S C I E N C E RESEARCH D E V E L O P M E N T

E U R O P E A N

C O M M I S S I O N

technical steel research

Properties and in-service performance

Improved classification of steel and composite cross-sections: new rules for local buckling in Eurocodes 3 and 4

Report

EUR 18404 EN

hi STEEL RESEARCH

EUROPEAN COMMISSION

Edith CRESSON, Member of the Commission responsible for research, innovation, education, training and youth

DG XII/C.2 — RTD actions: Industrial and materials technologies — Materials and steel

Contact: Mr H. J.-L. Martin Address: European Commission, rue de la Loi 200 (MO 75 1/10), B-1049 Brussels — Tel. (32-2) 29-53453; fax (32-2) 29-65987

European Commission

technical steel researcl Properties and in-service performance

Improved classification of steel and composite cross-sections:

new rules for local buckling in Eurocodes 3 and 4

J. B. Schleich ProfilARBED-Recherches

66, rue de Luxembourg L-4221 Esch/Alzette

B. Chabrolin CTICM

Domaine de St Paul BP1

F-78470 St-Rémy-les-Chevreuse

F. Espiga Ensidesa & Labein

Cuesta de Olabeaga, 16 E-48013 Bilbao

Contract No 7210-SA/319/519/934 1 July 1993 to 30 June 1995

Final report

Directorate-General Science, Research and Development

1998 EUR 18404 EN

LEGAL NOTICE

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Cataloguing data can be found at the end of this publication.

Luxembourg: Office for Official Publications of the European Communities, 1998

ISBN 92-828-4466-8

© European Communities, 1998

Reproduction is authorised provided the source is acknowledged.

Printed in Luxembourg

PRINTED ON WHITE CHLORINE-FREE PAPER

Acknowledgements

This research project n° P3198 which has been sponsored by C.E.C., the Commission of the European Community, has been performed from 01.07.1993 to 30.06.1995 by the working group composed of :

Profil AJRBED (coordinator) (CE. C. Agreement 7210-SA/519) CTICM (partner) (CE. C. Agreement 7210-S A/319) ENSIDESA-LABEIN (partner) (CE.C. Agreement 7210-SA/934)

We want to acknowledge first of all the financial support from the Commission of the European Community, as well as the moral support given during this research by all the members of the CE.C Executive Committee F6 "Steel Structures".

Many thanks are also due to all, who by any means may have contributed in this research :

ProfilARBED-Recherches RPS Department (Luxembourg): . MM. Chantrain Philippe, Conan Yves and Mauer Thierry, . MM. Klòsak Maciej and Linn Cao Hoang (as trainees),

CTICM (France) : . MM. Chabrolin Bruno, Galea Yvan and Bureau Alain,

ENSIDESA and LABEIN (Spain) : . MM. Anza Juan, Espiga Fernando,

RWTH- LfS (Germany) : . M. Feldmann Markus,

EPFL - ICOM (Switzerland) : . M. Couchman Graham.

Table of Contents

List of Symbols 6

List of Figures 10

List of Annexes 11

References 12

1 Introduction 13

1.1 Obj ectives of the Research Proj ect 13

1.2 Ways and Means 13

1.3 Final Report 14

2 Bibliography 14

3 Definition of Cross-Sections Classification 14

4 Review of Rules for classification 17

4.1 General review 17

4.2 Review of Elastic Global Analysis 22

4.3 Review of Plastic Global Analysis 23

5 Numeric Simulations 23

5.1 Numerical Simulations for the Influence of ε Parameter 23

5.2 Numerical Simulations of Steel cross-Sections Classifications for My

Loading 26

6 New Proposals for Rules of Classification 33

6.1 Proposal for Steel Sections 33

6.2 Proposals for Composite Sections 36

7 Conclusions 38

8 List of Annexes 39

Annex 1 41 Annex 2 45 Annex 3 57 Annex 4 67 Annex 5 75 Annex 6 91 Annex 7 117 Annex 8 127 Annex 9 161 Annex 10 171 Annex 11 221 Annex 12 245

List of Symbols

1. Latin symbols

a distance between forces

A, Atotal area of gross cross-section

Aeff area of the effective cross-section

Avz shear area of cross-section about z-z axis

A\veb area of m e w e D of H or I cross-section (= A - 2btf)

b flange width of H or I cross-section

b effective length of element part

c half of flange width (= b/2)

d distance between zero moment points (= di + d2)

d web depth of H or I cross-section (= h - 2(tf + r))

d ι part of the distance between zero moment points

d 2 part of the distance between zero moment points

dCRM specific web depth defined by CRM (= h - tf)

dy flange displacement

E modulus of elasticity or Young Modulus

e\i shift of centroidal axis of effective cross-section submitted to bending moment

eN shift of centroidal axis of effective cross-section submitted to uniform compression

Est tangent modulus for strain hardening

F concentrated load

Fi concentrated load (for which the 1st plastic hinge occurs)

fmax maximum deflection in a span

Fu ultimate concentrated load

fu ultimate tensile strength

fy yield strength

f y β yield strength of the flange

fy w yield strength of the web

h overall depth of the cross-section

I moment of inertia of cross-section

Iy moment of inertia of cross-section about y-y axis

Iz moment of inertia of cross-section about z-z axis

iz radius of gyration of cross-section about z-z axis

kf specific factor in formula

kr specific factor in formula

kçj buckling factor for outstand flanges

kvi specific factor in formula

kv2 specific factor in formula

L system length

£ span length

L ι length of flange overthickness (finite element modelization of fillet radius)

L2 length of web overthickness (finite element modelization of fillet radius)

Lj, LLT> LLTB distance between two adjacent lateral bracing

LSpan span length

M bending moment

Mb.Rd design resistance moment for lateral-torsional buckring

Mcr elastic critical moment for lateral-torsional budding (Eurocode 3, Annex F)

Mcrit elastic initial moment for local and/or global buckling

Meff elastic effective bending moment resistance from the effective cross-section

Mei elastic bending moment resistance (=Wei fy)

Mel.Rd design elastic moment resistance of the cross-section

Mexp. experimental bending moment

Mmax maximum bending moment in Μ-φ curves

MPa = N/mm2

Mpi plastic bending moment resistance of cross-section (= Wpi fy)

Mpi.N reduced design plastic resistance moment allowing for the axial force

Mpi.Rd design plastic moment resistance of the cross-section

M<¡¿ design bending moment applied to the member

Mu ultimate bending moment in M-m curves

Mv.Rd- Mpi.v.Rd design plastic resistance moment reduced by shear force

My bending moment about yy axis

My.sd design bending moment about yy axis applied to the member

Mz bending moment about zz axis

Ν normal force; axial load

Nfl flange resistance part of axial compression load in combined N-My loading

Npi design plastic resistance of the gross cross-section (= A fy)

Nsd> Nx design value of tensile force or compressive force applied to the member

O Other sections

Ρ point load Pc specific term in formula

Pei point load related to Mei

Pu specific term in formula

q distributed load

qj distributed load for which the 1st plastic hinge occurs

qu ultimate distributed load

R Rolled sections

R rotation capacity of plastic hinge r radius of root fillet

180 rad radian ( = unit for rotations; 1 radian = degrees = 57,3 degrees)

π Rav available rotation capacity of plastic hinge

RCRM available rotation capacity of plastic hinge from CRM model

Rexp. available rotation capacity of plastic hinge from experimental results

RFeld. available rotation capacity of plastic hinge from Feldmann model

RKS available rotation capacity of plastic hinge from Kemp's simplified model

Rreq required rotation capacity of plastic hinge

SLS Serviceability Limit States

t design thickness, nominal thickness of element, material thickness

tf flange thickness of H or I cross-section

tw web thickness of H or I cross-section ULS Ultimate Limit State Vba.Rd design shear buckling resistance of web

VpLRd design shear plastic resistance of cross-section

Vsd design shear force applied to the member

Vy shear force parallel to yy axis (=parallel to flanges of I or H sections)

V z e i shear force parallel to zz axis related to Mei

V z shear force parallel to zz axis (=parallel to web of I or H sections)

w deflection of member

W Welded sections

W e external work done by the load

Weff elastic section modulus of effective class 4 cross-section

W¿ internal work absorbed by the structure

w 0 initial plate imperfection magnitude

Wpi y plastic section modulus about y-y axis

y major axis of H or I cross-section

ζ minor axis of Η or I cross-section

2. Greek symbols

α, β length factor (giving the position of point load)

αϊ load factor for the first plastic hinge ocu load factor at collapse ß s yield strength of reinforcement bars

%LT reduction factor for lateral-torsional buckling

Δ moment redistribution

Δσ stress increase in strain-hardening domain

235 -, ε coefficient = I (with fy in N/mmz)

V fy

e s t strain at the end of yield plateau

e u ultimate strain corresponding to fu

Ey yield strain corresponding to fy

γ factor for uniform distributed load

Ya< YRd partial safety factors

YMl partial safety factor for resistance of member to buckling

ΥΜφ partial safety factor for available inelastic rotation <pav

Y M R partial safety factor for available rotation capacity R a v

η load factor after the occurence of the first plastic hinge

φ inelastic rotation of plastic hinge

φ 3 ν characteristic value of available inelastic rotation of plastic hinge

(Ppl elastic rotation of plastic hinge related to Mp i

cpreq required inelastic rotation of plastic hinge

(p ro t maximal rotation of plastic hinge obtained by the intersection between decreasing part of

experimental (Μ-φ) curves from 3-point bending tests and the plastic moment level Mpi of

the profile

(pj^t = (pav, characteristic value of available inelastic rotation of plastic hinge

λ length factor for adjacent spans

λ load factor

XQ effective lateral slendemess

5TLT reduced s lenderness of m e m b e r according to lateral-torsional buck l ing

λρ plate slendemess

λρί plate i slendemess μ^Ι ratio of support to span bending moments v Poisson coefficient (=0,3 for steel) θρ. 6p.req* öreq = 9req» required inelastic rotation of plastic hinge

θρί, θρ2 inelastic rotations of plastic hinges ρ reduction factor for determination of effective width σ, σχ normal stress ocrit elastic critical buckling normal stress σ0 amplitude of residual stresses τ shear stress Tcrit elastic critical buckling shear stress ξ length factor (giving the position of point load) ψ bending moments ratio for a member (or parts of it) between lateral bracings

3. Drawing symbols

-o-

7

perfect hinge

plastic hinge

simple support (with vertical reaction)

simple support (with vertical and horizontal reactions)

fully fixed support

List of Figures

Figure 1 : Definition of the classification of cross-sections

Figure 2 : Influence of ε and initial slendemess limits

Figure 3 : Deflection history of beam 2 including unloading branch

Figure 4 : Calculated and measured moment resistances

Figure 5 : a) Flange b) Web

Figure 6 : Flange simulations

Figure 7 : Web simulations

Figure 8 : Boundary conditions

Figure 9 : Modelization of fillet radius

Figure 10 : Border class 3 &4 elastic cross-sections : (meaning Mei.Rd is reached)

Figure 11 : EC3 and simulated borders between classes 3&4 and classes 2&3 versus EPE profiles

Figure 12 : EC3 and simulated borders between classes 3&4 and classes 2&3 versus HE AA profiles

Figure 13 : Moment-rotation curve for 3-points bending beam

Figure 14 : Summary of formulas from Feldmann's model to evaluate inelastic rotations of steel plastic hinges

Figure 15 : Table for the EC4 classification of cross-sections

10

List of Annexes

Annex 1: Complete Set of Distributed Documents

Annex 2: Final report (Excerpts) Simplified Version of Eurocode 3 for Usual Buildings

Annex 3 : Document 3198-1-1 (Ref. 15) Excerpts)

Local buckling rules for structural steel members Annex 4 Document 3 263 -1 -27 (ProfilARBED)

Available rotation capacity of plastic hinges Ravanabie - Tests results and models

Annex 5 Document 3 263 -1 -27 (ProfilARBED Available rotation capacity of plastic hinges Ravaiiabie - Tests results and models

Annex 6 Document 3198-1-18 (ProfilARBED) Stability of composite bridge girders near internal support

Annex 7 Document 3263-2-12 (CTICM) Required rotation capacity for a 15% reduction of elastic peak moment

Annex 8 Document 3263-2-15 (CTICM) Required rotation capacity for continuous beams

Annex 9 Document 3 263 -1 -29 Mr. Couchman' s thesis (excerpts) Design of continuous beams allowing for rotation capacity

Annex 10 Document 3198-3-3 (LABEIN) Technical report n° 4 Numerical simulations of class 2&3 limit and class 3&4 limit

Annex 11 Exploitation of Labein numerical simulations (ProfilARBED) presented in Document 3198-3-3 (Annex 10)

Annex 12 Document 3198-2-10 (CTICM) Some numerical tests for checking the influence of yield strength on limiting b/t ratios

n

References

The hst of the numbered documents distributed in the scope of this research project is given in Annex 1.

Ref. 1: Eurocode 3, ENV 1993-1-1, Design of Steel Structures, Part 1.1, General Rules and Rules for Buildings, CEN European pre standard.

Ref. 2: Eurocode 4, ENV 1994-1-1, Design of Composite Steel and Concrete Structures, Part 1.1, General Rules and Rules for Buildings, CEN European pre standard.

Ref. 3: "Improved classification of steel and composite cross-sections - new rules for local buckling in Eurocodes 3 and 4", CE.C. agreements 7210-S A/519/319/934, Technical report n° 1, ProfilARBED-Recherches, Luxembourg, March 1994.

Ref. 4: "Improved classification of steel and composite cross-sections - new rules for local buckling in Eurocodes 3 and 4", CE.C. agreements 7210-SA/519/319/934, Technical report n° 2, ProfilARBED-Recherches, Luxembourg, September 1994.

Ref. 5: "Improved classification of steel and composite cross-sections - new rules for local buckling in Eurocodes 3 and 4", CE.C agreements 7210-SA/519/319/934, Technical report n° 3, ProfilARBED-Recherches, Luxembourg, April 1995.

Ref. 6: "Improved classification of steel and composite cross-sections - new rules for local buckling in Eurocodes 3 and 4", CE.C. agreements 7210-SA/519/319/934, Technical report n° 4, ProfilARBED-Recherches, Luxembourg, September 1995.

"Local buckling rules for structural steel members", by Bild S. and Lulak G.L., Journal of Constructional Steel Research, n° 20 (1991), published in 1992.

"Available rotation capacity in steel and composite beams", by Kemp A.R. and Decker N.W., "The structural Engineer", volume 69, n°5/5, March 1991.

"Promotion of plastic design for steel and composite cross-sections: new required conditions in Eurocodes 3 and 4, practical tools for designers (rotation capacities of profiles,...)", CE.C agreements 7210-SA/520/321/935, Draft of Final Report, ProfilARBED-Recherches, Luxembourg, February 1996.

RWTH Thesis of Mr. Feldmann M. :"Zur Rotationskapazität von I-Profilen statisch und dynamisch belasteten Träger" (Aaachen; Heft 30; 1994; ISSN 0722-1037).

EPFL Thesis n°1308 (1994) of Mr. Couchman G., Lausanne, EPFL, 1995 : "Design of continuous composite beams allowing for rotation capacity."

"Rotation Capacity of wide-flange beams under moment gradient", by Lukey A.F. and Adams P.R., Journal of the Structural Division, ASCE Vol. 95, n° ST 6, pp. 1173-1188, June 1969.

"Experimentelle ermittlung der Rotationskapazität biegebeanspruchte I-Profile", by Roïk K. and Kuhlmann U., Stahlbau 56, n° 12, December 1987, pp. 353-358.

Projekt P169 "Untersuchung der Auswirkungen unterschiedlicher Streckgrenzen -Verhältnisse auf das Rotations- und Bruchverhalten von I-Trägern"; von Sedlacek G., Spangemacher R., Dahl W., Hubo R. und Langenberg P.; Studiengesellschaft Stahlanwendung e.V-Forschung für die Praxis; 1992.

Ref. 15 : "Elastisch-Plastisches verhalten von Stahlkonstruktionen, Anforderungen und Werkstoffkennwerte"; Sedlacek G., Spangemacher R., Dahl W. und Langenberg P.;EGKS-F6 Projekt 7210-SA/113; Abschußbericht 1992.

Ref. 16 : "Elasto-plastic behaviour of metallic frameworks- Interaction between strength and ductility" ; by D'Haeyer R., Delooz M., Defoumy J.; ECSC agreement 7210-SA/204; Draft of final report 1992.

Ref. 17 : Schaumann P.,Steffen Α.: Verbundbrücken auf basis von Walzträgern, Versuch 1 -Einstegiger Verbundträger, Nr. A 88199, Versuch 2 - Realistischer Verbundbrückenträger, Nr. A 89199-2, im Auftrag von Arbed Recherches, Luxembourg.

Ref. 18 : CM66 - Additif 80, DPU P22-701 (French code), "Règles de calcul des constructions en acier".

Ref. 19 : "Elasto-Plastic Behaviour of Steel frame works", by Gérardy J.C. and Schleich J.B., ECSC Agreement 7210-SA/508; Draft of Final Report, 1992.

12

Ref. 7:

Ref. 8:

Ref. 9:

Ref. 10 :

Ref. 11 :

Ref. 12 :

Ref. 13 :

Ref. 14 :

1. Introduction

1.1 Objectives of the research project

In each specification detailing the design of structural steel members there are usually rules about the local buckling. These rules are based on the combination of cross-sectional dimensions (slendemess of different parts of profiles, b/t for the web and the flange) and on the yield point; for these combinations a critical level is defined over which local buckling appears (classification of cross-sections). Thus, this classification does not take into account the real stresses of the cross-sections which are rarely equal to the yield point. Besides, for high strength steels (yield point = 460 MPa), these rules have been extrapolated without verification and because of their definition, they discriminate these steels.

For a designer the usual procedure is to choose a cross-section 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, beam-column).

Therefore the local buckling rules play an important part in the design of structural steel and composite members.

In this research we propose to evaluate the local buckling problem for all main steel grades (S 235, S 355 and S 460 steels) with a more realistic approach based on tests results and numerical simulations. This approach should take into account the existing stresses in members submitted to global buckling (cross-sections loaded by centered'and also eccentric compression) and should take into account the real boundary conditions of the cross-sections (for instance in a composite cross-section the collaborating concrete slab influences greatly the stability of the steel beam web).

The aim of this research is to improve the classification of steel and composite cross-sections in Eurocode 3 (Ref. 1) and Eurocõde 4 (Ref. 2) by a more realistic approach. The practical result of this research consists in new rules of classification of cross-sections which will be introduced in both Eurocodes 3 and 4 with the support of expertises. In such a way the competitivity of steel and composite (steel-concrete) cross-sections will be improved and these sections will not be evaluated too conservatively as it is done presently because of lack knowledge in the field of local buckling problems.

1.2 Ways and means

(1) The following financially independent partners participated in the research project:

ProfilARBED - Recherches, Luxembourg : Mr. Chantrain Ph.

CTICM, France : MM. Chabrolin B., Galea Y., Bureau A.

LABEIN and ENSIDESA, Spain : MM. Anza J., Espiga F.

(2) The technical coordination was handled by ProfilARBED - Recherches Department "Recherches et Promotion technique Structure (RPS)". It was decided that only one common ECSC report had to be written by ProfilARBED for each period. Each report included the contributions done by the different partners during different four research periods (Ref. 3, Ref. 4, Ref. 5 and Ref. 6).

(3) During this research project, the main works were distributed between partners as follows:

- ProfilARBED : management of the project,

generalities, study of tests results (see chapters 3,4 and 6),

- CTICM : . generalities (see chapters 4 and 6),

numerical simulations for the influence of ε parameter (see chapter 5.1),

- LABEIN : . generalities (see chapter 6)

numerical simulations of the border between class 3 & class 4 cross-sections (see chapter 5.2),

13

1.3 Final report

The present final report compiles all results of works done in the scope of this research project. This final report presents :

in chapter 2, bibliography,

in chapter 3, the definition of cross-sections classification,

in chapter 4, the review of rules about cross-sections classification, in general, for steel sections, for composite sections, in case of elastic global analysis, in case of plastic global analysis,

in chapter 5, numerical simulations of : flange and web, to highlight the influence of ε parameter, the border between Class 3 & Class 4 cross-sections,

in chapter 6, new proposal for rules of cross-sections classification, for steel sections, for composite sections.

2. Bibliography

(1) Collection of information according to Eurocode 3 (steel) and Eurocode 4 (composite steel and concrete ) has been performed by all the partners : bibliography, technical reports, papers, results from tests, statistical evaluations, conclusions of previous or in progress researches, existing rules or new proposals of rules, development of calculation models

(2) For convenience a specific numbering has been introduced for the documents distributed in the scope of this research project. The list of the numbered documents distributed up to December 1995 is given in Annex 1. The convention ofthat numbering is proposed as follows (for example 3198-2-4) :

'number of the project" ("3198"),

'number of the partner" ("1", "2" or "3" respectively related to ProfilARBED, CTICM or LABEIN),

'number of the paper in the chronological order of distribution".

3. Definition of cross-sections classification

(1) A classification of cross-sections has been introduced into several Design codes (Ref. 1, Ref. 2, ...) identifying :

conditions for global analysis of the structure to determine the effects of actions (for instance, internal forces and moments, deflections, rotations,...) and,

criteria to be used for ultimate limit state (ULS) checks of cross-sections and members.

Global analysis of structures involves either elastic global analysis, or elastic global analysis with specified limits of moment redistribution, or plastic global analysis. Moment resistance of a critical cross-section is then determined on either a plastic stress-block or elastic basis.

14

(2) In Eurocodes 3 (Ref. 1) and 4 (Ref. 2), four classes of cross-sections are defined, as follows :

Class 1 plastic cross-sections are those which can form a plastic hinge with the rotation capacity required for plastic analysis.

Class 2 compact cross-sections are those which can develop their plastic moment resistance, but have limited rotation capacity.

Class 3 semi-compact 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 hable to prevent development of the plastic moment resistance.

Class 4 slender cross-sections 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.

The characteristics of each class of cross-section are illustrated in case of simply supported beam in Figure 1.

(3) The four classes of cross-section are defined according to (see Annex 2 for Eurocode 3 rules) :

the slendemess of its compression elements (width-over-thickness ratios of web (d / tw) or flange (b/tf)),

the yield strength of the steel (fy) and,

the applied loading : separate or combined axial forces (Nx) and/or bending moments (My, Mz), all inducing normal stresses (σχ); the classification of cross-sections is not affected by shear forces (Vy, Vz).

(4) A structure or a substructure comprising members with class 1 plastic sections may be analysed plastically using plastic resistance moments (Mpi), whereas those containing class 2 compact sections should be analysed elastically with limited potential for moment redistribution, but will also develop the plastic resistance moments (Mpi) (see Figure 1).

A structure or a substructure including members with class 3 semi-compact sections may be analysed elastically using elastic resistance moments (Mei), whereas those containing class 4 slender sections should be analysed elastically with moment of inertia (I) of "complete" cross-section but will only develop elastic "effective" resistance moment (Meff) issued from the effective cross-section which takes into account local buckling.

15

Definition of the classification of cross-section

^ u i i l i J J ] ] ^ ^ M

'Sections" &

Classes

Behaviour model Design resistance Available rotation

capacity of plastic hinge

Global analysis

of structures

"Plastic" Classi

Mpi M

local buckling

φ Failure mechanism after resistance maintained over plateau of displacements

PLASTIC across full section

ƒ Plastic moment resistance

important plastic

or, elastic (with

moment redistribution)

'Compact'

Class 2

Mpi M

\ local

buckling Φ

PLASTIC across full section

Failure after reaching plastic resistance, but no plateau of displacements

limited

Plastic moment resistance

elastic (with limited

moment redistribution)

4 M

"Semi-compact"

Clas 3

Mpi Mel

local buckling Ψ

ELASTIC across full section

f̂

Δ V -y

none

Failure before plastic resistance is reached Stress limit of yield

elastic (without moment

redistribution)

"Slender" Class 4

Mpi Mel

i M

"~~ local buckling Φ

ELASTIC across effective section

' f ν y

L none

Failure before elastic resistance is reached Stress limit of yield

elastic (without moment

redistribution)

Figure 1 : Definition of the classification of cross-sections

16

4. Review of rules for classification

4.1 General review

(1) In general, the development of rules for local buckling has not received the same amount of attention as the evaluation of strength or overall stability of members. However the number of both analytical and experimental studies of local buckling has increased recently.

Therefore it is interesting to present a comparative review of the criteria for classification of cross-sections (local buckling rules for normal stresses, σχ) for several specifications or standards (see the distributed paper n° 3198-1-1 (Ref. 7) : "Local buckling Rules for Structural Steel Members"). The different compared specifications are : ISO-TC 167-SC1, Eurocode 3, national standards from Canada, USA, Germany, Switzerland, United Kingdom and Australia.

Those rules of local buckling always depend on siendemesses (b/tf or d/rw) and on yield strength (fy) of different parts of cross-sections. That paper highlights the differences between the various prescriptions: several examples are provided in Annex 3 issued from that paper.

In view of the non negligible differences between specifications rules, it is important to make a critical assessment of those rules.

(2) The classification of cross-sections provides greater flexibility for designers which should lead to improved efficiency and consistency. Unfortunately it has also-led to many new questions and uncertainties.

(3) The following points which concern steel and composite cross-sections falling in class 1 or 2 of Figure 1, are illustrations of questions and uncertainties about classification rules (Ref. 8):

a) Conditions for satisfying the classification of Figure 1 in steel and composite Codes have been based largely on a qualitative evaluation of experiments on local buckling of steel sections, without emphasising the actual rotation capacity required for plastic analysis or redistribution of moments from an elastic analysis, which will vary from one structural arrangement to another. A need therefore exists to quantify required rotation capacity as a function of the percentage redistribution of moments and the structural arrangement.

b) Increased emphasis is being given in design to avoid brittle or sudden mode of failure, if the conditions assumed in the structural design no longer apply or are subject to gross error. It would be therefore suitable to adopt relatively conservative requirements in Figure 1 which will lead to more ductile behaviour, if Oils can be achieved at little extra cost.

c) The use in recent Codes of local buckling as a limiting criterion for rotation capacity of steel sections is convenient and relatively simple as shown in Annex 2 (for Eurocode 3), but it is in disagreement with experimental evidence showing that lateral-torsional buckling, even at low slendemess ratios, is also a crucial limitation on rotation capacity, particularly where it interacts with flange buckling. In the case of local buckling of flanges, other factors such as moment gradient and combined axial force have a significant effect on inelastic rotation, but are not considered in Codes.

d) End connections in steel and composite members and cracking of concrete in composite members influence the distribution of moments and should be assessed consistently with the member properties in considering rotation capacity. It would be suitable that the contribution of both the members and their end connections to the available inelastic rotation prior to strain-weakening should be considered on a consistent basis, rather than providing separate classifications.

