
hot finished
any
a
cold formed
(using fyb *))
any
generally as
(except as below)
any
thick welds and
b/tf < 30
yy
c
h/tw < 30
zz
c
Hollow sections :
+
+
Welded box sections :
ζ
+
+
Angles, channels, tees and solid sections :
L,
f
/
iL· ι
* ■ ■s >
any
ι
Note : *) fyb is the basic yield strength of the flat steel material before cold forming
135
[5.5.1.5 (i)] (6) The buckling length Lb O^b.y» Lb.z) of a compression member with both ends effectively
[5.5.1.5 (2)]
held in position laterally may conservatively be taken as equal to its system length L; or
alternatively, the buckling length may be determined using informative Annex E of
Eurocode 3.
Buckling lengths of columns in a non­sway mode are provided in table VII.5 for different
boundary conditions.
Table VH.5
Buckling length of column : Lb
Buckling
length Lb
System
2L
x.Sd
x.Sd
N,sd
0,7 L
-H
N x.Sd
[5.8.3]
0,5 L
(7) For angles in compression (Nx.sd) connected with appropriate fixity (at least two bolts if
bolted) the eccentricities may be neglected if following effective slenderness ratios λ ^
are used to determine the design buckling resistance (Nb.Rd) of compression angles.
Kfí.v =0,35+0,7 λν
buckling about the ν ν axis:
λβΓί.χ = 0,50 + 0,7 Xy
buckling about the yy or zz axis:
Ι λ ε ί ί . ζ = 0 , 5 0 + 0,7λ ζ
where λ ν , λ γ , λ ζ are non-dimensional slenderness ratios respectively about w axis, yy
axis and zz axis (axes are defined in table 0.1).
136
[table 5.5.2]
If
Table VH.6
Reduction factors χ = f( λ )
χ for buckling curve
λ
ao
a
b
c
d
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
1,0000
0,9859
0,9701
0,9513
0,9276
0,8961
0,8533
0,7961
0,7253
1,0000
0,9775
0,9528
0,9243
0,8900
0,8477
0,7957
0,7339
0,6656
1,0000
0,9641
0,9261
0,8842
0,8371
0,7837
0,7245
0,6612
0,5970
1,0000
0,9491
0,8973
0,8430
0,7854
0,7247
0,6622
0,5998
0,5399
1,0000
0,9235
0,8504
0,7793
0,7100
0,6431
0,5797
0,5208
0,4671
1.1
1,2
1,3
1,4
1,5
1,6
1,7
1,8
1,9
2,0
0,6482
0,5732
0,5053
0,4461
0,3953
0,3520
0,3150
0,2833
0,2559
0,2323
0,5960
0,5300
0,4703
0,4179
0,3724
0,3332
0,2994
0,2702
0,2449
0,2229
0,5352
0,4781
0,4269
0,3817
0,3422
0,3079
0,2781
0,2521
0,2294
0,2095
0,4842
0,4338
0,3888
0,3492
0,3145
0,2842
0,2577
0,2345
0,2141
0,1962
0,4189
0,3762
0,3385
0,3055
0,2766
0,2512
0,2289
0,2093
0,1920
0,1766
2,1
2,2
2,3
2,4
2,5
2,6
2,7
2,8
2,9
3,0
0,2117
0,1937
0,1779
0,1639
0,1515
0,1404
0,1305
0,1216
0,1136
0,1063
0,2036
0,1867
0,1717
0,1585
0,1467
0,1362
0,1267
0,1182
0,1105
0,1036
0,1920
0,1765
0,1628
0,1506
0,1397
0,1299
0,1211
0,1132
0,1060
0,0994
0,1803
0,1662
0,1537
0,1425
0,1325
0,1234
0,1153
0,1079
0,1012
0,0951
0,1630
0,1508
0,1399
0,1302
0,1214
0,1134
0,1062
0,0997
0,0937
0,0882
Vlf,C¿,2
Resistance to torsionnal buckling and to flexural-torsional buckling
[5.5.1.1 (3)] (1) In some cases the torsional or flexural­torsional buckling modes may govern. Reference
may be made to the Annex G of Eurocode 3 which is not officially available yet
137
VILd Particular verifications at ULS for class 4 monosymmetrical cross-section
(1) This chapter concerns monosymmetrical cross-sections (channels ([), T-sections (T) and
angles (L): see table 0.1) which are class 4 in uniform compression.
(2) Monosymmetrical class 4 effective cross-section subject to uniform compression induces
a shift of the centroidal axis eN (see chapter V). An additional bending moment ΔΜ due to
that eccentricity of the centroidal axis eN shall be taken into account:
[5.3.5 (7)]
AMSd = Nx.Sd e N
(3) The criteria presented in this chapter VILd may be used for uniaxial and biaxial bending.
VTLd. 1 Resistance of cross-section to Ncompression
[54.4(3)]
[5.4.8.3]
(1) For members of class 4 monosymmetrical cross-section submitted to axial compression,
the design values of the compressive force Nx.sd combined with bending moment AMsd
shall satisfy in each cross-section:
interaction (NX;Sd,AMy,Sd,AMz.Sd) £ 1
where the interaction formula is given in chapter LX.d. 1.4„
AMy.sd = Nx.sd eNy » is the additional bending moment about major axis due to the
eccentricity of the centroidal axis y (eNy) of the effective
cross-section subject to uniform compression Nx.sd,
AMz.sd = Nx.sd eNz, is the additional bending moment about minor axis due to
the eccentricity of the centroidal axis ζ (eNz) of the effective
cross-section subject to uniform compression Nx.sdVn.d.2 Stability Of member tO Ncompression
(1) For members of class 4 monosymmetrical cross-section submitted to axial compression,
the design value of the compressive force Nx.sd combined with bending moment AMsd
shall satisfy:
[5.5.4 (5)]
[55.2(7)]
[5.5.4(6)]
interaction (N xSd ,AM y<Sd ,AM zSd )< 1
where the interaction formula is given in table IX.6 (see chapter IX.d.2.2),
AMy.Sd = Nx.sd e-Ny (see VII.d.l),
AMz.sd = Nx.sd eNz (see VHd.1).
(2) If there is an eccentricity of the centroidal axis about major axis y (eNy), then it induces an
additional bending moment about major axis (AMy.sd)· m that case, if the appropriate nondimensional slenderness λτ Τ >0,4 (see chapter VIILe.2), then lateral-torsional buckling is
a potential failure mode and a supplementary check has to be taken into account as follows
interaction (N X S d ,AM y Sd ,AM z Sd ) < 1
where the interaction formula is given in table IX.7 (see chapter LX.d.2.2),
AMy.sd = Nx.sd eNy (see VILd. 1),
AMz.sd = Nx.sd eNz (seeVn.d.l).
138
VILe
[54.3 (3)]
Particular verifications at ULS for angle connected by one leg
(1) In these particular cases the effects of eccentricities in the connections may be neglected
thanks to the following considerations of chapter VII.e. Those considerations should also
be given in a similar way to other types of sections connected through outstands such as
T-sections (T) and channels ([).
VILe. 1 Connection with a single row of bolts
VILe.1.1
Resistance of cross-section to Ncompression
[6.5.2.3 (2)] (1) Angles in compression (Nx.sd) connected by a single row of bolts in one leg may be
treated as concentrically loaded with the following requirements:
N x.Sd <ΞΝu.Rd
where Nu.Rd is the design ultimate resistance of net cross-section (see chapter VLc. 1).
VILe.1.2
Stability of member to Ncompression
[6.5.2.3 (3)] (1) For angles in axial compression connected by a single row of bolts in one leg , the design
value of the compressive force Nx.sd shall satisfy:
Ν x.Sd < N b.Rd
but Ν b.Rd < N u.Rd
where Nb.Rd is the design buckling resistance of the compression angle
(see chapter VII.c.2.1 (2)), where the gross cross-sectional area of the angle
(A) is used,
Nu.Rd is the design ultimate resistance of net cross-section (see chapter VI.c.l),
where the net area of the angle (Anet) is used.
VII.e.2 Connection by welding
VU.e2.1
Resistance of cross-section to N.
compression
[6.6.10 (3)] (1) Angles in compression (Nx.sd) welded by one leg may be treated as concentrically
loaded with the following requirements:
Ν x.Sd < N u.Rd
where NuRd is the design ultimate resistance of cross-section (see chapter VI.c.2).
ΥΠ, e,2,2
Stability of member to Ncompression
[6.6.10 (3)] (1) For angles in axial compression welded by one leg , the design value of the compressive
force Nx.sd shall satisfy:
Ν x.Sd < N b.Rd
where ND.Rd is the design buckling resistance of the compression angle
(see chapter Vfl.c.2.1 (2)), where the gross cross-sectional area of the angle
(A) is used.
139
VIII
MEMBER S IN BENDING (V ; M ; (V,M))
Vlll.a
Generalities
(1) For each load case (see chapter ΠΓ) the global analysis of the frame (see chapter IV)
determines the design values for the following effects of actions which are applied to
members in bending and which shall be checked at serviceability limit states and at
ultimate limit states :
- For SLS : . vertical deflections (δν),
. vibrations (f)
- For ULS : separate or combined shear forces and bending moments :
ζ
r£
M z.Sd
&
-y ι
χ -
φ
\
vz.sd
M
M,
ysd
(2) The flow-chart FC 8 presents the general procedure to check I-section members in
bending at SLS and at ULS (see the following page).
(3) The table VIII. 1 provides a list of the checks to be performed at Ultimate Limit States for
the member in bending (V; M; (V,M)). A member shall have sufficient bearing capacity if
all the checks are fulfilled according to the loading applied to that member. For instance,
in the case of loading nr φ , all checks from φ ( 1 ) to (D(3) have to be satisfied. Several
checks in the table VQI.l concern particular cases with specific conditions. All the checks
have both references to Eurocode 3 and to the design handbook.
The table VIII. 1 proposes the following loadings applied to the member:
φ Shear force Vsd ·' Vy.sd or V^sd
Uniaxial bending moment Msd : My.SdOrMz.sd
Biaxial bending moments (My <M . M^sd) : My.sdandMz.sd
@ Interaction of shear force and uniaxial bending moment (Vsd> Msd )'·
(Vz.sd andMy.sd) or (Vy.Sd andM^sd)
Interaction of shear forces and biaxial bending moments (Vsd> My.sdt M^sd) '■
(Vz.sd and My.sd) and (Vy.sd and M^sd)
140
Flow-chart (FC 8) : Design of I members in uniaxial bending (Vt:Mv:(VzMv)) or (Vv:Mz:(VvMzì)
rows:
Determine SLS load cases
Determine ULS load cases
i
ULS checks
I —
SLS checks
ïι Select beam size (A, I, We; Wp/) and steel grade (fy) ΐ'
ί
1
1-
ι
Determine the design shear forces (Vz.Sd ; Vy.sd)
and design bending moments (My.sd ; Mz.sd)
from the global analysis of the structure
1 ~
Select stronger section
}
3
3
Determine vertical deflections
and vibrations
4
S
c
6
7
Calculate shear resistance of cross-section
ion Vp£Rd J
(Classify the cross-section in bending; if class 4 calculate Weff J
yes
no
.Vz.Sd
Calculate Vba.Rd shear
buckling resistance of the web]
/tw<(
yes.
10
11
12
Calculate design bending moment
resistance of cross-section McRd
for the class of the cross-section
13
14
|
yes
Consider Mv.Rd = McRd
(i
culate Mv.Rd)
Si
15
s
16
s
17
«8
18
C Calculate Mcy.Rd for the class of the cross-section J
19
20
yes
< y l s d £ 0,5 Vb¡j£>—
no
•
Consider Mvby.Rd
equal to Mcy.Rd
Calculate the slenderness ratio λ ί τ
no
f
Calculate Mvby.Rd reduced
by shear buckling
21
22
23
24
25
yes
Calculate Mb.Rd
design lateral-torsional
buckling resistance moment
yes
'!·.
-J-
»(Adopt section if both series ULS and SLS checks are fulfilled^
141
J 29
30
List of checks to be performed at ULS for the member in bending
according to the applied internal forces and/or moments(V;M;(V,M))
References :
Shear force V v :
Table VIILI
(Î)
(1)
[54.6 (1)]
V s d — Vpi.Rd (design plastic shear resistance of the cross­section)
(2)
[6.5.2.2(1)]
[6.5.2.2(2)]
(3)
[5.6.1 (1)]
[5.6.3 (1)]
(5)
[5.4.5.1 (1)]
[5.4.5.1 (2)]
[55.2(7)]
(2)
[55.2 (1)]
(3)
[5.4.8.1]
[5.4.8.2]
[5.4.8.3]
(2)
[55.4 (1)]
[5.5.4 (3)]
[5.54(5)1
(3)
[55.2(1)]
Γ55.4 (2)1
[5.5.4(4)]
[554(6)]
Resistance of cross-section to Vzsd, if web with a group of
table VTII.6
fastener holes near the end of a beam :
V z .Sd ^ Veff.Rd (design value of the effective resistance to block shear)
Vm.d.l (3)
Stability of web to Vzsd, ifdltw > 69ε :
table Vm.7
Vz.Sd — VbaUd (design shear buckling resistance)
Vm.d.2 (5)
Resistance of cross-section to MsdM s d ^ McRd (design uniaxial bending moment resistance of the cross­section)
VDI.e.1 (1)
Stability of member to Mysd >if ^LT > 0,40 :
Vm.e.2 (3)
My.Sd — Mb.Rd (design lateral­torsional buckling resistance moment of the member)
VDI.e.2 (4)
Biaxial bending moments (My sd. Mz <? ,/) :
(1)
[55.2 (7)]
Vm.d.l (1)
Uniaxial bending moment Mw :
(1)
[55.2(7)]
Resistance of cross-section to Vsd ·"
(4)
Resistance of cross-section to (Mysd, Mzsd)'·
vm.f.1 (i)
interaction (My.sd , Mz.sd) ^ 1, for class 1 and 2 cross­section
for class 3 cross­section
for class 4 cross­section
LX.d.1.2 (2)
Stability of member to (My_sd, Mzsd)'·
vm.f.2 (i)
interaction (My.sd » Mz.sd) ^ 1, for class 1 and 2 cross­section
for class 3 cross­section
for class 4 cross­section
table ΓΧ.4
table LX.5
table LX.6
Stability of member to Mysd ,if λυτ > 0,40 :
Vni.e.2 (3)
My.Sd — Mb.Rd (design lateral­torsional buckling resistance moment of the member)
Vffl.e.2 (4)
rx.d.1.3 (1)
LX.d. 1.4(1)
Vni.e.2 (3)
Stability of member to (My.sd, Mz.sd), if %>LT > 0,40
(potential lateral-torsional buckling) : VUI.f.2 (2)
interaction (My.sd , Mz.sd) ^ 1, for class 1 and 2 cross­section
for class 3 cross­section
for class 4 cross­section
142
table ΓΧ.4
table IX.5
table ΓΧ.7
List of checks to be performed at ULS for the member iiι bending
according to the applied internal forces and/or moments!V;M;(V,M))
References :
Œ) Interaction of shear force and uniaxial bending moment (VIA. M*A ):
Table VIILI
vm.d.i (1)
If Vsd ^ 0,5 Vpi.Rd then interaction (Vsd, Msd ) is not considered and
154.7(2)]
checks nr φ and nr @ of this table Vffi. 1 shall be performed,
with the following check nr @ (6).
vm.g.i (i)
vm.g.i.i (i)
If Vsd > 0¿ Vpijid then interaction (Vsd, Msd ) has to be considered
154.7(3)]
and
following checks shall be carried out :
(1)
Resistance of cross-section to Vsd'·
vm.d.i (i)
154.6(1)]
V sd ^ Vpi.Rd (design plastic shear resistance of the cross-section)
[65.2.2(1)]
(2)
[65.2.2(2)]
[5.45.1 (1)]
[54.5.1 (2)]
[55.2(7)1
Resistance of cross-section to Vzsd, if web with a group of
fastener holes near the end of the beam :
(3)
Resistance of cross-section to Msd'.
(4)
Resistance of cross-section to (Vsd, Msd)'·
M sd — MViRd (design plastic resistance moment reduced by shear force)
(6)
Vm.e.2 (4)
vm.g.i.i(i)
table Vm.7
Stability of web to (Vzsd, MySd), ifd/tw > 69ε :
One of the three following checks ((5.1), (5.2), (5.3))
shall be fulfilled :
(6.1) If My.Sd ^ Mf.Rd (design plastic moment resistance of
cross-section with the flanges only)
then V z .sd ^ V ^ R d (design shear buckling resistance of the web)
(6.2) If M y .sd > MtRd and V z . S d <, 0,5 V ^ d
then My.Sd — McyJld (design uniaxial bending moment resistance
of the cross-section)
[5.6.7.2(3)]
Vm.e.2 (3)
Stability of member to Mysd, if λυτ > 0,40 :
My.sd — Mh.Rd (design lateral-torsional buckling resistance moment of the member)
154.7(3)]
[5.6.7.2(2)]
vm.e.i (1)
Msd ίΞ M c .Rd (design uniaxial bending moment resistance of the cross-section)
(5)
[5.6.7.2(1)]
Vffl.d.l.(3)
V^sd ^ Veff.Rd (design value of the effective resistancetoblock shear)
[55.2(1)]
[5.6.1 (1)]
table Vm.6
(6.3) If My.sd > Mfju and V^sd > 0,5 VbaLRd
then Mysd ^ design moment resistance reduced by shear
buckling (interaction (Vz.sd, My.Sd))
and, My.sd ^ M ^ j ^
and, Vz.sd ^ V^ju
143
Vm.g.2(3)®
Vm.d.2 (5)
Vm.g.2(3)@
vm.e.i (1)
vm.g.2(3)(D
Vm.g.2(3)(3)
Vm.e.l (1)
Vm.d.2 (5)
Table VIILI
List of checks to be performed at ULS for the member in bending
according to the applied internal forces and/or moments(V;M;(V,M))
Interaction of shear forces and biaxial bending moments
(Vsd, Myjsd, Mz.sd) :
[5.4.7 (2)]
References
If Vsd ^ 0,5 Vpiüd then interaction (Vsd, My.sd, Mz.sd) is not considered Vm.d.l (1)
and checks nr φ and nr ® of this table VU. 1 shall
be performed, with the following check nr (§) (7).
vm.g.i(i)
[54.7 (3)]
If Vsd > 0¿ Vpijid then interaction (VSd, My.Sd, Mz.Sd) is to be
considered and following checks shall
be carried out :
(1)
[6.5.2.2 (1)]
(2)
(3) Stability of member to Mysd ,if λυτ > 0,40 :
[55.2 (1)]
(4)
[55.4(1)]
[5.5.4(3)]
[554(5)1
[55.2 (7)]
vm.d.i (3)
Vm.e.2 (3)
M y .Sd — Mb.Rd (design lateral­torsional buckling resistance moment of the member)
Vm.e.2 (4)
Stability of member to (Mysd, Mz.sd)'·
vm.f.2 (i)
interaction (My.sd » Mz.sd) ^ 1 » for class 1 and 2 cross­section
for class 3 cross­section
for class 4 cross­section
table IX.4
table ΓΧ.5
table ΓΧ.6
(5)
Stability of member to (My.sd, Mz.sd), if ALT > 0,40
Vm.e.2 (3)
(potential lateral-torsional buckling) : Vm.f.2 (2)
table LX.4
interaction (My.sd » Mz.sd) ^ 1, for class 1 and 2 cross­section
for class 3 cross­section
table ΓΧ.5
for class 4 cross­section
table LX.7
(6)
Resistance of cross-section to (Vgd, Mysd, Mzsd)'·
vm.g.i.2
interaction (My.sd , Mz.sd) ^ 1, for class 1 and 2 cross­section
Vm.g.l.2(2)
Vffl.g.l.2(3)
for class 3 cross­section
Vm.g.l.2(3)
for class 4 cross­section
where design resistance moments are reduced by shear forces but
limited by appropriate values of moment resistance according to :
[55.4 (2)]
[55.4 (4)]
[5.5.4 (6)]
[5.4.8.1]
[5.4.8.2]
[5.4.8.3]
vm.d.i (i)
Resistance of cross-section to Vsd, if web with a group of fastener table Vm.6
holes near the end of a beam :
V s d — Veff.Rd (design value of the effective resistance to block shear)
[65.2.2(2)]
[55.2(7)]
Resistance of cross-section to Vzsd'·
V z .sd — VpURd (design plastic shear resistance of the cross­section)
[54.6 (1)]
vm.g.i.i (i)
[5.4.7 (3)]
Mv.Rd (design plastic resistance moment reduced by shear force),
vm.g.i.i (i)
[5.45.1 (1)]
[54.5.1 (2)]
With My.Rd — M c jRd (design uniaxial bending moment
resistance of the cross­section),
VIILe.l (1)
in other words, with Mvy.Rd ^ McyRd and Mvz.Rd ^ Mcz.Rd
(checks nr © to be continued)
144
List of checks to be performed at ULS for the member in bending
according to the applied internal forces and/or moments(V;M;(V,M))
Table VOLI
[5.6.1 (1)]
[5.6.7.2(1)]
References
Interaction of shear forces and biaxial bending moments
(continuation)
(Vsd, Mysd, Mzjsd)
(7) Stability of web (Vzsd, MySd), ifdftw > 69ε :
table Vni.7
One of the three following checks ((6.1), (6.2), (6.3))
shall be fulfilled :
VULg.2(3)®|
(6.1) If Mjf.sd ^ Mf,Rd (design plastic moment resistance of
cross­section with the flanges only)
t h e n V z .Sd — VbaJld (design shear buckling resistance of the web)
[5.6.7.2(2)]
(6.2) If My.sd > MfJU and V^sd $ 0,5Vba.Rd
then My.sd — M C y«d (design uniaxial bending moment resistance
of the cross­section)
[5.6.7.2(3)]
(6.3) If My.sd > Mfju and Vz.Sd > 0,5Vba.Rd
then My.sd ^ design moment resistance reduced by shear
buckling (interaction (V^sd, My.sd))
and, My.sd ^ Mcy­Rd
and, Vz.sd ^ Vbajw
Vm.d.2 (5)
VTJI.g.2(3)¿)
vm.e.i (1)
vm.g.2(3)(D
vm.g.2(3)(D
vm.e.i (1)
Vin.d.2 (5)
VIILI) Verifications at SLS
Vlll.b.l Deflections
[4.2]
(1) Steel structures and components shall be so proportioned that deflections are within limits
agreed between the clients, the designer and the competent authority as being appropriate
to the intended use and occupancy of the building and the nature of the materials to be
supported.