17

(4) Other questions are still open about the rules for classification of cross-sections :

about the sensitivity to local buckling of a cross-section, the relative influence between the different parts of a cross-section should be considered (see new proposals in chapter 6):

according to present mies of Eurocode 3 (Ref. 1, 5.3.2(6)), a cross-section is normally classified by quoting the highest (least favorable) class of its compression elements (flanges and web which are classified separately) but in reality the web and flanges behave in interaction; for a particular case, Eurocode 3 takes into account this interaction (Ref. 1, 5.3.4(5)) : with a class 2 compression flange, a class 3 web may alternatively be treated as an efffective class 2 web with a reduced effective area;

the rules of classification should be defined in function of relative stiffening of the different parts of a cross-section (steel webs and steel flanges, concrete slab and steel webs and flanges).

about class 4 cross-sections: are the criteria correct or too conservative ?

the formulas for the classification of different parts of the cross-section (border between class 3 and class 4) (see results of numerical simulations in chapter 5.2),

the formulas for the calculation of effective cross-sections,

the partial safety factor ym ι.

about the influence of yield strength: the reduction factor, ε = (235/yield strength)0·5, severely discriminates high strength steels and seems to provide conservative results (see results of numerical simulations in chapter 5.1).

4.1.1 General review of steel sections

(1) The following 90 available 3 point bending tests results which are provided in Annex 4 are used to review the rules of Classification of steel cross-sections :

15 tests from Lukey and Adams (USA) (Ref. 12),

20 tests from Roik and Kuhlmann (Bochum, Germany) (Ref. 13),

26 tests from Sedlacek (RWTH Aachen, Germany) (Ref. 14 and 15),

29 tests from CRM (Liège, Belgium) (Ref. 16),

In the tables of Annex 4 the values of Feldmann's (Ref. 10 and Figure 14) and Kemp's (Ref. 8) models are calculated with measured characteristics (geometry, steel grade) (see chapter 6.1.1 for explanation of available inelastic rotation (φ3ν ) and available rotation capacity (RaV) of plastic hinges) :

Φ rot (= Φ3γ) (Teldmann's model),

Ravailable (= Rav) predicted Kemp's simp. (= Kemp's simplified model),

Ravailable (= Rav) predicted Feldmann (= Feldmann's model),

(2) Tests results provided in Annex 4 are compared in Annex 5 with Eurocode 3 (Ref. 1) rules for classification of cross-sections (see definition of rules in present chapter 3 or in chapter 5.3 of Eurocode 3) : experimental rotation capacities are related to width-over-thickness ratios and to yield

¡235 points (b/(tfÆ) for flanges and d/(tw£) for webs, where ε = I , with fy in N/mm2).

V f y

18

(3) The conclusions for hot-rolled sections tests are :

all webs were in class 1 and no tests results were available with slender webs (see Annex 5 (4/14))

several tests results with "EC3 slender" flanges are conflicting with assumptions of Eurocode 3 (EC 3) rules (see Annex 5 (5/14)) :

high rotation capacities (> 6) for class 2, class 3 and even class 4 cross-sections according to EC3 (see Annex 5 (11/14)),

rotation capacities > 0 for class 3 and class 4 cross-sections according to EC3 (see Annex 5 (9/14 and 10/14));

because delivering available rotation capacities greater than zero, all tested cross-sections should be considered in EC3 class 1 or class 2 for which, per definition, the plastic bending moment resistance has been reached and passed over.

(4) The Annex 5 (9/14 to 11/14) shows the influence of ε factor on b/tf limits of Eurocode 3:

235 either, the present ε = I—— factor seems to be too much conservative if the initial slendemess

limits (b/tf) for the lowest steel grade S235 are considered correct,

or, the initial slendemess limits (b/tf) for the lowest steel grade S235 are too much conservative if ε factor is considered correct.

In practice both parameters (£ factor and initial slendemess limits (b/tf) for the lowest steel grade S235) are too much conservative. Numerical simulations highlight this reality respectively in chapter

f ι ι "\ 235 235

5.1 and 5.2. ε factor should be changed to a less restrictive relation like 31 or4l and the t V f y V f y j

initial slendemess limits (b/tf) for S 235 steel grade should also be increased (see Figure 2).

19

b / t f Influence of ε = f [235^

30 τ U 'y ) and initial slendemess limits

2 8 - , new initial slendemess limit (b/tf) [235" for the lowest steel grade S235 with ε = ——

Κ) o

*1 f KJ

=9 Β re s o re o « 9 ro w 3 α

c E. S" 3 α

ετ

26

24

2 2 -present initial limit (b/tf) in EC3 for class 1 compressed flange

1 4 - ε = i ; present EC3 mie for classification

fy [Mpa] 12

235 275 315 355 395 435

4.1.2 General review for composite sections

(1) Some full scale tests on composite bridges using hot-rolled steel girders have been carried out in Bochum (Germany) in 1990 with the following main purposes (see Ref. 17 and Annex 6) :

buckling behaviour of rolled shapes in composite sections in negative moment regions (local buckling in steel webs; global instability with lateral-torsional buckling),

application of high strength steel (S 460) in bridge building,

structural behaviour of the deck when using prefabricated concrete elements as composite formwork,

demonstration of construction principles.

(2) Figure 3 shows the deflection of a test specimen below the hydraulic jack as function of the ratio M/Mpi. That test specimen of 20 m length was composed of 3 beams HE A 900 (S 460 steel grade) and of a 28 cm thick concrete slab.

Taking into account the moments due to self-weight of the specimens the curves start at a value of M/Mpi = 0,14.

The experimental investigations demonstrated a ductile behaviour of the beams, although geometric and material properties have been chosen unfavourable for this type of composite bridge. In both tests the full plastic moment capacity predetermined by calculations could be verified by the tests.

1.1­1

1.0­

0.9­

u.o­

u. / ­

Q. O.b­

Ξ 0.5­

U.4"

U.o*

self­ ° ·2"

weiaht­i ^ _

o.o-

1

Mexp.2/Mpl­1 Π1/1 I 014 |

ι y

' _ - —

* ­=­—­­

/ b ï ü ·-■■■■-'-■■-·" ■■-■ t

ι ­

¡fc= · VH, j. - > . ¿ |

PI ι 1 ! 1 1 —

w 100 200 300 400

Deflection [mm]

500 600 700

Figure 3 : Deflection history of beam 2 including unloading branch

(3) According to Eurocode 4 (Ref. 2) the design of composite sections in hogging moment regions is linked to a classification based on the slendemess of steel elements in compression (here : web or lower flange).

As the cross-section is classified according to the least favourable class of its elements in compression, the test specimens had to be classified class 3 due to their width-over-thickness ratio of the web (d/tw = 48). Using an effective web with a reduced width the section could be lifted into class 2. This method leads to a moment resistance, MpijRd» which lies in between the elastic moment for a class 3 section governed by yield of the steel bottom flange, MeiJid» and the plastic moment Mpij^d, for a class 2 section (see Figure 4).

21

Moment resistance

Class 3 : Mei.Rd [kN.m]

Class 2 with effective web : Mpiüd [kN.m]

Class 2 : Mpi.Rd [kN.m]

Test result M e x p . [kN.m]

Test specimen

15580

18410

20860

21145

Figure 4 : Calculated and measured moment resistances

(4) In the test a typical class 2 behaviour of the beams could be observed. The theoretical plastic moment Mpi has been reached. The deflection history reflects an impressive ductile behaviour of the composite sections together with a high rotation capacity.

These results point out that the classification system of Eurocode 4 leads to conservative and uneconomic results especially in those cases where actual conditions do not fit the assumptions taken into account when drafting the code regulations.

(5) For the given parameters all influences like vertical stiffeners at the support, the restraint by the cracked concrete slab, combined stressing due to shear and bending and the non-linear elasticity of the bottom flange have been taken into consideration when calculating the buckling load of the beam with a computer program. With a more sophisticated calculation the cross-section can be classified into class 2.

Furthermore a proposal has been worked out for the maximum width-over-thickness ratios for steel webs in composite beams taking into account the restraint of the concrete slab (see chapter 6.2 and Figure 15).

4.2 Review of elastic global analysis

(1) The clause 5.2.1.3 (3) (which is a principle) of Eurocode 3 may be unsafe in certain cases :

" 5.2.1.3 Elastic global analysis (...)

(3) Following a first order elastic analysis, the calculated bending moments may be modified by redistributing up to 15% of the peak calculated moment in any member, provided that :

a) the internal forces and moments in the frame remain in equilibrium with the applied loads, and

b) all the members in which the moments are reduced have class 1 or class 2 cross-sections (see 5.3)".

In Annex 7 (document 3263-2-12 : see Ref. 9), the restrictions of that rule are precised for 2 common examples :

A- 2 spans continuous beam with a uniform distributed load;

B- 2 spans continuous beam with a point load.

In example A, the required rotation capacity is shown to be quite limited (Rrequired < 0.6) and so class 2 cross-sections may be used in that case.

22

In example Β, for certain cases (concentrated loads,...) the required rotation capacity may be very large (Rrequired > 4) and the use of class 2 or even class 1 cross-sections with a 15% redistribution of peak moment may be unsafe regarding the rotation capacity.

4.3 Review of plastic global analysis

(1) The clause 5.3.3 (4) (which is a principle) of Eurocode 3 may be unsafe in certain cases :

" 5.3.3 Cross-section requirements for plastic global analysis (...) (4) For building structures in which the required rotations are not calculated, all members

containing plastic hinges shall have class 1 cross-sections at the plastic hinge location."

Indeed it may happen that class 1 cross-sections would have not enough available rotation capacity Rav compared with rotation requirements depending on the percentage of redistribution of moments and the structural arrangement

The two following examples (2 spans beam with a concentrated load) illustrate this problem in using :

Feldmann's model to evaluate available plastic rotation (pav and available rotation capacity Rav (see Figure 14 (Ref. 10)), and

formulae of required rotation capacity (cpreq. Rreq) ror continuous beams (see Annex 8 (1/13 to 6/13) and Ref. 9):

Example 1 : IPE 400, S 460 : class 1 cross-section

α = 0,37 λ = 1 L = 5,8m 7M(p = 7MR = 1.5 Bending about major axis My.sd Vsd/VpLRd = 0,59

(Pav = 0,067/1,5 = 0,045 rad < (fteq = 0,069 rad (not fulfilled !)

Rav = 2,708/1,5 = 1,805 < Rreq = 2,792 (not fulfilled !)

Example 2 : IPE 60 0 A, S 355 : class 1 cross-section

α = 0,33 λ = 1 L=l l ,0m γΜφ = YMR =1.5 Bending about major axis My.sd Vsd/Vpl.Rd = 0,47

(pav = 0,077/1,5 = 0,051 rad < (fteq = 0,0794 rad (not fulfilled !)

Rav = 3,293/1,5 = 2,195 < R req= 3,389 (not fulfilled !)

5. Numerical simulations

5.1 Numerical simulations for the influence of ε parameter

5.1.1 General

The influence of the parameter ε has been investigated. According to Eurocode 3 (Table 5.3.1 in Ref. 1) (see also present Annex 2), the influence of yield strength on the limiting ratios for the classification of cross-sections is taken into account through the parameter ε :

ε = I with fv in N/mm^ Vfy y

(Except for circular hollow sections where ε2 is used)

23

In order to check the relevance of this proportionnality in ε for high strength steels, CTICM has carried out a parametrical study with the help of numerical simulations. 24 numerical simulations with ANSYS program allowed us to look for the influence of the following parameters (see Annex 12) :

Steel grade (S 235 and S 460)

Geometrical imperfection

Residual stresses

A web under pure bending (supported on both sides) and (half) a flange under pure compression (supported on one side) have been studied.

simply supported simply supported

simply supported

Figure 5 : a) Flange b) Web

We give hereafter more details concerning the simulations for the flange and for the web.

5.1.2 Flange

5.1.2.1 Purpose

The main purpose of these simulations is to check the adequacy of the formula for ε :

ε = 235 (fy in N/mm2)

In order to take into account the relative influence of residual stresses for steel grade S 460, it has been assumed that the following modified fornitila could be proposed.

ε = 235

1/n

^ f y ; with η = 2, 3,4

We consider the limiting ratio for Class 1: b / t < 10 ε. For steel grade S 235, ε is equal to 1,00 whatever the value of η is. Therefore if we choose b = 100 mm, the plate thickness is : t = 100/10 = 10 mm. This case can be considered as the reference case.

Then for steel S460, we have to keep the same limiting ratio, but now ε factor is not equal to 1,00. So we modify the plate thickness t as follows :

ί=100/(10.ε) where ε is calculated for the various values of n.

Thus we can simulate the behaviour of plates for steel grades S 235 and S 460 with the same limiting ratio, and only the relative influence of residual stresses is highlighted by assuming that the level of the residual stresses is the same for both steel grades : half the yield strength of steel grade S 235 (σ0 = 117,5 MPa).

5.1.2.2 Results

Figure 6 shows the curves obtained from the numerical simulations. This figure demonstrates that residual stresses have a non negligible influence in so far as the curve "n = 2" is not the closest curve to the reference curve, the curve "n = 3" is closer than the curve "n = 2". For instance, the value η =

24

1 00-

0.90-

0 20-

0.(

N1 Up

I

/ / / / / / / / / )0

II

// ¡r

•" **

/ I

=<5^>. " '

"*^C^

4 A

Reference curve : 5235

Steel grade : S460

I I

wo/b = 6/1000

^­~._

1.00 2.00 3.00 4.00

^ ~ _

5.00

ü a

6.00

L.

, —

• 2

«3

• 4

7.00 8.

d /dy

30

Figure 6 : Flange simulations

5.1.3 Web

5.1.3.1 Purpose

The process is the same except for the limiting ratio : b / t < 72 ε (instead of < 10 ε). Numerical simulations have also been made for S 460 with η = 2,3 and 4.

5.1.3.2 Results

Curves obtained from numerical simulations are plotted in Figure 7. The conclusions are approximatively the same as those given for the flange, even if the difference does not seem to be so large.

1 00-

0.90-

0.80-

0.70-

0.60-

0 50-

0.40-

0 30-

0 20-

0.10-

000

M/Mpi

0.00

/ /

A« II

¡1 h

¡1

V

v ^

R

Steel grade : S460 I , ¡ , !

esidual stresses : 117,5 MF

- wo/b = 3/1000

3a

1.00 2.00 3.00

\ ^^

I

7T7n = 2

n = 4 ^ n - ·

4.00 5.00 6

ψ/ΦρΙ

00

Figure 7 : Web simulations

25

5.1.4 Conclusions

These simulations have allowed us to show that the relative influence of the residual stresses is not negligible. But these are only comparisons. More general conclusions are not possible yet, for the following reasons :

residual stresses depend on the shape of the cross-section

simulations should be done for steel grade S 355

other stress distributions should be tested

5.2 Numerical simulations of steel cross-sections classification for My loading

5.2.1 Introduction

(1) These series of simulations (see Annex 10) have been carried out by LABEIN with the aim of providing information about realistic class limitations in order to verify the limits provided by Eurocodes 3 (limits on flange slendemess c/tf and on web slendemess d/tw ) for the borders between :

- class 3 & class 4 cross-sections (meaning that elastic bending moment resistance, Mel is reached), - and class 2 & class 3 cross-sections (meaning that plastic bending moment resistance,

Mpi is reached) The limits between classes 3 & 4 cross-sections and between classes 2 & 3 cross-sections have been evaluated according to linear and non linear numerical simulations and for both S 235 and S 460 steel grades.

(2) The numerical simulations studied the cases of 3-point bending beams for which a big amount of tests results is available. The finite element modelling for the simulations has been calibrated on those tests results and is proven to give realistic results (Ref. 9; Ref. 19). Such simulations of beams demonstrate the effect of web-flange interaction on the cross-section classification.

5.2.2 Finite element modelling

Cross sections:

Span:

Meshing:

Element type:

Load application:

Boundary conditions:

IPE A 500, HE A 200, HE AA 300, HE A 280, IPE 300, ΓΡΕ A 400 and HE A 450 with modified flange and web thickness (tf and tw).

6m

616 elements and 1913 nodes

S8R-Abaqus (parabolic 8-node shell element)

Central concentrated load with the vertical displacement of the central section upper flange nodes linked together

Vertical supports: both ends Lateral restraints: both ends, central section, and Lj_TB m

accordance with the specified rules in CM66 (Ref. 18).

Figure 8 : Boundary conditions

26

Analysis conditions: * h-tf = cte

* Fillet radius modelled by means of the following length of overthicknesses in each case :

IPE A 500 : Li = r=21mm

L2 = 4/5.r+ tf average/2 = 26,805 mm tf average = 20,01 mm

HE A 200: Li = 4/5.r + tw average/2 = 16,26 mm tw average = 3,72 mm ]_2 = r= 18 mm

HE AA 300 : Li = 4/5.r + tw average/2 = 25,35 mm L2 = r = 27 mm

HE A 280 : Li = 4/5.r + tw average/2 = 23,20 mm L2 = r = 24 mm

IPE 300: Li=r=15mm L2 = 4/5 r + tf average/2 = 17,35 mm

EPE A 400 : Li = r=21mm L2 = 4/5 r + tf average/2 = 22,80 mm

HE A 450 : Li = 4/5.r + tw average/2 = 27,35 mm L2 = r = 27 mm

These values have been taking into account in order to evaluate the elastic moment Mei and the plastic moment Mpi in terms of Li, L2 (see Figure 9) and the section geometry.

* For chosen values of (h - tf) and b for a profile, numerical simulations follows an iterative process to determine the web thickness tw corresponding to a chosen flange thickness tf (or vice versa) and allowing to reach :

- either, the elastic bending moment resistance of cross-section Mei ,for the border between class 3 & class 4,

- or, the plastic bending moment resistance of cross-section Mpi , for the border between class 2 & class 3.

tf

ψ + 2a

¿

• ^

>

Ί i

•

' !Γ~

a >

a ^ -

1

1 τ-|4

lW

( -e—

ν ï

L2

t» + 2a a

h

Figure 9 : Modelization of root fillet

27

5.2.3 Results

(1) On the basis of these numerical simulations presented in details in Annex 10 (see working document 3198-3-3) and exploited in Annex 11, Eurocode 3 present rules for classification of steel cross-sections submitted to bending about major axis yy (My), are shown to be too conservative for all values of flange slendemess (c/(tfe)) and web slendemess (d/(twe)), in cases of borders between class 3 & 4 cross-sections (Mel is reached) and borders between class 2 & 3 cross-sections (Mpi is reached).

(2) The class 3 & 4 limits for S235 and S460 steel grades obtained in the simulation are shown in figure 10 (issued from Annex 11). In Figure 10, the simulations gouvemed by shear buckling failure mode (see chapter 1.2 of Annex 11) have been excluded (see specific fines between concerned points). A new border for class 3 & 4 cross-sections is proposed. Present rules of Eurocode 3 are provided. The safety reserve between present rules and new proposal is highlighted by hatching : for flange slendemess a safety cefficient of 1,7 to 2,3 can be obtained, for web slendemess a safety coefficient of 1,3.

(3) In the upper graphs of Figures 11 and 12 (issued from Annex 11) the results shown in Figure 10 are presented with characteristic values of (c/(tfe) ; d/(twe)) for standard IPE and HE AA, hot-rolled profiles and for both S235 and S460 steel grades.

(4) In the lower graphs of Figures 11 and 12 similar results from simulations are presented for class 2 & 3 limits and for S235 and S460 steel grades, The safety coefficient of 1,8 to 3,2 can be obtained for flange slendemess and probably a safety coefficient of 1,5 for web slendemess. Although additional analysis would be required.

(5) For standard profiles, flange slendemess (c/(tfe)) is clearly relevant (see Figure 12 with HEAA profiles for the worst cases of slender flanges) whereas web slendemess (d/(twe)) is not determinant for classification (see Figure 11 with EPE profiles for the worst cases of slender web).

(6) At present state, following improved rules could be proposed :

1 4- |

1 - 1

-

fy

Present EC 3 rules

class 2

web

d/ t w <

83 ε

flange

C/tf<

11ε

New proposals

class 2

web

d/tw<

124 ε

flange

C/tf<

35ε d

6,2tw

28

ι + ι

_ / I - I

fy

Γ Ν,Μ,

Present EC 3 rules

class 3

web

d/tw£

124 ε

flange

cAf<

15 ε

New proposals

class 3

web

d/tw£

165 ε

flange

c/tf<

max (25 ε; 35 ε -8tw

(7) But more developments should be necessary to reach general and safe conclusion and to define precisely new improved limits.

29

o

σο'

Λ Η* Ο

Cd ο 1 Ο. ΓΒ "1

Q, Ρ CA C«

Rp

ST

-ι o VI VI

Zt. O 3

CA

? Ρ

s 3

ora

2 2. S α

N M .

CA

1 re

f9

α

Linear Analysis - Border Class 3&4 elastic cross-sections (= MtlRd is reached)]

tf .ε

45 τ 1 *

L 3L

"S 235 - Simulation border "S 460 - Simulation border with EC3 epsilon factor -Simulations excluded by shear buckling failure mode ■ Proposed limits of simulations Proposal of new border class 3&4 EC3 limits border class 3&4 EC3 limits border class 2&3 EC3 limits border class 1&2 Simulations numbers related to points of enclosed tables ML

EC3 borders and simulated borders (linear analysis) between Class 3&4 versus IPE profiles!

c

tfT

45 τ

S S 235 - Simulation border S 460 - Simulation border with EC3 epsilon factor (n = 2) Simulations excluded by shear buckling failure mode b y ;

nula Proposed limits of simulations Proposal of new border class 3&4 EC3 limits border class 3&4

—EC3 limits border class 2&3 -EC3 limits border class 1&2

A IPE profiles-S 235 □ IPE profiles - S 460

100 120 140 160 180 200 220 240 260

EC3 borders and simulated borders (non-linear analysis) between Class 2&3 versus IPE profiles |

A S 235 - Simulation border "S 460 - Simulation border with EC3 epsilon factor -Simulations excluded by shear buckling failure mode Proposal of new border class 2&3 EC3 limits border class 3&4 EC3 limits border class 2&3 EC3 limits border class 1&2

A IPE profiles-S 235 Π IPE profiles - S 460

Border meaning that MpLRd is reached

20 40 60 80 100 120 140 160 180 200 220 240 260

Figure 11 : EC3 and simulated borders between classes 3&4 and classes 2&3 versus IPE profiles

31

EC3 borders and simulated borders (linear analysis) between Class 3&4 versus HE AA profiles |

c tji

45 χ

A S 235 - Simulation border "S 460 - Simulation border with EC3 epsilon factor fn = 2) -Simulations excluded by shear buckling failure mode by.'

lula Proposed limits of simulations Proposal of new border class 3&4 EC3 limits border class 3&4 EC3 limits border class 2&3

-EC3 limits border class 1&2 A HE AA profiles-S 235 Π HE AA profiles - S 460

80 100 120 140 160 180 200 220 240 260

EC3 borders and simulated borders (non-linear analysis) between Class 2&3 versus HE AA profiles |

A S 235 - Simulation border "S 460 - Simulation border with EC3 epsilon factor - Simulations excluded by shear buckling failure mode Proposal of new border class 2&3 EC3 limits border class 3&4 EC3 limits border class 2&3 EC3 limits border class 1&2

A HE AA profiles - S 235 D HE AA profiles - S 460

Border meaning that MplRd is reached

20 40 60 80 100 120 140 160 180 200 220 240 260

Figure 12 : EC3 and simulated borders between classes 3&4 and classes 2&3 versus HE AA profiles

32

6. New proposals for rules of classification

6.1 Proposal for steel sections

6.1.1 In case of plastic global analysis

(1) For plastic global analysis new proposals using the concept of rotation capacity of plastic hinges include the properties of resistance to local buckling of the cross-sections if lateral-torsional buckling is prevented according to Eurocode 3 rules for instance (see Ref. 9).

(2) The proposed method (see Ref. 9) which compares required and available inelastic rotations (φ) or rotation capacities (R) for each formed plastic hinges, is an alternative to the use of width / thickness limits (rules for classification of cross-sections) existing in Eurocodes 3 and 4 (Ref. 1 and Ref. 2), for the verification of sufficient ductility of plastic hinges.

Eurocode 3 (Ref. 1) provides general rules concerning rotation requirements of plastic hinges:

" 5.3.3 Cross-section requirements for plastic global analysis (...) (2) At plastic hinge locations, the cross-section of the member which contains the plastic

hinge shall have a rotation capacity of not less than the required rotation at that plastic hinge location.

(3) To satisfy the above requirement, the required rotations should be determined from a rotation analysis.

(4) For building structures in which the required rotations are not calculated, all members containing plastic hinges shall have class 1 cross-sections at the plastic hinge location."

(3) Using plastic analysis, required inelastic rotation (Dreq can be determined, and computer programs can give directly these values for all plastic hinges in the frame.

On the other hand, tests results, numerical simulations and, now, analytical models and formulae allow to determine an available inelastic rotation (pav of plastic hinge for a given cross-section (see Figure 13). These studies are carried out especially on 3-point bending systems.

The following limit states criterion for ductility in bending (to be checked in all cross-sections) naturally yields for checking the validity of the plastic state of a frame under a given loading:

<Preq^ <_îay_

ΎΜφ (6.1)

where TMcp is a partial safety factor to allow for the uncertainties of (Dav model (see Ref. 9).

It can be shown that the available inelastic rotation (Dav depends only on local parameters :

material properties (yield strength, ultimate strength,... ),

shape and dimensions of the cross-section,

internal forces at the location of the plastic hinge.

This criterion (6.1) that only applies till now to class 1 and class 2 cross-sections according to present Eurocode rules, is sufficient in so far as the available inelastic rotation (pav can be given by formulae or in tables according to Feldmann's model (Ref. 10). The inelastic rotation can be expressed in radians. The basis of Feldmann's method is the modelling of the non-linear buckling phenomena in the yielding zones by a plastic folding mechanism that allows to determine the rotation capacity in a reliable way. Formulae have been developed for inelastic rotation (pav and the rotation capacity Rav of I-profiles and Η-profiles in bending about the major and the minor axes, for bending with and without shear only as weh as for bending with axial force (see Figure 14, Ref. 9 and Ref. 10). Practical tables and recommendations to evaluate the ductility of plastic hinges are given in Ref. 9.

33

(4) A great number of authors have preferred to talk about "rotation capacity" (R av, Rreq) instead of "inelastic rotation" ((pav. <Preq) by introducing an elastic rotation of reference (φρι) (determined at Mpi level) in order to adimensionalize the problem.

It is understood that the available rotation capacity R a v related to the available inelastic rotation (p a v

in the hinge is to be determined for a 3-point bending system and is given by (see Figure 13) :

where L

<Ppl

M p i

E

I

R _<Pav av <Ppl

<Ppl = _ Mpl.L

2. E.I

is the length of the beam,

is the sum of the elastic rotations (determined at Mpi level) at the ends of the beam,

is the plastic resistance moment of the cross-section,

is the modulus of elasticity of steel,

is the moment of inertia of the cross-section.

φ / 2

y

l F

1 «Pav

L

Φ / 2

/ t

M i

M p i -i

ΦΡΙ

<Pav

Γ"*»

I I

, - , Ι

: Φ

Φι«

Figure 13 : Moment-rotation curve for 3-points bending beam

So, the validity and the consistency of a plastic analysis can also be checked in all cross-sections by the following limit states criterion for ductility in bending which is equivalent to criterion n°(6.1):

Kreq * ~ YMR

(6.2)

where Rreq =<Preq/<Ppl and.

7MR is a partial safety factor allowing for the uncertainties of R a v model.

6.1.2 In case of elastic global analysis

(1) On one hand realistic numerical simulations of 3 point-bending beams (see chapter 5.2) highlight the excess of conservatism for the present classification of steel I or Η cross-sections submitted to bending moment about major axis My. New particular proposals have been made for borders between classes 2 & 3 and classes 3 & 4 but more developments are necessary to reach general and safe proposals.

(2) On the other hand a proposal concerning the classification of I-cross-sections taking into account the interaction between web and flanges has been submitted to the partners in the project but presently there is no practical issue from this proposal. This proposal is derived from Feldmann's model (Ref. 10) and it consists in a classification of a cross-section depending on combined check of the web slendemess and the flange slendemess. Therefore the class of a cross-section should not be anymore only determined by the weakest wall component that is presently checked separately.

34

Feldmann's model : evaluation of inelastic rotations φΓ§[ (= φ 3 ν ) of plastic hinges for I or Η steel profiles and for different load cases.

i Δ f Δ " ξ . ί . (Ι-ξ).ί ι

f f Φ (PD i ji φ Γ Ο Ι

4k, ( f y f l + A a ) b h

^4Ebt w

5h 2 + j ( f y . w h t w ) + 4 f y - w b t f t w h A a - f y # w h t w

ç L­/· 5I φ

ill

(pi) _ 0,2 2t f

" ' i _ — | b

M p i ­( h ­ 2 t f ) t w f . L

y.w

b 3 ( f y . f l +Aa)

/"-a

TT çTT\ ( p l ) _ _ 4 k v i k V 2 k | _ Ψ rot

Ν -»δ-

1 ( f y i l + A a ) b h

4Ebt w

5h' + J(fy.whtwJ +4fy wbtf twhAa-f y wht

w

Ξ-^ N

en 1 02

or (D(p l ) = 0 Γ

Ψ rot

ω( ρ 1 )

­ 2

4k f

t f tfMpLN t f ( N f l - P c )

where

( f y i l + A o ) b h '

9rot ( = 9av) m radians

Δσ = 15 k N / c m 2

kf =

h P„h '

^4Ebt w

5h 2

pu h j

+ J(fy.wht w ) + 4fy w btf twΙιΔσ - f y wht w

ly.ñ

40

1,3 2 — , with fy β is in k N / c m , + 0,25

k v l = ^a^°'

75200^

ly J

, with f y is in Ν / m m ,

0,35

kv2 = 1,5­0,38|f

Mpij«^ according to Eurocode 3 (Ref.l), for instance,

Nfl ^ t f - t M ^ f y i i A

t M =ib~

f

1 -N

V Ν

Pi

ρ _ 4 E b t w

5h 2

Pu=bt f ( fy . f l+Ac) .

igure 14 : Summary of formulas from Feldmann's model to evaluate inelastic rotations of steel plastic hinges

35

6.2 Proposals for composite sections

(1) For elastic and plastic global analysis of continuous composite beams : on the basis of experimental results (see chapter 4.1.2 (Annex 6 and Ref. 17)) a proposal for Eurocode 4 has been worked out for maximum width-over-thickness ratios for steel webs in compression and in combined pending-compression, in composite beams taking into account the restraint of the concrete slab (see Figure 15).

(2) For plastic global analysis of continuous composite beams, a specific design method has been proposed by Mr. Couchman G. (Ref. 11) for continuous composite beams (see Annex 9 (1/9 and 2/9)).

That design method based on the idea of rotation capacity :

includes the influences of all relevant parameters, is applicable to beams with plastic (class 1), compact (class 2) or semi-compact sections (class 3), gives a uniform margin of safety for all cases, is suitable for everyday use by the practising engineer.

That design method allows considerable increases in beam load capacity for beams with compact or semi-compact critical sections. At present stage, more developments (tests results, numerical simulations, statistical evaluations,...) are necessary to exploit that method for continuous composite beams in order to elaborate design aids or charts which win help designers in their daily works.

(3) The software Compcal developed at EPFL has been used to compute the available rotation capacities of beams with plastic and compact sections by introducing Kemp's model (Ref. 8). A wide variety of parameters have been chosen to study the influential factors to the available rotation capacities. These factors were :

slendemess of the cross-section, reinforcement at the support, represented by the ratio of hogging to sagging plastic resistance moment (the plastic moment ratio), structural steel characteristics, slip between the steel and concrete, degree of shear connection between the steel and the concrete, ductility of shear connectors, span lengths, number of spans, ratio of adjacent span lengths, type and arrangement of loading, propping of the beam during construction.

The influence of parameters which are related to the composite section, or length of beam in hogging, on available rotation capacity has been shown, and a single variable can be used to represent all such individual parameters (see Annex 9 (3/9) and (4/9)). All parameters which affect available rotation capacity should therefore be allowed for in a design model, which is not the case for simplified methods of analysis such as those proposed in Eurocode 4 (Ref. 2).

(4) For semi-compact composite sections, the available rotation capacity depends on cross-section properties but also on the arrangement of spans and loads (see Annex 9 (5/9)). The Kubo and Galambos model is used (see Annex 9 (6/9)).

(5) In the scope of the design method proposed by Mr. Couchman G. (Ref. 8), the required rotation 0req (= 9req) n as been graphically represented as a function of moment redistribution (Δ). The use of such curves allows for the parameters which affect the rotation capacity required by a beam to achieve a given moment redistribution. These parameters are:

Elastic moment ratio (μ^ι) and span type (external or internal). These two parameters affect the basis form of Oreq vs. Δ curves. Plastic moment ratio (μρΐ), which affects values of moment redistribution but not the form of Oreq vs- Δ curves. Degree of shear connection and construction method (propped or unpropped). These two parameters may necessitate modification to the value of moment redistribution which is given by a Oreq vs. Δ curves.

( See Annex 9, (7/9) to (9/9)).

36

EC4 Classification of cross­sections :

Web in compression and bending­compression

Webs : (Internal elements perpendicular to axis of bending)

X X

tw ­î e

Code Class Compression Combined bending and compression

Stress distribution in element (compression positive)

ad

ÊT

EC4

New proposal

d/tw < 33.£

d/tw<38.e

when α > 0,5 : d/t < 396 ε/(13α-1) when a < 0,5 : d / t<36e/a when a > 0,5 : d/t < 684 ε / (17a+ 1) when a < 0,5: d/t<36 ε / a

EC4

New proposai 2

d/tw ^ 38.£

d/tw<44.£

when α > 0,5 : d/t < 456 ε/(13α-1) when α < 0,5 : d/t<41,5e/a when a > 0,5 : d/t < 730 ε/(15,6a+1) when a < 0,5: d/t < 41,5ε/a

Stress distribution in element (compression positive) t :

J p

EC4

New proposai 3

d/tw ^ 42.8

d/tw<49.£

whenψ>-l : Μ<42ε/(0,67 + 0,33ψ) when\|/<-l : (1Λ<62ε(1-ψ)Λ/(Ι^)

when α > 0,5 : d/t < 730 ε/(15,6a+1) when a < 0,5: d / t<4L5e /a

= Λ/2357ζ fy [MPa] 235 275

0,92

355

0,81

460

0,71

Figure 15 : Table for the EC4 classification of cross­sections

37

Conclusions

On the basis of test results and numerical simulations, this project highlighted the tremendous conservatism of present mies for cross-sections classification defined by Eurocodes 3 and 4 : no interaction between web and flanges is considered for steel profiles, no influence of stiffening effect from concrete slab is considered for composite sections, the dependence on fy (included in parameter ε) is not enough precise,...

New rules for cross-section classification are proposed on the basis of tests results and numerical simulations :

- for Eurocode 3:

in case of elastic and plastic analysis : significant improvements are suggested in view of tests results and of related simulation results of I or Η hot-rolled sections submitted to bending about major axis My (see chapters 6.1.2 & 5). On the other hand improvements of ε factor that takes into account the influence of yield strength, are presented according to results of realistic numerical simulations (see chapter 5.1). But more developments (tests results, simulations,...) are necessary to confirm such proposals and extend them to other cross-sections and to other load cases;

in case of plastic global analysis : a new concept is presented with a limit states criterion for ductility of plastic hinges, including terms of inelastic available rotation cpav or available rotation capacity Rav (see chapter 6.1.1). But till now this criterion is only proposed for class 1 & class 2 cross-sections according to present conservative Eurocode 3 rules;

for Eurocode 4:

in case of elastic and plastic analysis : improved values of present limits are proposed for steel webs in compression and in combined bending (My) - compression for composite cross-sections in view of tests results (see chapter 6.2).

in case of plastic global analysis : a specific design method is suggested for continuous composite beams for present class 1, class 2 & class 3 cross-sections (see chapter 6.2). But more developments (tests results, simulations, ...) are necessary to exploit that method and produce practical design aids.

38

List of Annexes

Annex 1 : Complete Set of Distributed Documents

Annex 2: Final report (Excerpts) Simplified Version of Eurocode 3 for Usual Buildings

Annex 3 : Document 3198-1-1 (Ref. 15) Excerpts)

Local buckling rules for structural steel members Annex 4 Document 3 263 -1 -27 (ProfilARBED)

Available rotation capacity of plastic hinges RaVaHabie - Tests results and models

Annex 5 Document 3 263 -1 -27 (ProfilARBED Available rotation capacity of plastic hinges Ravanabie - Tests results and models

Annex 6 Document 3198-1-18 (ProfilARBED) Stability of composite bridge girders near internal support

Annex 7 Document 3263-2-12 (CTICM) Required rotation capacity for a 15% reduction of elastic peak moment

Annex 8 Document 3 263 -2-15 (CTICM) Required rotation capacity for continuous beams

Annex 9 Document 3263-1-29 Mr. Couchman's thesis (excerpts) Design of continuous beams allowing for rotation capacity

Annex 10 Document 3198-3-3 (LABEIN) Technical report n° 4 Numerical simulations of class 2&3 limit and class 3&4 limit

Annex 11 Exploitation of Labein numerical simulations (ProfilARBED) presented in Document 3198-3-3 (Annex 10)

Annex 12 Document 3198-2-10 (CTICM) Some numerical tests for checking the influence of yield strength on limiting b/t ratios

39

Complete set of distributed documents Pate: 14.12.95)

Concerning: ECSC research project:

** Ρ 3198: "Improved classification of steel and composite cross-sections New rules for local buckling in Eurocodes 3 and 4" **

3198-1. From ProfilARBED-Recherches :

3198-1-1 paper issued from the JCSR Journal ("Journal of Constructional Steel Research") nr. 20 (1991) published in 1992: "Local Buckling Rules for Structural Steel Members", S. Büd et G.L. Lulak.

3198-1-2 paper issued from the journal "Construction métallique" nr. 1,1991: "Application de 1'Eurocode 3: Classement des sections transversales en

I", MM. A. Bureau et Y. Galea.

3198-1-3 a few pages issued from a publication of the TC 11 technical committee on the classification of composite cross-sections according to Eurocode 4

3198-1-4 proposal of "Improvements of Eurocode 4" (TC 11-7-92) presented to the TC 11 technical committee about thew maximum width-to-thickness ratios for steel webs in composite cross-sections.

3198-1-5 paper issued from the JCSR Journal ("Journal of Constructional Steel Research") nr. 23 (1992) published in 1992: "Plate slendemess limits for high strength steel sections", K.J.R. Rasmussen and G.J. Hancock.

3198-1-6 paper issued from "Structural Engineering International" n° 4 / 91: "Local buckling and moment redistribution in class 2 composite beams", R.P. Johnson and S. Chen.

3198-1-7 paper issued from "IABSE Proceedings P-125/88": "Strength of continuous composite beams designed to Eurocode 4", R.P. Johnson and C.K.R. Fan.

3198-1-8 paper issued from the JCSR Journal ("Journal of Constructional Steel Research") nr. 27 (1993) published in 1993: "Effect of reinforced concrete between the flanges of the steel profile of partially encased composite beams", R.Kindmann, R. Bergmann, L.-G. Cajot and J.B. Schleich.

3198-1-9 Background Documentation to Eurocode 3, Annex D, Document D.01, April 1990: "Background document for design rules specific for high strength steels according to EN 10113", Eurocode 3 Editorial Group.

3198-1-10 Report n° MT 187 from CRIF - Liège, November 1993, about stub-column tests and buckling tests on steel H section with fy = 550 Mpa: "Essais de flambement sur profilés de classe 4 en acier à très haute résistance,"J. Janss.

3198-1-11 paper issued from the JCSR Journal ("Journal of Constructional Steel Research") nr. 31 (1994) published in 1994: "Local Buckling of I-Sections Bent about the Minor Axis", by Bradford M.A. & Azhari M.

42

Complete set of distributed documents (Date: 14.12.95)

3198-1-12 RPS report n° 108/91, Journées ATS 1991: "Acier HLE pour ponts mixtes à portées moyennes de 20 à 50 m", ,by MM. Schleich J.B. & Witry Α.

3198-1-13 RPS report n° 106/91, about tests on composite bridges: "Verbundbrücken auf Basis von Walzträgern - Versuch Nr. 2: Realisticher Verbundbrückenträger", by MM. Schaumann P. & Steffen A, "HRA" engineering office.

3198-1-14 Study submitted to the Editorial Panel of Eurocode 4, in September 1989: "Comparison between Eurocodes 3 and 4 of classification for local buckling of class 1 and 2 sections", by Kemp A. R.

3198-1-15 working document from ProfilARBED: "Proposals for improvements of EC3 and EC4: classification of cross-sections", by Chantrain Ph. & Klosak M.

3198-1-16 paper issued from the journal "Construction métallique", n° 1-1995 : "L'influence des défauts de planéité de l'âme des profilés reconstitués soudés sur leur résistance en flexion et compression", M. Braham, R. Maquoi, N. Rangelov & C. Richard.

3198-1-17 paper issued from the JCSR Journal ("Journal of Constructional Steel Research") nr. 32 (1995) published in 1994: "Resistance of Plate Edges to Concentrated Forces", B. Johansson & O. Lagerqvist,

3198-1-18 paper presented in Odense conference (1991): "Stability of composite bridge girders near internal support", P. Schaumann & J.B. Schleich.

3198­2. From CTICM

3198-2-1 list of papers / references from Journal of Constructional Steel Research, Revue Construction Métallique and Background Documentation.

3198-2-2 additional information of the paper 3198-1-2 about the classification of cross-sections according to Eurocode 3: practical tables for steel sections with the steel grade S 460 (yield strength = 460 N/mm2).

3198-2-3 paper issued from the journal "Construction métallique" nr. 1,1983: "Etude de la possibilité d'un relèvement des élancements limites de

parois en calcul plastique", Plumier A. et Richard C.

3198-2-4 paper issued from the JCSR Journal ("Journal of Constructional Steel Research") nr. 7 (1987) published in 1987: "Inelastic local buckling of fabricated I-beams", M.A. Bradford.

3198-2-5 paper issued from the JCSR Journal ("Journal of Constructional Steel Research") nr. 19 (1991) published in 1991: "Local buckling of semi -compact I-beams under biaxial bending and compression", M.A. Bradford.

43

Complete set of distributed documents (Date: 14.12.95)

3198-2-6 paper issued from the JCSR Journal ("Journal of Constructional Steel Research") nr. 26 (1993): "Research on elastic buckling of columns, beams and plates: focussing on formulas and design charts", CM. Wang, S. Kitipornchai and K.M. Liew.

3198-2-7 official offprint of the document 3198-1-2: paper issued from the journal "Construction métallique" nr. 1-1991: "Application de l'Eurocode 3: Classement des sections transversales en I", MM. A. Bureau et Y. Galea.

3198-2-8 work paper from CTICM: "Some numerical tests for checking the influence of yield strength on limiting B/T ratios for webs in pure bending","Working document for ECSC Project n° 3198".

3198-2-9 work paper from CTICM: "Note on possible modifications of EC3 classification'V'Working document for ECSC Project n° 3198", 20th March 95.

3198-2-10 work paper from CTICM: "Some numerical tests for checking the influence of yield strength on limiting b/t ratios","Working document for ECSC Project n° 3198", June 1995.

3198-3. From LABEIN

3198-3-1 work paper from LABEIN: "New series 5: Class 2&3 limit, Class 3&4 limit", Ensidesa - Labein, June 1995.

3198-3-2 work paper from LABEIN: "Class 3&4 limit from simulation results", Ensidesa -Labein, September 1995.

3198-3-3 work paper from LABEIN (given in December 1995): "Improved classification of steel and composite cross-sections -New rules for local buckling in Eurocodes 3 and 4: Technical report n° 4 , Period from 01.01.95 to 30.06.95 (Ensidesa -Labein contribution)".

44

Annex 2

Final report (excerpts)

(10 pages)

"Simplified version of Eurocode 3 for Usual buildings" (ECSC agreement 7210-SA/513) : Chapter V rules for

classification of cross-sections

45

Eurocode 3 rules for cross-sections classification :

Table V.2 Determinant dimensions of cross-sections for classification

Webs (internal elements perpendicular to axis of bending)(see tables V.3 and V.4) :

- I h . 1 s

Axis of bending t w~^

! '

f ' iL

. . d ir tw 1

t '

*)

tw"* fc - ι

d=h-3t ( t - t fs t . )

Rolled sections Welded sections

- Outstand flanges (see tables V.3, V.4 and V.5)

+SL»

rx c **) i. α ι ι

c **) Ί r ι Rolled sections Welded sections

- Internal flange elements (internal elements parallel to axis of bending)(see table V.6)

*

Axis of bending

τ . b ,

Rolled sections

i. « b » ^ b . E

Welded sections

Circular tubes and angles (see table V.7)

t= ■Hf*·

*) For a welded section the clear web depth d is measured : . between welds for section classification . between flanges for shear calculations (see chapter VBT)

**) For welded sections the outstand dimension c is measured from the toe of the weld.

46

Eurocode 3 rules for cross-sections classification

Table V.3 Classification of cross-section : limiting width-to-thickness ratios for class 1 & class 2 I cross-sections submitted to different types of loading

Types of

loading

Stresses distribution for

class 1 & class 2

Class 1 Web

d / t < ' MW

Flange c/u<

Class 2 Web

d / t ^ ' M W —

Flange c / t f<

N, compression

IE=] fy N 33ε R

I + I W

10ε R 38ε

9ε W

11ε

10ε

Ί + Ι fy

Mv à -xMy

R 10ε R 11ε

72ε 83ε

W 9ε W 10ε

M, Η#-7' R 10ε i? 11ε

W 9ε W 10ε

Ncomp. " My

^ ~ l fy

ad

■5 a > 0,5 :

396ε

1 3 a - 1

/? 10ε

Ψ 9ε

a > 0,5 :

456ε

1 3 a - 1

ƒ? 11ε

W 10ε

ι + ι fy

Ntens. - M y ~7pp Ν Λ α<0,5

36ε α

R 10ε

W 9ε

α<0 ,5 :

41,5ε α

R 11ε

W 10ε

•Ncomp. " Μζ

1 + ' fy Μ 2 R 10ε Λ 11ε 33ε 38ε

W 9ε W 10ε

3 fy

Ntens. - Μ ζ t HL g~A2 R 10ε/α R Ι ΐ ε / α

W 9 ε / α W 1 0 ε / α Values of d, tw , c, and tf are defined in table V.2

+ : stresses m compression - : stresses in tension

R = rolled sections ; V « welded sections

fy (N/mirP) 235 275 355

ε = = Λ/235~7ζ ε(ίί ΐ<40πητι) 0,92 0,81 ε (if 40 mm < t < 100 mm) 0,96 0,84

420 0,75 0,78

460 0,71

0,74

47

Eurocode 3 rules for cross-sections classification

Table V.4 Classification of cross-section : limiting width-to-thickness ratios for class 3 I cross­sections submitted to different types of loading

Types of

loading

Stresses distribution for

class 3

Class 3 Web

d / t < ' M W —

Range

ç/Vj

N compression

I + I fy

m U

N R

42ε

I + l W

15ε

14ε

Mv

3EZH fy

ί η -*\My R 15ε

124ε

W 14ε

Μ, Ηί-7 R 23εν057

W 21E­JOJÏ

Ν comp. " My

+

' l l f

-á

3 ψ%

Mv ψ>-1

42ε

R 15ε

( Ιψ&Ι < |fy| ) 0,67 + 0,33ψ W 14ε

Ntens. - Mv

Ι -t- I ( lfy/ψΙ < ffy| )

-*JVly

fy1

ψ < - 1

62ε(1 - Ψ)Λ/-Ψ

R 15ε

V7 14ε

N comp. ~ Μ ζ

r­Ff 3 Μ7

i?

42ε

23ε^(&)

W 2le^(b)

Ntens. - M , t b

I X Ν M, R 23eÆ,<a)

w 21e*Jk¿(a)

Values of d, tw , c, and tf are defined in table V.2

+ : stresses in compression - : stresses in tension

R = rolled sections; W = welded sections

kø is defined in table V.5

. = fi35ÏT}

fy ( N / m m Q

ε (if t < 40 mm)

ε (if 40 mm < t < 100 mm)

235 275

0,92

0,96

355

0,81

0,84

420

0,75

0,78

460

0,71

0,74

48

Eurocode 3 rules for cross-sections classification :

Table V.5 Buckling factor ka for outstand flanges

Ψ

-1,0

-0,9

-0,8

-0,7

-0,6

-0,5

-0,4

-0,3

-0,2

-0,1

-0,0

+0,0

+0,1

+0,2

+0,3

+0,4

+0,5

+0,6

+0,7

+0,8

+0,9

+1,0

ka

0,85

0,82

0,78

0,75

0,72

0,69

0,67

0,64

0,61

0,59

0,57

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)

i Tension / z j

/ !

/ ­ : '

/

. < h .

+ /

/Compression /

.J

(a)

(

_J **.

(b)

Compression

, G l

r

/

/

+ /

/

L, J"2

+ /

ka

23,80

20,05

16,64

13,58

10,86

8,48

6,44

4,74

3,38

2,37

1,70

1,70

1,31

1,07

0,90

0,78

0,69

0,61

0,56

0,51

0,47

0,43

Stress distribution (compression positive)

σ2<-_-.

TensionV \

1

•f

1

1

\

+\Compression

1 » \ - *■

\

+ \

' ■ · . .

(c)

(

J

' \

(d)

Compression Ì

. σ 2 Ì

' " \

\

A \

, , \σ

.

\

+ \

Note 1 : ψ = σ2 Ι σι and 1 σ2 I ^ Ι σι 1

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).

49

Eurocode 3 rules for cross-sections classification

Table V.6 Classification of cross-section : limiting width-to-thickness ratios for internal flange elements submitted to different types of loading

Type of loading

N, compression

Stresses distribution

+ ] f y

N + ;

classes 1,2 and 3

internal flange

R

O

(b-3tf)/tf < 4 2 ε

b/ t f < 4 2 ε

class 1 class 2

internal flange

R

M O

0-3ί£)Αί<33ε

b/ t f <33ε

R

O

0>-3ΐ£)Λ£<38ε

b / t f <38ε

class 3 +

Ρ I I

I 1 1

1

r

1 1 1

fy internal flange

R

O

(b-3tf)/tf < 4 2 ε

b / t f < 4 2 ε

Values of b and tf are defined in table V.2

+ : stresses in compression - : stresses in tension

R = rolled hollow sections O = other sections

= Λ/235?Ίζ

fy (N/mm2)

ε ( ί ί ί ί <40mm)

ε (if 40 mm < tf < 100 mm)

235 275

0,92

0,96

355

0,81

0,84

420

0,75

0,78

460

0,71

0,74

50

Eurocode 3 rules for cross-sections classification :

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

■N compression

M and,

(N,M)

Stresses distribution

1

+ 1

>

fy

+

class 1

h / t <

10 ε

class 2

h / t <

11ε

class 3

h / t < 15 ε and

D + h ^ 11 c

< 11,5 ε 2t

see table V.3 (classes 1 and 2) and table V.4 (class 3) with limiting

width-to-thickness ratios concerning outstand flanges.

Tubular sections

Type of loading

^compression »

M and,

(N,M)

O' d

V

Values of h, b, t and d are defined in table V.2

z = ^235lîy

class 1 class 2 class 3

d / t <

50 ε2 70 ε2

90 ε2

+ : stresses in compression

fy (N/mm2)

zQît<4Qmm)

ε(ΐί40πΜτι<ι< 100 mm)

ε2(1ίΐ<40ιηπι)

ε2 (If 40 mm < t < 100 mm)

235

1

1

1

1

275

0,92

0,96

0,85

0,92

355

0,81

0,84

0,66

0,70

420

0,75

0,78

0,56

0,60

460

0,71

0,74

0,51

0,55

51

Eurocode 3 rules for cross-sections classification

Table V.8 Effective cross-sectional data for symmetrical profiles (class 4 cross-sections)

Members in compression (N)

gross cross-section ­pi

effective cross-section

v b ®

υ©

φ b

t. ε

b

1

56,8

1

ι.ε 18,6

il

! il

Aeff

A — -t

+

«—ò* —T- ■-■ +

+

Ν Φ b 1

τ·ρ<Λ

"tt t. ε 56,8 I I I I *·

Aeff

tL.zzcu

Members in bending (My, Mz)

Φ 1

t. ε 138,8

b 1

ί.ε 18,6

—t T-b©

Η °-4%fb©

-=t0-6-p©4-b( 2" ® Wdt

b®T F' -*Mz

-3 Φ Jb_ 1 ΐ .ε '21,4

© .zXp®\ Weff

Φ 1 t-ε 138,8

b 1

fb©

T'p®-b© tt

°-6-p®-i-b©

Weff

t. ε 56,8

= Λ/2357Γ3 fy (N/mm*)

ε (if t < 40 mm) ε (if 40 mm < t < 100 mm)

235 275

0,92 0,96

355

0,81 0,84

420

0,75 0,78

460

0,71 0,74

52

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 kø corresponding to the stress ratio ψ (see [table 5.3.2] and [table 5.3.3] of EC3),

b) calculation of the plate slendemess λρ given by :

b / t

2 8 , 4 ε ^

in which t is the relevant thickness of the elements, kø is the buckling factor corresponding to the stress ratio ψ,

ε = 235

(with fy in N/mm2),

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 = for equal-leg angles,

b = h or for unequal-leg angles.

c) calculation of reduction factor ρ with the following approximation ([formula (5.11)] of EC3) :

when λ ρ < 0,673: p = l

(λρ-0,22) when λρ > 0,673 : ρ = v _2—'-

d) determination of the effective width beff.

53

<

e. e O »­h

α c *

ss co

ci Pu

F £ gr et"

<: Ν)

>

II D ­r *

í / ­ ν

^ et cr

c*

>

II

CI

o cr. o 3 Ei

et Ρ

ΙΛ

ο f­,

Ui

[>

O Ol Ui o

SJ U i

lo 4*.

to o U i O

r~> U i

£ 3

O

VO Ui

VO O

co Ui

CO O J

0 0

o ­ J 0 0

«O ON

-o 4^

TENSION

ο "(Λ Ν ) Ο

ρ "ί/ι f") o>

O

0 0 o\

O " j > Π\ -O

o

σ\

O J > tO 4S.

O "4a. >—> vo

o J > r-> O

ρ ίο -.1 4s.

ρ "ίο 4». Ui

p κ-» 4*.