(2) The deflections should be calculated making due allowance for any second order effects,
the rotational stiffness of any semi­rigid joints and the possible occurrence of any plastic
deformations at the serviceability limit state.
(3) The values given in table Vm.2 are empirical values. They are intended for comparison
with the results of calculations and should not be interpreted as performance criteria.
[4.2.2(1)] to
[4.2.2 (3)]
(4) The design values for the vertical deflections (δν) (see chapters ΙΠ and IV) should be
lower limiting values given in table VIII.2. Those limiting values are illustrated by
reference to the simply supported beam shown in table VHI.3.
145
Table VIII.2
Recommended limiting values for vertical deflections
ECCS n°65
table 4.2
Conditions
Limits
Omax
roofs generally
roofs frequently carrying personnel other than for maintenance
floors generally
floors and roofs supporting brittle finish or non­flexible partitions
floors supporting columns (unless the deflection has been
included in the global analysis for the ultimate limit state)
where omax can impair the appearance of the building
L/200
L/250
L/250
L/250
L/400
L/250
L = span of the beam; for cantilever beams : L = twice cantilever span
Discharge of rainwater:
slope of the roof less than 5%
slope of the roof less than 3%
ECCS n°65
table 4.1
Table Vffl.3
δ2
L/250
L/300
L/300
L/350
L/500
­
check that rainwater cannot collect in pools
additional check that incremental collapse cannot
occur due to the weight of water
Vertical deflections to be considered
­Tnax
Omax = δι + 02 ­ δο
State 0
δο
li
♦ δο
δι
'max
θ2
146
= the sagging in the final state
relative to the straight line
joining the supports,
= the pre­camber (hogging) of
the beam in the unloaded state
(state 0),
= variation of the deflection of
the beam due to permanent
loads (G) immediately after
loading (state 1),
= variation of the deflection of
the beam due to the variable
loading (Q) (state 2).
[4.3]
ECCS n°65
table 44
Vni.b.2 Dynamic effects ­ vibrations
(1) The vibrations of structures on which the public can walk shall be limited to avoid
significant discomfort to users.
(2) The design values for the effects of actions (vertical deflections (δν) and natural frequency
(f)) (see chapters ΙΠ and IV) should be limited by the values given in table Vm.4. Those
limiting values may be relaxed where justified by high damping values.
Recommended limiting values for floor vibrations
Table VIIL4
f>fe
δν < δι + Ô2
Type of floor
lowest natural frequency
U [Hz]
limited total deflection
δι + θ2 [mm]
Floors over which people walk regularly
(offices, dwellings,...)
Floors which are jumped or danced on
in a rhythmical manner (gymnasium,
dance hall,...)
3
28
5
10
#
fe =
fe
E
I
L
m
α
1 α Í1T
27lWm
[HZ1
natural frequency
modulus of elasticity
second moment of area
span
mass per unit length
coefficient of frequency of the basis mode vibration
gr^zzzz^
α = 9,869
α = 22,37
α = 3,516
α =15,418
V111 .c C lassification of cross-section
(1) At ultimate limit states the resistance of cross­section may be limited by its local buckling
resistance. In order to take into account that limitation the different elements (flange, web)
of the cross­sections shall be classified according to the rules defined in chapter V.
(2) For cross­sections submitted to bending moments (My.sd, Mz.sd) the classification may
specifically be determined according to the procedure defined in chapter V.d.2.
147
vm.d
Verifications at ULS to shear force VSd
VIII.d.l Resistance of cross-section to VSd
[5.4.6 (1)]
(1) For members submitted to shear force the design value of the shear force V Sd (Vz.sd>
Vy.sd) at each cross-section shall satisfy :
[form. (5.20)]
V
z.Sd -
V
pf.z.Rd -
A
vz77ã—
ΪΜ0
Vy.sd^Vp,.y.Rd-Avy-^-
where V p£z R d , Vp/;y.Rd
vz 'J "* *v y
/ \ y
Z
are the shear areas about ζ and y axes, given in table VHL5,
L
is the yield strength (see table Π.4),
YmO
is a partial safety factor (see table 1.2).
y
[5.4.6 (8)]
are the design plastic shear resistances about ζ and y axes,
(2) Fastener holes need not be allowed for in shear verifications provided that:
[form. (5.21)]
where Av
fu
is the shear area (see table Vm.5),
is the yield strength (see table Π.4),
is the ultimate tensile strength (see table Π.4).
When Av.net is less than this limit, an effective shear area of (fu/fy).Av.net may be assumed.
[6.5.2.2 (l)] (3) Near the end of a member with a group of fastener holes in webs the "block shear" failure
shall be prevented by using appropriate hole spacing.
The design value of shear force (Vz.sd) applied to the web shall satisfy :
ECCS n°65
z.Sd
form. (5.18)
where Veff.Rd
Av.net
fu
γΜ2
<veff.Rd
_ 0,6 f u Av.net
YM2
is the design value of the effective resistance to block shear,
is the effective shear area (see table VIII.6),
is the ultimate tensile strength (see table Π.4),
is a partial safety factor (see table 1.2).
148
[54.6(2)]
Table VHX5
Cross­sections
Shear area Av for cross­sections
Vz.sd
Loading
*)
Load
parallel
to web
A v.7. —
t«,—' L
A ­ 2btf + (tw + 2r)tf
*U
Rolled
Load
parallel
A v .y —
to
flanges
I and H
y.sd I
1 V
Vy­sd
4
2btf + (tw + r) tw
r
¿ür
r
tf
-w
|V 2 .Sd
sections
*t4
Load
parallel A v . z —
to web
(h ­ 2tf) t.
Welded
*)
Load
parallel
Ay y —
to
flanges
tw,
Load
parallel
to web
Αγ,ζ
—
A ­ 2btf + (tw + r)tf
t
HUf
JU
'z.Sd
1
*)
Rolled rectangular
hollow sections
of
uniform thickness
Load
parallel
to depth
Αγ.ζ
—tf
X
A ­ (h ­ 2tf) tw
*)
Rolled channel
sections
lVy.Sd
'z.Sd
Ah
b+ h
—
*)
Load
parallel
Av.y —
to
breadth
±±+
|Vy.Sd
Ah
b+ h
i-^-i
*)
Circular hollow sections
and tubes of uniform thickness
Vsd
2A
π
*)
Plates and solid bars
Note : *) A is the total cross­sectional area
149
Vsd
.Vsd
ECCS n°65
table 5.34
Table Vm.6
Determination of Ay.net for block shear resistance
ai
Lv
a2
Av.net = t ( Lv + Li + L 2 - (n do))
Li = 5,0 do ^ ai
L2 = 2,5 do < a2
t = web thickness
n = number of fastener holes on the block shear failure path
do = hole diameter
Vm.d.2 Stability of web to Vz.Sd
[5.6.1 (l)]
[5.6.1 (4)]
(1) If webs are submitted to shear force Vz.sd and if their ratio
exceeds the limits given
in table VHI.7 then they shall be checked for resistance to shear buckling and transverse
stiffeners shall be provided at supports.
Table VHI.7 Limiting width-to-thickness ratio related to the shear buckling in web.
Potential shear buckling
Profiles
to be checked if webs have
a) For unstiffened webs :
tw
± >69ε
t.k w
F—r
AL·
b) For stiffened webs :
30e->/k7
The value of kT is defined in table VIII.9
fv (N/mm2)
= ^235T
ε (if < 40 mm)
ε (if 40 mm < t < 100mm)
150
235
275
355
420
460
0,92
0,96
0,81
0,84
0,75
0,71
0,78
0,74
(2) Nearly all hot-rolled I and H sections do not need to be checked for shear buckling.
[5.6.2(1)]
(3) The shear buckling resistance nay be verified using either :
- the simple post-critical method, or
- the tension field method.
The first method is presented hereafter.
[5.6.2(3)]
(4) The simple post-critical method can be used for webs of I-section girders, with or without
intermediate transverse stiffeners, provided that the web has transverse stiffeners at the
supports.
(5) For webs with d/tw exceeding limits of table VHI.7, the design value of the shear force
Vz.sd shall satisfy :
15.6.3(1)]
z.Sd
^ V ba.Rd = d t w
where Vba.Rd
d
tw
Xba
Table VHX8
x
ba
ΎΜΙ
is the design shear buckling resistance,
is the web depth (see table Vffl.7),
is the web thickness,
is the simple post-critical shear strength (see table VIII.8).
where *yw
fv
is the yield strength of the web,
is the web slenderness.
Xw
Simple post-critical shear strength τ\,α
<0,8
Xw
f
Xba
*yw
V3
0,8<Xw<l,2
<1,2
- ^ ( 1 , 5 - 0,625Ü)
£yw ΓΛ9
V3 Xw
(6) The web slenderness λw should be determined from :
= ^235/ fy , given in table Vfll.7,
where ε
kx
Table VUI.9
a/d
<1
is the buckling factor for shear given in table VIII. 9
where a is the clear spacing between transverse stiffeners
Buckling factor for shear kx
>1
^ r
f
4+
5,34
W
^s
v
5,34 +
Sd
v
W
5,34
Sd
AL
a
151
Vlll.e
Verifications at ULS to bending moment Msd
VIII.e.1 Resistance of cross­section to MSH
(1) For members in bending and in absence of shear force, the design value of the bending
moment M$d (My.sd, Mz.sd) shall satisfy at each section without holes for fasteners :
Mcjid depending on classes of cross­section :
class 1 or 2
class 3
class 4
[5.4.5.2(1)]
My.Sd ^ M c y .Rd
Mz.sd ^ Mcz.Rd
­M
­W"­>f>
­
­M
­
M
pf.z.Rd
­
W
^
M C y ü d , MczJtd
M
p£y.Rd, Mpiz.Rd
M e £y.Rd, M e £ z J i d
y
­ ­
Rd
"
YMO
W
f y
ei.zfy
= Mef.z.Rd =
­
YMO
where Mc.Rd
Mef
_ M
_
ivl
efl.y.Rd ­
W
eff­y
_
­
M
M
YMO
eff.z.Rd
_Weff.zfy
­
YMI
is the design moment resistances of the cross­section,
are the design moment resistances of the cross­section about
major (yy) and minor (zz) axes,
are the design plastic moment resistance of the cross­section
about y and ζ axes,
are the design elastic moment resistance of the cross­section
about y and ζ axes,
Wp£y,Wp£Z
Wecy, W e£z
are the elastic section modulus about y and ζ axes,
Weff.y, W e ff.z
are the effective section modulus about y and ζ axes
(see chapter V),
is the yield strength (see table Π.4),
YMO» YMI
y
YMI
are the design effective moment resistance of the cross­
section about y and ζ axes,
are the plastic section modulus about y and ζ axes,
Meff.yJld, M e ff. z .R d
f
are partial safety factors (see table 1.2).
ECCS n°65 (2) In the presence of holes for fastener the following simple approach is proposed :
5.3.2(3)
­ no deduction of holes in the compression zone and,
­ deduction of holes in tension zone.
Otherwise Eurocode 3 should be consulted in [5.4.5.3].
152
Vm.e.2 Stability of member to My ci
[5.5.2]
153.2(8)]
(1) A beam with full restraint to the compression flange does not need to be checked for
lateral­torsional buckling.
(2) I and Η­sections, channels, angles, T­sections and rectangular hollow sections are
susceptible to lateral­torsional buckling in respect of bending about their major axis
(My.sd) but not about their minor axis (Mz.sd)-
[53.2(7)]
(3) For members with appropriate non­dimensional slenderness |λυτ ^0,40| no allowance for
lateral­torsional buckling is necessary. The value of Xur is defined hereafter.
(4) For laterally unrestrained members in bending, the design value for the bending moment
about major axis (My.sd) shall satisfy :
MbUd depending on classes of cross­section :
[form. (5.48)]
My.Sd ^ Mbjw
class 1 or 2
class 3
class 4
_ XLI W p f . y f y
_ 5CLT W ef.y f y
_ 3CLT Wrff.y f y
YMI
YMI
YMI
Wp¿y
s the design buckling resistance moment of members in bending,
s the reduction factor for lateral­torsional buckling,
s the plastic section modulus about major axis (yy),
We£y
s the elastic section modulus about major axis (yy),
Weff.y
s the effective section modulus about major axis (yy) (see chapter V),
fy
s the yield strength (see table Π.4),
YMI
s a partial safety factor (see table 1.2).
where Mb.Rd
XLT
(5) The value of χυ for the appropriate non­dimensional slenderness XLT may be determined
from :
X L T =f(X L T) =
1
1
T2
φτ^Τ+Λφτ^Γ­λτ^Γ
,but
XLT^1
where <|>LT =0,5[l+a LT (X L T ­ 0 , 2 ) + λ υ τ ]
CCLT
is the imperfection factor for lateral­torsional buckling;
OCLT should be taken as :
­ for rolled section
OCLT = 0,21 (buckling curve a in table VH.2)
­ for welded section au = 0,49 (buckling curve c in table VII.2).
The reduction factor for lateral­torsional buckling XLT is given in function of XLT and the
type of section in table Vm.10.
153
(6) The non-dimensional slenderness Xu may be determined from :
ΧΙΊ depending on the classes of cross-section
class 1 or 2
class 3
class 4
[5.5.2(5)]
_ | w p r . y fy
\
Mcr
Χυτ
i
(Wrf.y fy
M CT
1
(W eff .y fy
M CT
where W p£y , W^y, Weff.y
are respectively the plastic, elastic and effective section
modulus about major axis (yy),
is the yield strength (see table Π.4),
fy
M cr
is the elastic critical moment of the gross cross-section for
lateral-torsional buckling.
(7) The elastic critical moment MCT for doubly symmetrical cross-sections with in plane end
moment loading may be taken as
'Je V
[Annex F]
M
-C
M
cr - M
[form. (F.5)]
π2
Ρ*
72
'LT
1
I w + 0,039 f£ T I t
where Ci
is a factor which may be taken from table VIII. 12 using also
table Vm. 11,
E
is the modulus of elasticity,
Iz
is the second moment of area about minor axis (zz),
CUT — kL is the effective length for out-of-plane (xz) bending (see table VUL 11),
L
is the length of the member between points which have lateral restraint,
k
is the effective length factor for out-of-plane (xz) bending
(see table Vm. 11),
kw
is the effective length factor for warping (see table Vm. 11),
Iw
is the warping constant,
It
is the torsion constant.
(8) In a member with a system length L, each portion C, between adjacent points with lateral
restraint, or from one end to the nearest point with lateral restraint, can be checked
separately (see table VIII. 12).
(9) For other types of cross-section and for other loading conditions on the member, the
Annex F of Eurocode 3 should be consulted.
154
ECCS ηβ65
table 5.25
_
ECCS ne65
table 5.23
= f(Xu) for lateral­torsional buckling
Table VUL 10 Reduction factor χ
' ' '
rolled sections (curve a) welded sections (curve c)
0,4
0,5
0,6
0,7
0,8
0,9
1,0
0,9528
0,9243
0,8900
0,8477
0,7957
0,7339
0,6656
0,8973
0,8430
0,7854
0,7247
0,6622
0,5998
0,5399
1,1
1,2
1,3
1,4
1,5
1,6
1,7
1,8
1,9
2,0
0,5960
0,5300
0,4703
0,4179
0,3724
0,3332
0,2994
0,2702
0,2449
0,2229
0,4842
0,4338
0,3888
0,3492
0,3145
0,2842
0,2577
0,2345
0,2141
0,1962
2,1
2,2
2,3
2,4
2,5
2,6
2,7
2,8
2,9
3,0
0,2036
0,1867
0,1717
0,1585
0,1467
0,1362
0,1267
0,1182
0,1105
0,1036
0,1803
0,1663
0,1537
0,1425
0,1325
0,1234
0,1153
0,1079
0,1012
0,0951
Effective length factors : k, kw
Table V m . l l
for different out­of­plane (xz)
bending end conditions
Æ
'λ
for different warping
end conditions
k =1,0
Ll·
^
■K
Λ
k = 0,5
155
a-
4d
ï
kw =1,0
kw = 0,5
ECCS n°65
table 5.24
Table VUI.12
Numerical values for Cj and definition of ψ
-0,75
-0,5
-0,25
0
FSd
FSd
ι
t
0,25
0,5
0,75
FSd = yPSk
ι
Vm.f Verifications at ULS to biaxial bending moment (My.sd> M^sd)
Vin.f. 1 Resistance of cross-section to (My «=H. Mz.Sd)
(1) For members submitted to biaxial bending moments, the design values of both bending
moments shall satisfy in each cross-section :
interaction (M y Sd , M z S d ) < 1
where the interaction formula is given :
- in IX.d.1.2 (2) for class 1 or 2 cross-sections,
- in DC.d.1.3 (1) (using Nx.sd = 0) for class 3 cross-sections and,
- in IX.d.1.4 (1) (using Nx.sd = 0) for class 4 cross-sections.
156
Vm.f.2 Stability of member to (My *A. Mz.Sd)
(1) For members submitted to biaxial bending moments the design values of both bending
moments shall satisfy :
interaction (My Sd, Mz.sd) £ 1
where the interaction formula is given :
­ in IX.d.2.2( 1) (table IX.4) (using Nx.sd = 0) for class 1 or 2 cross­sections,
­ in IX.d.2.2 (1) (table IX.5) (using Nx.sd = 0) for class 3 cross­sections and,
­ in IX.d.2.2 (1) (table IX.6) (using Nx.sd = 0) for class 4 cross­sections.
(2) If the appropriate non­dimensional slenderness |λυτ >0,40| (see chapterVm.e.2) then
lateral­torsional buckling is a potential failure mode and a supplementary check has
to be taken into account as follows :
interaction (M yS d, M zS d) ^ 1
where the interaction formula is given in :
­ in IX.d.2.2 (1) (table IX.4) (using Nx.sd =0) for class 1 and 2 cross­sections,
­ in IX.d.2.2 (1) (table ΓΧ.5) (using Nx.sd =0) for class 3 cross­sections,
­ in IX.d.2.2( 1) (table DÍ.7) (using Nx.sd =0) for class 4 cross­sections.
Vm.g
[5.4.7(2)]
Verifications at ULS to combined f VSd, MSt] )
Vlll.g.l Resistance of cross­section to (VSd, MSd)
(1) If the design value of the shear force
V,sd^,5Vpi.,Rd
and,
V y .sd^0,5V p ,.y.Rd
where
Vpiz.Rd, VpC.yAd
are the design plastic shear resistance about minor
(zz) and major (yy) axes (see table VUL 13),
no reduction needs to be made in the resistance moments. With this condition the design
value of bending moment Msd shall be verified according to chapter Vm.e or chapter
Villi respectively in case of uniaxial bending or biaxial bending.
[5.4.7(3)]
y¡l!,g,l,l Shear force VSd and uniaxial bending MSd
(1) For the resistance of cross­section submitted to combined shear force (V^sd or Vy.sd ) and
uniaxial bending moment (My.sd or Mz.sd) if the design value of shear force
VSd>0,5Vp(.Rd (high shear),
then interaction between shear force and bending moment shall be considered. In this case
the design value of bending moment Msd shall satisfy at each cross­section :
[form. (5.22)]
Msd <. M V­Rd ,but
where
M V.Rd <M c.Rd
is the reduced design plastic resistance moment allowing for the
shear force (see table Vffl.13),
Mc.Rd is the design plastic moment resistance of the cross­section
(see chapter VIII.e.1).
MVJW
157
Table VIII.13 Reduced design plastic resistance moment Myjtd allowing for the shear force
If high shear: VSd > 0,5 VptRd
Applied bending moment
/
M V.y.Rd
My.sd ,for cross-section with equal flanges:
Wp,yV
PzAyJ f,
My.sd, for cross-section with unequal flanges:
Mv.y.Rd=(l-Pz)
Mz.sd, for any cross-section:
M
v.z.Rd = ( l - p y )
,s
^lw
J YMO
Wpf.y fy
YMO
Wpr.z fy
ΎΜΟ
= W pf ——, for class 1 or 2 cross - sections
YMO
but M v .Rd ^ M c . R d
= Wei ——, for class 3 cross - section
YMO
fv
= Weff ——, for class 4 cross - section
YMI
where
Wpf is the plastic section modulus of cross-section
f
Pz =
Pv =
V
^ V p f. z .Sd
(
y.Sd
V
Y
pf .y.Sd
withV pi . z . Rd =
Ay.z fy
YMO^3
A
-1
■**y V
f
V
J
- for shear areas (Av.z, Av.y), see table VIII.5
- tw
is the web thickness
-fv
is the yield strength (see table Π.4)
YMO is a partial safety factor (see table 1.2).
VIII.g.12 Shear force V Sd and biaxial bending moment MSd
(1) For the resistance of cross-section submitted to combined shear forces (Vz.sd and Vy.sd)
and biaxial bending moments (My.$d and Mz.sd), if the design value of the shear forces
V S d>0.5V p f.Rd
(high shear),
it is proposed that following interactions between shear forces and bending moments shall
be satisfied at each cross-section according to the class of cross-section.
158
(2) For class 1 or 2 cross-sections, the proposed interaction formula is
ECCS n°65
table 5.15
Mv.Sd
ι«
M V.y.Rd
M z.Sd
MV.z.Rd
where Mv.yjw and My.z.Rd
<. 1
aie m e
reduced design plastic resistance moments allowing
for shear forces (see table Vm.13),
α and β are constants, taken as follows :
α = 2, (3=1
- for I and Η sections :
α = 2, β = 2
- for circular tubes :
- for rectangular hollow sections : α = β = 1,66
α = β = 1,73
- for solid rectangles and plates :
(3) For class 3 and class 4 cross-sections, the proposed interaction formula is :
where My.y.Rd and My.z.Rd are the reduced design plastic resistance moments allowing
for shear forces (see table VIH. 13),
Vm.g.2 Stability of web to (Vz.Sd, My.Sd)
(1) If webs are submitted to combined shear force V^sd and bending moment My.sd and if
they have ratio - - exceeding the limits given in table Vm.7 then they shall be checked
for resistance to shear buckling.
(2) The interaction of shear buckling resistance and moment resistance is shown in
table Vm.14 according to the simple post-critical method.
Table Vffl.14
Interaction of shear buckling resistance and moment resistance
with the simple post-critical method
VplRd··
Vba.Rd-7
0,5Vba.Rd
M
M
159
f.Rd
M
pl.Rd
(3) The web may be assumed to be satisfactory if one of the three following checks
(CD, (D or (3))(according to the loading level of Vz.sd and My.sd) shall be satisfied:
(D If the design value of bending moment
My.Sd<MfRd
where
Mf.Rd is the design plastic moment resistance of a cross­section consisting
of the flanges only; proposal for cross­section with equal flanges:
class 4
class 1,2 or 3
Mf.Rd
"r t beff
"en ι.
= btf(h­tf)­2­ =
((b + beff)tf il_^j-((b-b e f f ) tf e M ) ] ^ -
YMO
j
, T^yJ ι ­φ tf
where b, tf, h (see
are flange
width, flange thickness, height of profile
table 0.1),
beff
is the effective width of the compression flange
M
e
L y Sd
lTl£ M ))
'
(see chapter V),
eM
is the shift of the centroidal major axis (yy) when the
­ ­­ ­
cross­section consisting of flanges only is subject to My.sd·
then the design value of shear force shall satisfy:
V
V
z.Sd ­
where
ba.Rd
Vba.Rd is the design shear resistance buckling of the web according
to simple post­critical method (see chapter Vm.d.2).
If the design values of bending moment and shear force
My.sd>M f.Rd and V z . Sd <0,50V ba .R d
then the bending moment shall satisfy :
M
y.Sd^Mcy.Rd
where Mcy,Rd is the design moment resistance of the cross­section
depending on classes of cross­sections (see chapter VEI.e.1).
If the design values of bending moment and shear force
M
y.Sd>Mf.Rd
and V z . Sd >0,50V ba .R d
then the bending moment and the shear force shall satisfy the three following checks:
where
MP£y.Rd
P i ï R d
is the design plastic moment resistance of the cross­section :
"
ΪΜ0
- and, J M y Sd < M c y Rd
- and,
z.Sd -
V
baJRd
160
IX
MEMBERS WITH COMBINED AXIAL FORCE AND BENDING
MOMENT ((N, M);(N, V, M))
IX.a
Generalities
(1) For each load case (see chapter ΠΓ) the global analysis of the frame (see chapter IV)
determines the design values for the following effects of actions which are applied to
members with combined axial force and bending moment and which shall be checked at
serviceability limit states and at ultimate limit states:
­ For SLS
:
. vertical deflections (δν),
. vibrations (f)
different combinations of axial forces, shear forces and bending
­ For ULS
moments :
r£
S
y"
&
Mz.Sd
5
Nx.sd
Λ
vz.sd
M
Nx.sd
y­sd
(2) The table IX. 1 provides a list of the checks to be performed at Ultimate Limit States for
the member submitted to combined axial force and bending moment (N, M). A member
shall have sufficient bearing capacity if all the checks are fulfilled according to the
loading applied to that member. For instance, in the case of loading nr (Î), all checks from
(J)(l) to (T)(5) have to be satisfied. Several checks in the table IX. 1 concern particular
cases with specific conditions. All the checks have both references to Eurocode 3 and to
the design handbook.
The table IX. 1 proposes the following loadings applied to the member:
φ Axial tensile force and uniaxial bending moment about major axis (Nx.sd, My.sd)
Axial tensile force and uniaxial bending moment about minor axis (Nx.sd > M^d)
Axial compressive force and uniaxial bending moment about major axis (Nxsd MySd)
(§) Axial compressive force and uniaxial bending moment about minor axis (Nr MA M7 $¿ )
Axial tensile force and biaxial bending moments (Nx¿d t Mysd> MzSd)
Axial compressive force and biaxial bending moments (Nxsd, My.sd, M^sd )
161
List of checks to be performed at ULS for the member submitted to
combined axial force and bending moment (N, M)
Table LX.1
φ
References :
Axial tensile force and uniaxial bending moment about major axis
(Nxsd My.Sd)'(1) Resistanc e of gross cross-section to Nxsd ·'
[5.4.3 (1)]
Nx.Sd — Np£Rd (design plastic resistance of the gross cross-section)
(2)
Resistanc e of the net cross-section to Nx.sd if holes for fasteners:
N x .Sd
[5.4.3 (1)]
[5.4.2.2]
(3)
— N u .Rd
(design resistance of the net cross-section considering the
net area of a member or element cross-section, A ^ )
[5.4.8.1]
[5.4.8.2]
[5.4.8.3]
[55.2 (7)]
VLcl (1)
VI.C.2 (1)
Resistanc e of cross-section to (Nx.sd, Mysd)·'
interaction (Nx.sd , My.sd) ^ 1 , for class 1 and 2 cross-sections
for class 3 cross-section
for class 4 cross-section
(5)
VI.b.2 (1)
Resistanc e of cross-section to Nxsd if angle connected by one leg:
see table VI. 1 : checks φ ( 3 ) and φ ( 4 )
(4)
Vl.b.l (1)
LX.d.1.1 (1)
LX.d.1.3 (2)
LX.d.l.4(2)
Out-of-plane (xy) stability of member to (Nxsd > Mysd)
if λ LT > 0,40 :
[5.5.3]
Meff.Sd — Mb.Rd
(design lateral-torsional buckling resistance moment of the
member)
[5.5.3 (3)]
[5.5.3 (4)]
with Meff.sd calculated with (Nx.sd, My.sd)
Vm.e.2 (3)
Vm.e.2 (4)
LX.d.2.1 (1)
Axial tensile force and uniaxial bending moment about minor axis
(Nx.sd>Mz.sd):
(1) Resistanc e of gross cross-section to Nxsd ■'
[5.4.3 (1)]
[5.4.3 (1)]
[5.4.2.2]
Nx.Sd — Np£Rd (design plastic resistance of the gross cross-section)
(2)
(4)
[5.4.8.1]
[5.4.8.2]
[5.4.8.3]
Resistanc e of the net cross-section to Nxsd if holes for fasteners:
Nx.Sd
(3)
VI.b.1 (1)
— N u .Rd
(design resistance of the net cross-section considering the
net area of a member or element cross-section, Ancl)
VI.b.2 (1)
Resistanc e of cross-section to Nxsd if angle connected by one leg:
VLcl (1)
see table VI. 1 : checks φ ( 3 ) and φ ( 4 )
VI.C.2 (1)
Resistanc e of cross-section to (Nxsd, Mzsd)·'
interaction (Nx.sd , Mz.sd) ^ 1, for class 1 and 2 cross-sections
IX.d.1.1 (1)
for class 3 cross-section
LX.d.1.3 (2)
for class 4 cross-section
LX.d.l.4(2)
162
Table IX. 1
D ($)
[54.3(1)]
List of checks to be performed at ULS for the member submitted to
combined axial force and bending moment (N, M)
Axial comDressive force and uniaxial bendine moment about maior axis References ; 1
(NxSdMy.Sd)'·
(1) Resistance of cross-section to NxsdNx.Sd — Np£Rd (design plastic resistance of the cross-section)
(2)
[5.5.1.1 (1)]
In-plane (xz) stability of member to NxsdNx.Sd ^ NbyrJld (design flexural buckling resistance of member)
(3)
Vn.c.2.1 (2)
Out-of-plane (xy) stability of member to NX£¿:
Nx.Sd — Nbz.Rd (design flexural buckling resistance of member)
[5.5.1.1 (1)]
Vn.cl (1)
General stability of member to Nxsd ■'
Nx.sd ^ design torsional buckling resistance of member
and, Nx.sd ^ design flexural­torsional buckling resistance of member
Vn.c.2.1 (2)
(4)
[Annex O]
[Annex G]
(5)
Resistance of cross-section and stability of member to Nx¿d if
class 4 monosymmetrical cross-section:
see table VILI : from checks φ ( 3 ) to φ ( 5 )
(6)
Resistance of cross-section and stability of member to Nxsd if
angle connected by one leg:
see table VILI : from checks φ ( 6 ) to φ ( 9 )
[5.5.2(7)]
(7) Out-of-plane (xy) stability of member to Mysd if λυτ > 0,40 :
[5.5.2(1)]
M y.Sd ίΞ Mb.Rd
(8)
(9)
15-5.2(7)]
15.5.4(2)]
15.5.4(4)]
[5-5.4(6)]
Vn.d.l (1)
Vn.d.2 (1)
Vn.d.2 (2)
Vn.e.1.1 (1)
Vn.e. 1.2 (1)
Vn.e.2.1 (1)
Vn.e.2.2 (1)
Vni.e.2 (3)
Vm.e.2 (4)
Resistance of cross-section to (Nxsd, Mysd)·'
interaction (Nx.sd, My.sd) ^ 1 , for class 1 and 2 cross-sections
for class 3 cross-section
for class 4 cross-section
[5.4.8.1]
[5.4.8.2]
[5.4.8.3]
15.5.4(1)]
[5.5.4(3)J
[5.5.4(5)]
(design lateral-torsional buckling resistance moment of the
member)
Vn.c.2.2
Vn.c.2.2
In-plane (xz) stability of member to (Nxsd, Mysd)-'
interaction (Nx.sd . My.sd) ^ 1 , for class 1 and 2 cross-sections
for class 3 cross-section
for class 4 cross-section
(10) Out-of-plane (xy) stability of member to (Nxjsd, My^d )
rx.d.i.i (i)
LX.d.1.3 (2)
LX.d. 1.4 (2)
LX.d.2.2(3)
table LX.4
table LX.5
table LX.6
LX.d.2.2 (2)
if λ w > 0,40 (potential lateral-torsional buckling): Vm.e.2 (3)
interaction (Nx.sd » My.sd) < 1 , for class 1 and 2 cross-sections
table LX.4
table LX.5
for class 3 cross-section
table LX.7
for class 4 cross-section
163
Table LX.l
0
Axial compressive force and uniaxial bending moment about minor axis References :
(1)
Resistance of cross-section to Nxsd'·
(Nx.sdMzSd)'·
Nx.Sd — Np£Rd (design plastic resistance of the cross-section)
[5.4.3 (1)]
(2)
(3)
N x .Sd — Nb y .Rd (design flexural buckling resistance of member)
General stability of member to Nxsd ·'
Nx.sd ^ design torsional buckling resistance of member
and, Nx.sd ^ design flexural-torsional buckling resistance of member
(5) Resistance of cross-section and stability of member to Nxsd if
class 4 monosymmetrical cross-section :
see table VILI : from checks φ ( 3 ) to φ ( 5 )
(6)
Resistance of cross-section and stability of member to Nxsd if
angle connected by one leg:
see table VILI : from checks φ ( 6 ) to φ ( 9 )
(7)
Resistance of cross-section to (Nx.sd, Mzsd)·'
interaction (Nx.sd , Mz.sd) ^ 1, for class 1 and 2 cross-sections
for class 3 cross-section
for class 4 cross-section
In-plane (xy) stability of member to (Nxsd, Mzsd)'
interaction (Nx.sd , M^sd) ^ 1, for class 1 and 2 cross-sections
for class 3 cross-section
for class 4 cross-section
[5.4.8.1]
[5.4.8.2]
[5.4.8.3]
(8)
[5-5.4 (1)]
[5-5.4(3)]
[5-5.4(5)]
Vn.c.2.1 (2)
Out-of-plane (xz) stability of member to Nxsd·'
(4)
[Annex G]
[Annex G]
Vn.c.1 (1)
In-plane (xy) stability of member to Nx.sdNx.Sd — Nb z .Rd (designflexuralbuckling resistance of member)
[5.5.1.1 (1)]
[5.5.1.1 (1)]
List of checks to be performed at ULS for the member submitted to
combined axial force and bending moment (N, M)
164
Vn.c.2.1 (2)
Vn.c.2.2
vn.c.2.2
Vn.d.l (1)
VII.d.2 (1)
Vn.d.2 (2)
Vn.e.1.1 (1)
VILe. 1.2(1)
VII.e.2.1 (1)
Vn.e.2.2 (1)
IX.d.1.1 (1)
IX.d.1.3 (2)
IX.d.1.4 (2)
LX.d.2.2 (3)
table IX.4
table IX.5
table IX.6
List of checks to be performed at ULS for the member submitted to
combined axial force and bending moment (N, M)
Table LX.l
(D
Axial tensile force and biaxial bending moments (Nxsd,Mysd,Mz.sd)'· References ;
(1) Resistance of gross cross-section to Nxsd ■'
Nx.Sd 5Í Np£Rd (design plastic resistance of the gross cross-section)
[54.3 (1)1
(2)
[54.3 (1)]
[54.2.2]
Resistance of the net cross-section to Nxsd if holes for fasteners:
Nx.Sd
(3)
— Nu.Rd
(design resistance of the net cross-section considering the
net area of a member or element cross-section, Anc[ )
Resistance of cross-section to Nxsd if angle connected by one leg.
see table VI. 1 : checks φ ( 3 ) and φ ( 4 )
[5.5.2(7)]
(4)
Meff.Sd — M b.Rd (design lateral-torsional buckling resistance moment of the member
[5.5.3 (3)]
[5.5.3 (4)]
(5)
[5.4.8.1]
[5.4.8.2]
[5.4.8.3]
(6)
[5.5.4(1)]
[5.5.4(3)]
15-54(5)]
[5-5.2(7)]
(7)
VI.b.2 (1)
VLcl (1)
VI.C.2 (1)
Out-of-plane (xy) stability of member to (NX£d, Mysd )
if λ L T > 0,40 :
[5.5.3]
VI.b.1 (1)
Vm.e.2 (3)
Vm.e.2 (4)
with Meff.sd calculated with (Nx.sd, My.sd)
LX.d.2.1 (1)
Resistance of cross-section to (Nxsd, Mysd, Mzsd )'■
interaction (Nx.Sd, My.Sd, Mz.Sd ) ^ 1
for class 1 and 2 cross­sections
for class 3 cross­section
for class 4 cross­section
LX.d.l
LX.d.1.2 (1)
LX.d.1.3 (1)
Stability of member to (Nxsd, Mysd, Mzsd )'■
rx.d.l.4(l)
vm.f.2 (i)
interaction (My.sd, Mz.sd)) ^ 1 for class 1 and 2 cross­sections
for class 3 cross­section
for class 4 cross­section
table LX.4
table IX.5
table LX.6
Stability of member to (Nxsd, Mysd, Mzsd ) if Λ-LT > 0,40 :
Vm.e.2 (3)
(potential lateral-torsional buckling) Vm.f.2 (2)
interaction (My.sd, Mz.sd)) ^ 1
for class 1 and 2 cross­sections
for class 3 cross­section
for class 4 cross­section
[5.5.4(2)]
[5.5.4(4)]
[5.54 (6)]
with My.sd reduced to Meff.sd ( as in check φ ( 5 ) )
165
table IX.4
table DÍ.5
table LX.7
LX.d.2.1 (1)
Table ΓΧ.1
List of checks to be performed at ULS for the member submitted to
combined axial force and bending moment (N, M)
References :
Axial compressive force and biaxial bending moments
(Nxsd> My.sd, M^si):
(1) Resistance of cross-section to Nxsd ■'
Nx.Sd — Np£Rd (design plastic resistance of the cross­section)
[54.3 (1)]
(2)
[5.5.1.1]
Vn.c.l (1)
Stability of member to Nxsd'
Nx.sd ^ minimum (Nbyjjd, Nbz.Rd)
Vn.c.2.1 (2)
(design flexural buckling resistances of member)
[Annex G]
[Annex G]
and, Nx.sd ^ design torsional buckling resistance of member
and, Nx.sd ^ design flexural­torsional buckling resistance of member
(3)
Resistance of cross-section and stability of member to Nxsd if
class 4 monosymmetrical cross-section:
see table VILI : from checks φ ( 3 ) to φ ( 5 )
(4)
Resistance of cross-section and stability of member to Nxsd if
angle connected by one leg:
see table VU.l : from checks φ ( 6 ) to φ ( 9 )
[5­5.2 (7)]
(5)
[5.5.2 (1)]
My .Sd — Mb.Rd (design lateral­torsional buckling resistance moment of the member)
(6)
(7)
[5.5.4(2)]
[5.5.4 (4)]
[5.5.4 (6)]
(8)
VILe. 1.1 (1)
VILe. 1.2 (1)
VII.e.2.1 (1)
VII.e.2.2 (1)
Vin.e.2 (3)
Vin.e.2 (4)
LX.d. 1.2(1)
LX.d. 1.3(1)
LX.d. 1.4(1)
Stability of member to (Nxsd, MySd, Mz¿d )'■
interaction (Nx.sd, My.sd, Mz.sd)) ^ 1
for class 1 and 2 cross­sections
for class 3 cross­section
for class 4 cross­section
[5.5.4(1)]
[5.5.4 (3)]
[5.54 (5)]
Vn.d.l (1)
VII.d.2 (1)
VILd.2 (2)
Resistance of cross-section to (NXmsd, Mysd, Mzsd )'·
interaction (Nx.Sd, My.Sd, Mz.Sd)) ^ 1
for class 1 and 2 cross­sections
for class 3 cross­section
for class 4 cross­section
[5.4.8.1]
[5.4.8.2]
[5.4.8.3]
[5­5.2(7)1
Stability of member to Mysd if &LT > 0,40 :
vn.c.2.2
VII.c.2.2
IX.d.2.2 (1)
table LX.4
table LX.5
table ΓΧ.6
Stability of member to (Nxsd, Mysd, Mzsd ) if %>LT > 0,40 :
Vm.e.2 (3)
(potential lateral-torsional buckling)
interaction (Nx.sd , My.sd, Mz.sd)) ^ 1
for class 1 and 2 cross­sections
for class 3 cross­section
for class 4 cross­section
166
IX.d.2.2 (2)
table LX.4
table ΓΧ.5
table ΓΧ.7
IX.b
Verifications at SLS
IX.b.l
Deflections
(1) About recommended limiting values for vertical deflections reference may be made to
chapter Vffl.b.1.
IX.b.2
Vibrations
(1) About recommended limiting values for floor vibrations reference may be made to
chapter Vm.b.2.
IX.C
Classification of cross-section
(1) At ultimate limit states the resistance of cross-sections may be limited by its local
buckling resistance. In order to take into account that limitation the different elements
(flange, web) of the cross-sections shall be classified according to the rules defined in
chapter V.
(2) For cross-sections submitted to combined axial load (Nx.sd) and bending moments
(My.Sd. Mz.Sd) the classification may specifically be determined according to the
procedure defined in chapter V.d.3.
I X.d
Verifications at ULS to (N,M)
(1) The verification of members submitted to combined axial force and bending moments
shall be performed with different (N, M) interaction rules about:
1)
the resistance of the cross-section (see chapter IX.d.l),
2)
the buckling of member (see chapter IX.d.2) and,
3)
the lateral-torsional buckling of the member (if potential)(see chapter K.d.2)
This principle of (N, M) interaction formulas is illustrated in table DÍ.2 in case of uniaxial
bending and compression (on the basis of the interpretation of Eurocode 3 explained in the
comment IX.d.2.2 (3)).
(2) All the (N, M) interaction formulas depend on the class of cross-sections.
(3) Uniaxial bending (M y sd or M z .sd) and biaxial bending (My.sd and Mz>sd) combined
with axial force N x .sd are presented in the following chapters.
IX-d.l Resistance of cross-section to (NSc], Msd)
IX.d.1.1 Uniaxial bending of class 1 or 2 cross-sections
[54.8.1 (1)] (1) For the plastic resistance of class 1 or 2 cross-sections submitted to combined axial load
(Nx.sd) and uniaxial bending moment (My.sd or M z .sd), the following criterion shall be
satisfied if the level of axial load n is high:
Msd^M N.Rd
[form. (5.23)]
where
MN.Rd is the reduced design plastic resistance moment allowing for the
axial force (MNyjui» MNZJW) (see table IX.3),
n
is the level of axial load (see table IX.3)
n=
N x.Sd
N pf.Rd
_
N x.Sd
Afj
ΎΜΟ
where Np£Rd is the design plastic resistance of the cross-section
[5.4.8.1 (3)] (2) If the level of axial load η is low (see table IX.3, for the limiting values of n), then the
reduction of the theoritical plastic resistance moment by the presence of small axial forces
is counter-balanced by the strain hardening and may be neglected.
In that case of low η reference may be made to chapter Vm.e.l for checking single Msd ·
167
Table IX.2 Principle of interaction formulas between axial force Nsd and bending moment Msd
1. For bending moment about maior axis M., *A without lateral­torsional buckling Γλτ τ < 0,4):
7.