O

oo O

ρ To .p. to

ρ Ì O o o

ρ I—"

(Λ O J

TENSION

ι Ο

fi ­ J

I

O

Is»

M CO

I

O f\

o ­ J

I

O ( 0 no Ui

ι O ί>1 rt\ to

I O ' >) (O Os

TENSION

ι O o co Ui

I O ι—ι - 1 1—»

1

o r~> ' 1 o

I O <s r» o

Ô

o M O J

ó o r­i 4^

I

O 1 0 'O H->

o

o o o

I

O 1 O (—) co

ι O Vi ~o co

ι O

IO 4

s»

Ui

I

O

M

o

I

O

^1

o

I

O

to ON

ι O r~> ­^1 CO

ι Ο

ο Μ 4 >

ρ (O O J

O

O O

o

ρ I—»

O

o

o o -o o

o J >

Ui

ρ Ν) OJ

­ J

to

o o o <_>

< l

o as co

σ\ o\

Os 4^

as to

Os O

Ui

oo Ui Os

Ui J >

Ul

to Ul

o 4^

oo 4*. Os it 4 ^

to 4*·

o OJ 0 0

OJ Os

t>J 4 ^

OJ

to

COMPRESSION

o o to J >

o o Ul l_>

o o ■o ~J

ρ I—»

O Os

ρ ι — · OJ Os

ρ ι—> Os VO

O

lo o 4*­

O

"to 4^ to

O

K>

co to

O

OJ

to U l

ρ OJ ­o to

o

to OJ

O

­ J

co

o Ul OJ

co

o "Os o J >

o Os -o ­ J

o "-o U i ­ J

ρ 0 0 J > .

Os

o vo 4 * Os

* ^^

COMPRESSION

o o OJ h-»

o

o Ui OJ

O

o ­ J

-o

o ι—· O

to

o I—·

to co

o I—»

Ui Os

o I—»

CO

as

o to I—*

­O

O

to Ul O

O

IO 0 0 U i

o OJ

to OJ

O

OJ Os OJ

o 4S. O ­O

o J >

Ul OJ

O

Ui O OJ

o Ul Ul ~J

ρ Os I—"

Ul

O

Os •O VO

o -o 4 *

co

o 0 0

to 4*.

o VO

o 0 0

* *—'

# **~*

* —̂'

# ■»—'

COMPRESSION

o

o ( — » U i

o

o OJ CO

O

O ON O)

o

o VO 1—>

o ι—»

to I — *

o h-»

Ui U I

o I — '

VO

to

Ο

to LO J >

o to Ui

to

O

to co o

O

O) O H-*

o OJ

to S )

o OJ ■ & .

U i

o OJ

os vo

O

OJ

vo J >

o J >

lo ►—'

o

è VO

o J > ^J VO

O Ul > — * ►—»

o Ul J > Ul

o Ul co to

o Os to Η-'

o Os Os OJ

o ^1

o VO

o

-o U l 0 0

O

co Η-» I—»

o 0 0 Os

oo

o vo OJ 1—'

* # # # * *

a

/—\ í

v2>

o1

Γ! s Vi

Κι

* β» 53"

S1

>­ι

S? VJ

K>

* «> Ö«

S1

Q

I O j

η o Cl

a? í-¡t o et 3

2 ν> O.

^

>

• ¿*> \—•

2 00

a ­­·

V * •

V, o.

>

«<*

Η o*

VO

§ & *s:3 i l 5«

g EL EL CI

o o ρ i­h

ie &

§ 1 Ο, C L

Cl <~s 3 CP

Is era <%

8 1* "^ «o

•̂s (ra g o g:

3 i &.0 o ** l-t .

B. s co g

^ 5 ^ έ S S

OÍ

Eurocode 3 rules for cross-sections classification

Table V.10 Examples of shift of centroidal axis of effective cross-sections

1. in case of monosymmetrical class 4 cross-sections submitted to uniform compression

(■N compression) ·

—Γί^ ί I I

It

113

Ti-j

'N

Ξ e Ν

x.Sd

2. in case of class 4 cross-sections submitted to bending (My.Sd)

Τ eMf : i : = :=D My.Sd

eMf ι A 1

: = !)'

My.Sd

Notes -1-1

-2-2

- elements

centroidal axis of gross cross-section

centroidal axis of effective cross-section

I i: :ι ι

non-effective zone of the element, taking into account the occurence of local buckling.

55

Annex 3

Document 3198-1-1 (Ref. 15) (excerpts)

(9 pages)

"Local buckling rules for structural steel members"

57

TABLE 1 Specifications

Reference no.

Country Specification Year Abbreviation

1 Canada J. L.Dawe, G.L.Kulak: Local 1981 DK Buckling of W Shapes used as Columns, Beams and Beam-Columns, Structural Engineering Report No. 95

5 — ISO/TC167/SC1, Steel Structures, 1990 ISO Materials and Design (N 236E)

6 — EUROCODE 3, Common Unified 1990 EC Code of Practice for Steel Structures

7 Canada CAN/CSA-S16.1-M89, Steel 1989 CSA Structures for Buildings—Limit States Design

8 USA AISC Specification for Structural 1989 AISC Steel Buildings (Allowable Stress Design and Plastic Design)

9 USA AISC Load and Resistance Factor 1986 LRFD Design Specification for Structural Steel Buildings

10 Germany DIN 18800Teill, Stahlbauten, 1990 DIN Bemessung und Konstruktion (Steel Structures; Design and Construction)

11 Switzerland SIA 161, Steel Structures 1979 SIA 12 United Kingdom BS 5950 Par t i , Structural Use of 1985 BSI

Steelwork in Building 13 Australia AS 4100, S A A Steel Structures Code 1990 AS

This table lists the specifications and identifies them by an abbreviation and a reference number.

58

TABLE 2 Sections

Section Element Contained in specification Case no.

I-shape

Box section

Rectangular HSS

Circular HSS

Tee section

Channel section

Angle section

Range in compression

Web in axial compression

Web in bending

Web in combined axial compression and bending Range in compression

Range in compression

Section in bending and/or compression Range in compression

Stem in bending

Range in compression

Leg in bending or compression

DK, ISO, EC, CSA, AISC, 1 LRFD, DIN, SIA, BSI, AS DK, ISO, EC, CSA, AISC, 2 LRFD, DIN, SIA, BSI, AS DK, ISO, EC, CSA, AISC, 3 LRFD, DIN, SIA, BSI, AS DK, ISO, EC, CSA, AISC, 4 LRFD, DIN, SIA, BSI ISO, EC, CSA, AISC, 5 LRFD, DIN, SIA, BSI ISO, EC, CSA, AISC, 6 LRFD, SIA, BSI ISO, EC, CSA, AISC, 7 LRFD, DIN, SIA, BSI, AS ISO, EC, CSA, AISC, 8 LRFD, DIN, SIA, BSI ISO, EC, CSA, AISC, 9 LRFD, DIN, SIA, BSI ISO, EC, CSA, AISC, 10 LRFD, DIN, SIA, BSI ISO, EC, CSA, AISC, 11 LRFD, DIN, SIA, BSI, AS

This table identifies the examined structural shapes and notes the loading condition for the various components of the cross-section.

59

TABLE 3

Format of Local Buckling Rules

Formats:

Maximum bit

Plate Sections

Dimensions Used in specifications

Circular Sections

a fy

235

fy

1

vZT

ι

a ··

β'

fy

235

Τ 1

Ty

1 δ ' —

fy

£,/ ,[MPa]

/y[MPa]

A[MPa]

/„[ksi]

ISO, SIA

EC, DIN, BSI, AS

CSA

DK, AISC, LRFD

Conversion factors:

Plate sections (= all sections except circular sections)

a

a= 1

β = 29­8934

y = 458­258

δ = 174­522

ß

0­0334522

1

15­3297

5­83814

7

0­00218218

0­0652328

1

0­380838

δ

0­00572994

0­171287

2­62579

1

Circular sections

a'

a' = 1

β' = 893­617

y ' = 210000

δ' = 30457­9

ß'

0­00111905

1

235

34­0839

y'

0­00000476190

0­00425532

1

0­145038

δ'

0­0000328322

0­0293394

6­89476

1

Note: α, β, γ, δ are arbitrary format factors.

This table gives the formats used by the various specifications. In addition, conversion factors are listed which relate one specification and format to another.

60

Section: Element:

TABLE 7.1 Pictorial Representation of Local Buckling Rules: Case No. 1

I-Shape Flange in compression

Rcf. No.

(see Table 1)

1

5

6

7

8

9

10

11

12

13

Legend:

Specification

DK

ISO

EC

CSA

AISC

LRFD

DIN

SIA

BSI

AS

Class 1 D

0

0

0.1 0.2 I . I ,

1 1 ' 5

0.3 0.4 0.5 0.6 CC l . l . l . l

1 ' 1 10 15 β

Δ

0

0

50 100 1 . 1

' l ' I 20 40

Class 2 ,-·

150 200 250 γ

60 80 100 δ

Class 3 Δ

61

Section:

Element:

TABLE 7.2

Pictorial Representation of Local Buckling Rules: Case No. 2

I-Shape

Web in axial compression

Ref. No.

(see Table 1)

1

5

6

7

8

9

10

11

12

13

Legend:

Specification

DK

ISO

EC

CSA

AISC

LRFD

DIN

SIA

BSI

AS

Class' 1 □

0

0

OS

I 1 ' 1

10 20

1.0 1.5

1 . 1

ι , ι ι .

30 40

ι 50

Δ Β

ζ ο α

... .' Ι

6 0 β

Δ Β

Λ Β

Ù Β

0

0

200

100

Class 2 ·

400 600

ι . ι i ! ι

200

Class 3

800

ι ι

300

Δ

Τ

δ

62

Section: Element:

TABLE 7.3 Pictorial Representation of Local Buckling Rules: Case No. 3

I-Shape Web in bending

Ref. No.

(see Table 1)

1

5

6

7

8

9

10

11

12

13

Legend:

Specification

DK

ISO

EC

CSA

AISC

LRFD

DIN

SIA

BSI

AS

Class 1 D

0

0

1 2 3 4 ι . f , ι , ι

Ι ' ι »' 50 100

5 α

ι ι

150 β

η —

0

0

500 1000 1500 2000

Ι ' Ι ' Ι ' ι 200 400 600 800

Class 2 · Class 3 Δ

2500 Υ Ι

ι loco S

63

TABLE 7.4a

Pictorial Representation of Local Buckling Rules: Case No. 4, Class 1

Section: I-Shape

Element: Web in combined axial compression and bending

α

3.0

2.5

2.0

1.5

1.0

0.5

0.0

-

-

-

-

-

—

-

—

—

-

-

—

-

ß

90.0

80.0

70.0

60.0

50.0

40.0

30.0

20.0

10.0

0.0 J L ­I I 1 L J 1_

0.0 0.1 0.2

η =

0.3 0.4 0.5 0.6 0.7 0.8

axial capacity in presence of moment

product of yield strength limes area

1400 —ι

.— 500

1200-.

— 400

1000 -

800 — 300

600 ­

— 200

400

— 100

200 ­

0.9 1.0

64

TABLE 7.4b

Pictorial Representation of Local Buckling Rules: Case No. 4, Class 2

Section: I-Shape Element: Web in combined axial compression and bending

α

3.5 -

3.0 ­

ß

110.0

_ 100.0

- 90.0

- 80.0

25

1.5

1.0

0-5

0.0

70.0

ZO ­ ­ 60.0

50.0

40.0

30.0

20.0

- 10.0

0.0 0.0 0.1 0.2 03 0.4 0.5 0.6 0.7 0.8

axial capacity in presence of moment

product of yield strength times area

1600 - 600

1 4 0 0 -

. - 5 0 0

1200

­ 400

1 0 0 0 -

800 ­ 300

600 -

400 ­ ■

200

- 100

200 ­

0.9 1.0

65

TABLE 7.4c Pictorial Representation of Local Buckling Rules: Case No. 4, Class 3

.Section: I-Shape Element: Web in combined axial compression and bending

5.0 -

4.5 —

4.0

3.5

3.0 -

2.5 - "

2.0 -

1.5

1.0

0.5 - -

0.0 0.2

n =

0.3 0.4 03 0.6 0.7 0.8 axial capacity in presence of moment

product of yield strength times area

0.9

900

800

- 700

600

— 500

— 400

— 300

— 200

— 100

1.0

66

Annex 4

Document 3263-1-27 (ProfilARBED)

(7 pages)

Available rotation capacity of plastic hinges Ravailable -Tests results and models. (Excerpts of Chapter 5)"

67

4. Comparison of Kemp's simplified model and Feldmann's model with experiments (3-point bending tests)

4.1 Tables of experimental data (RWTH Aachen, CRM Liège, Roik-Kuhlmann, Lukey-Adams)

68

ON

NO

Ref. ■ ■ : · : : , : ■ : ■ : . ■ . . . ■ ; : : , : : : : ■ , , : ■ , : : '

■■ ... V : ^ : S : : i ï : i:ï i

illfifftii!

·. ■: Lukoy§f|

; :Lukeyj|: í

Lukoy

■ VLukoyW;·

Lukey

Lukoy

Lukey

Lukey

Lukoy

Lukoy

Lukey

Lukey

Lukoy

Lukoy

Lukey

RoTk & Kuhlm

Roîk & Kuhlm

RoiK & Kuhlm

RoTk & Kuhlm

Rolk&Kühirn:

Roîk & Kuhlm

Roik&:kuhlm

Rolk & Kuhlm

Roîk & Kuhlm

Rom&Kuhim:

Roîk & Kuhlm

Rolk & Kuhlm

Roîk & Kuhlm

:Roîk;&: Kuhlm

Roîk & Kuhlm

RoikAKùhim

Rotk& Kuhlm

■■.■y.:::-,·-".**.*---:-:·:-·

N°

l l l l ■■■;■'■:■.■>■■'■:■­■■■■.■■'■;■:■.:■:■:■.­

A2

82

B3 ■:ο:::νί>'.:-::-;:-.-,-:τ:::'::::::::

C2 C3 C5

::^:::::-:'^_·-·ο:-:>';^:-:-.·

■ . ■ . . ■ . ■ . ■ : ■ . - ■ . ■ . · ■ . . ■ . . ■ . . . '

D3

D5

D6

:.;?ΛΪ; F. ■::::::■:;::;

E3

Ë5

E6

Î;V:;Ï:Î:ÎSÎ;:-Ï'Ï?

::'Î

1 o

4

5

6 .■o::::­:v::::::̂ .:::vw:::::

ÍAíSífi/íSíSí­í:

8

9

10

13

1 4

15

16

17

Ï 9

20

21

Design. h

[mm]

250,4

250,4

200,2

200,2

250,4

250,4

250,4

201,7

201,7

201,7

201,4

201,7

201,4

201,4

201,4

294,0

294,0

277,0

274,0

275,0

296,0

296,0

295,0

295,0

260,0

259,0

258,0

169,0

220,0

298,0

299,0

299,0

b

[mm]

203,5

176,0

73,9

86,1

73,7

85,9

89,9

67,4

67,5

67,5

67,1

87,9

87,9

87,9

87,9

141,0

150,0

160,0

160,0

160,0

160,0

160,0

160,0

170,0

141,0

150,0

160,0

160,0

160,0

160,0

160,0

160,0

tw

[mm]

7,6

7,6

4,4

4,4

4,6

4,6

4,6

4.6

4,6

4,7

4,6

4,8

4,6

4,6

4,6

5,0

5,0

6,0

5,0

4,0

5,0

5,0

5,0

5,0

5,5

5,5

5,5

5,5

5,5

6,0

6,0

6,0

t f

[mm]

10,8

10,8

5,3

5,3

5,3

5.3

5,3

4,8

4,9

4,9

4,8

4,9

4.8

4,8

4,8

8,0

8,0

8,0

8,0

8,0

8,0

8,0

8,0

8.0

10,2

10,0

10,4

10,2

10,0

10,0

10,0

10,0

r

[mm]

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

d

[mm]

208,9

208,9

169,6

169,6

219,9

219,9

219,9

172,0

172,0

171,9

171,9

171,9

171,8

171,9

171,8

266,1

266,1

248,5

247,2

249,3

269,0

269,0

268,0

267,2

228,6

228,0

226,1

139,1

189,4

265,5

266,5

266,5

fy flange

[Mpa]

284,9

284,9

373,0

373,0

373,2

373,2

373,2

285,6

301,5

280,8

305,5

301,5

281,2

289,6

305,5

245,0

245,0

298,0

298.0

298,0

298,0

298,0

298,0

245,0

346,0

346,0

346,0

346,0

346,0

346,0

346,0

346,0

fy web

[Mpa]

309,3

309,3

396,5

396,5

352,0

352,0

352,0

369,6

373,4

382,3

380,3

373,4

366,1

376,5

380,3

226,0

226,0

270,0

262,0

270,0

262,0

262,0

262,0

226,0

737,0

737,0

737,0

737,0

737,0

363,0

363,0

363,0

b/(tf.e )

20,8

18,0

17,6

20,5

17,7

20,6

21,6

15,4

15,8

15,1

16,0

20,4

20,0

20,4

20,9

17,7

18,8

22,1

22,1

22,1

22,1

22,1

22,1

.21,3

16,5

17,9

18,3

18,7

19,0

19,0

19,0

19,0

α

0,5

0.5

0.5

0,5

0,5

0,5

0.5

0.5

0,5

0,5

0,5

0,5

0,5

0.5

0.5

0,5

0.5

0.5

0,5

0,5

0,5

0,5

0,5

0,5

0,5

0.5

0,5

0,5

0,5

0.5

0.5

0,5

α . d/

(tw. ε )

15,7

15,7

24,8

24,8

29,3

29,3

29,3

23,3

23,4

23,3

23,5

22,6

23,2

23,5

23,5

26,1

26,1

22,2

26,1

33,4

28,4

28,4

28,3

26,2

36,8

36,7

36,4

22,4

30,5

27,5

27,6

27,6

Flange

Class

2

1

1

2

1

2

2

1

1

1

1

2

2

2

2

1

1

3

3

3

3

3

3

2

Web

Class

2

2

2

Section

Class

2f

1

1

2f

1

2f

2f

1

1

1

1

2f

2f

2f

2f

1

1

3f

3f

3f

3f

3f

3f

2f

2w

2W

2W

Ref.

N°

Lukey

■ï­^ÏükeyapI

Lukey

Lukey

Lukey

Lukey

Lukey

^■PL^keyltlf

Lukey

■;;:v.;­;LÜkey3§?|;

■ ;i;;:;;Lukèy:::­í:|

^ | ; L Ï k è y | | |

^pL^keypl; ;

M lukey MM:

:;:;;||Lukèyy|i

Roîk & Kuhlm;

RoIk&Kùhirh:

RoTk & Kuh Im!

Roîk & Kuhlm;

Roîk& kuhlm;

Rólk&kühlm

ROTH ii:Kùhtm

Roîk & Kuhlm­

Roîk & Kuhlm

Roîk & Kühlm.

Roîk & kuhlm;

Roîk & Kuhlm:

Roîk & Kuhlm

RoTk "&: Kuhlm

RoTk&KuHÎmi;

Roîk &Kuhlm

Roîk & Kuhlm

Test

N·

:'>χ-:-ί:;:'::0:-;::::·:·:-:::::·:';:

A1 A2 Θ2 B3 C2 C3

. - . v . . . - . · . · . . ■ ■ .■--.■■;■:;.■■ : ■ : ■ ■ ■ : ■ ■ ■ ■ : : ■ : - . ■ . ■ : : ■ . · ■ . ■ - ■ ■ . · . ■

C5

D1

■;: :v.D3 M

D5

D6 ■::::"::;:ί:ο:-:::-.-:-:-:-:-::-::::·:'':

E1 E3

&WË5Ü?

E6

JB§iM 2

4

5

6

l l l f i f ­8

9 ;> :v :v : · : ' : ' : ­ : ■■:■■'. ν · . ­ . ' · ; ­ : ­ :

10

13

14

15

16

17 ; 19 ■ ■

20

WÏWÈ

L

[ m m ]

3479,8

2946,4

1036,3

1254,8

960,1

1168,4

1239,5

1752,6

2646,7

2270,8

883,9

2479,0

3718,6

3190,2

1239,5

3404.0

3704,0

2540,0

2636,0

2716,0

1796,0

2196,0

2598,0

2802,0

3000,0

3200,0

3508,0

2304,0

'2204,0

2000,0

2402,0

2804,0

A

[ m m * 2 ]

6228,1

5635,8

1709,5

1838,4

1963,5

2091,7

2134,4

1624,2

1628,1

1646,2

1618,1

1864,6

1816,6

1812,4

1821,2

3650,0

3790,0

4130,0

3850,0

3600,0

3960,0

3960,0

3960,0

4120,0

4190,0

4310,0

4630,0

4080,0

4300,0

4870,0

4870,0

4870,0

iy

[mmM]

71827311,0

63318047,1

10675688,5

11898980,2

18124108,0

20051209,7

20693576,9

9786987,5

9823136,0

9899431,8

9682127,0

11885863,5

11631423,2

11592901,9

11644005,2

55100000,0

58040000,0

55210000,0

52450000,0

51430000,0

62240000,0

62240000,0

61780000,0

65070000,0

51200000,0

52780000,0

57150000,0

22110000,0

38970000,0

77120000,0

77700000,0

77700000,0

lz

[ m m ]

49,3

41,8

14,5

17,5

13,4

16,3

17,3

12,4

12,4

12,4

12,3

17,3

17,4

17,3

17,3

32,6

35,1

37,1

38,4

39,7

37,9

37,8

37,9

40,6

34,3

36,8

39,9

42,1

40,6

38,2

38,1

38,1

Ll/(lz . ε )

38,8

38,9

45,1

45,1

45,0

45,1

45,1

77,9

120,4

99,9

41,1

81,3

117,2

102,2

40,8

53,3

53,9

38,6

38,7

38,5

26,7

32,7

38,6

35,2

53,0

52,7

53,3

33,2

32,9

31,8

38,2

44,6

λο

29,1

25,2

35,6

41,5

39,0

45,4

47,6

46,8

65,2

54,6

28,7

63,4

80,3

75,7

37,2

44,7

48,0

38,7

42,0

47,3

31,1

37,5

43,6

37,2

53,5

57,6

59,3

28,6

33,8

31,0

36,8

42,3

Mpi

[MN.mm]

0,1839

0,1637

0,0451

0,0498

0,0598

0,0656

0,0676

0,0971

0,1020

0,1257

0,1187

0.1155

0,1305

0,1305

0,1300

0,1132

0,1756

0,1801

0,1921

0,1078

0,1509

0,1939

0,1947

0,1947

ΦρΙ

[rad]

0,0218

0,0187

0,0113

0,0135

0,0081

0,0096

0,0103

0,0070

0,0076

0,0068

0,0070

0,0p72

0,0044

0,0054

0,0064

0,0057

0,0121

0,0128

0,0138

0,0132

0,0100

0,0059

0,0071

0,0082

<p(r

exp.

[rad]

0,2600

0,2720

0,1200

0,0920

0,1120

0,0780

0,0680

0,0640

0,0000

0,0470

0,1360

0,0980

0,0540

0,0330

0,0830

0,0564

0,0535

0,0858

0,0603

0,0331

0,0598

0,0624

0,0501

0,0315

0,0615

0,0486

0,0499

0,1386

0,0952

0,0708

0,0614

0,0594

ΦΪ? Feldmann

[rad]

0,1877 0,1916 0,0793 0,0782 0,0600 0,0591 0,0589 0,1120 0,1042 0,1173 0,1020 0,1057 0.1117 0,1071 0,1002

0,0686 0,0678 0,0737 0,0655 0,0585 0,0603 0,0603 0,0606 0,0660 0,0936 0,0920 0,0945 0,1608 0,1113 0,0741 0,0738 0,0738

R available exp.

M 11.9 14,5 10,6 6,8 13,8 8,1 6,6 3,9 0,0 2,2

15,9 4,3 1.0 1,9 7.2

8.1 7.0

12,6 8,6 4,6 13,6 11,6 7,8 5,5 5,1 3,8 3.6 10,5 9,5

12,0 8.6 7.2

predicted Kemp simp.

Η

8.9 11,0 6,6 5.2 5.7 4.6 4.2 4.4 2.6 3,5 9.1 2.8 1.9 2,1 6,1

4,7 4.2 5.8 5.1 4,3 8,0 6.1 4.8 6,1 3,6 3,2 3,1 9.1 7.1 8.1 6.2 5,1

predicted Feldmann

["]

9,0 10,7 7,4 6,1 7.4 6.1 5.8 8.0 4.7 6.6

13.5 5.2 3.9 4.2 9,7

7,5 6,8 7,3 6,2 5,4 9,1 7.4 6.3 8.9 4,6 4,3 4,1 7,3 6,6 7.5 6.5 5,5

Ref.

■yœïïWmfm ■■ ::::>;:^ν:νΐ·.·;.-: :; ::·-ν. :■:.■ ·.­.:■

RoTk & Kuhlm

:Roîk;& Kuhlm

Roik&Kuhlm

RWTH .■;■:■;■:■:■:■:■:■:χ ;-.■■:-:·: .■.■:■:■:■:■:·:■: :·;■■;■:-:■

RWTH

RWTH

RWTH

RWTH

RWTH

RWTH

RWTH

: : : : v v : : - · : : ; : : · , : - . . · ; - : : : ' : : · · . : : · ; . ·

■ . ■ - . - . : ; . ; . ; ■ . : . - . - : , : . ■ . ■ . . . ' . . . ■ - ; . ; . ; . ; . ; . ; . , ; ; .

:|&ERVyTrM nwTH RWTH

■ ■ : ■ : ■ . ■ : ■ ■ ■ . ■ : : ■ . ■ : ■ ■ · : ■ ■ ■ ■ . ■ : ■ : ■ · : ■ : ■ : ■ : - . ■ : : ■ . ■ : ■ : ■ : ; ■ :

RWTH

RWTH

RWTH . . . - . -y.-.;. ..■.■-■.:.·.■.■.;.■.;.■.;.:-:.....;.;.:.■.

;"■';':■:': ■:■■■:■. ■■:■:■:- '■'■'■ ■: y <■:■:■:■:'■:■:■:'■

RWTH

RWTH

RWTH

;ÍI::F#H1|1;

Í M H W I H ^

m.mm?m RWTH

RWTH

RWTH

Test

l § r © l :-·:ί:::':ίί-¥: ■::.::■:■:,

;.:.:::.ϊ'. Y':::.:V::/' :

22 , ■ ■ : ■ . : : ■ ; ■ ■ ■ . , : · . - ; , ■ : · , : · : ■ ■ ■ : ■ : ■ ; ■ ■

Mm 1 24

ΐϊ;.ΐ:ϊ-:ί:.:::,ί::;ΐΐ-: « « « ï d15a3m

d16t53m

d19a4m

d20b4m

d01a4m

d09a3m

d10b3m

EA22312

IÍ$233j.

fÍA22Íl EA22412

EA2243

ΕΑ224Λ4ί: iMÆmMMy

EB28312

Ifltïlli HIS! iEB28412

EB2843

íEBlifíí;

EÂ28312

pApfi :;ΕΑ2'834;;

EÀ284Ì2

|ëA2843:

5EÀ2844';

Design.