Ν
'
L
NcRdi
(Nb.Rd)mi
'min
Instability
without lateral­
torsional buckling
TnctîiKilitT/
r£
</\>
i¿—
χ —
Hl—x
y"
Nx.sd
2
M y.Sd
lack of resistance
Mcy.Rd
2. For bending moment about major axis M y KA with lateral­torsional buckling (λτ τ > 0,4):
Ν
ζ
NcRd ' ■
.-y
r—χ
χ—
y"
3
N x.Sd
y.Sd
Instability due
to lateral­torsional buckling
(Nb.RdX
Mb.Rd
z
Mcy.Rd
3. For bending moment about minor axis M 7 ΖΛ :
Ν
NcRd f
M z.Sd
Ê
Instability
■ ■ ■ ■ - *
X
(Nb.Rd>'min
—
y"
ι
I
lack of resistance
MCZ.Rd
168
z
'
^
N x.Sd
Reduced design plastic resistance moment MNjtd allowing for axial load
class 1 or 2 cross­sections
Table IX.3
[54.8.1 (4)]
Values of
the limit a
Rolled and welded
I­sections
H
• (η π A - 2 b t f > j
L
= min 0,25;
:
'£¡L'
bjz
. ( __ A - 2 b t Lf
a = min n0,50;
Values of
the limit a
Hollow sections
a = 0,25
κΉ
A-
τ=
a=
A-2bt
- ζ
η
W r ——
ΎΜΟ
1,04(1-n^iWp,^
V
;
P
ΪΜ0
M N .R d = 1,26(1 - n ) W p f
YMO
M N y jid= L 3 3 ( l - n ) W p r . j
ΎΜΟ
M
Nz.Rd =
For a plate without
bolt holes
Notes :
-(fff)3
—
ΎΜΟ
If high level of axial load Nx.sd:
if|n>a|
A-2bt
2A
r
A 2b
a = min 0 , 2 5 ; - L
2A )
V
-y
MNZJW =
M N .R d =
a = min 0,25;
h^H
1-n^j
Wp'-y
1-a,
MNy.Rd =
2A J
V
- ζ
y-
If high level of axial load Nx.sd:
if|n>a|
1-n
ht
Λ Γ
0,5 + —
A
MN.Rd = Mp£Rd[l-n 2 ]
_Nx.sd
Afy
ΎΜΟ
- Limitation of M N J W obtained from this table:
MN.Rd ^ M p£Rd
(design plastic resistance moment)
MNy.Rd^M p f , y .Rd=
in other words,
M
Nz.Rd ^ Mpf.z.Rd =
169
W ,
fy
γΜο
Wprzfy
γΜ()
ΎΜΟ
IX.d.i.2 B iaxial bending of class 1 or 2 cross-sections
[54.8.1(H)] (1) For the plastic resistance of class 1 or 2 cross­sections submitted to combined axial load
(Nx.sd) and biaxial bending moments (My.sd and Mz.sd) the following criterion may be
used :
α
M
[form. (5.35)]
y.Sd
M
+
ß
z.Sd
<1
M
L^Ny.Rd_
_ Nz.Rd_
where MNy.Rd and MNZ.Rd are reduced design plastic resistance moments allowing
for axial load (see table IX.3)
α and β are constants taken as follows:
­ for I and Η sections:
; β = 5n, but β > 1
α =2
­ for circular tubes:
α =2
;ß = 2
1,66
­ for rectangular hollow sections: α = β =
2,buta=ß < 6
1­1,13η
­ for solid rectangles and plates: α = β = 1,73 + 1,8 η 3
(2) In case of biaxial bending moments without axial load it is proposed to use the
following criterion:
α
M y .sd
Wpi.yfy
ß
+
z.sd
Wpf.Zfy
[ ΎΜΟ
where Wp£y,Wp/;z
ΎΜΟ
α and β
[5.4.8.2]
M
<1
ΎΜΟ
are plastic section modulus about major axis (yy) and minor
axis (zz),
is the yield strength (see table II.4)
is a partial safety factor (see table 1.2) ,
are constants taken as follows:
- for I and Η sections:
- for circular tubes:
- for rectangular hollow sections:
- for solid rectangles and plates:
α =2 ; β=1
α =2 ;β=2
α =β= 1,66
α = β = 1,73
IX.d.i.3 Bending of class 3 cross-sections
(1) For the elastic resistance of class 3 cross-sections submitted to combined axial load
(Nx.sd) and biaxial bending moments (My.sd and Mz.sd) the following criterion shall be
satisfied:
[form. (5.38)]
(2) In case of uniaxial bending moment combined with axial load ((Nx.sd + My.sd) or
(Nx.sd + Mz.sd)) and in case of biaxial bending moments without axial load (My.sd +
Mz.sd), it is proposed to use the above criterion (IX.d.1.3 (1)).
170
[54.8.3]
[form. (540)]
IX.d.1.4 Bendino of class 4 cross-sections
(1) For the elastic resistance of class 4 cross-sections submitted to combined axial load
(Nx.sd) and biaxial bending moments (My.sd and Mz.sd) the following criterion shall be
satisfied:
N x.sd
^eff
where A^f
Weff
βΝ
, My-sd + Nx-sd eNy
ΎΜΙ
W eff.y
|
Mz-Sd + N x sd e Nz ^ 1
W eff.z'
ΎΜΙ
YMI
is the effective area of the cross-section when subject to uniform
compression (single Nx.sd)·
is the effective section modulus of the cross-section when subject only to
moment about the relevant axis(single My.sd , single Mz.sd) ·
is the shift of the relevant centroidal axis (eNy, eNz) when the cross-section
is subject to uniform compression (single Nx.sd) ·
(2) In case of uniaxial bending moment combined with axial load ((Nx.sd + My.sd) or
(Nx.sd + Mz.sd)) and in case of biaxial bending moments without axial load (My.sd +
Mz.sd), it is proposed to use the above criterion (IX.d.1.4 (1)).
IX.d.2 Stability of member to (Nsd,Msd)
[5.5.3]
ΙΧΛ2.1 Stability of member to (N,ension, Mysd)
(1) If the non-dimensional slenderness of the member λτ^τ >0,40 (see chapter Vin.e.2), the
member subject to combined major axis bending (My.sd) and axial tension (Nx.sd) shall be
checked for resistance to lateral-torsional buckling as follows (if bending moment and
axial force can vary independently (vectorial effect)):
Meff.Sd <M b.Rd
where Mb.Rd is the design buckling resistance moment (see chapter VITLe.2)
Meff.sd is the effective design internal moment obtained from:
Meff.Sd = Wcom
°com.Ed
where Wcom is the elastic section modulus for the extreme compression
fibre,
<*com.Ed is the net calculated stress (which can exceed fy) in the
extreme compression fibre determined from:
Ύ)
ÍMy.Sd^
ψ vee
V "com y
where \j/Vec is the reduction factor for vectorial effects
'com .Ed
Vvec=0,8.
171
[5.54]
IX.d.2.2 Stability of member to (Ncompression, Msd)
(1) For members subject to combined axial compression and biaxial bending moments (Nx.sd
+ My.sd + Mz.sd) the stability is guaranteed if the requirements (concerning the case © )
described in tables IX.4 to IX.7 are satisfied:
- in table IX.4 for class 1 or 2 cross-sections,
- in table IX.5 for class 3 cross-sections and,
- in tables IX.6 and IX.7 for class 4 cross-sections.
(2) As given in the tables IX.4 to IX.7, when the non-dimensional slenderness of the member
^LT >0,40| (see chapter VQI.e.2), supplementary specific formulas also need to be
satisfied to take into account the potential failure mode of lateral-torsional buckling of the
member.
(3) The cases of uniaxial bending moment combined with axial compression ((Nx.sd + My.sd)
or (Nx.sd + Mz#sd)) are not fully explained in Eurocode 3. Therefore it is proposed to use
the rules for biaxial bending with the buckling reduction factor xmin (%min = minimum
(%y, Xz); where (x y , χ ζ ) are the buckling reduction factors about y axis and ζ axis) (see
rules concerning the case (2) in tables IX.4 to IX.7).This principle of (Ν, M) interaction
formulas is illustrated in table IX.2.
(4) According to Eurocode 3 Background Document 5.03 (/8/), another interpretation
could be proposed for uniaxial bending combined with axial compression: this proposal
may be less conservative because the factor xmin is replaced by the buckling reduction
factors Xy or χ ζ according to the relevant bending axis. Moreover the stability out of the
bending plane should be also checked (buckling resistance of the member to single axial
compression) (see rules concerning the case (3) in tables IX.4 to IX.7).
172
Table IX.4
Internal
forces
and
moments
Interaction formulas for the (NM) stability check of members of class 1 or 2
If X L T > 0 , 4 :
potential lateral-torsional buckling
needs supplementary checks:
General formulas to be always satisfied:
φ Eurocode 3 formulas for biaxial bending (My.sd, M z .sd) and axial compression N x .sd:
k v M Sd ,
Nx.sd
Nx.sd
+ My.Sd
+ Mz.sd
nin * ■
ΎΜΙ
W pr.y
k
zMz.Sd
c l
Ν x.Sd
X z A Jy_/„
ΎΜΙ
ΎΜΙ
ΎΜΙ
k
LTMy.sd
k z M z .Sd
w
A
XurWp/.y
ΎΜΙ
<1
ΎΜΙ
Eurocode 3 formulas for uniaxial bending and axial compression:
use the formula for biaxial bending and axial compression (see φ ) introducing the
relevant bending moment equal to zero and using Xmin buckling reduction factor.
Other proposal of interpretation of Eurocode 3 formulas for uniaxial bending and axial
compression (see comments in IX.d.2.2) using buckling reduction factors x y or χ ζ according to
the relevant bending axis:
Nx.sd
f,
3CyA-
Nx.sd
+ My.Sd
,
ΎΜΙ
and,
k M
y y-Sd
L
W pf.y
Ν x.Sd
f
^
ΎΜΙ
ΎΜΙ
k
LTMy.Sd
XLTWpí.y
f
<1
ΎΜΙ
N x . S d <> N b . z . Rd = χ ζ A
ΎΜΙ
Nx.Sd
Nx.sd
XzA^
+ Mz.sd
ΎΜΙ
and,
,
k M
z z.Sd
w
»
c l
<
N x S d <,Nb.yRd = X y A—*ΎΜΙ
where: x m m = minimum ( χ γ ; χ ζ ), where
χ ν and χ ζ are given in chapter VII.c.2.1
XLj and %LT are given in chapter VHI.e.2
ky = 1 -
μ>
*fSd but k y <, 1,5 ; where \ny = Xy(2ßMy
­ 4) +
Wer.y
*y'"y
^zN*Sd
X z Af y
Wpr.y­Wef.y
but k z <, 1,5 ; where μ ζ = λ ζ ( 2 β Μ ζ ­ 4 ) +
Wpf,z­Wef,z^
weC.z
b u t μ y <0,90,
but μ ζ < 0,90,
k
LT = l ­ ^ T " * ; S d but k L T £ 1,0; where μ ί Τ = ( 0 , 1 5 . λ ζ . β Μ 1 τ ) - 0 , 1 5 but μ ί Τ ^0,90,
3CzAfy
where ßM (ßMy, ßMz) is the equivalent uniform moment factor related to the shape of the bending
moment (My.sd, M z .sd):
βΜ=1,8­0,7ψ
­1£ψ$ 1
βΜ=1,3
173
βΜ=1,4
Table ΓΧ.5
Internal
forces
and
moments
Interaction formulas for the (NM) stability check of members of class 3
General formulas to be always satisfied:
IflLT>0,4:
potential lateral-torsional buckling
needs supplementary checks:
φ Eurocode 3 formulas for biaxial bending (My.sd, M z .sd) and axial compression Nx.sd^
Nx.sd
+ My.sd
+ Mz.sd
k„M
y»'y.Sd
Ν x.Sd
AA
, k z M Z-Sd
<1
Ν x.Sd
k
LTMy.Sd
~
™
XurWef.y
X z A Ì
w eC.z '
YMl
ΎΜΙ
ΎΜΙ
Eurocode 3 formulas for uniaxial bending and axial compression:
-min
y
ΎΜΙ
k
+
ΎΜΙ
zM z .sd
<1
Wefz-^
ΎΜΙ
use the formula for biaxial bending and axial compression (see φ ) introducing the
relevant bending moment equal to zero and using % m i n buckling reduction factor.
Other proposal of interpretation of Eurocode 3 formulas for uniaxial bending and axial
compression (see comments in IX.d.2.2) using buckling reduction factors %y or χ ζ according to
the relevant bending axis:
Nx.sd
Nx.sd
,
ΎΜΙ
and,
y y-Sd
Nx.sd
f
X z A Ì
^
We'.yr1-
XyA-^
+ My.sd
k M
,
ΎΜΙ
ΎΜΙ
k
LTMy.sd
f
XurWef.y
—™
iï
ΎΜΙ
Nx.Sd<Nb>z.Rd=xzAΎΜΙ
Nx.sd
Nx.sd
%
+ Mz.sd
and,
Z
!
A^
ΎΜΙ
Nx-Sd < N b
k M
z z.Sd
we
Rd
«^
ΎΜΙ
= χ A—*ΎΜΙ
where: %min = minimum ( %y ; χ ζ ) , where
χ γ and χ ζ are given in chapter VII.c.2.1
X LT and %LT are given in chapter Vm.e.2
k y = 1-!~Z * S d but k y < 1,5 ; where μ γ = Xy (2ß M y ­ 4) but μ γ <0,90,
XyAfy
_ ι μζΝ,
kz = 1_r>z-^sd but ^ < ι , 5 ^ η 6 Γ β μ ζ = λζ(2βΜ2-4)οϋΐμζ<0,90,
xy
XzZAAI
k L T = l ­ ^ T ^ * S d but k L T < l , 0 ; where μ ι τ = ( 0 , 1 5 . λ ζ . β Μ υ Γ ) ­ 0 , 1 5 but μ ^ <0,90,
XzAfy
where ß ^ (ßMy, ßMz) is the equivalent uniform moment factor related to the shape of the bending
moment (My.sd, M z .sd):
βΜ=1,8­0,7ψ
­1<ψ< 1
ßM=l,3
174
ßM=l,4
Table IX.6
General interaction formulas for the (NM) stability check of members of class 4
Internal
forces and moments
General formulas to be always satisfied:
(Ï) Eurocode 3 formulas for biaxial bending (My.sd, M z .sd) and axial compression N x .sd:
Nx.sd +
My.Sd +Mz.Sd
N x.Sd
Xmin"·
min^eff
|
k
y(My,sd+Nx,SdeNy)
kz(MzSd+N;t.SdeNz)^1
|
W eff.z
Weff-y ^ ΎΜΙ
ΎΜΙ
ΎΜΙ
Eurocode 3 formulas for uniaxial bending and axial compression:
use the formula for biaxial bending and axial compression (see CD) introducing the
relevant bending moment equal to zero and using Xmin buckling reduction factor.
Other proposal of interpretation of Eurocode 3 formulas for uniaxial bending and axial
compression (see comments in IX.d.2.2) using buckling reduction factors x y or χ ζ according to
the relevant bending axis:
Ν x.Sd
Nx.sd + My.Sd
XyA
y^eff
τ
k
y( M y.Sd
Nx.sdeNy)
weff.y YMI
ΎΜΙ
and,
+
N x - S d < N b - z . R d = χ ζ A eff -
YMI
k
Nx.sd
Nx.sd + Mz.sd
Xz^eff
2A
and,
M
, z( z.Sd + N x S d e N z )
YMI
YMI
Nx.sd < N b . y . R d = Xy A e f f - ^ YMI
where: x m i n = minimum ( Xy ; χ ζ ), where
%y and χ ζ are given in chapter VII.c.2.1
Xu and XLT are given in chapter VHI.e.2
ky = 1 kz =
y
x,
y but ky <, 1,5 ; where μ γ = X y (2ß M y ­ 4 ) but μ γ <0,90,
Xy Aeff f y
ι_μζΝΧ.Μ
Xz A eff r y
but kz s l 5
. w h e r e μ ζ = Χ ζ ( 2 β Μ ζ ­ 4 ) but μ ζ <0,90,
where ßM (ßMy, ßMz) is the equivalent uniform moment factor related to the shape of the bending
moment (Msd + Nx.sd eN):
^P
7
βΜ=1,8­0,7ψ
ßM=l,4
­l^yál
where Aeff, Weff.y, Weff.z, eN.y, eN.zare effective properties of cross­section defined in chapter V.
175
Table ΓΧ.7 Supplementary interaction formulas for the (NM) stability check of members of class 4
Internal
forces and moments
If λτ^τ > 0,4: potential lateral-torsional buckling needs supplementary checks
CD Eurocode 3 formulas for biaxial bending (My.sd, M z .sd) and axial compression Nx.sd^
Nx.sd +
My.Sd + Mz.sd
j kLT(My.sd + N x . Sd e N y )
Nx.sd
Xmin Aeff
|
k z (M z . S d + N x . S d e N z ) "
W eff.z
XirWeff.y
YMI
YMI
ΎΜΙ
Eurocode 3 formulas for uniaxial bending and axial compression:
use the formula for biaxial bending and axial compression (see (f)) introducing the
relevant bending moment equal to zero and using Xnun buckling reduction factor.
Other proposal of interpretation of Eurocode 3 formulas for uniaxial bending and axial
compression (see comments in IX.d.2.2) using buckling reduction factors x y or χ ζ according to
the major axis:
Nx.sd
Nx.sd + My.sd
, kLT(My.Sd+Nx,SdeNy)^i
Xz^-eff"
ZA
YMI
where: Xmùi = minimum ( Xy ; χ ζ ), where
XLTWeff.y-^
YMI
x y and χ ζ are given in chapter VII.c.2.1
XLT and XLT are given in chapter Vm.e.2
ky=lk =1
- ^ but k y <1,5 ; where \iy = Xy(2$My -4) but μ γ <0,90,
Xy Aeff f y
μ ζ Ν , ,Sd but k < 1,5 ; where μ = λ ( 2 β - 4 ) but μ <0,90,
z
ζ
ζ
Μζ
ζ
Χζ Aeff fy
k
L T = l - ^ T A N x f S d but k L T <1,0; where μ ί Τ = ( 0 , 1 5 Ä z . ß M . L T ) ­ 0 , 1 5 but μ ί τ < 0 , 9 0 ,
XzAeff t y
where ßM (ßMy, ßMz) is the equivalent uniform moment factor related to the shape of the bending
moment (Msd + N x .sd e ^ :
VOLLV
βΜ=1,8­0,7ψ
ßM=l,4
ß M = 1,3
­1<ψ< 1
where Aeff, Weff.y, Weff.z, en.y, eN.z are effective properties of cross­section defined in chapter V.
IX.e
Verifications at ULS for (N$d >Vsd)
(1) If the design values of shear force
Vz.Sd<0,50Vp,z.Rd
and,
Vy.sd<0,50Vpry.Rd
where V p f z jid, VPfy.Rd are the design plastic shear resistances about minor (zz) and
major (yy) axes (see table K . 8 ) ,
no reduction of the tension or compression resistances is needed. With this condition
the design value of axial force N x .sd shall be checked separately according to chapter
VI (tension) or chapter VII (compression).
176
IX e. 1
[54.9 (3)]
Resistance of cross-section to (NSd,Vsd)
(1) For members submitted to combined axial force Nx.sd and shear force (Vz.sd or Vy.sd) if
V M > 0,50 V p i Rd then it is proposed that each cross-section shall satisfy the following
criterion:
Nx.sd*N V.Rd
where Nv.Rd
is the reduced design resistance of the cross-section allowing for
shear force (see table IX.8).
Reduced design resistance Ny.Rd allowing for shear force
Table IX.8
Combined loading
If high shear: Vsd > 0,5.Vpi^d
Nx.sd + V^sd
N Vz.Rd
NX.Sd + Vy.Sd
Ν Vy.Rd
2 _V 2 l Sd—!
where:
k
Py =
Vpfz.Rd
J
y.Sd
-1
pfy.Rd
(A - p z AVtZ)f3
ΎΜΟ
(A - Py Ay.y)f3
YMO
Aν · ζ f y
with Vp í z . R d = 7=•iu\r
YMOV3
withV
^
Rd
=
A
f
W5
If (Nx.sd)tension,
A = gross cross-section or net section (Anet) (see chapter VI),
If (Nx.Sd)compression, A = gross cross-section for class 1,2 or 3 cross-section, or effective
cross-section (Aeff) for class 4 cross-section (see chapter VU).
IX.f
Verifications at ULS to (Nsd ,Vsd,Msd)
(1) The verification of members submitted to combined axial force, shear forces and bending
moments shall be performed with different (N,V,M) interaction rules about:
1) the resistance of the cross-section (IX.f.l),
2) the stability of web (LX.f.2)
(2) All the (N,V,M) interaction formulas depend on the class of cross-sections.
(3) Uniaxial bending (My.sd or Mz.sd) and biaxial bending (My.sd and Mz.sd) combined
with shear forces (Vz.sd and Vy.sd) and with axial force Nx.sd are presented in the
following chapters.
177
DC.f. 1 Resistance of cross­section to (NSd,VSd,MSd)
(1) If the design values of shear force
V z . S d <0,50V p f z . R d
and,
V y . S d <0,50V p i y R d
where Vp¿z R d, Vpfy.Rd are the design plastic shear resistances about minor (zz) and
major (yy) axes (see table ΓΧ.9),
no reduction needs to be made in combination of bending moment and axial force.
With this condition the members shall be verified to combined (Nsd, Msd) loading
according to chapter DC.d.
[5.4.9 (3)]
(2) If the design values of shear force VSd > 0,50 V p i Rd (high shear), the design resistance
of the cross­section to combinations of moment and axial force should be calculated
according to Eurocode 3 with a reduced yield strength (1 ­ p)fy for the shear area (Ay)
2\
where ρ =
2V Sd
V. V pi.Rd
­1
The interaction formulas (N,V,M) proposed in the following chapters (IX.f.1.1 to IX.f.1.4)
in case of high shear are simplifications replacing f, by (l­p)f, for the sections
properties of cross­sections: A, Aeff, Wp£We/­,Weff.
IX f.1.1 Uniaxial bending of class 1 or 2 cross-section
(1) For the plastic resistance of class 1 or 2 cross­sections submitted to combined axial force
(Nx.sd) with shear forces and uniaxial bending moments ((Vz.sd and My.sd) or
(Vy.sd and Mz.sd)), if the design values of shear force
V z . S d >0,50V p f 2 . R d
Vy. S d >0,50V p i y . R d or, (high shear)
then relevant interaction between (Nsd,Vsd,Msd) shall be considered.
In this case the design value of bending moment Msd shall satisfy the following criterion
if the level of axial load η is high:
M S d <M N i V ­ R d
where MN.vjtd is the reduced design plastic resistance moment allowing for the
axial load and shear force (MN.v.y.Rd (about major axis (yy) bending),
MN.V.z.Rd (about minor axis (zz) bending))(see table IX.9),
is the level of axial load (see table IX.9)
η
n= J^*LΝ pf .Rd
Ν x.Sd
Afy
ΥΜΟ
where Np¿Rd is the design plastic resistance of the cross­section
(2) If the level of axial load η is low (see table IX.9, for the limiting values of n), then the
reduction of the plastic resistance moment may be neglected and the applied bending
moment Msd combined with shear force Vsd shall be verified according to the rules given
in chapter VEI. g. 1.1. On the other hand the axial load Nx.sd shall be verified in
combination with shear forces Vsd according to chapter IX.e.
178
[5.4.9]
Table IX.9 Reduced design plastic resistance moment Mfj.yjid allowing for axial load and
shear force for class 1 or 2 cross­sections
If high level of axial load Nx.sd:
Rolled and
Values of the limit a
if[n~>äl
welded I­sections
K^H
b
MN.v.y.Rd = Í ^ ^ ^ ^ W
y-í^
P y
• (η o< A ­ 2 b t f ^
= min^0,25;^^J
V
• Γη <n A ­ 2 b t f >
L
= nun 0,50;
MN.V.z.Rd= (1­Py) 1 ­
V
A
j
Hti'
hollow sections
J
1­a
(1­Py)
1­a
r
a = 0,25
η
L
y-
a = min 0,25;
-y
u-b.
ζ
J
- 3=- - z
a=
MN.v.Rd= 1,04 1 ­ p ­
A-2bt^
MN.VJW=
A-2bt
2A
-n =
n
Λ
A-2bt
MN.V^.Rd =
J
U6(l-η-p)Wpf
ht
0,5 + —
V
A j
MN.VJW
w
n c
pr.y
= MpÉRd [1-n 2 ]
^ A£y
YMO
- The values of p z and p y are given on the following page.
- Limitation of MN.vjtd obtained from this table:
Mw.vjid ^ M pCRd (design plastic resistance moment)
M
N.V.y.Rd ^ Mpf.yJtd
or, in other words,
M
N.V.z.Rd - Mpf . z . Rd
179
f
W rpt
0.7
_ Wpiyfy
γ^—
_ W pf . Z f y
ΎΜΟ
v
ΎΜΟ
YMO
MN.v.y.Rd= 1 , 3 3 ( 1 - n - p z ) W p r .
For plate without
bolt holes
notes;
YMO
1,7
(1­P)
S=
WpfX
ν
If high level of axial load Nx.sd:
if|n~>äl
Values of the limit a
. fA„c
YM0
YMO
—
YMO
Pz =
2-^Sd.
pf.z.Rd
v
Pv =
-1
V
y.Sd
, withV
f-2-Rd
=
A
fly
^v.z
·
'
YMOV3
ι
-1
pi.y.Rd
IX f. 1.2 Biaxial bending of class 1 or 2 cross-section
[54.8.1(H)] (1) For the plastic resistance of class 1 or 2 cross-sections submitted to combined axial force
(Nx.sd), with shear forces and biaxial bending moments ((Vz.sd and My.sd) and
(Vy.sd and Mz.sd)), if the design values of shear force
V z .sd>0,50Vp fz . Rd
and,
Vy.sd>0,50V pry . Rd
then the following criterion is proposed:
ία
My.Sd
M,N.V.y.Rd J
r-
M z.Sd
L M N.V.z.Rd .
ß
<1
where MN.y.y.Rd and Mjsj.v.z.Rd are the reduced design plastic resistance moment
allowing for the axial load and shear force (see table IX.9),
α and β are constants taken as follows:
­ for I and Η sections:
­ for circular tubes:
­ for rectangular hollow sections:
­ for solid rectangles and plates:
[5.4.8.2]
α = 2 ; ß = 5n, b u t ß > l
α=2 ;β=2
1,66
, buta=ß < 6
α
1­1,13η'
α = β = 1,73 + 1,8 η 3
IX.f.1.3 B ending of class 3 cross-section
(1) For the elastic resistance of class 3 cross­sections submitted to combined axial load
(Nx.sd), shear forces (Vz.sd, Vy.sd) and biaxial bending moments (My.sd and Mz.sd) the
following proposed criterion shall be satisfied in case of high shear( V Sd > 0,50 Vp[ R d ) :
ECCS n°65
table 5.16
where ρ = maximum value of (pz, py)
where p z and p y are given in previous chapter IX.f.1.1
(2) In case of uniaxial bending moment combined with axial load and shear force the
following criteria shall be satisfied in case of high shear(VSd > 0,50 V pf R d ):
­ for bending about major axis (Nx<sd, Vz.sd and My­sd):
180
for bending about minor axis (Nx.sd, Vy.sd and Mz.sd):
[54.8.3]
IXf.l .4 Bending of class 4 cross-section
(1) For the elastic resistance of class 4 cross-sections submitted to combined axial force
(Nx.sd), shear forces (Vz.sd, Vy.sd) and biaxial bending moments (Mysd and Mzsd) the
following proposed criterion shall be satisfied in case of high shear( V Sd > 0,50 Vp/· R d ):
Nx.sd
f
v
A
eff/
, My.Sd + Nx.Sd e N y
f
W cc
YMO
1
M s.Sd + N x .sd CNZ
< 1 - -p
y
ΎΜΟ
YMO
where ρ = maximum value of (p z , p y ),
where p z and p y are given in previous chapter IX.f. 1.1
where Aeff is the effective area of the cross-section when subject to uniform
compression (single Nx.sd),
is the effective section modulus of the cross-section when subject only to
Weff
moment about the relevant axis(single My.sd , single Mz.sd) ,
is the shift of the relevant centroidal axis (eNy, eNz) when the cross-section
is subject to uniform compression (single Nx.sd) ·
(2) In case of uniaxial bending moment combined with axial load and shear force the
following criteria shall be satisfied:
eN
for bending about major axis (Nx.sd, Vz.Sd and My.sd):
and,
- for bending about minor axis (Nx.sd, Vy.Sd and Mz.sd):
N
x.sd , M z.Sd+N x .s d e Nz
—Fy-+
ry—*1-P>
L
eff"~
YMO
W
eff.z~
YMO
181
IX.f.2 Stability of web to (Nx.sd, Vz.Sd, My.Sd)
[5.6.7.2]
(1) If webs are submitted to combined axial load Nx.sd, shear force Vz.sd and bending
moment My.sd and,
if they have ratio
exceeding the limits given in table VHI.7,
w
w
then they shall be checked for resistance to shear buckling.
(2) The interaction of shear buckling resistance and moment resistance is shown in table
VUL 14 according to the simple post-critical method.
(3) The web may be assumed to be satisfactory if one of the three following checks ((T),
(2) or (3)) according to the loading level (Vz.sd, My.sd) shall be satisfied:
[5.6.7.2 (i)]
(T) If the design value of bending moment
My.Sd g MN.f.Rd
where
M N . ^ is the reduced design plastic moment resistance of a cross-section
consisting of the flanges only and allowing for axial force; proposal for
cross-section with equal flanges:
- for class 1, 2 or 3
M
N.f.Rd=btf(h-tf)
YMO
^tf
-T4
e
N x.Sd
"2btffy^
v.
My.sd - for class 4:
M )
MN.f.Rd =
YMO
>
f
((b + beff)tf ^ y L ) - ( ( b - b e f f ) t f eM) fy
YMI
1—
Nx.sd
r(b+b eff )t f f y ^|
^
YMI
JJ
where b, tf, h are flange width, flange thickness, height of profile (see table 0.1),
beff
is the effective width of the compression flange (see chapter V),
eM
is the shift of the centroidal major axis (yy) when the cross-section
consisting of flanges only is subject to My.sd·
then the design value of shear force shall satisfy :
V
z.Sd ^
where
V
ba.Rd
Vbajid is the design shear resistance buckling of the web according
to simple post-critical method (see chapter Vffl.d.2).
182
[5.6.7.2(2)]
(D //"the design values of bending moment and shear force
My.sd > M N.f.Rd and V . s d ^ C U O V ^ d
then the bending moment shall satisfy :
My.Sd ^ M N . y j R d
where M ^ R J is the redudced design resistance moment of the cross-section
allowing for axial load depending on classes of cross-sections
(see chapter LX.d.1.1, LX.d.1.3, LX.d.1.4).
[5.6.7.2 (3)]
If the design values of bending moment and shear force
My.sd > M N.f.Rd and
v^^sov^
then the bending moment and the shear force shall satisfy the three following checks
where MN.p£y.Rd is the reduced design plastic resistance moment of the cross-section
allowing for axial load :
M
N.pf.y.Rd = MN.y.Rd (see table LX.3)
- and, My sd ^ M N y R d
- and,
z.Sd - VbaJld
183
TRANSVERSE FORCES ON WEBS (F ; (F,N,V,M))
χ
Generalities
X.a
(1) For each load case (see chapter ΙΠ) the global analysis of the frame (see chapter IV)
determines the design values for the following effects of actions which are applied to the
web of members and which shall be checked at ultimate limit states:
transverse forces with separate or combined internal forces and moment acting in the
plane of the web:
'U
Fsd
ζ
&
...y
χ -
- -χ
Ρ
Νx.Sd
I
f
'z.Sd
Ν.x.Sd
My.Sd
hd\
[5.7.1(2)]
[5.7.1(1)]
[5.7.1(6)]
(2) The transverse forces Fsd may be applied in different ways:
- either,
through one flange
- or,
to one flange and transferred through the web directly to the other flange.
(3) The resistance of an unstiffened web to transverse forces applied through a flange, should
be cheked for all the three following modes of failure (see table X. 1) :
. crushing of the web close to the flange, accompagnied by plastic deformations of the
flange (see chapter X.c.2),
. crippling of the web in the form of localised buckling and crushing of the web close to
the flange, accompagnied by plastic deformation of the flange (see chapter X.d. 1),
. buckling of the web over most of the depth of the member (see chapter X.d.2)
(4) In addition the effect of the transverse force on the moment resistance of the member
should be considered : resistance to local buckling (see chapter X.b) and yield criterion
(see chapter X.c.1)
Table X.1
ECCS n°65
table 5.36
Failure modes due to load introduction
Crippling
Crushing
1
=T
ι
1
ι
Buckling
I
I
184
.
I
I
I
.
I
Classification of cross-section
(1) The effects of significant transverse compressive stresses on the local buckling resistance
of a web shall be taken into account in design. Such stresses may arise from transverse
forces on a member and at member intersections.
(2) The presence of significant transverse compressive stresses may effectively reduce the
maximum values of the depth­to­thickness ratios d/tw for class 1, class 2 and class 3 webs
below those given in chapter V, depending on the spacing of any web stiffeners.
(3) A recognised method of verification should be used. Reference may be made to the
application rules for stiffened plating given in ENV 1993­2 Eurocode 3: Part 2 (which is
in preparation).
X.b
[5.3.6]
X.C
X.c.1
Resistance of webs to (F,N, V,M j
Yield criterion to (F,N,V,M)
[54.10]
(1) The web of a member subject to a transverse force in the plane of the web (see table X.2)
in addition to any combinations of moment and axial force on the cross­section, shall at all
points satisfy the criteria given in table X.3.
[Fig. 5.4.3]
Table X.2
Stresses in web panel due to bending moment, axial force and transverse force |
Fsd
f
Nx.sd
M
y­sdf
'Γ
A'
C.
'Β
*My . S d N
x.Sd
.D
Fsd
(a) Layout
σ. ΟΖΕΕΡβ* σ,
ED
f
o*., . a n m * σχ­1
Β
>x.l
öb
=:
4
D?
σF^C
2
*· a m p o , ° ^
b) Stresses in element E
(c) Stresses in panel ABCD
185
σχ.2 σ^,η
oh
(where ab=f3rn.fy)
(d) Equivalent stresses
Yield criteria to be satisfied by the web
Table X.3
IfVsd < 0,5.V p £ R d (low shear)
[form. (5.42)]
Class 1 or 2
(plastic distribution
of stresses)
[form. (5.41)]
Class 3 or 4
"xm.Ed
fy Ι ΎΜΟ
-a
»z.Ed
+ f ν Ι ΎΜΟ
2
°x.Ed
fy/ΎΜΟ
r
°xm£d
f y /ΎΜΟ
-k
>z.Ed
f
y / ΎΜΟ
-a
°z.Ed
fy / ΎΜΟ
°x.Ed T g z.Ed
<1-β m
fy/YMoJLfy/YMO
<1
IfVsd > 0,5.V p £ R d (high shear)
[form. (5.44)]
Class 1 or 2
(plastic distribution
of stresses)
[form. (5.43)]
CJ
°z.Ed
xm.Ed
fy/ΎΜΟ
fy/ΎΜΟ
2
Class 3 or 4
f
'x.Ed
y / ΎΜΟ
r
»z.Ed
fy/ΎΜΟ
where σχ Ed
σ xm.Ed
JΓ gZ.Ed
ί γ /ΎΜθ1ίLVYM
γ /ΎΜθ
>x.Ed
f y / Ύ MO
I*
Ed
ΎΜΟ
^l­ßm­Ρ
<l-p
is the design value of the local longitudinal stress due to moment and axial
force at the point (see table X.2),
σ ζ £d is the design value of the stress at the same point due to transverse force
(see table X.2),
c
xm.Ed i s m e design value of the mean longitudinal stress in the web(see table X.2)
°x.Ed »
°ZSÂ
an
d G xrn £d ®& taken as positive for compression and as negative for
tension
ßm = Mw.Sd/Mp/:w.Rd>
where
M w Sd is the design value of the moment in the web,
M p£w . Rd = 0,25t w d 2 f y /Y M0 .
p=(2VSd/Vp£Rd-l)2,
and
k
f r
o
for
is obtained as follows :
crxmiEd/ azJEd < 0: k = 1 - ß m
°xm.Ed/ azEd > 0: if ß m < 0,5:
ifß m >0,5:
186
then k = 0,5 (1 + ßm)
thenk=l,5(l-ßj
[5.7.3 (i)j
X.C.2 Crushing resistance to F
(1) The design crushing resistance Ry.Rd of the web of an I, H or U section (see table X.4)
should be obtained from:
[form. (5.71)]
where sg is the length of stiff bearing determined by dispersion of load through
solid steel material which is properly fixed in place at a slope of 1:1,
(see table X.5); no dispersion should be taken through loose packs,
f
Sy is given by :
[form. (5.72)]
l
f
w *yw
>2
n
Of.Ed
yf / ΎΜΟ
where
bf £ 25tf,
Of.Ed is the longitudinal stress in the flange.
(2) At the end of a member sy should be halved.
[5.7.3 (3)]
Load introduction
Table X.4
Fsd
t*
•
Msd
1
-
1
1
ι
- -*
\
r
u
1
1
* - ■
J
V
Msd
ss
1
1i
d
bf
=
[Fig. 5.7.2]
=
=
=
^—-
=
Length of stiff bearing, ss
Table X.5
r
\
I
I
Ss
*
Ss
45°
V
/
sT
Ss
187
¥
1
r^
\
Ν
1
[5.7.4(1)]
X.d
Stability of webs to (F ; (F,M))
X.d.l
Crippling resistance to (F : (FM))
X.d.1.1 Crippling resistance to F
(1) For S 235 up to S 420 steel grades the design crippling resistance Ra.Rd of the web
(see table X.4) should be obtained from:
[form. (5.77)]
where ss
tw
tf
d
E
tyw
is the length of stiff bearing from table X.5.
buts s /d<0,2,
is the thickness of the web,
is the thickness of the flange,
is the depth of the web between the flanges,
is the modulus of elasticity,
is the yield strength of the web.
(2) For S 460 steel grade only the design crippling resistance RaRd of the web should be
obtained from the formula given X.d. 1.1 (1) but replacing the factor 0,5 by 0,6.
[Annex D]
X,dJ ,2 ÇrjppWng resistance \o (FM)
Table X.6
Interaction formula of crippling resistance and moment resistance
F
ii
Ra.Rd -
y s
0,5 R a . R d / /
/
/
ss/
/
x
s / /
s^>^
/ /
V//////,
M
0,5 M c . Rd
[5.7.4 (2)]
M c . Rd
(1) Where the member is also subject to bending moments, the following criteria should be
satisfied (see table X.6):
F
Sd - R aJld
Msd^McJld
F
[form. (5.78)]
sd . M Sd ^ 1,5
a.Rd M cJRd
R
where Mcj^d is the design noment resistance of the cross-section (see chapter Vin.e. 1).
188
!
X,d,2
[5.7.5 (i)]
Buckling resistance to F
(1) The design buckling resistance Rbjw of the web of an I, H or U section (see table X.4)
should be obtained by considering the web as a virtual compression member with an
effective breadth beff obtained from :
[form. (5.79)]
b^f = ­yV+S?
The table X.7 gives values of beff for different cases of loads.
[5.7.5 (3)]
(2) The buckling resistance should be determined from chapter VII.c.2.1 using buckling
curve c and PA = 1 (table VII.6).
[5.7.5 (4)]
(3) The buckling length of the virtual compression member should be determined from the
conditions of lateral and rotational restraint at the flanges at the point of load application.
[figure 5.7.3]
Table X.7
Effective breadth ¿>gryfor web buckling resistance
beff = h
s8
!! ν
Β
-f¿-->
1
'
beff
'
I
- -t
I
=)
=1
I
-t-
beff1
a ss
H-H
Η
a
beff
>eff
3
I'
= Vh 2
beff = 2
2
+ a
butb e ff < h
beff=}Vh 2 +s 2 +a+|
I·
189
butbeff <Vh 2 +S s 2
X.e
Stability of webs to compression flange buckling
Table X.8
Compression flange buckling in plane of the web
Nf
Nf"1
ΤΠΙΙΙΙΙΙΙΙΓ
My.Sd
My.Sd
(
)
^--iüüiülüJ
Nf"
[5.7.7]
ECCS n° 65
table 5.20
Nf"
(1) To prevent the possibility of the compression flange buckling in the plane of the web (see
table IX.8), the thickness ratio d/tw shall be lower than the value given in table X.9.
Table X.9
Steel grade of
flange
d/tw
Maximum width-to-thickness ratio d/tw
S 235
S 275
S 355
S 420
S 460
360
300
240
200
185
(2) I and H hot-rolled sections never meet such problem of compression flange buckling in
plane of the web.
190
XI
CONNECTIONS
Xl.a
Generalities
[6.1.1 (i)]
(1) All connections shall have a design resistance such that the structure remains effective and
is capable of satisfying all the basic design requirements given in chapter La.
(2) The partial safety factors TM concern the resistances of bolts (7Mb), of welds (TMw). of
members and cross-sections (ΎΜΟ» ΎΜΙ» ΎΜ2) and the slip resistance of preloaded bolts
(7Ms.ser)· Their numerical values are provided in table 1.2.
[6.1.2 (1)] (3) The forces and moments applied to connections at the ultimate limit state shall be
determined by global analysis in conformity with chapter IV.
[6.14 (l)] (4) Connections may be designed by distributing the internal forces and moments in
whatever rational way is best, provided that:
(a) the assumed internal forces and moments are in equilibrium with applied forces and
moments,
(b) each element in the connection is capable of resisting the forces or stresses assumed
in the analysis, and
(c) the deformations implied by this distribution are within the deformation capacity of
the fasteners or welds and the connected parts.
[6.2 (l)]
(5) Members meeting at a joint shall normally be arranged with their centroidal axes
intersecting at a point.
[6.2 (2)]
(6) Where there is eccentricity at intersections this shall be taken into account in the design of
the joint and the member.
[6.2 (3)]
(7) In the case of bolted connections of angles and tees with at least two bolts per connection,
the setting out lines of the bolts may be regarded as the centroidal axes for the purpose of
intersection at joints.
Xl.b
Bolted connections
[6.5]
[6.5.1.1]
Xl.b. 1
Positioning of holes
(1) The positioning of holes for bolts shall be such as to prevent corrosion and local buckling
and to facilitate the installation of the bolts.
(2) The minimum and maximum distances between bolts and recommended distances (as
used in table XL 6 for the bolt bearing resistances) are given in table XL 1. Those values
are valid for structures not exposed to weather or other corrosive influences.
XI.b.2
Distribution of forces between bolts
[6.5.4 (l)]
(1) Where the design shear resistance Fvjid of a bolt (see chapter XI.b.5.2.1) is less than
the design bearing resistance Fbjtø (see chapter XI.b.5.1), the distribution of internal
forces between bolts at the Ultimate Limit State shall be proportional to the distance from
the centre of rotation (see table XI.2).
[6.54 (2)]
(2) In other cases of bearing type connections the distribution of internal forces between
bolts at the Ultimate Limit State may be plastic (see table XI.3).
191
ECCS n'65
table 6.2
Designation of distances between bolts
Table XLI
1,2 do ^ ei < maximum ( 12t ; 150 mm)
1,5 do ^ e2 ^ maximum ( 12t ; 150 mm)
2,2 do ^ pi ^ maximum ( 14t ; 200 mm)
3,0 do < p2 ^ maximum ( 14t ; 200 mm)
PL
Pi
ef
Recommended distances
Bolts
Recommended distances in mm
shear joint
e2
M 12
Pi >P2
40
P2
M 16
55
40
30
M 20
70
50
40
M 24
80
60
50
M 27
90
70
55
M 30
100
75
60
M 36
120
90
70
e2
+—4
ei
pi
tension or compression joint
_Çi
30
e2
25
The designations e2 and p2 also apply when distances measured are not in the direction of
stress.
In case of smaller values of tq. and p2 see Eurocode 3 Part 1.1 (J2I).
[figure 6.5.7]
Linear distribution of loads between fasteners
Table XI.2
Ρ
Msd
Ρ
Ρ
Ρ
0,5 F h.sd
)
Vsd
F h.Sd
F
h.Sd ­
M Sd
5p
f
F
v.Sd ­
192
'MSd^2
5p
+
m
ECCS n" 65
table 64
| Table XI.3
Possible plastic distribution of loads between fasteners. Any realistic
combination could be used, e.g.
—'—*¿—
———
Fv.sdv
1
Ρ
Ρ
ρ
Ρ
MSd
t
Γ-Γ l. C l
-—
ψ
Ρ K D.l
—i
)
Vsd
_
FvSH —
h.Sd
v
Sd~rb.Rd
= ^ . ­ 2 Ρ b.Rd
,
2p
VFh.Sd + Fv.Sd ^Fb.Rd
[6.5.9 (1)]
[Annex J]
[Fig. 6.5.8]
XI.h.3 Prving forces
(1) Where bolts are required to carry an applied tensile force, they shall be proportioned to
also resist the additional force due to prying action, see table XI.4.
Prying forces
Table XI.4
N = FN + Q
Q.
N = FN + Q
f
J
I
[6^.3]
. Q = prying force
2 FΝ
XI.b.4 Categories of bolted connections
(1) The design of a bolted connection loaded in shear or in tension shall conform with one of
the following categories: see table XI.5.
193
[table 6.5.2]
Table XI.5
Categories of bolted connections
Shear connections
Remarks
Criteria
No preloading required.
Fv.sd
^
F v .Rd
All grades from 4.6 to 10.9.
Fv.Sd
<
Fb-Rd
Category
A
Bearing type
Β
Slip-resistant at SLS
Fv.Sd.ser ^
F v . R d.ser
Fv.Sd
^
F v .Rd
Fv.Sd
<
Fb.Rd
Fv.Sd
^
Fs.Rd
Fv.Sd
<
Fb.Rd
C
slip-resistant at ULS
Preloaded high strength bolts.
No slip at the serviceability limit
state.
Preloaded high strength bolts.
No slip at the ultimate limit state.
Tension connections
Category
D
Non-preloaded
Criteria
E
Preloaded
Key:
Fy.Sd.ser
Fy.sd
F v Jid
Fbjid
Fsudser
FsRd
Ftsd
Ft.Rd
[6.5.5. (2)]
[6.5.5. (3)]
=
=
=
=
=
—
=
=
FtSd
^
FLRd
Ftsd
^
Fuid
Remarks
No preloading required.
All grades from 4.6 to 10.9.