HE 220 Β

HE 220 Β

HE 220 Β

HE 220 Β

HE 220 Β

HE 220 Β

HE 220 Β

HE 220 Β

HE 220 A

HE 220 A

HE 220 A

HE 220 A

HE 220 A

HE 220 A

HE 280 Β

HE 280 Β

HE 280 Β

HE 280 Β

HE 280 Β

HE 280 Β

HE 280 A

HE 280 A

HE 280 A

HE 280 A

HE 280 A

HE 280 A

h

[ m m ]

299.0

299,0

299,0

220,0

219,5

220,4

220,0

219,1

217.3

218,7

217,4

208,3

209,0

211,0

215,5

210,3

214,0

278,0

276,8

282,0

283,0

281,2

284,5

255,2

266,0

269,0

276,1

275,6

275,0

b

[ m m ]

170,0

183,0

190,0

220,5

220,6

219,2

219,4

218,8

218,6

219,0

218,4

220,5

220,0

221,0

225,5

225,5

222,0

279,0

279,3

279,0

281,3

283,3

284,0

280,0

280,0

280,0

280,5

281,0

281,0

t w

[ m m ]

6,0

6,0

6,0

9,8

10,0

9.4

9.7

9,8

9.4

9,6

9,4

7,5

7,5

7,4

7,5

7,5

7,5

10,5

10,9

10,8

11,5

11.4

11,5

7,8

8.0

7,5

8.8

9.0

9.3

t f

[ m m ]

10,0

10,3

10,2

15,5

15,7

16,0

15,9

16,3

16,2

16,3

16,1

10,5

10,5

11,0

11,0

11,0

10,7

17,6

17,8

17,7

17,4

17.4

17,4

12,7

12,6

12,8

12,6

12,6

12,7

r

[ m m ]

18

18

18

18

18

18

18

18

18

18

18

18

18

18

24

24

24

24

24

24

24

24

24

24

24

24

d

[ m m ]

266,5

265,5

266,5

153,0

152,1

152,4

152,2

150,5

148,9

150,1

149,2

151,3

152,0

153,0

157,5

152,3

156,6

194,8

193,2

198,6

200,2

198,4

201,7

181,8

192,8

195,4

202,9

202,4

201,6

fy flange

[Mpa]

346,0

346,0

346,0

274,5

274,5

525,2

525,2

486,2

486,2

278,5

278,5

282,5

282,5

282,5

420,5

420,5

420,5

248,5

248,5

248,5

489,0

489,0

489,0

276,5

276,5

276,5

504,0

504,0

504,0

fy web

[Mpal

363,0

363.0

363.0

348,5

348,5

541,3

541,3

531,7

531,7

286,1

286,1

308,0

308,0

308,0

437,5

437,5

437,5

252,5

252,5

252,5

539,0

539,0

539,0

311,5

311,5

311,5

535,0

535,0

535,0

b/(tf. ε )

20,2

21,1

22,2

15,4

15,2

20,5

20,6

19,3

19,4

14,6

14,8

23,0

23,0

22,0

27,4

27,4

27,8

16,3

16,1

16,2

23,3

23,5

23,5

23,9

24,1

23,7

32,6

32,7

32,4

α

0,5

0,5

0,5

0,5

0.5

0,5

0,5

0,5

0,5

0,5

0,5

0,5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0,5

0,5

0,5

0,5

0,5

0,5

0.5

0,5

0.5

0,5

α . d/

( tw.8)

27,6

27,5

27,6

9.5

9,3

12,3

11,9

11,5

11,9

8,6

8,8

11,5

11,6

11,8

14,3

13,9

14,2

9,6

9.2

9,5

13.2

13,2

13,3

13,4

13,9

15,0

17,4

17,0

16,4

Flange

Class

2

2

3

1

1

2

2

1

1

1

1

3

3

3

3

3

3

1

1

1

3

3

3

3

3

3

4

4

4

Web

Class

Section

Class

2f

2f

3f

1

1

2f

2f

1

1

1

1

3f

3f

3f

3f

3f

3f

1

1

1

3f

3f

3f

3f

3f

3f

4f

4f

4f

o o a ,­* 5" c η α. O 3 3 η

•α w οο η

Ref.

-;i§Ä!|: :' : ■'*-*■ .-.:ΐ":^:,'·:·.·:':-:':'ίVii

Roîk & Kuhlm

Roìk&Kuhlm

Roîk & Kuhlm

RWTH

RWTH

RWTH

RWTH

RWTH

i :RWTH 1 1

V W F W T H Ì Ì !

; RWTH ;

RWTH.

FTWTH

RWTH i

: RWTH ;

;{RWTH ;:■

RWTH

RWTH

RWTH

RWTH

RWTH

RWTH

V RWTH V

RWTH

RWTH '

i|^.RWTH'§f|

■■RWTH

y RWTH :

RWTH

:';

:;-'

:ν.".:

:;:0"'- "

Test

N° '

22

1ΐ23Ί | | 24 :

d15a3m d16b3m

d19a4m d20b4m dQ1a4m d02b4m d09a3m d10b3m

EA22312 EA2233 EA2234

EA22412 EA2243 EA2244

EB28312 EB2833 EB2834

EB28412

EB2843 EB2844

EA28312

EA2833 EA2834

EÄ28412

EA2843

EA2844

L

[mm]

2406,0

2500,0

2700,0

1200,0

1200,0

1200,0

1200,0

3500,0

3500,0

3500,0

3500,0

1200,0

3000,0

4000,0

1200,0

3000,0

4000,0

1200,0

3000,0

4000,0

1200,0

3000,0

4000,0

1200,0

3000,0

4000,0

1200,0

3000,0

4000,0

A

[mmA

2 ]

5070,0

5440,0

5550,0

8965,8

9086,0

9063,5

9080,6

9238,7

9098,8

9204,1

9051,5

6313,4

6308,1

6538,7

6690,4

6651,4

6473,4

12864,6

13066,6

13034,3

13138,0

13162,2

13249,2

9398,9

9476,8

9487,9

9771.0

9829,2

9953.1

iy

[mmA

4 ]

81880000,0

89370000,0

91670000,0

79392787,4

79868183,0

80913958,9

80451046,0

81009884,5

78889079,5

80636429,0

78551976,4

51675380,0

51956689,3

55109126,6

58837226,0

55745432,0

55967617,2

185917989,8

186322650,9

193112959,9

194512289,5

192846398,8

198501630,5

118473640,1

129214326,4

133426416,4

141503655,1

141419405,3

141966284,6

Iz

[mm]

41,0

44,9

46,7

55,6

55,6

55,7

55,6

55,5

55,7

55,7

55,6

54,6

54,4

55,0

56,1

56,3

54,9

70,4

70,4

70,2

70,2

70,8

70,9

70,4

69,8

70,3

68,9

68,9

68,7

Ll/(lz ε )

35,6

33,8

35,1

11,7

11,7

16,1

16,1

45,3

45,2

34,2

34,3

12,1

30,2

39,8

14,3

35,7

48,7

8,8

21,9

29,3

12,3

30,5

40,7

9.3

23,3

30,9

12,7

31,9

42,6

λβ

36,7

36,5

39,6

5,4

5,2

11,2

11.1

26,7

27,2

13,6

13,9

9.2

22,1

27,5

14,4

33,6

45,5

4,3

10,2

13,7

10,2

24,3

31,7

7,9

19,8

26,4

16,9

39,9

50,5

Mpi

[MN.mm]

0,2044

0,2218

0,2270

0,2276

0,2299

0,4296

0,4286

0,4050

0,3969

0,2294

0,2241

0,1520

0,1523

0,1601

0,2476

0,2405

0,2370

0,3627

0,3661

0,3720

0,7450

0,7428

0,7560

0,2742

0,2872

0,2922

0,5511

0,5526

0,5574

ΦρΙ

[rad]

0,0071

0,0073

0,0079

0,0082

0,0082

0,0152

0,0152

0,0417

0,0419

0,0237

0,0238

0,0084

0,0209

0,0277

0,0120

0,0308

0,0403

0,0056

0,0140

0,0183

0,0109

0,0275

0,0363

0,0066

0,0159

0,0209

0,0111

0,0279

0,0374

<p(r

exp.

[rad]

0,0705

0,0488

0,0408

0,0472

0,0810

0,1260

0,0858

0,1344

0,1638

0,2268

0,2376

0,2616

0,2578

0,2656

0,1696

0,0885

0,0626

0,1702

0,4780

0,3835

0,1339

0,2632

0,3053

0,2563

0,3071

0.1385

0,1493

0,1846

0,1609

9(r

Feldmann

[rad]

0,0731

0,0737

0,0728

0,4124

0,4304

0,1549

0,1624

0,1898

0,1805

0,3931

0,3826

0,2440

0,2427

0,2389

0,1371

0,1427

0,1371

0,3354

0,3589

0,3419

0,1504

0,1501

0,1488

0,2010

0,1938

0,1791

0,0901

0,0927

0,0972

R available

exp.

Η

9,9

6,7

5,2

13,1

22.5

20,0

13,2

6,4

7,8

18,9

19,8

32,9

12,0

9.3

15.4

2.8

1,5

45,4

34,1

20,5

15,8

9,5

8,3

50,4

19,0

6,4

16.5

6,4

4.1

predicted

Kemp simp.

[­1

6,3

6,3

5,6

111,9

116,3

37,4

37,7

10,1

9,8

27,8

27,0

50,1

13,5

9.7

25,5

7,2

4,5

153,8

42.7

27.4

43,0

11.7

7,8

62,3

15,8

10,3

20,1

5,5

3,9

predicted

Feldmann

[ · ]

6,4

6.3

5.7

50,4

52,3

10,2

10,7

4,6

4,3

16,6

16,1

29,0

11,6

8.6

11.4

4.6

3,4

60,2

25,6

18,6

13,7

5,5

4.1

30.4

12.2

8.6

8,1

3,3

2,6

^ 1

CFW

cai

ill'' kk kk

wmmmt Test

Ν"

IflllfS :>:v::;W

::i:

::::Ö'

::::ï?:

19

ii -k-k

21

SSíííí:

IP« kMZßM..

29

30

kikifXOMSk

Design.

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

HE 200 Β

[mm]

199,7

200,4

198,0

197,6

199,7

204,3

198,4

200,5

198,2

198,5

203,1

195,8

204,5

198,6

197,5

197,5

198,9

204,5

197,6

198,6

197.8

198,2

198,2

198,8

199,1

197,9

203,6

201,8

197,4

[mm]

201,5

200,4

200,2

200,5

201,5

199,9

200,4

200,6

200,4

200,3

199,8

200,1

199,9

200,2

200,3

200,3

200,3

200,0

200.3

200.4

200.2

200.4

200,2

201,2

201,1

200,8

200,2

199,9

200,3

tw

[mm]

9,5

9,6

9.5

8.8

9.5

9.4

9,5

9.5

9,0

9,1

9.6

10.0

9,4

9,2

8,8

8,8

9,2

9,4

8,9

9,2

9.1

9.1

9,0

9,3

9.2

9,0

9,4

9.1

8.7

t f

[mm]

15,1

14,6

14.7

14,6

15,1

14,9

14,7

14,6

14,5

14,7

14,5

15,3

14,8

14,8

14,6

14,6

14,8

14,8

14,5

14,8

14,7

14,8

14,8

15,0

15,1

14,5

15,1

14,3

14,5

[mm]

18

18

18

18

18

18

18

18

18

18

18

18

18

18

18

18

18

18

18

18

18

18

18

18

18

18

18

18

18

[mm]

133,5

135,2

132,6

132,4

133,5

138,5

133,0

135,3

133,2

133,1

138,1

129,2

138,9

133,0

132,3

132,3

133,3

138,9

132,6

133,0

132,4

132,6

132,6

132,8

132,9

132,9

137,4

137,2

132,4

fy flange

445,0

261.0

375,0

303,0

445,0

409.0

375,0

261,0

303,0

375,0

261,0

375,0

409,0

375.0

303,0

303,0

375,0

409,0

303,0

375,0

375,0

375,0

375,0

445,0

445,0

303,0

409,0

261,0

303,0

fy web

[Mpa]

462,0

291,0

421,0

342,0

462,0

426,0

421,0

291,0

342,0

421,0

291.0

421,0

426,0

421,0

342,0

342,0

421,0

426,0

342,0

421,0

421,0

421,0

421,0

462,0

462,0

342,0

426,0

291,0

342,0

b/(tf.e )

18,4

14.5

17.2

15,6

18,4

17,7

17,2

14,5

15.7

17.2

14,5

16,5

17,8

17,1

15,6

15,6

17,1

17.8

15,7

17,1

17,2

17,1

17,1

18,5

18,3

15,7

17,5

14,7

15,7

0,5

0,5

0,5

0,5

0,5

0,5

0.5

0.5

0,5

0.5

0.5

0.5

0.5

0.5

0,5

0,5

0,5

0,5

0,5

0,5

0.5

0,5

0,5

0,5

0,5

0,5

0,5

0,5

0,5

a . d/

(tw. e )

9,9

7.8

9.3

9.1

9,9

9,9

9.4

7.9

8,9

9,8

8,0

8,6

9,9

9.7

9,1

9,1

9.7

9.9

9,0

9,7

9,7

9,8

9,9

10,0

10,1

8,9

9.8

8.4

9,2

Flange

Class

Web

Class

Section

Class

o o 3 w­*

3' c

SL o 3 3

» Χ} ta

OO (Ό

Rel.

N° : : ;

::;i:yXy:y:>.;­;:;:v;

:.::::y

;v:

::;:;.;x':.

CFM

CFM

CFM

CFM

CFM

CFM

CFM

CFM

CFM

CFM

CFM

CFM

CRV1

CFM

CFM

CFM

Í ' ;SIÍC¿W|Í| |

CR/

CFM

CFM

CFM

CFM

CFM

CFM ■ · · ■ ­ ■ ; ­ ■ ■ : ■ · ' ­ ; ■ : ­ · : : ■ . · : ■ ; ■ : ­ . ­ : ■ : ­ : ■ : ■ : ■ : : ■ : ■ : ■ : ■ : ■ : ■ :

CFM

CFM

:IiCFMi;i:|i;

CFM

CFM '.'.:!^< ί̂̂ ::;-:·:-:::';':;.:·:ν:·:ν:':ν:-:":'

Test :·:ϊ:ϊ Sï: i íí i i i . : ï:'­

'χ-:::;:':ΐ:-:-::::-:·::;-:::'::-:;:':::

■;-:^:::Xy::;::;:;;;i;:::;:;'.::^:

::ί>:>-:ΐ:ί:ΐ:<::-:ί!:::!^:;:

2

3 4 5

IllUßlllt ::­'"'\

::;:::;::;: '

: '

::':ν.

:::.;::::::

7

8 q

10

11

12

13

15

17

:'\.;:;i8:,V.

19

20

21

•:.W- O 0 kk:

23

24

25

26

27 * 28

29

30

list Í | : ;3j ' | |

L

[ m m ]

3000,0

3000,0

3000,0

2000,0

2000,0

2000,0

2000,0

2000,0

2800,0

2390,0

3000,0

3000,0

3000,0

3000,0

3000,0

3000,0

3000,0

3000,0

3000,0

3000,0

3000,0

3800,0

3800,0

3800,0

3800,0

3800,0

3800,0

3800,0

2000,0

A

[ m mA

2 ]

7973,7

7773,3

7765,7

7614,6

7965,2

7875,4

7775,4

7763,0

7612,5

7705,8

7743,7

8053,2

7839,2

7758,8

7607,9

7607,9

7756,1

7842,2

7587,4

7764,8

7696,4

7735,8

7721,4

7884,0

7905,2

7621,4

7954,1

7571,4

7551,9

iy

[ m mA

4 ]

56803440,0

55586421,0

54355584,0

53562004,0

56979367,0

58709275,0

54804294,0

55742352,0

53880942,0

54594652,0

56901292,0

54875046,0

58809476,0

54926922,0

53376779,0

53391822,0

55164066,0

58755458,0

53407445,0

54796179,0

54042000,0

54568645,0

54423749,0

55971068,0

56289825,0

53832413,0

58935499,0

55455652,0

53175832,0

lz

[ m m ]

50,9

50,2

50,4

50,8

50,9

50,2

50,4

50,3

50,6

50,6

49,9

50,4

50,2

50,6

50,7

50,7

50,6

50,2

50,6

50,6

50,6

50,7

50,7

50,9

50,9

50,7

50,4

50,2

50,8

Ll/(iz . ε )

40,6

31,5

37,6

22,4

27,0

26,3

25,1

20,9

31,4

29,8

31,7

37,6

39,4

37,5

33,6

33,6

37,4

39,4

33,6

37,4

37,5

47,3

47,4

51,4

51,3

42,5

49,7

39,9

22,4

λβ

21,3

11,8

18,1

10,0

14,7

13,8

12,5

8,1

13,7

15,0

12,1

16,7

20,2

18,2

14,5

14,5

18,3

20,2

14,6

18,2

18,4

22,6

22,7

26,6

26,6

18,0

24,3

15,5

10,1

Mpi

[MN.mm]

0,2867

0,1655

0,2356

0,1873

0,2876

0,2666

0,2371

0,1658

0,1881

0,2356

0,1672

0,2356

0,2661

0,2370

0,1868

0,1867

0,2373

0,2659

0,1869

0,2364

0,2340

0,2357

0,2350

0,2834

0,2846

0,1881

0,2680

0,1634

0,1860

ΨρΙ

[ rad ]

0,0318

0,0212

0,0308

0,0166

0,0240

0,0216

0,0206

0,0142

0,0233

0,0307

0,0214

0,0313

0,0329

0,0314

0,0255

0,0255

0.0313

0,0344

0,0255

0,0314

0,0309

0,0398

0,0390

0,0467

0,0466

0,0322

0,0419

0,0272

0,0170

Φ!? exp.

[ rad ]

0,3582

0,4968

0,3752

0,4734

0,4260

0,3814

0,4794

0,5718

0,3537

0,3513

0,4606

0,4207

0,3071

0,3636

0,2935

0,3265

0,2987

0,2856

0,3655

0,4056

0,3881

0,3422

0,3410

0,2853

0,3144

0,2738

0,2391

0,4148

0,4750

ΦΑ? Feldmann

[rad]

0,2387

0,4964

0,3097

0,3641

0,2366

0,2525

0,3085

0,4866

0,3747

0.2868

0,4823

0,3516

0,2514

0,2925

0,3645

0,3645

0,2890

0,2514

0,3700

0,2924

0,2887

0,2858

0,2832

0,2319

0,2277

0,3757

0,2555

0,4429

0,3572

R available

exp.

[­1

11,3

23,4

12,2

28,4

17,7

17,6

23,3

40,4

15,2

11,4

21,5

13,5

9.3

11.6

11,5

12,8

9.5

8.3

14.4

12.9

12,5

8,6

8,7

6,1

6,8

8,5

5,7

15,3

28,0

predicted

Kemp simp.

[­1

14,2

34,3

18,1

44,3

24,7

27,2

31,7

60,2

27,6

24.1

33,2

20,4

15,3

17,9

25,1

25.1

17,8

15,3

25,0

17,9

17,7

13,0

12,9

10,1

10,2

18,2

11,6

22,9

43,5

predicted

Feldmann

[­1

6,6

23,3

10,0

21,9

9,8

11,7

15,0

34,4

16,1

11.7

23.0

11,5

7.8

9.5

14.6

14,6

9,4

7,8

14,8

9.5

9.3

7.3

7,3

5,1

5,0

11,9

6,2

16,6

21,4

Annex 5

Document 3263-1-27 (ProfilARBED)

, (14 pages)

Available rotation capacity of plastic hinges Ravailable -Tests results and models. (Excerpts of Chapter 5)"

75

5. Classification of cross sections

76

5.1. Experimental data compared to EC3 rules.

Experimental data: 3-point bending tests, My $¿ (Lukey/Adams, Roik/Kuhlmann, RWTH CRM);

EC3 classification for cross-section submitted to My gj : flange in compression, web in bending;

Selection of experimental data:

a) the sensitivity of cross-section to local buckling is estimated without LTB effects =>

experimental results presented in chapter 5.1 exclude cases with Lj/(i2-e) > 60; b) The cases with Mpi influenced by high shear forces (Vsd > 0.5 Vpj R d ) are also not

considered in this chapter, because the values of available rotation capacities depend on the model chosen to evaluate M p l V R d (< MpI) and because those values of available rotation capacity (RaV) are always very hign;

c) Roik and Kuhlmann tested welded sections, so they are not taken into account, because we only consider limits of EC3 classification for hot-rolled sections.

Graphs:

Φ page 109 (1 graph)

Ravailable = f(ctd/twE) where cc=0.5 (pure bending), ε = f(fy.Web) © page 110(1 graph)

Ravailable = f(b/tfs) where ε = fCfy.flange) Q) pages 111-113 (3 graphs)

b/tfs = f(ad/tws) where Rav is measured between [0,2]

(2,6] (μ class 2 ?)

> 6 (= class 1 ?) © pages 114-116 (3 graphs)

b/tf=f(fy) where fy = f(fy.flange) with RaV between [0,2]

(2,6] (= class 2?)

> 6 (= class 1 ?) © pages 117-119 (3 graphs)

ad/tws = f(fy) where fy = f(fy.web) with RaV between [0,2]

(2,6] (= class 2?)

> 6 (= class 1 ?)

N.B. in the last 6 graphs © and ©, the plain curves represent the relation ε = Í-—

77

Conclusions:

a) No test results were available with slender webs for hot-rolled sections: all webs are in class 1 (see αόΥτ̂ ε values in graphs ©,© and ©).

b) In graph ©, it can be seen that tests furnished results conflicting with assumptions of EC3 rules: - high rotation capacities (> 6) for EC3 class 2 and class 3 cross-sections, - rotation capacities > 0 for EC3 class 3 and class 4 cross-sections.

c) In general, for elastic global analysis of a structure, the resistance of cross-sections and members are directly related to the class of the concerned cross-sections: plastic resistance is allowed for class 1 and 2 cross-sections, elastic resistance for class 3 cross-sections and effective resistance for class 4 cross-sections. If tests results on 3-point bending beams give available rotation capacity greater than zero, then we can say that the concerned cross-sections are in class 1 or 2, because the plastic bending moment resistance have been reached and passed over. In view of the 3 graphs N0©, the influence of ε factor which is enclosed in rules of EC3 classification has been highlighted.

d) The main preliminary conclusion of those 3 graphs © is that ε could be forgotten (ε=1) for flange classification fb/tf) in case of hot-rolled cross-sections which are in class 1 or 2 for fy=235 MPa. More development (tests or simulations) are necessary to help draw final conclusion.

78

- J VO

R av. experiments

45,0

40,0

35,0

30,0 - -

25,0 - -

20,0

15,0

10,0

5,0

0,0

Θ

■ ■

3ΐ -■t :"■

ί ■ ■ ■

10 15 20 25

class 1

30

class 2

RExp

35 40

alpha.d/(tw.epsilon)

00

o

R av. experimentó

45

40 —

35 - -

30

25

20 - -

15

10 - -

Θ

10

class 1

■

■ ■

: ί ■ ■

■ ■ ■

cl. 2 class 3

■

H ■

class 4

■ RExp

15 20 25 30 35

b/(tf.epsilon)

b/(tf.eps¡lon)

35 ­ r R av. [0; 2]

10 15 20 25 30 35

■ RWTH

Θ

40

alpha.d/(tw.epsilon)

oo to

b/(tf.epsilon)

35 R av. [2; 6]

10 15 20 25 30 35 40

alpha.d/(tw.epsilon)

0 0

b/(tf.epsilon)

40 - r

35

30

25

20

15

10

class 3

class 2

5 - -

class 1

Θ

D - θ -

oD ü

CP

J ►«Ρ

R av. [6; infini]

10 15 20 25

■

D

♦

Lukey

RWTH

CPM

30 35

alpha.d/(tw.epsilon)

00 ■p.

b/tf

35,0 ­ r

30,0

25,0 ­ ­

20,0

15,0

10,0 ­ ­

5,0 ­ ­

0,0

235 285 335

R av. [0; 2]

RWTH EA2244

R= 1.5 Mu/Mp|= 1.04

385

class 4

class 3

class 2

class 1

435

fy flange

©

oo

b/tf

35 - r

30

25 - -

20

15 - -

10 - -

235

®

285 335

R av. [2; 6]

RWTH EA2243 R=2.8MU/Mp|=1.09

RWTH EA2844 R=4.1MU/Mp]=1.12

385 435 485

class 4

class 3

class 2

class 1

535

fy flange

00 ON

b/tf

35 - r

30

25 - -

20

15 - -

10 - -

5 - -

R av. [6; infini]

RWTH EA2833 R= 19.0 Mu/Mp|= 1.13

RWTH EA2843 R=6.4Mu/MpI=1.16

RWTH EB2843 R=9.5MU/Mp|=1.27 RWTH EB2844

R= 8.3 Mu/Mp|= 1.25

class 4

class 3

class 2

class 1

235 285 335 385 435 485 535

fy flange

®

alpha.d/tw

23

40,0

35,0

30,0

25,0 - -

20,0 - -

15,0

10.0 - -

5,0 - -

0,0

235,0 285,0 335,0 385,0

R av. [0; 2]

435,0 485,0

class 2

class 1

535,0

fy web

Θ

alpha.d/tw

oo oo

40

35

30 - -

25 - -

20

15 - -

10

5 - -

R av. [2; 6]

235 285 335 385 435 485 535 585 635 685

class 2

class 1

735

fy web

alpha.d/tw

oo VO

40

35

30

25

20 - -

15 - -

10 - -

235

R av. [6; infini]

285 335 385 435 485 535 585 635 685

class 2

class 1

735 fy web

Θ

Annex 6

Document 3198-1-18 (ProfilARBED)

(24 pages)

"Stability of composite bridge girders near internal support", by Schaumann P. and Schleich J.B.

91

STABILITY OF COMPOSITE BRIDGE GIRDERS NEAR INTERNAL SUPPORT

Dr.-Ing. Peter Schaumann Ing.-Büro HRA, Kohlenstraße 70, D-4630 Bochum 1, Germany

Dipl.-lng. J.B. Schleich ARBED Research, P.B. 141, L-4002 Esch-sur-Alzette, Luxembourg

Summary

The design of composite bridges in the hogging moment reaion ¡ς n n u 0 m ^ ^ and global instabilities. Stability failure occurs in form C ^ S ^ M Z ^ V ^ V ' '0Ca l

form of lateral-torsional buckling of the lower flange Comoosite hr iS ' f WGb a n d i n

beams have a very compact steel section. In this case cune'nt Sestamies S f***" ^ results. Two full scale tests on composite bridge q i rdersht¡ S I

s ^ to uneconomic

Germany, in 1990. In the tests the hogging momem reaion of Γ Τ h B

°C h u m

· simulated. The experimental investigatef d e m o n s Ä a d Ä ^ ,

1" ^

specimen, although geometric and material properties teve been chn^n T™ °L t h e t6St

type of composite bridge. The full p.astic K ^ Tornent t S S ^ S ^ · ^ ^ calculations could be verified by the tests. Considering the t e s t S K Î n d ï ? Γ * ^ recently developed computer program some additional proposals for tïe d S ™ of * °

f­ *

bridges in the hogging moment region are presented p r

°p O S a l s f o r t n e d e s i

9n of composite

Introduction

The stressing at internal supports of continuous beams is characterized by a combina tion of high shear forces and high hogging moments. comoma­

Thus, the design of composite bridges in this region is governed bv effects nf iora i a „ , global instabilities. Local instability occurs in form of plate buck"no ¡ η ΐ Ρ 1 1 . κ whereas global instability is a result of lateral-torsiona'. bucWing of he l o w Î f l a n S ' Both effects are interdependent. Normally these stability p9

roÎlemS TeaJeTe provision of stiffeners and bracings. H require tne

Recent improvements in rolling technology result in the production of verv laroe hot rolled steel sections with high strengths applicable for briäge structures Roa br idou based on rolled shapes connected to a concrete roadway'pla*"have a ver^ compac steel section. In this case current design codes lead to uneconomic results with œqard to detailing and overall design. " n r e 9 a r a

In 1989 a research program was initiated by ARBED, one of Europeans oreatP^t min™ mills, to optimize design and construction of composite bridges ustaa hS rSSri J Ï Ï Ï girders Within the scope of this research work two'ful. scale tests ™ Ä L b £ £ g,rders have been earned out in Bochum, Germany, in 1990. In the tes*the hooaino moment reg.on of continuous beams has been simulated (see Fig. 1) n o99'ng

92

Specimen 1 Specimen 2

25CO

mmsswM^^M

I ífc_

9500 9500

Fig. 1 Elevation and cross-sections of the full scale tests

Composite bridges using hot-rolled steel girders

The research program, which is still in progress, is titled 'Composite bridges based on hot-rolled steel girders'. The main purposes of research are:

Buckling behaviour of rolled shapes in composite sections in negative moment regions

• Application of high strength steel (FeE 460) in bridge building

• Structural behaviour of the deck when using prefabricated concrete elements as composite formwork

• Demonstration of construction principles

The application of special hot-rolled steel beams as one part for composite girders led to the developments of new structural concepts for bridges. This modular design system using completely prefabricated and finished steel beams and precast concrete elements acting compositely with the in situ concrete deck slab facilitates the development of more economical erection procedures for bridges.

93

Fig. 2 shows a typical cross-section of multi-beam composite bridge deck with universal beams. The rolled l-shapes with a depth up to 1100 mm, a width up to 450 mm and a flange-thickness up to 50 mm are suitable for composite bridges of spans up to about 45 m. With regard to generally approved technical standards yield strengths of 460 N/mm2 can be provided.

13.60

Fig. 2 Typical cross-section of a composite bridge with hot-rolled steel beams

The workshop finishing of the beams like

• cambering,

• welding of vertical stiffeners at the supports,

• welding of the stud shear connectors,

• preparation of the field connections (bolted type) and

• corrosion protection

is provided by ARBED. The prefabricated beams are delivered directly to the construction site. AH connections are of bolted type to facilitate the work on site.

In order to improve the construction procedures of this type of bridge a contracting company WIEMER&TRACHTE was involved in the research program. One reason to use prefabricated concrete elements in bridge construction is linked to the following advantages:

• They represent an efficient bracing against wind loads and provide lateral stability for the rolled shapes during erection.

• By their high quality due to the advanced fabrication technologies they quickly provide a safe working area on top of the steel structure.

• The erection time on the building site can be considerably reduced.

94

On September 12th

, 1990, a symposium was organized at the Ruhr­University Bochum

with the support of TradeARBED Germany.The first part of this symposium contained a

number of lectuëVs from designers, scientists and contractors to present related,

existing bridge structures, the purposes of research and other aspects like overall cost

saving and maintenance of composite bridges. In the presence of more than

200 guests from industry, science and administrations the second of two full scale

ultimate load tests was carried out.

Test program

The total test program consists of different tests on five test specimen. This paper

deals with two full scale tests on composite bridge girders in which the hogging

moment region of continuous beams has been simulated [1]. The test specimen had a

length of 20 m; the first one with a single beam, the second one, which had a total

weight of nearly 1001, comprising three beams in the cross section.

The rolled shapes were chosen to be HE 900 A, FeE460. The thickness of the slab

was 28 cm. A typical span of a multi­span beam with this cross­section is about 20 m.

In the tests hogging regions of length half of this span were used to create moment

gradients more adverse for the stability problems than it would occur in practice.

The cross­sections and the material strengths are given in Fig. 3.

2S »1417 [ M l H/««'l

Specimen 1

! ; ) ' / .' > ' X -\ / / / / / / \ -\ / , / k \ / . ' I -

y»oo>.— t l , . " ê l

^

< D

\ l l » Ι Ο Ι IP, .S43I

Π .» .02 1(1, ·

I Is » , .SOI

3 0 1 . 10 I f , . L I O

I, /

taei

7*12 Iß,.· H i l

Specimen 2 s s

I

C7 012 iPsxC98 I

U ■ > ·Ι-τ

61*16 (β «637N/mm2l

A ; ι . ■ \ ■ V . ; i l l . ; ) I—? · > Λ 1

Λ/-'/.- S ΛΧ\.' / ­· /.- Ζ Tu ■■ > ν χ7

7*« ips»í.sai

A—r-

Fig. 3 Cross­sections of the test specimen

ffT3

\3or-30 I I , - « I 1

95

The preparation of the test specimen, especially the concrete work was done by

WIEMER&TRACHTE. The second specimen contained three main girders in the cross-

section spaced at 2,30 m. The rapid and easy erection with prefabricated concrete

elements with a thickness of 8 cm used as permanent formwork to the in situ deck slab

was monitored.

Fig. 4 shows both the static system and the load during concreting and in the tests.

The test specimens were supported at midspan and tied down at one end. The load

was introduced at low increments at the opposite cantilevering end.

1 i i i J I I M I I i i i

_Ζ3Γ JIT

50 _9,50_ 10,00

+- 20.00

a) Static system: Steel beam Load: Self-weight and concrete slab

ΖΖΖΖΖΖΖΖΖΞΖΞΖΖΖΖΖΖΖΖΖ22ΖΖΖΖΞΖΖΖ

rzn 9,50 4- 9,50

■s-t-50

Fig. 4 b) Static system: Composite beam Load: Test load

In Test 1 the load has been introduced directly to the concrete slab in the axis of the

steel web whereas in Test 2 the load has been introduced by a very rigid transverse

beam to simulate a line load at the cantilever end (see Fig. 5).

96

Fig. 5 Load introduction at test 2

At more than 100 measuring points deflections and strains were registered. The data transmitted by the measuring indicaters have been recorded by PCs. The progression of cracks and crack width in the concrete slab hás been recorded.

By these tests the stability of the lower flange under compression, the buckling of the web and the rotation capacity under negative moments were studied.

In both tests the ultimate load was determined by the local buckling of the web aocvc-the midspan support. For the purpose to emphasize this effect the web was markeo b\ a mesh. Fig. 6 shows the buckling pattern of the web at ultimate load leve!.

97

Fig. 6 Plasticity and local buckles at interior support specimen 2

Fig. 7 and 8 show the deflections of the beams below the hydrolic iack durino th* ultimate load tests as function of the ratio M / M p , Taking into a c ^ u n L moment due to self-weight of the specimens the curves start at a value of M/M -0 .14

self-weighft i

0.0 0 100 200

" ^ ^ 4 w

Fig. 7

300 400 500 Deflection [mm]

Deflection history of beam 1 including deloading branch

600 700

98

Fig. 8

300 400 Deflection [mm]

Deflection history of beam 2 including deloading branch

700

The experimental investigations demonstrated a ductile behaviour of the beams, although geometric and material properties have been chosen unfavourable for this type of composite bridge. In both tests the full plastic moment capacity predetermined by calculations could be verified by the tests.

The tests proved that no complementary transverse girders are required to guarantee the stability of hot-rolled steel sections at ultimate load levels adjacent to internal supports of multi-span composite bridges.

Comparison between test results and design ace. to Eurocode 4

According to Eurocode 4 [2] the design of composite sections in hogging moment regions is linked to a classification based on the slendemess of steel elements in compression (here: web and lower flange). The classification system defines four classes:

. Class 1 - plastic moment resistance with high rotation capacity

Class 2 - plastic moment resistance with limited capacity

« Class 3 - yield strength in the extreme compression fibre, local buckling

• Class 4 - resistance is governed by effects of local buckling

99

As the cross-section is classified according to the least favourable class of its elements in compression, the test specimens had to be classified Class 3 due to their width-to-thickness ratio of the web (d/t=48). Using an effective web with a reduced width the section could be lifted into Class 2. This method leads to a moment resistance, M . Rd, which lies inbetween the elastic moment for a Class 3 section governed by yield of'the steel bottom flange, Me( Rd, and the plastic moment, M ( Rd, for a Class 2 section.

Moment resistance

Class 3: Mel?Rd [kNm]

Class 2 with eff.web Mpl,Rd EkNml Class 2: Mpl,Rd [kNm3 test result Mexp [kNm]

Specimen 1

5460

6295

7300

7190

Specimen 2

15580

18410

20860

21145

Table 1 Calculated and measured moment resistances

In the test a typical Class 2 behaviour of the beams could be observed. The theoretical plastic moments Μ , have been reached. The deflection history reflects an impressive ductile behaviour ofthe composite sections together with a high rotation capacity.

Fig. 9 Eigenform ofthe test specimen calculated by the program BDK2 [3]

These results point out that the classification system of Eurocode 4 leads to conserva­tive and uneconomic results especially in those cases where actual conditions do not fit the assumptions taken into account when drafting the code regulations. For the

too

given parameters all influences like vertical stiffeners at the support, the restraint by the cracked concrete slab, combined stressing due to shear and bending and the non­linear elasticity of the bottom flange have been taken into consideration when calculating the buckling load of the beam with a recently developed computer program [31 (see Fig. 9). With a more sophisticated calculation the cross-section can be classified into Class 2.

Furthermore a proposal has been worked out for the maximum width-to-thickness ratios for steel webs in composite beams taking into account the restraint of the concrete slab (see Fig. 10).

Webs: (Internal elements perpendicular to axis οΓ bending)

A i U e l banolng

Class

Stress distribution in clement (compression positive)

Stress distribution in element (compression positive)

Bending

+

d/t s 72e

d/t ί 83e

Π ΑΛ

i/235/ζ

d/t i 124«

Compression

d/t s3Se

d/t s 44«

d/t s' 49e

Combined bending and compression

when <r > 0,5: d/t í 6S4e/(17a

when o < 0,5:

d/t s 36 e/a

* D

when a > 04:

d/t s 730e/( 15,6α + 1)

when a < 0,5:

d/t i 41,5e/a

Γ τ

L Y ι— cr —I—

235

when ψ > -1:

d/t s 49e/(0,7 + 0,3 ψ)

when ψ i - 1 :

d/t s 62e (1-ψ) / ( -ψ ) '

275

0,92

355

0.81

Fig. 10 Maximum width-to-thickness ratios for steel webs in composite beams

According to Eurocode 4 the design buckling resistance moment M b R d of á laterally

unrestrained beams shall be taken as

Mb.Rd = XLT * M

pl,Rd ' < V W

ιοί

for Class 1 and Class 2 cross-sections with TR d= 1.10,

where

χ. T is the reduction factor for lateral-torsional buckling

Μ , R d is the plastic resistance moment.

According to Eurocode 4 values of xL T for rolled sections (buckling curve a) may be

determined for the appropriate slendernesslLT from

XLT-^LT+^-V)^2.

where tpLT = 0.5 · [1 + 0.21 · (XLT - 0.2) + 7LT2] .

The slendemess is taken from

\Τ - (VV172

where

M ■ is the value of M , R d when the γ Μ factors γ3 ,7C and γβ

are taken as 1.0,

M cr

is the elastic critical moment for lateral-torsional buckling.

1.0'

0.9

0.8

■o

rx

0.7-I

0.6

ξ 0.5 rr

ή 0.4 0.3

0.2H

0.1 0.0

0.83

0.73

■ testno.2(1.014)

'testno.1 (0.985)

0.0

buckling curve a

ace. to EC4 P.1

lateral­torsional

buckling ace. to DIN 18800

0.3(3 ­ i — ι — ι — ι — ι — ι — ι — I — ι — ι — ι — ι — ι — : — ι — ι — ι — ι — ι — Ï

1.0 1.5 2.0 0.5 ~l 1 1 1 1 ! Γ"

2.5 _ 3.0

\T

Fig. 11 Lateral­torsional buckling Comparison between design curves and test results

102

A simplified analytical model for the determination of Mc r taking into account profile

deformation in the beam and crack formation in the concrete slab [3,4] has been used

when calculating 1 L T . This simplified method has been inserted into Eurocode 4

as Annex Β. 1.

The ultimate loads measured in the tests demonstrated, that the regulations for lateral­

torsional buckling according to Eurocode 4 are too conservative.

Considering the test results the following equation for the reduction factor xL T is

suggested according to the lateral­torsional buckling curve ofthe German DIN 18800

P.2[5] (see Fig. 11).

X L T = 1 forXL T<0.4

XLT = t i +

\ τ2 η

)Γ1 / η f o r

\ T > 0.4

where η = 2.5.

References

[1] Schaumann, P., Steffen, Α.: Verbundbrücken auf Basis von Walzträgern, Versuch 1 ­Einstegiger Verbundträger, Nr. A 88199, Versuch 2 ­ Realistischer Verbundbrückenträ­ger, Nr. A 89199­2, im Auftrag von ARBED Recherches, Luxembourg (unpublished)

[2] Eurocode 4 Editorial Group: Eurocode 4, Design of Composite Steel and Concrete Structures, Part 1 ­ General Rules and Rules for Buildings, Revised draft, Issue 1, Commission ofthe European Communities, Oct.1990

[3] Kina, J.: Zum Biegedrillknicken von Verbundträgern, Technisch­Wissenschaftliche Mit­teilungen des Instituts für Konstruktiven Ingenieurbau der Ruhr­Universität Bochum, Heft 91­6,1991

[4] Roik, K., Hanswille, G., Kina, J.: Zur Frage des Biegedrillknickens bei Stahlverbundträ­gern, STAHLBAU 59, H. 11, S. 327 bis 333, 1990

[5] DIN 18800 ­ Stahlbauten ­ Teil 2, Stabilitätsfälle, Knicken von Stäben und Stabwerken, Ausgabe 11.90, Beuth Verlag, Berlin, 1990

103

Vorschläge zur Ergänzung bzw. Änderung von Eurocode 4 P.1

auf der Grundlage des ARBED-Forschungsprogramms

Verbundbrücken auf der Basis von Walzträgern

Erstellt von Dr.-lng. P. Schaumann

Bochum, im Januar 1991

HAENSEL · ROIK · ALBRECHT & PARTNER

Dr.-lng. J. Haensel Dr.-lng. G. Albrecht Dr.-lng. P. Schaumann WVssonscricnilcriei· Barsler: Prof. Dr.-lng. Dr.-lng. E A K. Roik

Kohlenstr. 70 * D-*630 Bochum 1

104

1. Einleitung

Im Rahmen des Arbed-Forschungsvorhabens 'Verbundbrücken auf Basis von Walzträgern' wurden zwei Großversuche an Verbundbrückenträgern durchge­führt.

Eine spezielle Fragestellung dieser Versuche war das Stabilitätsverhalten der Verbundträger im Bereich negativer Momente. Dabei ging es einerseits um das lokale Beulverhalten des Stahlträgersteges' und andererseits um das globale Biegedrillknickproblem des Stahlträgeruntergurtes, die miteinander in Interaktion stehen.

Im folgenden werden Verbesserungsvorschläge 'für' die Bemessungsverfahren nach Euroebde 4 1 vorgestellt';'.· die durch die";Ve'rsuchsergebnisse begründet sind. Die Verbesserungsvorschiäge betreffen'die.

. Kapitel 4.3 (Classification of cross-sections of beams) und dort speziell das Kapitel 4.3.3 (Classification of steel webs')

und . Kapitel 4.6 (Lateral-torsional buckling of composite beams

for buildings).

Für die Klassifizierung der Stege von Verbundträgerquerschnitten wird eine ergänzende Tabelle vorgeschlagen. Beim Nachweis des Biegedrillknickens wird vorgeschlagen, die Reduktionsfaktoren der Biegedrillknickkurve der DIN 18800 Teil 2 2 anzupassen.

Eurocode 4 Editorial Group: Eurocode 4, Design of Composite Steel and Concrete Structures, Part 1 - General Rules and Rules for Buildings, Revised draft, Issue 1, Commission of the European Communities, Oct.1990 DIN 18800 - Stahlbauten - Teil 2, Stabilitätsfälle, Knicken von Stäben und Stabwerken, Ausgabe 11.90, Beuth Verlag, Berlin, 1990

105

2. Klassifizierung der Querschnitte

2.1 Generelles Vorgehen

Der Eurocode 4 P.l bietet zunächst die Klassifizierung der staählernen Gurte und Stege über Grenzverhältnisse von Breite zu Dicke an. Prinzipiell möglich ist die Höherstufung von gedrückten stählernen Querschnittsteilen' über das Verfahren der wirksamen Querschnitte (Ch. 4.3.1 (4)).

Die vorliegende Fassung bietet für die Klassifizierung des Querschnitte Anwendungsregeln, die z.B. für den Brückenbau unzureichend sind. Grundsätz­lich sind hier Möglichkeiten vorzusehen, die eine differenzierte Berück­sichtigung des Beulverhaltens einzelner Querschnittsteile gewährleisten. Dabei müssen z.B. auch individuelle Randbedingungen, Belastungen und Beul­steifen berücksichtigt werden können.

An dieser Stelle sei beispielhaft auf die Ermittlung wirksamer Querschnitte in der DIN 18800 Teil 2 Abs. 7 verwiesen.

2.2 Nachweis für den Steg mit Hilfe von d/t-Verhältnissen

2.2.1 Ergänzungsvorschlag

In Ergänzung der Tabelle 4.2 (Maximum width-to-thickness ratios for steel webs) wird für Verbundträger, die schubfest mit einer aufliegenden Stahlbe­tonplatte verbunden sind, die nachfolgende Tabelle 4.2a vorgeschlagen.

2.2.2 Begründung

Im Bereich negativer Momente befindet sich der Steg von Verbundträgern überwiegend im Druckbereich. Die Einstufung in die Querschnittsklassen 1, 2, 3 oder 4 ist erforderlich.

Im Eurocode 4 P.l sind in der Tabelle 4.2 die maximal zulässigen Verhält­nisse von Breite zu Dicke für die Stege von Stahlträgern angegeben. Diese Tabelle ist vollständig aus Eurocode 3 übernommen worden. Der Tabelle liegt die Annahme allseitig gelenkiger Lagerung des Steges zugrunde.

106

Webs: (Internal elements perpendicular to axis of bending)

Class

Stress

distribution

in element ■

(compression

positive)'

Stress

distribution

in element,

(compression

positive)

= V235/Ç

A*U o( banalnq

Bending

-ï

—

.

d/t ¿ . 72e

d/ t ^ 83e

Á

Tcr

d/t <. 124e

Compression

d/t s 3 8 e

d/t ¿ 44e

d/t 5 49e

235

Combined bending

and compression

ad +

when a > 0,5:

'"" d/t : '¿'684e7(17a + 1)

when a < 0,5:

d/t ¿ 36 e / a

when c > 0,5:

d/t i 730e/(15,6a + 1)

when a < 0,5:

" d / t <;41,5e/a

j ^cr

when ψ > -1 :

d/t 5 49e/(0,7 + 0,3 ψ)

when ψ s - 1 :

d/t s 62e (1-ψ) ^(-ψ)'

275

0,92

355

0,81

EC4 - Table 4.2a Maximum width-to-thickness ratios for steel webs in composite beams

107

Bei Verbundträgern, die schubfest mit einer aufliegenden Stahlbetonplatte verbunden sind, kann der Steg an der Oberseite als eingespannt betrachtet werden. Die Klassifizierung unter Zuhilfenahme der Tabelle 4.2 liefert somit Stegschlankheiten, die zu sehr auf der sicheren Seite liegen.

Daher wurde vom Verfasser die leicht veränderte Tabelle 4.2a erarbeitet, die den speziellen Bedingungen der Verbundträger, insbesondere die Steifigkeit der aufliegenden Betondeckenplatte berücksichtigt.

Die Begrenzung der d/t-Verhältnisse ist eine Begrenzung der Stegschlankhei­ten mit dem Ziel, den Steg entsprechend seiner Beulgefährdung zu klassifi­zieren. Die Beultragspannung ist eine Funktion des bezogenen Schlankheits­grades

*PB mit ?lki

- (Viki*1 Ί

der ideellen Beul normal ..= k/· 18980 · (t/d)2

Spannung [kN/cm2].

Die veränderten d/t­Verhältnisse der Tabelle 4.2a ergeben sich bei Auf­

rechterhai tung des bezogenen Schlankheitsgrades aus den Beul werten k­, für

die allseitig gelenkig gelagerte Platte und den Beulwerten ko für die ein­

seitig eingespannte Platte wie folgt:

(dAhable 4.2a = W ^ T a b l e 4.2 * (k2/

kl)

Die Berechnung der Beulwerte k0 erfolgte, soweit die Lösungen nicht aus der

^Literatur entnommen werden konnten, mit Hilfe eines FEM­Rechenprogramms;

.·'" siehe auch nachfolgendes Bild.

\ ι

108

buckling factors k for different boundary conditions

1.18

­1.0 ­0.8 ­0.6 ­0.4 ­0.2 ­0.0 0.2

stress ratio γ

Beulwerte

I

Druck (elastisch und plastisch)

Biegung (elastisch) T¡> = -1.0

Biegung (plastisch) α = 0.5

al lsei t ig gelenkig

k l

4.0

23.94

10.1

einseitig . eingespannt

k2

5.41

23.88

10.1

109

3. Biegedri11 knicken

3.1 Änderungsvorschlag

Beim Nachweis gegen Biegedrillknicken (lateral­torsional buckling) darf das

Tragmoment bei Querschnitten der Klassen 1 und 2 nach Eurocode 4 Ch. 4.6.3

zu

Hb,Rd = *LT '

Hpl,Rd ' (V?Rd>

mit X[_j ­ Reduktionsfaktor für Biegedrillknicken

und Mp­j Rd ­ plastische Grenztragfähigkeit

und Y R d = 1,10.

ermittelt werden. Der Reduktionsfaktor xLT ergibt sich in-Abhängigkeit des

bezogenen Schlankheitsgrades X Q

Anstelle.'des in der vorliegenden Fassung von Eurocode 4 P.l angegebenen

Reduktionsfaktors

X L T = [<PLT +

^LT2 " %

2^ " ^

mit _ Ψυ = 0,5 · [1 + aLT . (XLT - 0,2) + λ , τ

2 ] und

aLT = 0,21 für Walzquerschnitte (Knickspannungskurve a)

wird gemäß DIN 18800 der Reduktionsfaktor

XLT = 1 für I L T < 0,4

XLT = [1 + \ΊΖη)Γ1/η für XLT > 0,4

mit η = 2,5

vorgeschlagen; siehe dazu auch folgendes Bild.

110

Lateral-torsional buckling 1.0

0.9-

0.8-

0.7-

5 0.6 D. 1 ο.5-| cc .d 0.4-Έ

0.3-

0.2-

'6.1-0.0 0.02· rr1-1"

lateral:torsional buckling

a c e t o DIN 18800

buckling curve a a c e t o EC4 P.1

- i — i — ι — ι — i — i — i — i — ι — i — i — i — i — i — i — i — i — ι — ι — ι — i — i — i — i — Γ

0.5 1.0 1.5 2.0 2.5 - - . . 3 . 0

λ LT 3.2 Begründung

Beim Nachweis gegen Biegedrillknicken wird der bezogene Schlankheitsgrad

L j bei Querschnitten der Klassen 1 und 2 zu

% ." (Mpl/

Hcr)*

mit Mpl - Mp-|_Rd mit -tø-Faktoren γδ, γς und γ5 gleich 1,0

und M c r - kritisches Biegedrillknickmoment (elastisch)

ermittelt.

Das kritisches Biegedrillknickmoment M c r wurde bei der Einordnung der Versuchsergebnisse wurde auf zwei verschiedenen Wegen ermittelt:

• Näherungsverfahren nach Eurocode 4 (Annex B) • Computerprogramm BDK2

In der nachfolgenden Tabelle sind die Zahlenwerte der bezogenen Schlankheiten, die mit den beiden Berechnungsmethoden ermittelt wurden, gegenübergestellt. Es ist erfreulich, daß das Näherungsverfahren ein

3 Kina, ύ\: Programmbeschreibung BDK2, Biegedrillknicken, Institut für Konstruktiven Ingenieurbau, Ruhr-Universität Bochum

111

Ergebnis l i e fe r t , das auf der sicheren Seite l iegt , während mit der

Computerberechnung eine günstigere Schlankheit ermittelt wurde. Es lohnt

sich also, genauer zu rechnen.

Die Tabelle enthält gleichzeitig die Reduktionsfaktoren xLj nach EC4 bzw.

nach DIN 18800, die sich in Funktion der bezogenen Schlankheit ergeben.

Methode zur Bestimmung des kr i t i sche Biegedrillknickmomentes

Näherung nach EC4

Computerprogramm BDK2

hi 0,95 '

0,90

curve a

Al

0,693

0,734

18800

XLT

0,788

0,'831

Obwohl' durch die Wahl der Versuchsparameter für den· Anwendungsbereich der

Walzträger·-im Verbundbrückenbau extreme Verhältnisse im Hinblick auf die

Schlankheit- des Untergurtes· vorgegeben wurden, konnte im Versuch nahezu

(98,5%) des vollplasti sehen'Momentes erreicht werden. Der Versuchsträger

zeichnete sich durch ein gutmütiges, d.h. duktiles Verhalten, und durch

hohe Rotationskapazität im Nachtraglastbereich aus. Dieses Ergebnis -wurde

durch einen weiteren Großversuch mit drei Hauptträgern voll bestät igt .

Lateral-torsional buckling Comparison to test result

•1 Π­, I . υ

0.9-

0.8-

0.7-

5 0.6-CL

1 0.5-

5 0.4-

0.3-

0.2-

0.1-

u.u-0

0.83 \ \

Γ

.0

0.73

I 'I

test no.1

(0.985)

"\ \

r Λ buckling curve a

ace. to EC4 P.1 V J

°·ί 0.5

30 1.0

I J 1 i I I I

1.5

r Λ lateral-torsional

buckling a c e t o DIN 18800

* " * ^ > ^ w

1 1 1 1 1 1 1 1 ^~\

2.0 2.5 1 1

3. λ LT

Die Bemessungswerte liegen gegenüber dem Versuchsergebnis deutlich auf der sicheren Seite. Das Bemessungsergebnis nach Eurocode 4 P.l l i eg t noch

112

einmal 12% unter dem Wert nach DIN 18800. Die Bochumer Versuchsergebnisse liegen auf einer Ebene mit den Ergebnissen von Johnson/Fan^ und untermauern daher die Forderung nach einer Veränderung der z.Zt. gültigen Reduktionsfaktoren des Eurocode 4 P.l.