Preloaded high strength bolts.
design shear force per bolt for the serviceability limit state
design shear force per bolt for the ultimate limit state
design shear resistance per bolt
design bearing resistance per bolt
design slip resistance per bolt for the serviceability limit state
design slip resistance per bolt for the ultimate limit state
design tensile force per bolt for the ultimate limit state
design tension resistance per bolt
XI.b.5 Design ULS resistance of bolts
(1) At the Ultimate Limit States the design shear force Fv.sd on a bolt shall not exceed the
lesser of:
- the design bearing resistance Fb.Rd (see chapter XI.b.5.1)
- the design shear resistance Fv.Rd (see chapter XI.b.5.2)
(2) At the Ultimate Limit States the design tensile force Ftsd» inclusive of any force due to
prying action, shall not exceed the lesser of:
- the design tension resistance
FLRd (see chapter Xl.b. 5.3)
- the design punching shear resistance Bp.Rd (see chapter XI.b.5.4)
(3) At the Ultimate Limit States bolts subject to both shear force and tensile force shall satisfy
the interaction criterion of chapter XI.b.5.5.
XI.b.5.1 Bearing resistance
[table 6.5.3] (1) The design bearing resistance shall be taken as:
b.Rd
b.Rk
(see table XI.6 for Fb.Rk)
Ύινη>
194
ECCS n" 65
table 6.6
Β T a b l e XI.6
Bearing resistance per bolt for recommended detailing for t = 10 m m in [kN]
e2
ffce-2.5af.dt
P2
e2 ei
; _PjL_I ; íub ;l7 o
a = mm JEL.
3d0 3d0 4 fu
pi
20
22
24
27
30
13
16
18
22
24
26
30
33
36
39
ei
20
27,5
35
37,5
40
45
50
60
P1.P2
30
40
50
55
60
674
75
90
e2
S 235
S 275
20
25
30
32,5
35
40
45
55
55,4
70,7
60,0
76,6
91,4
99,0
101,8
110,2
110,8
120,0
121,5
131,6
166,2
180,0
S 355
754
104,0
1413
149,8
150,8
153,8
163,1
165,4
168,8
178,9
2262
76,9
81,5
124,4
126,9
134,5
138,5
S 420
S 460
96,2
98,1
136,4
147,7
185,6
189,4
200,8
30
40
50
55
60
70
75
90
40
55
70
75
80
90
100
120
25
30
40
45
50
55
60
70
S 235
83,1
106,7
136,4
151,3
166,2
182,3
204,5
2492
S 275
S 355
90,0
113,1
115,6
145,2
147,7
185,6
163,9
205,9
180,0
226,2
197,4
270,0
248,1
221,6
278,4
S 420
115,4
122,3
148,1
157,0
189,4
200,8
210,1
222,7
230,8
244,6
253,1
268,3
284,1
301,1
3462
366,9
ei
40
55
70
75
80
90
100
120
P1.P2
50
70
85
95
100
115
130
150
e2
S 235
S 275
35
50
60
65
70
80
90
110
108,0
117,0
198,0
214,5
269,5
243,0
270,0
324,0
234,0
263,3
147,0
180,0
195,0
245,0
216,0
S 355
144,0
156,0
196,0
294,0
330,8
292,5
3674
351,0
441,0
S 420
150,0
200,0
250,0
275,0
300,0
337,5
375,0
450,0
S 460
159,0
212,0
265,0
291,5
318,0
357,8
3974
477,0
Bolt diameter d [mm]
Hole diameter do [mm]
12
compact
detailing
recommended ei
P1.P2
values
e2
S 460
high bearing
230,8
244,6
3392
- For steel grades greater than S 235 the values of fu are issued from table IL 2
(prEN 10113) and are valid for plate thickness not greater than 40 mm.
- For intermediate values of α the value of FbjUc may be determined by linear interpolation.
- For different plate thickness t in [mm] multiply the values given in the table by — .
1
195
XI.b.5.2 Shear resistance
XI.b.52.1 General case
[table 6.5.3] (1) The design shear resistance of a bolt shall be taken as:
_ F v.Rk
v.Rd ­
(see table XI.7 for Fvjuc)
YMb
ECCS n° 65
table 6.7
Table XI.7
Shear resistance per bolt and per shear plane in [kN]
Γ zip
Fv.Rk ­Ci.f u b.A s
where
Ci = 0,6
C2 = 0,5
for strength grades 4.6, 5.6 and 8.8
for strength grades 4.6, 5.8, 6.8 and 10.9
Shear in tl ire adec i portion oftheb olt
13
16
18
20
22
22
24
24
26
27
30
30
33
36
39
84,3
157
245
303
353
459
561
817
20,2
37,7
47,1
75,4
58,8
73,5
117,6
72,7
90,9
145,4
84,7
110,2
105,9
169,4
137,7
220,3
134,6
168,3
269,3
196,1
245,1
392,2
78,5
122,5
151,5
176,5
229,5
280,5
408,5
Bolt diameter d [mm]
12
Hole diameter do [mm]
Tensile stress2 area of
bolt As [mm ]
Shear resistance grade
per bolt and per 4.6
5.6
shear plane
8.8
F v.Rk in [kN]
10.9
25,3
40,5
42,2
XI.b.52.2 Long Joints
[6.5.10. (1)]
(1) Where the distance Lj between the centres of the end bolts in a joint is more than 15 d,
where d is the nominal diameter of the bolts, the design shear resistance Fv.Rd of all the
bolts calculated as specified in chapter XI.b.5.2.1 as appropriate shall be reduced by
multiplying it by a reduction factor ßLf, given by (see table XL 8) :
[form. (6.11)]
[Fig. 6.5.10]
ßLf=l
PU
Table XI.8
L:­15d
3
but 0,75 < ß L f < 1,0
PU
2 0 0 d
Long joints
I
I
I
I
I
I
I
I
Lj
15 d
65 d
196
•ι
; I I 1 1 1 I ii I 1 I I I I ; = 3 ·
Ι
ι
ι
ι
ι
ι
l i l i l í
ι
ι
ι
ι
ι
ι
l i l i l í
ι—i
ΧI.h J.3 Tension resistance
[table 6.5.3] (1) The design tension resistance of a bolt shall be taken as follows:
[6.5.5. (3)]
ECCS ne 65
table 6.8
F
_ Ft.Rk
t.Rd -
(see table XI.9 for Ftjik)
YMb
Table XI.9
Tension resistance per bolt in [kN]
: mm-
—
Bolt diameter
d [mm]
grade
Tension
4.6
5.6
8.8
10.9
resistance
F t.Rk in [kN]
Ft.Rk =0,9.f ub .A s
12
16
20
22
24
27
30
36
30,3
37,9
60,7
75,9
56,5
70,7
113,0
141,3
88,2
109,1
136,4
218,2
272,7
127,1
158,9
254,2
317,7
165,2
202,0
252,5
403,9
504,9
294,1
110,3
176,4
220,5
206,6
330,5
413,1
367,7
588,2
735,3
XI.hS.4 Punching shear resistance
[65.5. (4)] (1) When the plate thickness tn is smaller than 0,5.d, the design punching shear resistance of
the bolt head and the nut, Bp.Rd shall be checked and evaluated as follows:
[form. (65)]
Bp.Rd = 0,6 π dm t.
YMb
where tp is the thickness of the plate under the bolt head or the nut
dm is the mean of the across points and across flats dimensions of the bolt head or
the nut, whichever is smaller, in other words dm is the mean diameter of
inscribed and circumscribed circles of bolt head or nut:
dm = minimum (dm bolt head» dm nut)·
XI.hJ.5 Shear and tension interaction
[6.5.5. (5)] (1) Bolts subject to both shear force and tensile force shall in addition satisfy the following
criterion which is illustrated in table XI. 10:
[form. (6.6)]
v.Sd
v.Rd
Ft Sd £1,0
l,4.F t Rd
Table XI. 10 Interaction formula of shear resistance and tension resistance for bolts
197
XI.b.6
[5.4.3 (1)]
[6.5.2.3]
[6.5.2.2]
[6.5.8]
ULS resistance of element with bolt holes
XI.b.6.1 Net section ULS resistance
see chapter VI.b.2
XI.b.6.2 ULS resistance of angle with a single row of bolt
see chapter VLcl
XI.b.6.3 Block shear ULS resistance
see chapter VDXd.l
XI.b.7
High strength bolts in slip-resistant connections at SLS
(1) When the slip resistance is needed at serviceability timit states the design for a
preloaded high-strength bolt shall be carried out as given hereafter. In the ultimate limit
state the bolt is considered as a bolt in shear and bearing without friction (see chapter
XI.b.5).
(2) In connections designed for slip-resistance at serviceability limit states the design
serviceability shear load should not exceed the design slip resistance V&Rd(3) The design slip resistance of a preloaded high strength bolt shall be taken as:
F
_
s.Rd -
r
s.Rk
(see table XI. 11 for Fs.Rk)
YMS. ser
(4) When the slip resistance is needed at ultimate limit state, see chapter [6.5.8] of
Eurocode 3 Part 1.1 (121).
ECCS n° 65
table 4.5
Table XI.11 Characteristic slip resistance per bolt and per friction interface for 8.8 and
10.9 bolts, where the holes in all the plies have standard nominal clearances
*"*.** = 0 ^ f u b A s
Bolt diameter
d [mm]
12
Tensile stress
area of bolt
As [mm2]
Fsjik for 8.8 bolts [kN]
FsRk for 10.9 bolts [kN]
surface class
class Α (μ = 0,5)
class Β (μ = 0,4)
μ = 0,2
class D
μ = 0,2
class D
20
22
24
27
30
36
84,3 157 245
303
353 459
561
817
9,4
16
17,6 27,4 33,9 39,5 51,4 62,8 91,5
11,8 22,0 34,3 42,4 49,4 64,3 78,5 114,4
description
surfaces blasted with shot or grit, with any
loose rust removed, no pitting.
surfaces blasted with shot or grit, and spraymetallized with aluminium or a zinc-based
coating.
surfaces blasted with shot or grit, and
painted with an alkali-zinc silicate paint
multiplication factor
2,5
2,0
class C (μ = 0,3)
surfaces cleaned by wire brushing or flame
cleaning, with any loose rust removed.
1,5
class D (μ = 0,2)
surfaces not treated.
1,0
198
[6.6.2.1]
XI.C
Welded connections
xic.i
Type of weld
(1) Welds are generally be classified as (see table XI. 12):
- fillet welds
- butt welds (with full or partial penetration)
ECCS n° 65
table 6.10
Table XI.12
Common types of welded joints
Type of joint
Type of weld
Butt joint
Tee-butt joint
Ύ.
I
Lap joint
Fillet weld
Full penetration
butt weld
single V
X
double bevel
double V
ï
single U
:
double U
Partial penetration
butt weld
single bevel '
double J
ΣΏ
double V
I
double U
XI.C.2 Fillet weld
[6.6.2.2. (l)] (1) Fillet may be used for connecting parts, where the fusion faces form an angle of 60° to 120'
[6.6.2.2. (2)] (2) Smaller angles than 60° are also permitted. However, in such cases the weld shall be
considered to be a partial penetration butt weld.
[6.6.2.2. (3)] (3) For angles over 120°, fillet welds shall not be relied upon to transmit forces.
199
XI.C.3
Design resistance of fillet weld
XI.C.3 J Throat thickness
[6.6.5.2. (1)] (1) The throat thickness, a, of a fillet weld shall be taken as the height of the largest triangle
which can be inscribed within the fusion faces and the weld surface, measured
perpendicular to the outer side of this triangle(see table XI. 13).
ECCS n° 65
table 6.13
Table XI.13
(a)
Throat thickness
Design sections of fillet welds
(b)
Design throat thickness aup for
submerged arc welding
[6.6.5.2. (2)] (2) The throat thickness of a fillet weld should not be less than 3 mm.
[6.6.5.2. (4)] (3) In the case of a fillet weld made by an automatic submerged arc process, the throat
thickness may be increased by 20% or 2 mm, whichever is smaller, without resorting to
procedure trials.
ECCS n°65
6.3.4.2 (4)
ECCS n°65
table 6.14
(4) The design force used for checking fillet welds should be taken as the resultant of the
forces to be transmitted by the weld (see table XL 14).
Table XI.14
Action effects in fillet welds
Sd
Cw.Sd
Vj_sd
VjTSd
Fw.Sd=VNÌ,Sd + VÌ. S d +V 2 / i Sd
200
Vj_sd
[6.6J.3]
ΧI.c.3.2 Design resistance
(1) The design resistance of a fillet weld shall be taken as follows
w.Rk
w.Sd
ECCS n° 65
table 6.15
(see table DC. 15 for
ΎΜν
FWR±)
Resistance of a fillet weld
Table XI.15
w.Rk
V3.ßv
■a.L
fu ­ nominal ultimate tensile strength of the weaker part joined
a ­ throat thickness
L ­ weld length
ß w ­ correlation factor
Weld resistance Fw.Rk in [kN] for 100 mm weld length
A Throat thickness
H a [mm]
S 235
ß w = 0,80
S 275
ß w = 0,85
S 355
ß w = 0,90
S 420
S 460
ß w = 0,95
ß w =l,00
3
4
77,9
79,5
94,3
91,2
103,9
91,8
106,0
125,7
121,5
122,4
5
6
7
129,9 155,9 181,9
132,5 158,9 185,4
157,2 188,6 220,0
151,9 182,3 212,7
8
9
1
10
207,8 233,8 259,8
211,9 238,4 264,9
251,5 282,9 314,3
243,1 273,5 303,9
153,0 183,6 214,2 244,8
275,4 306,0
­ For different weld lengths L in [mm] multiply the values given in the table by I
12
311,8
317,9
377,2
364,6
367,2
J.
­ For steel grades greater than S 235 the values of fu are issued from table Π.2 (prEN 10113)
and are valid for plate thickness not greater than 40 mm.
XLc.4
Design resistance of butt weld
[6.6.2.3 (l)] (1) A full penetration butt weld is defined as a butt weld that has complete penetration and
fusion of weld and parent metal throughout the thickness of the joint.
[6.6.2.3 (2)] (2) A partial penetration butt weld is defined as a butt weld that has joint penetration which is
less than the full thickness of the parent material.
[6.6.6.1 (l)] (3) The design resistance of a full penetration butt weld shall be taken as equal to the design
resistance of the weaker of the parts joined.
[6.6.6.2 (1)] (4) The design resistance of a partial penetration butt weld shall be determined as for a deep
penetration fillet weld.
[6.6.6.2 (2)] (5) The throat thickness of a partial penetration butt weld shall be taken as the depth of
penetration that can consistently be achieved.
201
XI.C.5 Joints to unstiffened flanges
[6.6.8 (i)] (1) In a tee­joint of a plate welded to an unstiffened flange of an I, H or a box section, a
reduced effective breadth shall be taken into account both for the parent material and for
the welds(see table XL 16).
[6.6.8 (2)] (2) For an I or H section the effective breadth beff should be obtained from:
't w + 2r + 7t f
beff = minimum
(3) If beff is less than 0,7 times the full breadth, the joint should be stiffened.
(4) For a box section the effective breadth beff should be obtained from:
2tw+5tf
beff = minimum·'
[6.6.8 (5)]
ECCS n° 65
table 6.16
, but b eff < b
design strength of plate
Lyp
[6.6.8 (4)]
r.2v f Λ
f
v*Py v yp J
design strength of member
where
[6.6.8 (3)]
t w + 2r + 7
(¿\
(*
K%VJ
y yp J
2t w + 5
\
, but b eff < b
f
(5) The welds connecting the plate to the flange shall have a design resistance per unit length
not less than the design resistance per unit width of the flange.
Effective breadth of an unstiffened tee joint
Table XI.16
*W|
t
l
Τ
W|
' i , \s
4
_
ΊΆ s\ -
'eff
s
s
!
^
V
!
I
S
^
J^
ι
1 t
ι ;Ι θ , 5 .
•PI
Xl.d
[6.5.13]
XLe
Xl.f
[6.11]
[Annex L]
Pin connections
Reference may be made to Eurocode 3 Part 1.1 (121)
Beam-to-column connections
Reference may be made to Eurocode 3 Part 1.1 (/2/)
[6.9]
[Annex J]
Design of column bases
Reference may be made to Eurocode 3 Part 1.1 (121)
202
10,5
. [tø
·
b eff
ΧΠ
DESIGN OF BRACING SYSTEM
XILa
Generalities
(1) The definition of bracing system and its braced frame is given in chapter Lb. 1 and
in table 1.1.
(2) Examples of bracing system and its braced frame are given in table XII.5.
XILa. 1 Flow-chart FC 12: Elastic global analysis of bracing system according to EC 3
(1) The flow-chart FC 12 aims to provide a general presentation of elastic global analysis of
steel bracing system according to Eurocode 3.
(2) The flow-chart FC 12 is nearly similar to the flow-chart FC 1 of chapter I about elastic
global analysis of steel frames according to Eurocode 3 because only two items are
different:
- the bracing system should be designed to resist supplementary loads and
supplementary effects of global imperfections issued from the frame which it
braces (see the part in comments on FC 12 and FC 1 concerning the
"Generalities about Eurocode 3") and,
- the classification of sway or non-sway bracing system should be established with
the same criteria but with specific conditions (see comments on rows 5 and 6 in
the part "Choice of the type of global analysis for ULS" of FC 12).
(3) The flow-chart FC 12 is divided in 3 parts:
Xn.a.1.1 general part (1 page)
Xn.a.1.2 details (1 page)
XILa. 1.3 comments (6 pages)
XII.a.1.1
Flow-chart FC 12: general
see the following page
XII.a.1.2
Flow-chart FC 12: details
see the second following page
203
Flow-chart fFC 12): Elastic global analysis of bracing system according to Eurocode 3 (General)
Actions
Predesign
SLS checks
Choice of the type of global analysis
for ULS
ULS global analysis of the bracing system
to determine the internal forces and moments (N, V, M)
ULS checks of members
submitted to internal forces and moments (N, V, M)
17
ULS checks of local effects
ULS checks of connections
204
Flow­chart [FC 12: Elastic global analysis oí bracing SXStem according to Eurocode 3 (Details)
Determination of load arrangements (ECl and EC 8) J
1
Load cases
for SLS [2.3.4.]
Load cases
for ULS [2.3.3.]
Predesign of members: beams & columns => Sections
with pinned ^nd/or rigid connections
Τ
notfulfilled
1
ULS checks
notfulfilled
SLS checks
[Chap. 5]
[Chap. 4]
i
J.
ι
,J
k
Global imperfections and global imperfections
of the bracing system
of the braced frame
[5.2.4.4.]
[5.2.4.3]
i
S.
Non-sway bracing system yes/Non­sway bracing system [52.5.2.]\no
Vsd <0,1
Sway bracing system
1
^Vx>0,5[A.fy/NSd]°­5V^í
\
[5.2.4.2. (4)]
yes/ 0,1 < ^ ^ 0 , 2 5 ν»£$
Ver
[5.2.6.2. (4)]
FIRST
O
lORDER ANALYSIS
ι SECOND ORDER ANALYSISi
t
Non­sway mode buckling
Sway mode buckling
length approach
length approach
[5.2.6.2(1) a)][5.2.6.2. (7)] :
[5.2.6.2(1) b)][5.2.6.2. (8)] :
with sway moments
with sway moments
amplified by factor
amplified by factor 1,2 in
l/(l­VSd/Vcr)
beams & connections
[5.2.6.2. (3)]
Çr
Non­sway mode iLb
l
±
=W
Mjembers imperfections
ι eo,d
[5.2.4.5.]
[5.2.6.2.(2)]
■0
eo.d where
necessary
[5.2.4.5.(3)]
'0
™ /members
with eo.d
eo.d in all
members
[5.2.4.5.(2)]
yes
Ψ
\ (
Sway mode Lb
Non­sway mode Lb
_
/
«c,
τ
ï
±
Classification of cross­tsections [Chap. 5ι3]
±
Checks of the in­plane stability: members buckling [Chap. 55.]
notfulfilled (n.f.h
i
Checks of the out­of­plane stability: members and/or frame buckling [Chap. 5.5.]
Checks of resistance of cross­sections [Chap. 5.4.]
15
η
yif.
Checks of local effects (buckling and resistance of webs) [Chap. 5.6 and 5.7]
Checks of connections [Chap. 6 and Annex J]
205
Pfl»
_J
XII.a.1.3
Comments on Flow-chart FC 12:
comments (1/6) on flow-chart FC 12:
[5.2.5.3 (7)]
[5.2.1.2(1)]
[5.2.1.2(2)]
[5.2.5.3 (6)]
* Generalities about Eurocode 3:
- Definition of a bracing system and its braced frame: see chapter I.b.l (table LI) and chapter
XILa.
- Where bracing system is a frame or sub-frame, it may itself be either sway or non-sway.
- All checks of (ULS) Ultimate Limit States and all checks of (SLS1 Serviceability Limit
States are necessary to be fulfilled.
- According to the classification of cross-sections at ULS (row 14; chapter V of the design
handbook) Eurocode 3 allows to perform:
. plastic global analysis of a structure only composed of class 1 cross-sections when
required rotations are not calculated [5.3.3 (4)] or,
. elastic global analysis of a structure composed of class 1. 2. 3 or 4 cross-sections
assuming for ULS checks, either a plastic resistance of cross-sections (class 1 and 2)
or, an elastic resistance of the cross-sections, without local buckling (class 3) or, with
local buckling (class 4 with effective cross-section).
- In order to determine the internal forces and moments (N. V. M) in a bracing system
Eurocode 3 allows the use of different types of elastic global analysis either:
a) first order analysis using the initial geometry of the structure or,
b) second order analysis taking into account the influence of the deformation of the
structure
- First order analysis (row 9) may be used for the elastic global analysis in the following casei
(types of bracing systems):
The/irsi order elastic global analysis of the bracing system should
take into account
the
the effects
the
the global
member
actions
horizontal
of global
vertical and imperfections imperfections
loads
imperfections horizontal
of the bracing of the bracing
from the
from the
loads of the
system
system
braced frame braced frame jracing system
types of
(a)
(b)
(a)
bracing system
(row 5)
(row 10)
1) non-sway
bracing systems
(path®)
2) sway bracing
systems (c)
(paths (2) and (3))
[5.2.5.3 (5)]
[5.2.5.3 (6)]
X
Notes : (a) actions issued from the frames which are braced by the analysed bracing system.
(b) the horizontal and vertical loads which are directly applied to the bracing
system.
(c) use of design methods which make indirect allowance for second-order effects.