Bochum, 25. Januar 1991

Dr.-lng. Peter Schaumann

Johnson, R.P., Fan, C.K.R.: Distorsional Lateral Buckling of Continuous Composite Beams. University of Warwick, Research Report CE 30, Jan. 1990

113

Statile und Stabilität

der

Baukonstruktionen

Elasto- und plasto-statische Β erechmings verfahren

drackbeansprachterTragwerke :

Nachweisformen gegen Knicken, Kippen, Beulen

Dr.-Ing. Christian Petersen

Professor an der

Hochschule der Bundeswehr

Müncheni

V

Friedr. Vieweg & Sohn Braunschweig/Wiesbaden

114

Talet 8.1

Oíucl

Beulwerte untersch ied l i ch g e l a g e r t e r R e c h t e c k p l a t t e n

Porameler

i g | Ψ= ì Sponnungs»erhallnis ύ

Se il en» s r hol Im i a : — b

Ψ=0

Gelenkige Quer rander

Lagerung der l o n g s r u n d t r .

\ΕΠ/ ¿ u s den Oiagrommen wird l 0 a b ­

geg rill r η ideale Beulspannung

" U i1 l

o ·««

r t 'K

b't

gelenkt ger t a n d

e ingespannter Rand

freier Rand

Oie j e w e i l s k l e i n s i e n BeuUerle min k 0 s ind lür 8 Lagerla l le in nachs tehender Tabel le auf ­

g e l i s t e t .

Φ

1

O.J

0. E

0.1

0.3

0.2

0.1

0

-0.1

-0.7

-0.3

-0.1

-0.5

-0 . Í

-0.7

-0.1

-0.9

- 1

-1.1

-1.7

-1.3

-1.1

-1.5

-1.5

-1.7

-1.0

-1.5

- 7

I

CD l

l . t t

1.59

5.E!

6.0J

5.59

7.10

7.11

1.55

5.19

10.57

11.96

13.10

15.13

17.10

19.73

71.51

73.JJ

76.35

75.93

31.67

JIJLJ

37.35

10.11

13.57

(6.17

50.76

53.7»

11

m 6.9t

7.71

1.70

9.90

10.63

11.16

17. t7

13.55

H.I5

16.31

11.77

• 70.73

' 73.73

75.57

7J.66

32.07

35.71

39.S6

11.06

(0.10

57.36

• 56.9!

61.99

6635

77.10

77.55

13.01

19.72

m

Θ 5 . l i

5.9t

6.57

7.35

7.10

1.31

1.19

9.51

10.71

11.13

13.11

13.73

11.57

15.99

17.67

19.51

71.67

73.9t

36.3!

21.31

31.6S

3t. lt

37.36

10.13

13.57

11.97

50.77

53.79

ΠΙ b ι

5.11

6.01

(.9t

9.07

1.77

9.59

10.56

11.73

13. Η

11.15

16.91

19.36

22.15

35.23

29.51

32.03

35.20

39.56

tt.CS

19.10

52 J 6

56.91

61.99

66.95

77.10

77.63

93.21

99J2

,,

Θ 0.125

0.567

0.550

....

Θ 0.125

1.71

9.31

10.93

13.03

15.05

17.

19.

IO

21

21.19

23.95

26.33

26.91

31.63

31.11

37.35

10.13

13.57

16.9t

50.53

5t .71

V

Ξ 1.71

1.33

1.3S

1.16

1.19

1.S3

1.57

1.51

1.65

l./O

1.71

1.79

1.15

1.90

1.96

7.02

2.09

7.15

2.23

2.30

3.3S

2Λ7

7.56

7.66

2.77

7.11

7.93

3.11

Θ 1.7!

1.57

1.97

7.13

7.96

3.t5

1.36

5.15

5.50

13.15

15.01

19.06

77.11

75.73

79.SI

37.02

35.69

39.56

13.61

17.96

57.36

56.9!

61.39

57.33

72.10

77.55

63.21

19.22

Oie T a l e l w e r t e und die Diagramme g e l t e n (ür μ ι 0 . 3

115

Annex 7

Document 3263-2-12 (CTICM)

(8 pages)

"Required rotation capacity for a 15% redistribution of elastic peak moment"

117

ECSC PROJECTS SA 319-SA 321 Working Document

Required rotation capacity for a 15% redistribution of elastic peak moment

Following an elastic analysis, the paragraph 5.2.1.3 of Eurocode 3 allows to redistribute up to 15% of the elastic peak moment, provided that

- the internal forces and moments remain in equilibrium with the applied loads

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

This procedure is equivalent to a plastic analysis, in which the redistribution of moments would be limited and compatible with the reduced rotation capacity of Class 2 cross-sections.

The question which comes up is : is this allowance of 15% redistribution always safe for Class 2 cross-sections, that is to say, is the required rotation capacity always small?

The required rotation capacities for two common cases are studied hereafter.

A - Two span continuous beam with an uniform distributed load

III k

I

q immun II I

ω L

1

© XL

I

1

λ<1

Figure 7

The first plastic hinge always occurs at the central support for :

8 M p | q l " ί 2 ( λ 2 - λ + 1 )

and the mechanism (2nd plastic hinge in span 1) for :

2 Mp|

118

Required plastic rotation

Between these two levels, at q = (1 + η) q ι, , the required plastic rotation in the first hinge is

η Mpi L ep.req = 3 0 + λ) £,

Øpjeq 's plotted versus λ and η.

θ <>­'

e<MplL

0.14 ■

0.12 ■

0.1 ­

0.0» ­

0.0s ­

0.04 ­

0.02 ­

\ \

V

\

- — -

\ X

Ί , ^χζ^ - ^

1

1 1 1

mechanism

ru---— 1

/ '—-H z^U-η

■ /

X ^ ­ 1 — t — ! Xj—hnC—

­ Ì — — —

- ' 1 1 J .

1

η

0.2

0.1765

0,15

0,12

0,1

0.05

0 0.1 0,2 0.3 0.4 0 3 0.6 0.7 0.8 0,9

λ Figure 2

Of course, this expression is valid if

q < q , j

that is to say if

η < 2 (3 + 2Λ/2) (λ2 ­ λ +1 ) ­ 1 (dotted line in figure 2)

The limiting value of η is plotted versus λ on the figure below. For example, with

1

^ =

f l 8 5 ­1 = a l 7 6 5

the valid range of λ is

λ < 0.2604 or λ 2:0,7396

(max value allowed by EC3)

119

0.5 ι

T Ì ΚΛ -

0,1765

C

\l Κ I s

I

• ­ ­ Ι ­

Ι

) 0

I I

1 1

l i l i l í

I I mcc hanism 1

1

\ Ι Α ι V I IHvl

­ i ­ h ^ K i I I I

I 1 0.2 ! °·

3

1 0.4 o.s

■

0.6 1

/

/

■A-

0,7 ι 0.9

/ 7^

1 1/

A /

1 1 1

1 1

0.9 1

0,2604 X

Figure 3

Required rotation capacity

The plastic rotation of reference φρ| is given by :

ΦρΙ= 2 El

0,7396

where

d = d ] + d 2

is assumed to be the distance between zero moment points at the current load level q

η)ςι = (1 +

i ^Kuiii[ij|iiJijJ^ * ^^liUiULu^A

XL Figure 4

L λ2 - λ + 1 d

i=

4 ­ T T Í T ­L

L 1 +λ=> and d = ^ ( l + T | ) — —

and

with

d2 ­ λ

d £ d j +XL

The rotation capacity at q = (1 + T|).qi is given by :

6 r

R 'p.req

req ΦρΙ

120

that is to say

r 8 1 +X X < V 2 ­ 1 Rreq = 3 ^

( 1 +^ 4 λ ( 1 + λ ) + λ

2 ­ λ + 1

Γ 8 λ (1 + λ)

λ > Λ / 2 ­ 1 Rreq = 3η ( 1 + η )

1 + λ3

Rreq is plotted on the figure below.

0.6

0,5

0,4

R req 0.3

0,2

0,1

1

\ ! 1 k K \

V I ^ \

V 'hO""­*·^.

1 l ' I 1 1 Ι Ι ¿­Η""" i

■ 1

mechanism

ν

^

\ Τ*Ν-^Γ,,Ή»^.

ι 1 '