206
comments (2/6) on flow-chart FC 12:
[5.2.1.2(3)]
[5.2.5.3 (6)]
Second order analysis may (row 9) be used in all cases (types of bracing systems):
The second order elastic global analysis of the bracing system should
take into account
the global
the
the
effects
the
member
actions
horizontal
of global
vertical and imperfections imperfections
loads
imperfections horizontal of the bracing of the bracing
system
from the
from the
loads of the
system
types of
braced frame braced frame bracing system
(a)
(b)
(row 5)
(row 10)
bracing systems
(a)
1 ) for sway
bracing systems
X
X
(path @ )
X(c)
X
(D)
2) for(path
bracing
systems in general
( path©) W)
[5.2.5.3 (5)] Notes : (a) actions issued from the frames which are braced by the analysed bracing system.
[5.2.5.3 (6)]
[5.2.4.5 (3)]
(b) the horizontal and vertical loads which are directly applied to the bracing system.
(c) members imperfections are introduced where necessary.
(d) the more complex possibility of second order global analysis of the frame
(path©) could be conservative because it allows the bypass of the "sway or
non-sway frame" classification and consequently :
- either the first order analysis might be sufficient,
- or, the introduction of member imperfections would not be necessary in all
members.
On the other hand, particular care shall be brought to the introduction of member
imperfections ( eo,d) which would be imposed for the global analysis in the
realistic directions corresponding to the deformations of the members for the
failure mode of the frame; that failure mode of the frame is related to the
combination of applied external loads; otherwise, with more favourable direction
of member imperfections, the second order global analysis might overestimate
the bearing capacity of the frame.
in the flow-chart FC 12 from path (Î) to path (6) (from left to right) the proposed
methods for global analysis become more and more sophisticated.
207
comments (3/6) on flow-chart FC 12:
* row 1:
ECl: Draft
EC 3: ENV 1993-1-1
Eurocode 1
Eurocode 3
EC 8: Draft
Eurocode 8
Basis of design and actions on structures
Design of steel structures, Part 1.1:
general rules and rules for buildings.
Design of structures for earthquake resistance
* rows 2.4:
[Chap. 5]
- ULS
means Ultimate Limit States
[Chap. 4]
- SLS
means Serviceability Limit States
* row 3:
This flow-chart concerns structures using pinned and/or rigid joints.
In the case of semi-rigid joints whose behaviour is between pinned and rigid joints,
the designer shall take into account the moment-rotation characteristics of the joints
(moment resistance, rotational stiffness and rotation capacity) at each step of the
design (predesign, global analysis, SLS and ULS checks). The semi-rigid joints
should be designed according to chapter 6.9 and the Annex J of Eurocode 3.
[4.2.1 (5)]
* row 4:
For SLS checks, the deflections should be calculated making due allowance for any
second order effects, the rotational stiffness of any semi-rigid joints and the possible
occurrence of any plastic deformations.
* row 5:
[5.2.4.4]
Global imperfections of the bracing system
Bracing system imperfections
initial bow imperfection
equivalent stabilizing force
11
' Τ τ Τ Τ Τ Τ Τ Τ Τ Τ Τ '
'
T
T
T
T
,
'
T
T
T
T
T
k
k
i
k
k
1
1
'
''
1
f
'
ι
XX XX
1
208
^ <
r 5q
comments (4/6) on flow-chart FC 12:
- Global imperfections of all the frames which are braced by the bracing system:
initial sway imperfections of the frame
equivalent horizontal forces
[5.24.3]
F2
i 4 t
/
/ *
i f
could be applied
in the form of
Fi
4 *
I
Λ
L·
φ (Fi + F2)
φ (Fi + F2)
classification of swav or non­swav bracing system:
* row 6:
A bracing system may be classified as non-sway if according to first order elastic global
analysis of the bracing system for each ULS load case, one of the following
[5.23.2]
criteria is satisfied;
either, a) in general :
[5.2.5.2 (3)]
Vsd__ ι
V,
af
< 0,1
aCT > 10
, condition which is equivalent to
design value of the total vertical load
elastic critical value of the total vertical load for failure in a sway mode
»cr*
( = π2ΕΙ / L2 with L, buckling length for a column in a sway mode; V cr of
a column does not correspond necessarily to VCT of the frame including that
column).
α cr
coefficient of critical amplification or coefficient of remoteness of critical
state of the frame.
The total vertical load includes the vertical loads applied directly to the bracing
system and the ones acting on all the frames which it braces.
where Vsd¡
[5.2.5.3 (8)]
or,
bi in case of bracing systems with beams connecting each columns at each storey level:
[5.23.2(4)]
δ.5> ,5.(V1+V2)
h­ΣΗ
209
]( ι . ^ + H2)
_
v
>*
comments (5/6) on flow-chart FC 12:
[5.2.5.3 (9)]
where H, V:
total horizontal and vertical reactions at the bottom of the storey.
δ:
relative horizontal displacement of top and bottom of the storey.
h:
height of the storey.
H, V, δ are deduced from a first order analysis of the bracing system submitted to:
­ the horizontal and vertical loads:
. applied directly to the bracing system and,
. acting on all the frames which it braces,
­ and, the global imperfections applied in the form of the equivalent
horizontal forces:
. from the bracing system (see comments on row 5) and,
. from all the frames which it braces (see below in the
comments on row 6).
Notes:
­ A same frame could be classified as sway according to a load case (Vsdl for
instance) and as non­sway according to another load case (Vsd2 for instance).
For multi­storeys buildings the relevant condition is —— = maximum
condition which is equivalent to
where
Vsdi
aCT = minimum (oten),
^ L or acrj are related to the storey i.
cri
5.2.4.2 (4)
* row 7:
λ>0,5
where λ :
Af,
­|0,5
ΝSd
»condition which is equivalent to
N
N Sd >
π
or equivalent to ε> —
2
;
Νcr·
non­dimensional slenderness ratio calculated with a buckling length equal
to the system length
yield strength
area of the cross­section
design value of the compressive force
elastic critical axial force ( = π2ΕΙ/ L2, with L = system length)
ε:
factor ( = L , '—Sá., with L = system length)
Nsd
5.2.6.2 (4)
Members imperfections may be neglected except in sway frames in the
cases of members which are subject to axial compression and which have
moment resisting connections, if :
* row 8: According to the definition of aCT introduced in comment on row 6:
Vsd
0,1<—aa.< 0,25
»condition which is equivalent to
4 < aa < 10
* row 9:
The actions to be considered in first order elastic global analysis and in second order
elastic global analysis are listed in the "generalities about Eurocode 3" (see the first
comments on flow­chart FC 12) in function of the type of bracing system.
210
commente (6/6) on flow-chart FC 12:
♦ rows 10.11.12 :
- p a t h © : Sway moments amplified by factor 1,2 in beams and beam­to­column
connections and not in the columns. The definition of "sway moments"
is provided in [5.2.6.2 (5)].
[5.2A5]
­ paths © and © : the introduction oí member imperfections eo,d
should be considered equivalent to the introduction of
distributed loads along the members :
e 04
NSd
Nsd
Nsd
ι
τ
τ
?
?
τ
?
J'
Nsd
Ρ"
equivalent to
L
».
M
q = 8.N S d .e 0 , d /L'
ÌQ = 4.N S d .e 0 ) d /L
with
Note : the equivalence of en,d and (q, Q) loading is proposed here for a practical point of
view but it is not included in Eurocode 3.
Verf
* row 11:
[Annex E]
For the meaning of the ratio ——, refer to comment on row 6.
"cr
* row 13: L ¡,, buckling length of members for sway or non­sway mode
NSd
*""
"**
NSd
Lb
* row 14:
* rows
The classification of cross-sections have to be determined before all the
ULS checks of members, cross­sections and webs (rows 15 to 18).
15.16.17.18.19:
The sequence of the Ultimate L imit States checks is not imposed and it is up
to the designer to choose the order of the ULS checks which are anyhow all
necessary to be fulfilled. On the contrary, the sequence of steps to select the
type of analysis is well fixed and defined in rows 5 to 8.
[5.5.13(6)]
* row 17:
When the member imperfections en,d are used in a second order analysis
(paths © or © ) , the resistance of the cross-sections shall be verified as
specified in chapter [5.4] but using the partial safety factor ymi in place of γ™
211
ΧII.b Static equilibrium
(1) Reference may be made to chapter IV.b.
XII.c
Load arrangements and load cases
Xn.c.l Generalities
(1) Load arrangements which may be applied to buildings are provided in chapter ÏÏLb.
(2) Load cases (see chapter ni.c) may be established according to two procedures to study
structures submitted to actions:
a general procedure presented in flow­chart FC 3.1 (chapter ΠΙ) or,
a particular procedure presented in flow­chart FC 3.2 (chapter ΙΠ) which is
applicable for non­sway buildings because such structure may be studied
by first order elastic global analysis.
(3) Two types of load cases shall be considered:
load cases for Serviceability Limit States and,
load cases for Ultimate Limit States,
where differences are related to combination rules:
see table ΙΠ.7 for SLS combinations of actions
see table ΙΠ.8 for ULS combinations of actions
(4) A bracing system should be designed to resist different loads and effects of global
imperfections from the braced frame and of the bracing system itself (see comments on
flow­chart FC 12 concerning the "generalities about Eurocode 3" (see chapter XILa. 1.3)
and see table ΧΠ.1).
Load arrangements of the bracing system
Table XII.l
a~) The horizontal loads from the braced frame:
I I I I II
TT
Ρ
c?
θ
^ 7 &
■A
■— ^
s?
b) The effects of global imperfections from the braced frame:
TT
p<BH-
A
&7
c) The vertical and horizontal loads of the bracing system:
&,
&,
d) The global imperfections of the bracing system:
see chapter XII.c.2
212
XII.C.2 Global imperfections of the bracing system
[5.24.3 (ï)] (1) The effects of imperfections shall be allowed for in bracing system design which are
required to provide lateral stability within the length of beams or compression members,
by means of an equivalent geometric imperfection of the members to be restrained, in the
form of an initial bow imperfection, or of the equivalent stabilizing forces according to
table ΧΠ.2.
(2) The numerical values for the stabilizing force Σ q are given in table ΧΠ.3 according to the
following model:
k.L
e 0 =­ 1 —
500
where
kr=J0,2+
where nr
k
^=J£r< '->
where a =
(but k r £ l ) ,
is the number of members to be restrained
500 δΓ
(3) Practical examples of such global imperfections are given in table ΧΠ.4 which presents
the case where the bracing system is required to stabilize a beam.
213
ECCS n°65
table 5.7
Bracing system imperfections
Table ΧΠ.2
initial bow imperfection
X
s
Nl.Sd
.
0
Ni.Sd
/
N2.Sd
N2.Sd
N3.Sd
N3.Sd
/
ECCS n° 65
table 5.8
equivalent stabilizing forces
ie°
μΠΓ
i.
i
4
i
i
i
'r
<1
'
<
I
'
XXXXX
»t-δς
>r
Table ΧΠ.3
\
^1
Values for the equivalent stabilizing force Σ q
nr=l
nr = 2
nr=3
nr=4
nr=5
Γ
ι
ι
*
]
ι
ι
χκζχ EEEE
KKKK
KZEÉ
χζζζ
\
XXXCX
ζ ΣΝ8(1
75,1 L
ζ ΣΝ 8ά
70,8 L
ζ ΣΝ 8ά
64,7 L
ζ ΣΝ 8ά
55,2 L
ζ ΣΝ8(1
96,6 L
ζ ΣΝ 8ά
89,6 L
ζ ΣΝ 8ά
80 L
ζ ΣΝ 5α
66 L
J
5q
η Γ = οο
<L/2500
L/2000
L/1500
L/1000
number of
panels
ζ ΣΝ 5α
ζ ΣΝ5α
67,2 L
71,8 L
ζ ΣΝ 8α
ζ ΣΝ8ά
63,8 L
67,9 L
ζ ΣΝ 8α
ζ ΣΝ5ά
62,2 L
58,8 L
ζ ΣΝ 8α
ζ ΣΝ5α
53,4 L
50,8 L
ζ N Sd
52,1 L
ζ N Sd
50 L
ζ N Sd
46,9 L
ζ N Sd
41,7 L
ζ ΣΝ 5α
60,3 L
ζ ΣΝ 5ά
57,5 L
ζ ΣΝ 8ά
53,4 L
ζ ΣΝ 8α
46,8 L
2
3
4
m
pw
f*f*m
5
6
7
Wffl PWffl p?<m*m
25/24
49/48
9/8
1,0
ι,ο
ι,ο
ζ
δς is the in-plane deflection of the bracing system due to Σq plus any external loads.
214
Bracing system imperfections (examples)
Table XII.4
©
N
\
2Sd
N
\r
N
2Sd
\
N
2Sd
2Sd
Φ
1
1
­ * * * ­ .
\
P
1
r
lSd
|P2Sd
,P2Sd
1
Ι
, r P 2Sd
V Iq+w
v
XXXX
- S\V—
Ä
ÀV
Ν
2Sd
y
(D
A Β
f Η ΜΗ tΗ Η Ι Η tΠ Η Η
215
lSd
Xll.d Bracing system stability
(1) Reference may be made to chapter IV.d.
XILe
First order elastic global analysis
(1) Reference may be made to chapter IV .e.
ΧΙΙ.Γ
Verifications at SLS
(1) Reference may be made to chapters IV.f.l and VIII. b.
XH.g
Verifications at ULS
ΧΠ.g. 1 Classification of the bracing system
Xll.g.1.1 Non-sway bracing system
[5.2.5.3 (7)] (1) Where bracing system is a frame or sub-frame, it may itself be either sway or non-sway.
(2) Examples of sway frames are mentionned in chater I.b.2.
(3) In order to define the criterion used to classify a bracing system as sway or non-sway
reference may be made to comments on rows 5 and 6 of flow-chart FC 12
(see chapter XILa. 1.3).
(4) As the criterion of sway or non-sway bracing system classification depends on the total
vertical load, a same bracing system could be classified as sway according to a load case
and as non-sway according to another load case. Therefore the criterion of sway or nonsway bracing system classification should be checked for each load case.
[5.1.2 (1)]
Xn.g.2 ULS checks
(1) The frames shall be checked at ultimate limit states for the resistances of cross-sections,
members and connections. For those ULS checks reference may be made to the following
chapters:
- Classification of cross-sections:
see chapter V
- Members in tension:
see chapter VI
- Members in compression:
see chapter VH
- Members in bending:
see chapter VEI
- Members with combined axial force and bending moments:
see chapter IX
- Transverse forces on webs:
see chapter X
- Connections:
see chapter XI
216
APPENDIX A :
List of symbols (1/6)
1.
Latin symbols
a
a
ad
at
aup
an
ai, a2
A
designation of a buckling curve
throat thickness of füllet weld
geometrical data of the effects of actions
geometrical data for the resistance
design throat thickness for submerged arc welding
designation of a buckling curve
distance between fastener holes and edge
accidental action; area of building loaded by external pressure of wind;
area of gross cross-section
effective area of class 4 cross-section
effective area of class 4 cross-section subject to uniform compression
(single Nx.sd)
effective area of class 4 cross-section subject to uniaxial bending
(single My.sd or single Mz.sd)
net area of cross-section
reference area for Cf (wind force)
tensile stress area of bolt
shear area of cross-section
effective shear area for resistance to block shear
shear area of cross-section according to yy axis
shear area of cross-section according to zz axis
designation of a buckling curve; flange width; building width
effective breadth
design punching shear resistance of the bolt head and the nut
designation of a buckling curve; outstand distance
altitude factor for reference wind velocity
dynamic factor for wind force
direction factor for reference wind velocity
exposure coefficient for wind pressure and wind force
Aeff
Aeff.N
Aeff.M
Anet
Aref
A8
Av
Ay.net
Av.y
Av.z
b
beff
BpJld
c
c ALT
cd
com
ce
Cf
Cpe
cr
ct
CJEM
Cd
Ci, C2
Ci, C 2, C3
d
d
dm
do
eN
eNy
eNz
eM
eo
eo,d
ei, β2
E
wind force coefficient
external pressure coefficient for wind pressure
roughness coefficient for determination of c e
topography coefficient for determination of c e
temporary (seasonal) factor for reference wind velocity
nominal value related to the design effect of actions
factors for determination of FyR^
factors for determination of MCT
designation of a buckling curve; web depth
bolt diameter
mean diameter of inscribed and circumscribed circles of bolt head or nut
hole diameter
shift of relevant centroidal axis of the class 4 effective cross-section subject to
uniform compression (single N x .sd)
shift of the y centroidal axis of the class 4 effective cross-section subject to
uniform compression
shift of the ζ centroidal axis of the class 4 effective cross-section subject to
uniform compression
shift of relevant centroidal axis of the class 4 effective cross-section subject to
uniaxial bending (single My.sd or single Mz.sd)
equivalent initial bow imperfection
design value of equivalent initial bow imperfection
distance between hole fastener and edge
modulus of elasticity or Young Modulus; effect of actions at SLS
217
List of symbols (2/6)
ECCS
ECSC
EC 1
EC 3
EC 8
Ed
Ek
fd
fe
fmin
fu
fub
f,
fyb
fyb
European Convention for Constructional Steelwork
European Community of Steel and Coal
Eurocode 1 (/11)
Eurocode 3 (/2/)
Eurocode 8 (¡3f)
design value of the effect of action
characteristic value of effects of actions at SLS
design natural frequency
natural frequency
recommended limit of natural frequency
ultimate tensile strength
nominal value of ultimate tensile strength for bolt
yield strength
basic yield strength of the flat steel material before cold forming
nominal value of yield strength for bolt
F, Fi, F2
FC
Fb.Rd
Fb.Rk
Fd
Ffr
Fh.sd
Fk
FpjRd
F8d
F8k
Fs.Rd
Fs.Rd.ser
Fs.Rk
Ft.Rd
FtRk
FLsd
Fv.Rd
Fv.Rk
Fv.8d
Fv.sd.ser
Fw
Fw.Rk
Fw.sd
g
G
Gd
Gk
h
ho
H
i
action (load, transverse force, imposed deformations,...)
flow-chart
design bearing resistance per bolt
characteristic value of bearing resistance per bolt
design value of action
friction force
force on bolt calculted from Msd and/or Fbjw
characteristic value of action
design punching shear resistance per bolt
design transverse force applied on web through the flange
characteristic value of transverse force
design slip resistance per bolt at the ultimate limit state
design slip resistance per bolt at the serviceability limit state
caracteristic slip-resistance per bolt and per friction interface
design tension resistance per bolt
characteristic value of tension resistance per bolt
design tensile force per bolt for the ultimate limit state
design shear resistance per bolt
characteristic value of shear resistance per bolt and per shear plane
design shear force per bolt for the ultimate limit state
design shear force per bolt for the serviceability limit state
resultant wind force
characteristic value of resistance force of fillet weld
design force of fillet weld
distributed permanent action; dead load
permanent action
design permanent action
characteristic value of permanent action
overall depth of cross-section; storey height; building height
overall height of structure
total horizontal load
radius of gyration about relevant axis using the properties of gross cross-section
second moment of area A
second moment of effective area Aeff (class 4 cross-section)
torsional constant
warping constant
second moment of area about zz axis
subscript meaning characteristic (unfactored) value
effective length factor
factor for lateral-torsional buckling with N-M interaction
buckling factor for outstand flanges
fyW
eff
k
k
kLT
k<j
yield strength of the web
218
List of symbols (3/6)
kw
Ky» Kz
Kr
ί
¿LT
L
Lb
LTB
Lv
m
max
min
M
Mb.Rd
MCT
M c .Rd
Me
Mef
Mf.Rd
MN.Rd
MN.V.Rd
MN.V.y.Rd
MN.V.z.Rd
MN.y.Rd
MN.z.Rd
Mpr
Mp£Rd
MpiwJld
Mp/:y.Rd
Mp£ z Jtd
MRd
Msd
My.Rd
Mw.sd
My
My.Sd
Mz
M z .sd
n
nc
nr
ns
N
NAD
Nb.Rd
Nb.y.Rd
Nb.z.Rd
N compression
Ncr
NcRd
effective length factor for warping end condition
factors for N-M interaction
roughness factor of the terrain
portion of a member
effective length for out-of-plane bending
system length; span length; weld length
buckling length of member
lateral-torsional buckling
distance between extreme fastener holes
mass per unit length
maximum
minimum
bending moment
design resistance moment for lateral-torsional buckling
elastic critical moment for lateral-torsional buckling
design resistance moment of the cross-section
torsional moment
elastic moment capacity
design plastic resistance moment of the cross-section consisting of the flanges
only
reduced design plastic resistance moment allowing for axial force N
reduced design plastic resistance moment allowing for axial force N and by shear
force V
reduced design plastic resistance moment about yy axis allowing for axial force
N and shear force V
reduced design plastic resistance moment about zz axis allowing for axial force
N and shear force V
reduced design plastic resistance moment about yy axis allowing for axial
force N
reduced design plastic resistance moment about zz axis axial force N
plastic moment capacity
design plastic resistance moment of the cross-section
design plastic resistance moment of the web
design plastic resistance moment of the cross-section about yy axis
design plastic resistance moment of the cross-section about zz axis
design bending moment resistance of the member
design bending moment applied to the member
design plastic resistance moment reduced by shear force
design value of moment applied to the web
bending moment about yy axis
design bending moment about yy axis applied to the member
bending moment about zz axis
design bending moment about zz axis applied to the member
number of fastener holes on the block shear failure path
number of columns in plane
number of members to be restrained by the bracing system
number of storeys
normal force; axial load
National Application Document
design buckling resistance of the member
design buckling resistance of the member according to yy axis
design buckling resistance of the member according to zz axis
normal force in compression
elastic critical axial force
design compression resistance of the cross-section
219
List of symbols (4/6)
N j . sd
Np£Rd
NRd
N8d
Nt.Rd
Ntension
N u .Rd
N x .Sd
Pl»P2
Ρ
q
qk
qref
Q
Qd
Qk
v¿k.max
Γ
R
Ra,Rd
Rb,Rd
Rd
Rk
Ry,Rd
S
S
Sd
Sk
Ss
S
sd
Sk
SLS
tf
h
tw
U
ULS
v
Vref
Vref.O
V
Vba.Rd
Vcr
V//Sd
v±8d
V
p£Rd
V p £yJld
Vp£zJid
v Rd
Vsd
Vy
Vy.Sd
Vz
design value of tensile force applied perpendicular to the fillet weld
design plastic resistance of the gross cross­section
design resistance for tension or compression member
design value of tensile force or compressive force
design tension resistance of the cross­section
normal force in tension
design ultimate resistance of the net cross­section at holes for fasteners
design internal axial force applied to member according to xx axis
distances between bolt holes
Point load
imposed variable distributed load
characteristic value of imposed variable distributed load
reference mean wind pressure
imposed variable point load
design variable action
characteristic value of imposed variable point load
variable action which causes the largest effect
radius of root fillet
rolled sections
design crippling resistance of the web
design buckling resistance of the web
design resistance of the member subject to internal forces or moment
characteristic value of Rd
design crushing resistance of the web
snow load
thickness of fillet weld
design snow load
characteristic value of the snow load on the ground
length of stiff bearing
effects of actions at ULS
design value of an internal force or moment applied to the member
characteristic value of effects of actions at ULS
Serviceability Limit states
design thickness, nominal thickness of element, material thickness
flange thickness
thickness of the plate under the bolt head or the nut
thickness of a plate welded to an unstiffened flange
web thickness
major axis
Ultimate Limit States
minor axis
reference wind velocity
basic value of the reference wind velocity
shear force; total vertical load
design shear buckling resistance
elastic critical value of the total vertical load
design value of shear force applied parallel to the fillet weld
design value of shear force applied perpendicular to the fillet weld
design shear plastic resistance of cross­section
design shear plastic resistance of cross­section according to yy axis (// to web)
design shear plastic resistance of cross­section according to zz axis (_L to flange)
design shear resistance of the member
design shear force applied to the member; design value of the total vertical load
shear forces applied parallel to yy axis
design shear force applied to the member parallel to yy axis
shear force parallel to zz axis
220
List of symbols (5/6)
Vz.sd
w
Wd
we
W
Weff
Weff.y
Weff.ζ
Wcf
We£y
We£z
Wpf
Wp£y
Wp£ 2
x, xx
Xk
y, yy
z, zz
Ze
design internal shear forces applied to the member parallel to zz axis
wind pressure on a surface
design wind load
wind pressure on external surface
welded sections
elastic section modulus of effective class 4 cross­section
elastic section modulus of effective class 4 cross­section according to yy axis
elastic section modulus of effective class 4 cross­section according to zz axis
elastic section modulus of class 3 cross­section
elastic section modulus of class 3 cross­section according to yy axis
elastic section modulus of class 3 cross­section according to zz axis
plastic section modulus of class 1 or 2 cross­section
plastic section modulus of class 1 or 2 cross­section according to yy axis
plastic section modulus of class 1 or 2 cross­section according to zz axis
axis along the member
characteristic value of the material properties
principal axis of cross section (parallel to flanges, in general)
principal axis of cross section (parallel to the web, in general)
reference height for evaluation of c e
2.
Greek symbols
α
α
α
ctcr
coefficient of frequency of the basis mode vibration
coefficient of linear thermal expansion
factor to determine the position of the neutral axis
coefficient of critical amplification or coefficient of remoteness of critical state of
the frame
non­dimensional coefficient for buckling
equivalent uniform moment factor for flexural buckling
equivalent uniform moment factor for lateral­torsional buckling
equivalent uniform moment factor for flexural buckling about yy axis
equivalent uniform moment factor for flexural buckling about zz axis
non­dimensional coefficient for lateral­torsional buckling
correlation factor (for a fillet weld)
partial safety factor for force or for action
partial safety factor for permanent action
partial safety factor for the resistance at ULS
partial safety factor for the resistance of bolted connections
partial safety factor for the slip resistance of preloaded bolts
partial safety factor for the resistance of welded connections
partial safety factor for resistance at ULS of class 1,2 or 3 cross­sections
(plasticity or yielding)
partial safety factor for resistance of class 4 cross­sections
(local buckling resistance)
partial safety factor for the resistance of member to buckling
partial safety factor for the resistance of net section at bolt holes
partial safety factor for variable action
relative horizontal displacement of top and bottom of a storey
horizontal displacement of the braced frame
design deflection
design vertical deflection of floors, beams,...
design horizontal deflection of frames
recommended limit of horizontal deflection
in plane deflection of the bracing system due to q plus any external loads
βA
ßM
PM.LT
ß\iy
ßvi z
ßw
ßw
YF
YG
YM
YMb
YMs.ser
YMW
YMO
YMI
YMI
YM2
YQ
δ
5b
5d
5dv
fød
OHmax
δη
221
List of symbols (6/6)
δς
ôu
ôvd
Svmax
δο
δι
δ2
Δ
ε
θ
λ
λχ
λ
Xeff.v
Xeff.y
Xefí.z
λυτ
deflection due to variable load (q)
horizontal displacement of the unbraced frame
design vertical deflection of floors, beams,...
recommended limit of vertical deflection
pre­camber (hogging) of the beam in the unloaded state (state 0)
svariation of the deflection of the beam due to permanent loads (G) immediatly
after loading (state 1)
variation of the deflection of the beam due to the variable loading (Q) (state 2)
displacement
1235
coefficient = I
(with fy in N/mm 2 )
V fy
rotation
slenderness of the member for the relevant buckling mode
Euler slenderness for buckling
non­dimensional slenderness ratio of the member for buckling
effective non­dimensional slenderness of the member for buckling about vv axis
effective non­dimensional slenderness of the member for buckling about yy axis
effective non­dimensional slenderness of the member for buckling about zz axis
non­dimensional slenderness ratio of the member for lateral­torsional buckling
Ρ
plate slenderness ratio for class 4 effective cross­sections
λν
non­dimensional slenderness of the member for buckling about w axis
Xy
non dimensional slenderness ratio of the member for buckling about yy axis
λζ
non dimensional slenderness ratio of the member for buckling about zz axis
μ
factor for F s R^ depending on surface class
μί
snow load shape coefficient
μΐ_τ
factor for N-M interaction with lateral-torsional buckling
\x.y
factor for N-M interaction
μζ
factor for N-M interaction
ρ
density
ρ
reduction factor due to shear force V 8 d
py
reduction factor due to shear force Vy.sd
ρζ
reduction factor due to shear force Vz.sd
σ
normal stress
Oq
numerical values for the stabilizing forces of a bracing system
Gx.Ed> tfxm.Ed> design values of normal stresses for web check with Von Mises criteria
<*z.Ed
τ
υ
φ
χ
XLT
Xmin
Xy
Xz
shear stresss
Poisson's ratio
initial sway imperfection of the frame
reduction factor for the relevant buckling mode
reduction factor for lateral-torsional buckling
minimum of xy and χ ζ
reduction factor for the relevant buckling mode about yy axis
reduction factor for the relevant buckling mode about zz axis
222
APPENDIX Β:
0.c
Table 0.1
I
Table 1.1
Table 1.2
Table 1.3
Table 1.4
Table 1.5
Table 1.6
Table 1.7
Table 1.8
Π
Table Π.1
Table Π.2
Table Π. 3
Table Π.4
Table Π.5
Table Π.6
Table Π.7
Table Π.8
ΙΠ
Table ΠΙ. 1
Table ΠΙ.2
Table ΠΙ. 3
Table ΠΙ.4
Table ΠΙ.5
Table ΠΙ.6
Table ΙΠ.7
Table ΙΠ.8
Table ΠΙ.9
IV
Table IV. 1
Table IV.2
Table IV. 3
Table IV.4
Table IV.5
Table IV.6
V
Table V. 1
Table V.2
Table V.3
Table V.4
Table V.5
Table V.6
List of tables (1/3)
SYMBOLS AND NOTATIONS
Dimensions and axes of rolled steel sections
INTRODUCTION
Summary of design requirements
Partial safety factor YM for the resistance
Definition of framing for horizontal loads
Checks at Serviceability Limit States
Member submitted to internal forces, moments and transverse forces
Planes within internal forces, moments (N8d, V8d» M8d) and transverses
forces F8d are acting
Internal forces, moments and transverse forces to be checked at ULS for
different types of loading
List of references to chapters of the design handbook related to all check
formulas at ULS
STRUCTURAL CONCEPT OF THE BUILDING
Typical types of joints
Modelling of joints
Comparison table of different steel grades designation
Nominal values of yield strength fy and ultimate tensile strength fu for
structural steels to EN 10025 and EN 10113
Maximum thickness for statically loaded structural elements
Maximum thickness for statically loaded structural elements
Nominal values of yield strength fyb and ultimate tensile strength fub for bolts
Material coefficient
LOAD ARRANGEMENTS AND LOAD CASES
Load arrangements (Fk) for building design according to ECl
Imposed load (qk, Qk) on floors in buildings
Pressures on surfaces
Exposure coefficient ce as a function of height ζ above ground
External pressure Cpe for buildings depending on the size of the effected area A
Reference height ZQ depending on h and b
Combinations of actions for serviceability limit states
Combinations of actions for ultimate limit states
Examples for the application of the combinations rules in Table ΙΠ.8.
All actions (g, q, P, s, w) are considered to originate from different sources
DESIGN OF BRACED OR NON-SWAY FRAME
Modelling of frame for analysis
Modelling of connections
Global imperfections of the frame
Values for the initial sway imperfections φ
Specific actions for braced or non-sway frames
Recommended limits for horizontal deflections
CLASSIFICATION OF CROSS-SECTIONS
Definition of the classification of cross-section
Determinant dimensions of cross-sections for classification
Classification of cross-section : limiting width-to thickness ratios for
class 1 & class 2 I cross-sections submitted to different types of loading
Classification of cross-section : limiting width-to thickness ratios for
class 3 I cross-sections submitted to different types of loading
Buckling factor k^ for outstand flanges
Classification of cross-section : limiting width-to-thickness ratios for internal
flange elements submitted to different types of loading
223
Page
6
11
12
23
24
25
26
27
28
30
31
32
33
34
35
35
36
40
41
42
43
43
44
45
46
46
48
47
57
58
59
60
65
72
73
74
75
76
Table V.7
Table V.8
Table V.9
Table V. 10
VI
Table VI. 1
Table VI.2
Table VI.3
Table VI.4
vn
Table VILI
Table Vfl.2
Table Vfl.3
Table VII.4
Table VIL5
Table Vfl.6
vm
Table vm.l
Table VIII.2
Table νΠΙ.3
Table Vm.4
Table Vni.5
Table Vfll.6
Table VIII.7
Table Vfll.8
Table VIfl.9
Table VIA. 10
Table Vm. 11
Table VIH. 12
Table Vm. 13
Table Vm. 14
LX
Table IX. 1
Table IX.2
Table IX.3
Table IX.4
Table IX.5
Table LX.6
Table ΓΧ.7
Table IX.8
Table IX.9
List of tables (2/3)
Classification of cross­section : limiting width­to­thickness ratios for angles
tubular sections submitted to different types of loading
Effective cross­sectional data for symmetrical profiles (class 4 cross­sections)
Limiting values of axial load Nsd for web classification of I cross­sections
subject to axial load Nsd and to bending according to major axis My.sd
Examples of shift of centroidal axis of effective cross­section
MEMBERS IN TENSION (Ntension)
List of checks to be performed at ULS for the member in tension (Ntension)
Gross and net cross­sections
Reduction factors ß2 and ß3
Connection of angles
Page
77
MEMBERS IN C OMPRESSION (Ncompression)
List of checks to be performed at ULS for the member in compression (NComp.)
Imperfection factor α
Value of Euler slenderness λι
Selection of buckling curve for a cross­section
Buckling length of column : Lb
Reduction factors χ = f (λ)
MEMBERS IN BENDING (V ; M ;( V,M))
List of checks to be performed at ULS for the member in bending according
to the applied internal forces and/or moments(V ; M ;( V,M))
Recommended limiting values for vertical deflections
Vertical deflections to be considered
Recommended limiting values for floor vibrations
Shear area A v for cross­sections
Determination of A vnet for block shear resistance
Limiting width­to­thickness ratio related to the shear buckling in web
Simple post­critical shear strength z\,a
Buckling factor for shear kT
Reduction factor %LT = f (XLT) for lateral­torsional buckling
Effective length factors : k, kw
Numerical values for Ci and definition of ψ
Reduced design plastic resistance moment My.Rd allowing for shear force
Interaction of shear buckling resistance and moment resistance
with the simple post­critical method
MEMBERS WITH COMBINED AXIAL FORCE AND
BENDING MOMENT ((N, M) ;(N, V, M))
List of checks to be performed at ULS for the member submitted to
combined axial force and bending moment (Ν, M)
Principle of interaction formulas between axial force Nsd
and bending moment Msd
Reduced design plastic resistance moment MNJM allowing for axial load
for Class 1 or 2 cross­sections
Interaction formulas for the (N,M) stability check of members of Class 1 or 2
Interaction formulas for the (N,M) stability check of members of Class 3
General interaction formulas for the (N,M) stability check of members of Class 4
Supplementary interaction formulas for the (N,M) stability check of members
of Class 4
Reduced design resistance Nyjid allowing for shear force
Reduced design plastic resistance moment MN.v.Rd allowing for axial load
and shear force for Class 1 or 2 cross­sections
224
78
79
80
84
85
86
87
91
94
94
95
96
97
103
107
107
108
110
111
111
112
112
116
116
117
119
120
124
130
131
135
136
137
138
139
141
List of tables (3/3)
TRANSVERSE FORCES ON WEBS (F ; (F,N,V,M))
Table Χ. 1
Table X.2
Table X.3
Table X.4
Table X.5
Table X.6
Table X.7
Table X.8
Table X.9
Failure modes due to load introduction
Stresses in web panel due to bending moment, axial force and transverse force
Yield criteria to be satisfied by the web
Load introduction
Length of stiff bearing, s s
Interaction formula of crippling resistance and moment resistance
Effective breadth beff for web buckling resistance
Compression flange buckling in plane of the web
Maximum width-to-thickness ratio d/tw
XI
CONNECTIONS
Table XL 1
Table XI.2
Table XI.3
Table XL 12
Table XL 13
Table XL 14
Table XL 15
Table XL 16
Designation of distances between bolts
Linear distribution of loads between fasteners
Possible plastic distribution of loads between fasteners. Any realistic
combination could be used, e.g.
Prying forces
Categories of bolted connections
Bearing resistance per bolt for recommended detailing for t = 10 mm in [kN]
Shear resistance per bolt and shear plane in [kN]
Long joints
Tension resistance per bolt in [kN]
Interaction formula of shear resistance and tension resistance of bolts
Characteristic slip resistance per bolt and friction interface for 8.8 and
10.9 bolts, where the holes in all the plies have standard nominal clearances
Common types of welded joints
Throat thickness
Action effects in fillet welds
Resistance of a fillet weld
Effective breadth of an unstiffened tee joint
ΧΠ
DESIGN OF BRACING SYSTEM
Table XII. 1
Table XII.2
Table XII. 3
Table XII.4
Load arrangements of the bracing system
Bracing system imperfections
Values for the equivalent stabilizing force Zq
Bracing system imperfections (examples)
Table XI.4
Table XI.5
Table XI.6
Table XI.7
Table XI.8
Table XI.9
Table XL 10
Table XL 11
Page
147
148
149
150
150
151
152
153
153
156
156
157
157
158
159
160
160
161
161
162
163
164
164
165
166
176
178
178
179
APPENDIX D
Table D.l
List of references to Eurocode 3 Part 1.1 related to all check formulas at ULS
225
191
APPENDIX C :
θ
List of flow-charts
Chapter Pages
Elastic global analysis of steel frames according to Eurocode 3
General
Details
Comments (6 pages)
I
I
14
15
16 to 21
(FC3A\
Load arrangements & load cases for general global analysis of the structure
HI
38
(FC3.2)
Load arrangements & load cases for first order elastic global analysis of
m
39
IV
IV
IV
50
51
52 to 55
©
the structure
Elastic global analysis of braced or non­sway steel frames according to EC 3
General
Details
Comments (4 pages)
(FC5.Ï) Classification of I cross­section
62
nFC5.2) Calculation of effective cross­section properties of Class 4 cross­section
ry
63
(FC6.I)
Members in tension (Ntension)
VI
82
(FC62\
Angles connected by one leg and submitted to tension
VI
83
(jFC lj Members in compression (Ncompression)
VU
90
(FC 8) Design of I members in uniaxial bending (Vz;My;(Vz,My)) or (Vy;Mz;(Vy,Mz))
VIII
102
XII
XII
XII
168
169
170 to 175
ÍFC 12jElastic global analysis of bracing system according to Eurocode 3
General
Details
Comments (6 pages)
226
H Table D.l
List of references to Eurocode 3 Part 1.1 related to all check formulas
at ULS
Typ ; References to Eurocode 3 Part 1.1 for ULS checks
Internal forces
moments, and
Physical phenomena
in function of classes of cross­sections ([5.3]) :
of
class 3
class 4
transverse forces check s classes 1 or 2 |
tension resistance (gross & net section)
R
Γ5.4.3 (1)1 + [5.4.2.2] + [6.5.2.3] + [6.6.10]
L Ntension.
[5.4.4 (1).(2)]
Í5.4.4 (1),(2)1 compression resistance
2. Ncompression R
S
[5.5.1.1 (1).(3)]
[5.5.1.1 (1),(3)1 Ν buckling of members
shear and block shear resistances
R
[3. V
[5.4.6 (1)1 + ,6.5.2.2]
shear buckling
Γ5.6.1 (1)1 + 15.6.3]
S
uniaxial bending resistance
■ι M
Í5.4.5.1 (2)1
[5.4.5.1 (1)1
R
15.4.5.1 (1)1
lateral­torsional
buckling (M y ) (LTB)
[5.5.2(1)]
[5.5.2
(1)]
[5.5.2(1)]
1
_JS
h . (M y ,M z )
[5.4.8.2(2)]
[5.4.8.3 (2)]
biaxial bending resistance
R [5.4.8.1 (11),(12)]
1
¡6. (V,M)
(V z ,M y )
7. (V,M y ,M z )
(V z , My, M z )
8. (N,M)
(Ntension^y)
(Ncomp.. My)
(Ncomp.. Mz)
9. (N,My,M Z )
10. (N, V)
11. (N, V,M)
(N,V z ,My)
112. (RV.My.Mz)
f5.5.4 (3H(4)1
[5.5.4 (5)+(6)l
S [5.5.4 (1)+(2)1
R
[5.4.7(3)]
S
[5.6.7.2 (1),(2),(3)]
[5.4.7
(3)]
+ formulas for (My. Mz)
R
formulas for (Vz, M y )
S
R
,5.4.8.2(2)1
Γ5.4.8.Π
[5.5.3 (2) to (5)]
S
[5.5.4 (3)+(4)]
S [5.5.4 (l)+(2)]
[5.5.4(1)]
[5.5.4(3)]
S
S
R
S
R
Il3. F,(F,N),(F,My),
(F,N,My)
lateral­torsional buckling (LTB)
N­M buckling + LTB
N­M buckling
biaxial bending & axial force resistance
(N­biaxial M) buckling + LTB
shear and axial load resistance
shear buckling
uniaxial bending &
shear and axial force resistance
[5.6.7.2 (1),(2),(3)] + formulas for (Ν,Μ) interaction (N­uniaxial M) resistance & shear buckling
[5.4.9 (3)] + formulas for (N,My,Mz) interaction
biaxial bending & shear and axial force
resistance
formulas for (N.V^My) interaction
(N­uniaxial M) resistance & shear buckling
[5.4.10(2)]
[5.4.10(1)] transverse force (+N, +M y ) resistance
[5.4.10(1)1
R
S
[5.7.3 (1)1
[5.7.4(1)] +[5.7.5 (1),(2),(3)]
S
[5.7.4 (2)] + formulas for M
[5.7.7 (1),(2)]
[5.7.7 (1),(2)]
[5.7.7 (1),(2)]
S
14. (F.VZ),(F,N,VZ),
R
(F,Vz,My),
(F.N,Vz,My)
S
tvpe of loadine:
[5.5.4 (5)+(6)]
[5.5.4 (5)]
uniaxial bending & shear buckling
uniaxial bending & axial force resistance
R [5.4.8.1(11) ,(12)] [5.4.8.2 (2)]
[5.4.8.3 (2)]
Γ5.5.4 (3H(4)1
[5.5.4 (5)+(6)l
S Í5.5.4 (1)+(2)1
[5.4.9 (3)] +1"ormulas for (N tension , Ncomp.)
R
[5.6.7.2]
S
[5.4.9 (3)] + formulas for (Ν,Μ) interaction
R
(N,Vz,My,Mz)
F
F
(F.My)
Γ5.4.8.3 (2)1
[5.5.3(2)to (5)]
biaxial flexural buckling
uniaxial bending & shear resistance
uniaxial bending & shear buckling
biaxial bending & shear resistance
[5.4.10(5)]
[5.4.10(4)]
[5.4.10(4)]
formulas for (N,V¿.MV) interaction
1.
2.
3. t o ' '.
8. t o l 12.
13. to 14.
=
=
=
=
=
crushing
crippling + buckling
crippling
compression flange induced buckling
transverse forces + shear V z (+N, +My)
resistance
(N­uniaxial M) resistance & shear buckling
tension members
com pression members
men ibers in bend i η
men íbers with c o m jined N­M
tran¡»verse forces or t webs
R = re sistance of cros s­sections ([5.4] )
5 = Stiìbility of members ([5.5]) or w e bs ([5.6],[5.7])
227
φ
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European Commission
EUR 16839 — Properties and service performance
Simplified version of Eurocode 3 for usual buildings
P. Chantrain, J.-B. Schleich
Luxembourg: Office for Official Publications of the European Communities
1997 — 227 pp. — 21.0 χ 29.7 cm
Technical steel research series
ISBN 92-828-1485-8
Price (excluding VAT) in Luxembourg: ECU 38
The aim of this ECSC research is to elaborate a simple but complete
document to design commonly used buildings in steel construction. This
document is completely based on Eurocode 3 and each paragraph totally
conforms to Eurocode 3. Only the design formulas necessary to design
braced or non-sway buildings are taken into account in this document. Tall
buildings (skyscrapers) and halls are not treated. The designers and steel
constructors are able to calculate and erect a commonly used steel
building with this design handbook. Therefore also the important load
cases from Eurocode 1 will be included in this document.
The working group of the research project was constituted of 10 European
engineering offices. Firstly that working group has carried out different
examples of calculation of braced or non-sway buildings according to
Eurocode 3, Part 1.1: check of existing steel structures and design of new
steel buildings. Afterwards, thanks to those examples of calculation the
needed design formulas of Eurocode 3 were highlighted and general
procedure of design was determined. The design handbook 'Simplified
version of Eurocode 3' is based on that experience.
The link of the working group to the drafting panel of Eurocode 3 was
guaranteed by the Professor Sedlacek of Aachen University.
Liaison has been ensured with both other ECSC research projects
No SA/312 and No SA/419 also dealing with Eurocode 3: respectively
'Application software of Eurocode 3: EC3-tools' (CTICM, France) and
'Design handbook for sway buildings' (CSM, Italy).
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