1 1.' 1 i 1' 1 ___L

I V . ^ I ! ! 1 ^U-"-T" 1

/' J—-—i"— i 1 1 ^ L —

—~^4-ΖΓ~ι—4^---" ̂ Γ 1 1"—¡— 1 _J

r ­ T 1 1 ! 1

·■

~~~ 1

— τ ­ I

!

η 0,1765

0,14

0,12

0,10

0.08

0,05

Noies

0 0.1 0,2 0,3 0,4 0.5 0.6 0.7 O.B 0.9 1

λ Figure 5

- The parts of curves above the dotted line are invalid because in this zone, the

mechanism occurs before the load (1 + n j .q i is reached.

For λ < Λ/2 ­ 1, d2 is limited at XL.

Conclusion

For this sketch, the required rotation capaci ty is quite limited ( R r e q < 0,6). Therefore,

Class 2 cross­sections may be used in that case.

121

Β ­ Two span continuous beam with a point load

I al

• ' L XL H

Figure 6

Two ranges for a are to be considered

a < ­\^2(λ+ 1) ­1 first plastic hinge at loading point

a > -\J2{X+ 1) ­ 1 first plastic hinge at central support

B.l a <\J2(\ + 11 ­ 1 first plastic hinge at loading point

The first plastic hinge occurs at

Mp| F ! = ­ ^ f ( a , X )

For F = (l +T\).F-{ :

The plastic rotation in this hinge is given by

Mp,L

öp.req = ~ ¡ E T ΚαΛ)·Η(α.λ,η)

The distance d between zero moment points each side of the plastic hinge is

d = L g(aXTi)

and the required rotation capacity is given by

1 f(a,X).h(a,X,T|) Rreq = 3 ς(α.λ.η)

where :

2 ( λ + 1 ) ί ( α ' λ ' ~ α ( α - 1 ) ( α (α+1 ) -2 ( λ+1 ) ]

a j l +η) 9(α.λ.η)= ^ T T J

τ1 + α _ 2 ( λ + 1 )

Γΐ(α.λ,η) = η (1 -α) [2λ + 3(1 - a J + J ( l - α ) ( λ + 1 -α) ]

122

All these expressions are valid if the mechanism is not reached, that is to say

and this condition becomes

a ( a + 1 )

Tl^Ca­HHl­ j ïXTÏ j l ­1

B.2 a > \]2(X +1) - 1 first plastic hinge at support

The first plastic hinge occurs at

Mp| F 1=­¡

£ Lf (a ,X)

For F = (1 + T\).F-[ :

The plastic rotation in this hinge is given by

Mp,L ep.req = ¿Ë~ ^^MO-.T])

The distance d between zero moment points each side of the plastic hinge is :

d = L g(a,X,T|)

and the required rotation capacity is given by

1 f(a,X).h(a.,X,T|) Rreq = 3 β (α.λ,η)

where :

2 ( λ + 1 ) fí

a^

=a ( l ­ a

2)

1 9(αΛ.η) = j ­ΤαΤ

1 +2 ( λ + 1 ) ( 1 + η )

Ιι(α,λ,η) = η α (1 ­α2)

All these expressions are valid if the mechanism is not reached, that is to say

Fu ρ Γ * 1 ­ η '1

and this condition becomes

2

*1 ­ 2 ( λ + 1 ) ­]

123

For the max value of η allowed by EC3 (ññc ­ 1 = 0,1765), R req is plotted on the figure below

for the λ values : 0,1 ­1 ­ 1,5 ­ 2. The range of a values is limited to [0,2 ­ 0,8].

req

12

11

10

9

8

/

6

5

4

3

2

1

n

•i l I Ml I t i l i I ι π ι u i 11 ¡Μ I I

JI lVl II U I I V J I I I I I URI iu Xi mi for UHKUI Ή II w.\ I'! 11 K1> \\\j \ I M \ s

Ml- K lJ 11 M l 11ΙΉ IM K

J i l

i i r - k . I l l 1 III 1

1

1 1 TT 1 1 II

-ll-li-l l M I 1 1 1 1 II

Τ Γ Ι Ι Ν I

ÏÏ;I" IL IL \ r r ι M \ Nv 11 M \\U :ΤΠΓ

IM I I I I I 1 Ml Μ Ι Ι Μ 1 M I M I 1 I II I 1 l| 1 MM M M

Tt IT I TT Τ Μ I ! j ! ^ _ _ , ! ¡ ~ — η = 0,1765 (15% Mpeak) |

V /

I I ι ι 1 ι I

2 λ

MX, 1 1 1 1.5_ ^

\J\IW. 'v WU$L2:J · · ­

τ _Sp^­2^L:­^___:::.̂ _t

' - . ι 1 "0.5

"ΓΛ ι

W W * . '­­­. : : : : l s

f ' r­­ . Ir~^H?lij44ddlLL_ ι"" ■■■■·. _­­F'T­I II I f f" · ■ ­L L LI TTT~rrB

:í^^^=====i!xt.

i i i i I I I I

0,2 0,25 0,3 0.35 0,4 0.45 0,5 0.55 0,6 0,65 0.7 0,75 0.8

α

Figure 7

Conclusion

For values of λ > l and if the point load acts in the left half­span of the first span, the

required rotation capaci ty may be very large (> 4), and the use of Class 2 cross­

sections with a 15% redistribution of peak moment may be unsafe regarding the

rotation capacity.

124

Example :

i XL

Section IPE 270 ­ L=10m ­ λ =1.5 ­ a = 0,27

Mp| =113,74 kN.m ­ I = 5790 cm4 ­ E = 21000kN/cm

2 ­ f y=235N/mm

2

The first plastic hinge occurs at the loading point for : F-j = ¿1,95 kN

The second plastic hinge (mechanism) occurs at support for : Fu - 73,3 kN

The peak moment is at the loading point.

The maximum allowable moment redistribution (15% of peak moment) is given for :

F = 61,95/0,85 = 72,88 kN

At this load level F :

plastic rotation : |θ ρ = 0,189 rad

length of equivalent isostatic beam : d = 640 cm

Mpi d nM Φρ1= " ¿ g =0,03

R req = 0,189/0,03 = 6,3

At SLS (F,j/1,5), the maximum deflection is in the first span

'max = 5

<0 5 c m

( « L

/2 0

° )

a) It should be noted that the available plastic rotation φΓΟ* calculated with the Feldmann's

model for IPE 270 is

φΓΟϊ = 0,11 rad

which is much lower that θρ r eq = 0,189 rad

Therefore, the 15% redistribution is not valid in that case, although IPE 270 is a Class 1 cross-section.

b) It would be possible to find Class 2 (near Class 3) cross-sections with the same characteristics Mpi and I as IPE 270 for which the rotation capaci ty or available plastic rotation could be still lower.

For example : h = 26 cm - b =17,5 - t w = 0,6 - tf = 0.85 - r = l , 5 —> Class 2 flanges and Class 2 web

125

Annex 8

Document 3263-2-15 (CTICM)

- (33 pages)

Required rotation capacity for continuous beams"

127

CENTRE TECHNIQUE INDUSTRIEL DE LA CONSTRUCTION METALLIQUE ROTREQ.DOC

ECSC Research Project P3263

REQUIRED ROTATION CAPACITY FOR CONTINUOUS BEAMS

Working document (based upon document 3263-2-13)

128

CONTENTS

Notations

1 Introduction

2 Two span beam with a concentrated load

2.1 Ultimate load 2.2 Required plastic rotation 2.3 Required rotation capacity 2.4 Examples

3 Two span beam with uniform distributed load

3.1 Ultimate load 3.2 Required plastic rotation 3.3 Same uniform distributed load for both spans 3.4 Examples

4 Two span beam with a concentrated load and an uniform distributed load

5 Two span beam - General case

5.1 Principle

5.2 Example

6 How to perform a plastic analysis using a elastic analysis program

6.1 Scope 6.2 Methodology 6.3 Example 6.4 Remarks

Annex : continuous beam with three spans

129

NOTATIONS

E

fy

ι

Mpi

L

ereq

Rreq

Young modulus

Yield strength

Second moment of area

Plastic moment resistance

Length of span

Required plastic rotation

Required rotation capacity

130

INTRODUCTION

This document gives formulas for the required plastic rotation 0 r e q and the required

rotation capaci ty R req when the collapse is reached by performing an elastic­

perfectly plastic analysis. The formulas are exact. Abacuses may help the designer to

determine either the required plastic rotation or the required rotation capacity.

Results issued from an analysis with PEP micro program are supplied in order to check

the validity of the formulas.

TWO SPAN BEAM WITH A CONCENTRATED LOAD

aL 0< a <1

/Γ 7Γ ¿i

XL

2.1 Ultimate load

Figure 1

Only one plastic mechanism may be reached. The ultimate load Fy can be

determined by using the kinematic theorem : the external work W e done by the load

as it moves through a virtual displacement Δ is equal to the internal work W¡ absorbed

at the plastic hinges as they rotate through corresponding angles (see figure 2).

We = W¡

with :

and :

W e = Fy Δ

νή = Μ ρ | ( θ ρ 1 + θ ρ 2 ) + Μρ|θρ 2

aL

θρΙ+βρΣ

Figure 2

From above expressions and geometrical considerations, we can obtain

Fu = M p l a + 1

L a (1 - a )

Note Fu does not depend on λ

131

Let us assumed that the first plastic hinge occurs for the load Fi. The expression of this

load can be derived by considering that the maximum moment in the beam

calculated from an elastic analysis, reaches the plastic moment Mp | . Two cases must

be investigated in so far as the maximum moment can be reached :

At the loading point if :

then

a < V 2 ( X + 1) ­ 1 (case A)

F,.üa 2 ( λ + 1 ) 1 " L a ( a ­ l ) [ a ( a + 1 ) ­ 2 ( λ + 1 ) ]

At the intermediate support if : a > Λ/2(λ+Τ) ­ 1

Mpi 2 ( λ+1 )

then FT = , r-1 L

a ( l ­ a2

)

[case B)

Mp l 4

For the particular case λ = 1 : Fi = , ; .. ; y ­ — ' L a (a ­ I ) [a/- + a ­ 4)

(α<Λ/2 (λ+1) ­ 1 = 1)

Figure 3

The ratio Fu/F] is an indicator for the interest of the plastic analysis. This ratio is easy to

be expressed from the previous expressions.

F,, (A ) a < Λ /2(λ+1) ­ 1

(Β) a > Λ/2(λ + 1) ­ 1

a (a + 1 )

7 Γ( α + 1 ) ( 1

­ 2 ( Γ Τ Τ , »

Fu ( a + 1 )

2

F i " 2 ( X + l )

For λ = 1 — =0.25(a+1) ( 4 ­ a ­ a2]

With the restriction : a*0 and a * Í

132

Fu / F l

λ

0.1

0.2

0.3

0,4

0.5

0,6

0.7

0.8

0,9

1,0

® Ι ï 1

® * 1 2

Figure 4

Noie : Figure 4 shows that the loading capacity is increased up to about 22% for

equal spans owing to a plastic analysis, when the concentrated load is near

the middle of the span.

2.2 Required plastic rotation 9 r eq :

The required plastic rotation may be obtained by integrating the bending moment

under the load increment (Fy ­ Fi) along the beam in which a perfect hinge has been

introduced ät the plastic hinge location. Therefore the required plastic rotation is the

rotation in the perfect hinge under (Fa ­ F­|).

I f a < V2^ + l ) ­1 (A : 1st plastic hinge at loading point)

If a

Ι ίλ =

oreq

>ν2(λ+ΐ) -

oreq

= 1

Mp,L . " 6EI

1

a2

(B:

Mp|L = 6ETla 2 +

Θ - M P , L

oreq - 6 E |

(a

■ 2 α + 2 λ + Ί a

1st plastic hinge at support]

2 α - 2 λ - 1 ]

+ 3)(1 -a) a

133

Figure 5 shows curves giving kr versus α for various values of the ratio λ. 0 r e q may be expressed by :

Ω I M P | L

e req ~ kr g | kr

3.0-,

0 0,1 0,2 0,3 0,4 0,5 0.6 0,7 0,8 0,9 1,0 a

Figure 5

2.3 Required rotation capacity Rreq :

The required rotation capaci ty is defined as the ratio of the required plastic rotation over the sum <pp| of the elastic rotations at the ends of an equivalent single beam (see figure 6) whose span is equal to the length between zero moment points each side of the first plastic hinge :

veq ^req ΦρΙ

If Leq is the length of the equivalent single beam, then :

M p l Leg ΦρΙ - 2E I

The equivalent length is taken as the distance between zero-moment points (see figures 6 and 7).

134

<s° A : First plastic hinge at loading point

α < - \ / 2 ( λ + 1) -1

Plastic Hinges ^ p |

( ι ί )

Equivalent single beam

Figure 6

Rreq = 2 3

a2 + 2a a (a

- 2 λ · + υ

1

®~ Β : First plastic hinge at support

α > Λ / 2 ( λ + 1) -1

Plastic Hinges Mpi

Equivalent single beam Λ"

Figure 7

Kreq 2

" 3 a2 +

1 2 a · 2 λ -- α + 2 λ

1

For the particular case λ = 1

Rreq 2 (a

3 + 3 ) ( 1 - a ) a ( a + l )

135

Rreq

— 7 See Figure 9

Figure 8

Note : The required rotation capacity can be very high if the load is close to the first support, but the Influence of shear force on plastic resistance has to be taken Into account.

7 0.8 0.9 1 - > a

Figure 9

The following examples allowed us to apply the previous formulas and to compare them with the results given by PEP micro.

136

2.4 Examples

Example 1

Data : IPE 270 ­ steel S235 (Mp) = 1,1374 106 daN.cm. I = 5790 cm

4)

L= 1000 cm λ=1

F=1000daN a = 0,35

ÍA)

First plastic hinge

Ultimate load

Fu /h

Required rotation

Required rotation capacity

Formulas

F] = 5669,24 daN

Fu = 6749,4 daN

1,1905

9 r e q = 0,096996 rad

Rreq

= 3.0723

PEP micro

Fi = 5669 daN

Fa = 6749 daN

1,1905

«req = 0,096993 rad

Rreq

= 3.0724

Table 1

Example 2

Data : IPE 270 ­ steel S235

L= 1000 cm λ = 0,5

F =1000 daN a = 0,8

m

First plastic hinge

Ultimate load

f u / h

Required rotation

Required rotation capacity

Formulas

F] = 11847,9 daN

Fu= 12795.7 daN

1,08

e r e q = 0,0037417 rad

R r e q = 0,1333

PEP micro

F] = 11847 daN

Fu= 12795 daN

1.08

0 r e q = 0,0037438 rad

R r e q = 0,1334

Table 2

137

TWO SPAN BEAM WITH UNIFORM DISTRIBUTED LOADS

0< λ<1 γ > 0

yq

Δ TS Ά λL

Figure 10

3.1 Ultimate load

Three possible mechanisms may occur (see figure 11), but we assumed that γ s 0. Therefore the mechanism C has not to be considered. Depending on the values of γ and λ, mechanism A or mechanism Β will be formed (see figure 12).

λ ί

Figure 11

The methodology is similar to the one described in § 2.1 for a concentrated load.

If γ λ 2 < 1 Mechanism A : The mechanism occurs in the first span

a) First plastic hinge on the intermediate support

The first plastic hinge occurs on the intermediate support if the following condition is fulfilled :

4(3-2V2)<if^|-<4(3 + 2V2)

or 0,6863 1 +γλ 3

Λο < . ' < 23, 1 + λ 3137

138

Let us assumed that q ] is the loading for which the first plastic hinge occurs, then :

8 M 0.1=·

pi 1 + λ L2 1 + γ λ3

so 3 υ _ 3 + 2Λ/2 1 + γλ 3

q] 4 1 + λ

b) First plastic hinge in the first span

If the first plastic hinge occurs in the first span, then :

128 M el

so

q i = _ ΙΛ

3 υ _ 3 + 2λ/2 qy~ 64

1 + λ 3 + 4 λ - γ λ 3

3 + 4 λ - γ λ 3 ' 1 + λ

If γ λ 2 > 1 Mechanism Β : The mechanism occurs in the second span

q U = ^ [ ( o + 4%/2)-4 λ2

a) First plastic hinge on the intermediate support

The first plastic hinge occurs on the intermediate support if the following condition is satisfied :

4(3-2V2)< V " , ! \ <4(3 + 2>/2) γ λ 2 ( 1 + λ )

1 + γ λ 3

0,6863 < — T T - 1 < 23,3137 or γ λ 2 ( 1 +λ)

Let us assumed that q-j is the loading for which the first plastic hinge occurs, then :

_ 8 M p l 1 + λ q i = L2 1 + ν λ 3

so q u _ 3 + 2>/2 1 + γ λ 3

<=Π ~ 4 γ λ 2 (1+λ)

139

b) First plastic hinge in the second span

If the first plastic hinge occurs in the second span, then

= ^ p j , 128 q i

" L2

γ λ 2 ( 4 . y Τ λ3 â

γ Α ( 4 γλ 2 (1+λΓ

so % _ 3 + 2 Λ / 2 1 + γ λ

3 ο

^1 6 4

~ γ λ2( 1 + λ )

If γ λ2 = 1 Plastic resistance is reached in the two spans at the same time.

y io

9

8

7

6

S

A

3

2

1

0

V V \

\

V \

Mechanism A

"Π"" ï Γ Γ Mechanism Β

s^

0 0.1 0.2 0.3 0.4 0.5 0,6 0.7 0.8 0.9 1 λ

Figure 12

Figure 13 shows the evolution of the ratio qulq] related to λ for various values of γ

(boxed values). The curve peaks indicate the limit between mechanism A and

mechanism B.

1 s

1 4 -

1 3 -

1 1 -

1 -

q u / q l

10,0

0,10

0,25 '

Meen ar

0,50

i ism, <\

4.00

Mec :nc

/

am sm B

3,00 2,00

\ \ Ν

/ 1.00

/ 0.75

s

■»

0.1 0 4 0.3 0,4 0.5 0.« 0,7 0,8 0,9

Figure 13

140

3.2 Required plastic rotation θΓβ„

Mechanism Α (γ λ2 < 1)

If the first plastic hinge occurs on the intermediate support, then

_Μρμ

*req­ E I λ + 1 3 + 2Λ/2

12 (1+γλ

3)]

If the first plastic hinge occurs in the first span, then :

'req

Mp l

L λ + 1

^♦D[I­<W^]

Mechanism Β (γ λ2 > 1)

If the first plastic hinge occurs on the intermediate support, then

If the first plastic hinge occurs in the second span, then

ereq = ^ . ^ [ - > - < ^ i , ^ ] 4 γ λ '

Figure 14 shows curves giving the factor kr versus the ratio λ for various values of the

ratio γ. The required plastic rotation 9 r e q can be obtained with :

Mp[L 9req ~ ^r f

141

Mp|L Øreq

= r̂ E I

kr

0,4

0,3

0,2

0,1

-'— "

___ — — '

__-

J

—

/ /

IC _ - -

y

—¡SO

___;

jrXZ

­"­"

ol /

"jT1

V / >

Γ}

/ Λ '

„ - -

Π—Γ

F = ^

- ~ " y

2 01

_.

/ Ν

^Λ

,~-—.

LT

/ '

[0 un

^—' - - r ? '

Jo ini

1.5- 1.0 f

Jf/ -5 rz_ £ .

/

Ιο,ΰορ· <Γ/ Χ ^

0,1 04 0,3 0.4 0,5 0.6 0,7 0,8 0,? 1 λ

Figure 14

Note : In the case of Increasing the loading from zero to reach the plastic mechanism, the first plastic hinge may occur in one of both spans. This first plastic hinge is accurately located at the maximum moment location which Is determined from an elastic analysis. While the loading Increases to reach the ultimate loading, the first plastic hinge moves to reach its final location at collapse, at the distance d given by :

A Mechanism A 0.414 XL

Mechanism Β

XL

Figure 15

142

3.3 Same uniform distributed load for both spans (y = 11

For this case, the mechanism A always occurs since : γ λ 2 < 1

The first plastic hinge always occurs at the intermediate support :

8 Mp| q i ί 2 ( λ 2 - λ + 1 )

The collapse is obtained for the load qu :

2 Mp| qu=—f- (3 + 2V2)

The second plastic hinge is located at the distance [y¡2 ­ 1) L = 0,414 L from the origin

of the first span.

The following ratio is an indicator

for the interest of the plastic

analysis :

2u=2±¿£ | λ 2. λ + 1 1 qi 4

l

\¿> ·

1.4 ·

r- 1,0 _

σ 3

1 -

0 0.1 0.2 0,3 0,4 0.5 0.6 0,7 0.8 0.9 1

λ

Figure 16

Required plastic rotation 6 r eq

MD | L 3 + 2 J2 _ r-Qreq = " f j

­— y ¿ (1+λ) ( λ * ­ λ ­11 +8^/2)

Mp|L or : e r e q = 0,4857 -=j- (1 + λ) ( λ

2 ­ λ + 0,3137 )

Mp|L The curve 0 r e q = f (a) is given at figure 17 : 9 r e q = kr Γ.

143

Λ Κ ^

Ω ■

Ο 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

λ

Figure 17

Required rotation capacity Rreq :

The required rotation capacity can be calculated by dividing 0 r eq by the sum of the

elastic rotations at the ends of a single equivalent beam. The lengtn of this beam can

be taken as the distance between zero-moment points of each side of the

intermediate support, when the collapse is reached.

0,4142 L

v j ? Equivalent single beam

Figure 18

If λ > Λ/2 - 1. a zero moment point is located between the second and the third

support of the beam. In this case, R r eq is given by :

Or:

*req = ί 3 ± 2 _ & £ λ ( λ 2 _ λ _ 1 1 + 8 ^ )

R r e a = 5,6618 λ ( λ 2 - λ + 0,3137 )

144

If λ ¿, Λ/2 ­ 1. the total length of the second span must be taken into account for the

calculation of φρ|. In this case, R r e q is given by :

Or:

Rreq 3 + 2^2 π

6 + λ) ( λ

2­

λ + 3

λ ­ η +

­ 2 Λ / 2

8 Λ /2 )

η 0 7 1 , (1+λ) ( λ2 ­ λ + 0,3137

Rr e q ­ ° '

9 7 1 4 λ + 0,1716

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

λ

Note

Figure 19

The maximal required rotation capacity of a two span continuous beam

subject to an uniform distributed load is 1,78.

145

3.4 Examples

The following table gives the results for values of the ratios γ and λ. These results are compared with results issued from PEP micro program.

For all the cases, the basic data are :

Cross-section IPE 270 Steel S235 Length of the 1st span L = 1000 cm

Note : Mpi = J, I374.106 daN.cm

El = 0,0935438

Wp| » = 484 c m 3 pl.y

fy = 235 MPa ly = 5790 c m 4

E = 210000 MPa

1

2

3

4

S

4

λ

0,35

0,80

0,80

1.00

1,00

1,00

y

5,0

2.0

0.1

0.5

2.0

1,00

Results from

Formulas PEP micro Formulas PEP micro Formulas PEP micro Formulas PEP micro Formulas PEP micro Formulas PEP micro

mechanism

A A Β Β A A A A Β Β

A-B A-B

q i (daN/cm)

10,115 10,114 8.092 8,091 12,476 12,525 12,132 12,131 6,066 6,066 9,099 9,098

3u (daN/cm)

13,258 13,269 10,358 10,366 13,258 13,269 13,258 13.269 6,629 6,635 13.258 13,269

Qu/qi

1,311 1.312 1.280 1.281 1,063 1,059 1.093 1,094 1,093 1,094 1,457 1,458

^req (rad)

0.013080 0,013127 0.015717 0.015779 0.020196 0,020820 0.005789 0.005850 0.005789 0,005850 0.028506 0.028583

Table 3

For example 3, the first plastic hinge occurs in the span.

Examples 4 and 5 are similar since the span lengths are equal and the loading ratios between the spans are inverse.

146

TWO SPAN BEAM WITH A CONCENTRATED LOAD AND AN UNIFORM DISTRIBUTED LOAD

In order to evaluate the relative influence of a concentrated load and an uniform

distributed load, the following example has been studied :

Cross­section

Steel grade

L= 10m

IPE 270

S235

172 172

7Γ

­4­

Δ

Figure 20

In figure 21, the required rotation at collapse 6 r e q is plotted as a function of the ratio :

μ = qL

0.06-

0.05-

0.04-

0.03.

0.02-

0.01.

n .

Oreq (rad)

" 1 1 1 1 1 —

μ­F/íqU 1 1 1 1 1

0 1 2 3 4 Ì 6 7 8 9 10

Figure 21

In the first part of the curve, the distributed load is predominant and the first plastic

hinge occurs on the intermediate support. In the second part, the concentrated load

becomes predominant and the plastic hinge occurs at the load location.

It is rather difficult to give formulas and abacuses to deal with this case for which

parameters could be :

­ Span length ratio

­ Relative position of the concentrated load

­ Ratio of the distributed loads for the first span and the second one

­ Ratio between the concentrated load on the distributed load

A more general method is given in the next paragraph.

147

5 TWO SPAN BEAM - GENERAL CASE

5.1 Principle

After an elastic analysis, we assume that the maximum moment M m a x exceeds the

plastic moment M p | at a distance ßL from the origin of the beam.

ßL 0< β <1

Δ" Έ Ά XL

Figure 22

So ΔΜ = M m a x - Mp| is to be redistributed. For this, the following diagram has to be added to the one resulting from the elastic analysis.

ßL

Figure 23

This diagram is obtained by considering the beam with a perfect hinge at the

maximum moment location, and moments ΔΜ applied each side of this hinge. The

required plastic rotation is the rotation in this hinge subjected to ­ΔΜ.

Therefore the expression of the required plastic rotation is given by :

Note This expression does not depend on the loading, but only on ΔΜ.

It must be checked that the plastic moment is not exceeded somewhere else in the

beam : the mechanism must not be reached.

If a is the loading factor of all the applied loads,

ΔΜ = Μ ρ | ( ^ ­ 1 )

al

ai

Loading factor for the first plastic hinge

Loading factor at collapse

148

Ω L· A f A l

ereq ~ Kr El

O.I 0.2 0.3 0.4 0.5 0Λ 0.7 0,8 0.9 1 β

Figure 24

Note : The required rotation capacity Rre„ can not be expressed directly from AM because it depends on the shape of the moment diagram.

This approach allows the designer to calculate the required plastic rotation for a load level between the first plastic hinge and the second one.

5.2 Example

Let us apply this method to a particular case of the load configuration given in Paragraph 4.

5200 daN

lllllllllllllllllllll 2.6 daN/cm

llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll Δ Δ Δ , 500 cm , 500 cm , 1000 cm , I I 1

Figure 25

An elastic analysis gives the moment diagram plotted in figure 26. The maximum moment is located a concentrated load location and it exceeds the plastic moment.

-812500

1218700 45500

Figure 26

149

We can calculate : ΔΜ = 1218700 ­ 1137400 = 81300 daN.cm

ΡθΓλ= 1,0 and β = 0,5:

ΔΜ L λ + 1 e

r e q= El 3 2

= a 0 1 7 8 3 r a d

AMLX+ 1 Note : PEP micro program gives : fyeq = ~τ~. j = 0,01786 rad

3 β

In order to check that the plastic moment is not reached in another cross­section after

moment redistribution, the following moment diagram must be added to the first one :

. 0.5 L

"Β Δ

-Δ M = -81300 ZãssstitiÈ

Hinge 1626Γ­

Figure 27

In the most critical section, that is :

On intermediate support : M^ = ­812500 ­ 2 χ 81300 = 975100 daN.cm < M p |

6 HOW TO PERFORM A PLASTIC ANALYSIS USING AN ELASTIC ANALYSIS PROGRAM

6.1 Scope

A plastic analysis of a frame can be performed with the help of a simple elastic

analysis program provided the following assumption is satisfied.

Assumption : The axial force and the shear force must not reduce the plastic

bending moment resistance.

6.2 Methodology

The method consists in a succession of elastic linear analyses. This method is known as

the step­by­step method. Each step is limited by the occurence of a new plastic hinge

: the plastic moment is reached in a new cross­section somewhere in the structure.

150

Figure 28 shows α flowchart of the methodology, considering that the loading

increases in proportion to a load factor a. This method allows the designer to

calculate the required plastic rotation in the plastic hinges just before collapse and

even for each step of the incrementation.

I Geometry ­ Steel Support conditions

Loading j

Initializations i = 0

Moment diagram : Mo = 0 Shear force diagram : Vo = 0 Displacements : Do = 0

ELASTIC ANALYSIS

If i = 0 : Support conditions are not valid

If ¡ > 0 : A plastic mechanism is reached

Results of the elastic analysis i

Moment diagram : m Shear force diagram : ν Displacements : d

Rotations in i h · hinges : θ

Elastic return

Suppression of the hinge

Research of the cross­section for which

Δα = (Mpi ­ Mi)/m is minimum

I 1 = 1+1

Load factor : a i = ai­1 + Δα

I State of the structure at the end of the step

(for the loading a i F)

Moment diagram : Mi = M(i­1) + Δα m Shear force diagram : Vi = V[i­1 ) + Δα ν Displacements: Di = D(i­l)+ Δα d

Rotations tn t h · hinges

Θ Ι = θ Μ + Δ α θ

A new hing· Is Introduced in th · structure

Figure 28

Note : This method is rather easy to apply to a simple struture (Simple frame.

Continuous beam, ... ) . For other cases, a specific program as PEP micro is

necessary.

151

6.3 Example

We propose to make a plastic analysis of a continuous beam with three spans, as

shown in figure 29. For this example, an elastic analysis program can be used, but the

elastic calculations could also be made with the help of a manual method. The results

can be compared with PEP micro results (see Annex).

Cross­section

Steel

IPE 270

S235

30 kN/m 25 kN/m

6 m 8 m 6 m

Figure 29

First step

Elastic calculation

­1368900 ­1233900

L· Å 748450 698610 586450

Figure 30­a

First plastic hinge

The moment is maximum on the second support, so :

1137400­0 Δα] = 1368900

• = 0,8309 and a] = a 0 + Δα

α-ι = 0,8309

Then, the state of the structure can easily be determined at the end of this step,

by multiplying the moment, the shear force, the deflections .... by the load a

So the moment diagram is :

­1137400 ­1025247

621887 580475 487281

Figure 30­b

At the end of the first step, there is no plastic rotation to calculate.

152

Second step

An hinge is introduced in the beam, a new elastic analysis is performed on the

following structure.

30 kN/m 25 kN/m

— 2 5

-1625000

/$.

1350000 1270000 457500

Figure 30-c

We must check that the rotation in the first hinge has the same sign as the

moment : θ = -0,04825 rad. So, Μ.Θ > 0. This means that there is no elastic return.

Four cross-sections must be investigated :

First span Δα = ( 1137400 - 0.8309 χ 748450) / 1350000 = 0,3819

Second span Δα = (1137400 - 0,8309 χ 698610) / 1270000 = 0,4385

Third support Δα = ( 1137400 - 0,8309 χ 1233900) / 1625000 = 0,0690

Third span Δα = ( 1137400 - 0,8309 χ 586450) / 457500 = 1,4210

Therefore, the second plastic hinge is located on the third support and it is

occured for a2 = a] + Δα] = 0,8309 + 0,0690

a 2 = 0.8999

At the end of step 2, the moment diagram is :

-1137400 -1137400

À 4 715037 668105

Figure 30-d

518848

At the end of step 2, the plastic rotation is the rotation in the hinge given by the

analysis : 9 r e q = Δαι θ = 0,0690 χ 0,04825 = 3,3292.IO3 rad

153

Third step

A new hinge is introduced in the beam and a new elastic analysis is performed

on the following structure.

Ill 30 kN/m

l i l í l l i l l l l l l l i l i l Í Ü i l l i l . . .

¡ i l il ill limili 25 kN/m

liilllllllllllllllllllllllllllilllllllllin -â-

0

1350000 2000000 1125000

Figure 30-e

The rotations in the hinges have the same sign as the moments :

1st plastic hinge (second support) : θ = ­0,066069 rad

2nd plastic hinge (third support) : θ = ­0,062368 rad

Now, only three cross­sections must be investigated in order to know where the

last plastic hinge occurs :

First span Δα = (1137400 ­ 715037) / 1350000 = 0,3129

Second span Δα = ( 1137400 ­ 668105) / 2000000 = 0,2346

Third span Δα = (1137400 ­ 518848) / 1125000 = 0,5498

Therefore the last plastic hinge occurs in the middle of the second span for the

load factor a3 = a 2 + Δα2 = 0,8999 + 0,2346

a 3 = 1,1345

­1137400 ­1137400

A A

1031747 1137400 782773

Figure 30-f

Plastic rotation in the hinge on the second support :

ereq = 3.3292.10

3 + 0.2346 χ 0,066069

e r e q = 0,018829 rad

Plastic rotation in the hinge on the third support :

ereq = °<

2 3 4 6 x 0.062368

6 r e q = 0.014631 rad

The plastic analysis is now complete in so far as a plastic mechanism is now

obtained in the second span.

154

6.4 Remarks

This methodology may be applied to some simple structures such as continuous

beams or simple frames. However the designer must take care to the behaviour of the

structure during such a design. Some particular phenomena may occur and they must

be taken into account. We give some examples hereafter.

D Elastic return

During the step-by-step procedure, the sign of the rotation in an hinge may

change. This can be explained by the fact that a plastic hinge takes an elastic

behaviour again. It is usually assumed that the plastic rotation remains. Then the

structure must be modified by suppressing the hinge and the elastic analysis is to

be made again.

2) M-N interaction

In a frame, a reduced moment resistance must be taken into account if the

influence of axial force can not be neglected. Then an interaction curve has to

considered. During the step-by-step method, moment and axial force may vary

in such a way that they remain on the interaction curve.

Moreover, axial elongation and rotation have to be evaluated with respect to

the normality law.

3) Modification of the type of plastic hinge

In a plastic hinge which has been formed without taking into the axial force, the

axial force can increase and it may reach a value that it is not without influence

on the moment resistance anymore.

The designer must keep in mind the real plastic behaviour of a steel frame, and it is

recommended to use a specific program such as PEP micro which account for all

these phenomenas. Figure 31 gives examples of the evolution of the forces in a cross-

section.

/N N/Npl

ν Interaction curve

Isteo'l >Jsieo3l

Elastic " V Return AS.

|Slep2f ^

^^~ rjirojj r

^Usieoíl

. ISlíDSl

Ν Χ Μ/ΜρΙ

Figure 31

155

ANNEX : Continuous beam with three spans

* * * * * * * * * * * * =* *=

C . T . I . C M . * P E P m i c r o * Vers ion 2.02 - 06/94 * * * * * * * * * * * * * *

DATA FILE : C:\PEP2\RON\BEAM.PEP

Date: 14/03/1995 Time: 9:30:03 Continuous beam

Coordinates 1 0.0. 2 600. 0. 3 1400. 0. 4 2000. 0.

incidence 1/1 to 4

characteristics 1 2 3 catalog IPEX 270

supports 1 χ y, 2 3 4 y

loads/member 1 uniform fy -30. loads/member 2 3 uniform fy -25.

analysis plastic stop failure

output after factor 0.

Plot deflected, events

End

GENERAL PARAMETERS Units : daN cm rad Number of nodes : 4 Number of members : 3 Modulus of elasticity : .210E+07 Yield strength : 2350.00 Coefficient de dilatation..: .120E-04

SIGN CONVENTION FOR THE FORCES Μ, Ν, Τ : action of the right part on the left part

in the initial local system of the member consequence : compression : N<0

tension : N>0

ANALYSIS PLASTIC ANALYSIS End calculations factor : COLLAPSE

156

Step: Load factor: .831 Units : daN cm rad

DISPLACEMENTS OF NODES (Global system of the frame)

Node X Disp Y Disp Ζ Rot

1 2 3 4

.O00O0E+00

.OOOOOE+OO

.00000E+00

.00000E+00

.00000E+00

.00000E+00

.OOOOOE+OO , 00000E+00

-.90954E-02 -.25814E-03 .14881E-02 .69430E-02

SUPPORT REACTIONS (Global system of the frame)

Node X Force Y Force

1 2 3 4

.00000E+00

.00000E+00

.OOOOOE+OO

.00000E+00

.55818E+04

.17821E+05

.16108E+05

.45225E+04

Ζ Moment

.OOOOOE+OO

.00000E+00

.OOOOOE+OO

.00000E+00

GLOBAL STATIC EQUILIBRIUM (Global system of the frame)

Horizontal Vertical Overturning 1 Overturning 2 Loads .00000E+00 -.44033E+05 -.42288E+08 -.42288E+08 Reactions .00000E+00 .44033E+05 .42288E+08 .42288E+08

MEMBER FORCES AND ROTATIONS (Initial local system of members)

Member Node/Pos

1

1

2

2

3

3

1 .400

2

2 .500

3

3 .600

4

M

M

M

Ν

.ΟΟΟΟΟΕ+00

.ΟΟΟΟΟΕ+00

.ΟΟΟΟΟΕ+00

.ΟΟΟΟΟΕ+00

.ΟΟΟΟΟΕ+00

.ΟΟΟΟΟΕ+00

.OOOOOE+OO

.ΟΟΟΟΟΕ+00

.ΟΟΟΟΟΕ+00

Τ

-.55818Ε+04 .40002Ε+03 •93728Ε+04

-.84483Ε+04 -.14020Ε+03 .81679Ε+04

-.79396Ε+04 -.46233Ε+03 .45225Ε+04

Μ

.00000Ε+00

.62182Ξ+06 -.11373Ε+07

-.11373Ε+07 •58042Ε+06

-.10251Ε+07

-.10251Ε+07 .48723Ε+06 .00000Ε+00

End.Rot.

.ΟΟΟΟΟΕ+00

.00000Ε+00

.00000Ε+00

.00000Ε+00

.00000Ε+00

.00000Ε+00

Glob.rot.

.ΟΟΟΟΟΕ+00

.00000Ε+00

.00000Ε+00

» » PLASTIFICATION Member 1 Position 1.000 Node 2 Type M Number of plastic events in the frame: 1

157

Step: Load factor: .900 Units : daN cm rad

DISPLACEMENTS OF NODES (Global system of the frame)

Node X Disp Y Disp Ζ Rot

1 2 3 4

.OOOOOE+00

.00000E+00

.OO0O0E+00

.00000E+00

.00000E+00

.00000E+00

.00000E+00 ,00000E+00

-.10629E-01 -.20574E-02 .20562E-02 .72981E-02

SUPPORT REACTIONS (Global system of the frame)

Node X Force Y Force

1 2 3 4

.OOOOOE+OO

.00000E+00

.00000E+00

.00000E+00

.62036E+04

.18993E+05

.17644E+05

.48536E+04

Ζ Moment

.00000E+00

.0000OE+OO

.00000E+00

.OOOOOE+00

GLOBAL STATIC EQUILIBRIUM (Global system of the frame)

Horizontal Vertical Overturning 1 Overturning 2 Loads .OOOOOE+00 -.47694E+05 -.45805E+08 -.45805E+08 Reactions .00000E+00 .47694E+05 .45805E+08 .45805E+08

MEMBER FORCES AND ROTATIONS (Initial local system of members)

mbe:

1

1 1

2

2

3

3

c Node/

1 .400

3 2

2 .500

3

3 .600

4

'Pos

M

M

M

Ν

.OOOOOE+00

.ΟΟΟΟΟΕ+00

.OOOOOE+00

.OOOOOE+00

.00000E+00

.00000E+00

.OOOOOE+00

.00000E+00

.00000E+00

Τ

-.62036E+04 .27566Ξ+03 .99945Ξ+04

-.89988E+04 .12835E+00 .89991E+04

-.86449E+04 -.54580E+03 .48536E+04

M

.00000E+00

.71135E+06 -.11373E+07

-.11373E+07 .66245E+06

-.11374E+07

--11374E+07 .51693E+06 .00000E+00

End.Rot.

.00000E+00

.OOOOOE+00

.00000E+00

.00000E+00

.00000E+00

.00000E+00

Glob.rot.

.00000E+00

.00000E+00

.OOOOOE+00

PLASTIC DEFORMATIONS IN MEMBERS

Member Node/Pos Plast.Rot. Plast.Elong.

1 2 -.33333E-02 .OOOOOE+00

» » PLASTIFICATION : Member 2 Position 1.000 Node 3 Type M Number of plastic events in the frame: 2

158

* Step: 3 Load factor: 1.137 Units: daN cm rad

DISPLACEMENTS OF NODES (Global system of the frame)

Node X Disp Y Disp Ζ Rot

1 2 3 4

.00000E+00

.00000E+00

.OOOOOE+00

.00000E+00

.00000E+00

.00000E+00

.00000E+00

.00000E+00

--15902E-01 -.12473E-01 -.23377E-02 .11692E-01

SUPPORT REACTIONS (Global system of the frame)

Node X Force Y Force

1 2 3 4

.OOOOOE+00

.OOOOOE+00

.00000E+00

.OOOOOE+00

.83406E+04

.23505E+05

.21799E+05

.66344E+04

Ζ Moment

.00000E+00

.OOOOOE+00

.00000E+00

.00000Ξ+00

GLOBAL STATIC EQUILIBRIUM (Global system of the frame)

Horizontal Vertical Overturning 1 Overturning 2 Loads .00000E+00 -.60279E+05 -.57891E+08 -.57891E+08 Reactions .OOOOOE+00 .60279E+05 .57891E+08 .57891E+08

MEMBER FORCES AND ROTATIONS (Initial local system of members)

Member Node/Pos

1

1

2

2

3

3

Η

Η

1 .400

2

2 .500

3

3 .600

4

M

M

M

N

.00000E+00

.00000E+00 •OOOOOE+00

.00000E+00

.00000E+00

.00000E+00

.00000E+00

.00O00E+O0

.OOOOOE+00

Τ

-.83406E+04 -.15174E+03 .12132E+05

-.11373E+05 .12835E+00 .11374E+05

-.10426E+05 -.18963E+03 .66344E+04

M

.00000E+00 •10191E+07

-.11373E+07

-.11373E+07 .11374E+07

-.11374E+07

-.11374E+07 .77338E+06 .00000E+00

End.Rot.

.0OOOOE+00

•OOOOOE+00

.00000E+00

.000O0E+00

.00000E+00

.00000E+00

Glob.rot.

.00000E+00

.00000E+00

.00000E+00

PLASTIC DEFORMATIONS IN MEMBERS

Member Node/Pos Plast.Rot.

1 2

2 3

-.19021E-01 -.14809E-01

PIast.Elong.

.OOOOOE+OO

.OOOOOE+00

» » PLASTIFICATION : Member 2 Position .500 Type M Number of plastic events in the frame:

159

UNSTABLE STRUCTURE

TESTING FOR ELASTIC RETURN AFTER COLLAPSE

UNSTABLE STRUCTURE

CONFIRMED FAILURE ...

REVIEW OF PLASTIC EVENTS

Factor : Factor : Factor :

.831

.900 1.137

Member : Member : Member :

1 2 2

Node : Node : Posit :

2 3

.500

Type: M Type: M Type: M

Total calculation time : .72 seconds

END

160

Annex 9

Document 3263-1-29 Mr. Couchman's thesis (excerpts) (Ref. 7)

(9 pages)

Design of continuous beams allowing for rotation capacity"

161

b) Semi-compact sections (according to EC4):

- Determine 0 a v . This is a function of the sections, spans and loads, and is either given by

equation 7.1 or taken as 2.0.

AVAILABLE ROTATION CAPACITY

ff'T · « I

REQUIRED ROTATION CAPACITY

Oreq

[,-t, f t

jy

: ï ! * i « i A î ï " : - : v . . .'

-LL

-■■'■•"I

- Ü .

M'

PLASTIC or

COMPACT

©a

0a

F.g.7.2

JÍJ

Fig. 7.3

-~-■ ­ ­ — ­ = ■ : ­

SEMI COMPACT

EQ7.1

or

2.0

M M M'

μβι = M

Hg. 7.5

Θ; av

Hg. 7.6 Θ

COMPARISON OF Θ

VERIFICATION OF SECTION

Resistance moments > applied redistributed moments

Figure 7.1 ­ Overview of design method.

162

no

Calculate Θ ¡

Revise section

no

Web or flange slender?

yes

no

Calculate Mp| and Mpl

Calculate MeI and Mei

Web or flange N. semi-compact? /

Determine Δ \

Identify

®req vs. ί(Δ) curve

Determine Δ from Θ req vs. f(A) curve

Allow for η and propping

Calculate applied moments

Resistance moments >

Applied moments

Design method not applicable

yes

Calculate

Determine Θ ■

yes Section OK

Preliminary calculations

J Λ

Available rotation capacity

Required > rotation

capacity

Comparison > of rotation

capacity

Verification ofthe

section

Figure 7.7- Flowchart ofthe detailed procedure used to verify that a given beam can support a given

load.

163

Table 7.1­ Identification of curves in Figure 7.2 tobe used for different combinations of Lu and a.

LLT=

(m)

a=0.50

a=0.75

a=1.00

1.0

1

4

8

1.25

2

6

10

1.5

3

8

11

1.75

4

9

13

2.0

5

10

15

2.5

7

11

17

3.0

8

13

­

3.5

9

15

­

4.0

10

16

­

5.0

11

17

­

6.0

12

­

­

8.0

14

­

­

10.0

14

­

­

Note : values are not given for cases where 0 a is less than 2.0, since plastic or compact sections will

normally exceed this value.

Available rotation capacity

of steel section alone (Θ )

1 2 3 20.0

18.0

16.0

14.0

12.0

10.0

8.0

6.0

4.0

2.0

0.0

/

y

^

ι =

— ■

/

^ ^

6

7

8

9

10

11

12

14

17

13

15

200 250 300

f = 235 N/mm2

350 400 450 500 550 600

Steel section (IPE)

Figure 7.2 ­ Relationship between Θα and steel section. Example for IPE sections with Fe E 235 steel.

Curves showing ©a as a function ofthe steel section are not only useful in simplifying the new design

method, they also enable the influence on ©a of changing the steel section or the proportion ofthe web

depth in compression to be appreciated. For curves 11 to 17 a small change in section size does not

164

produce much change in ©a, whereas this is not the case for curves 1 to 7. For a given curve there is generally an increase in © a as steel section size increases, but because section size is related to span length the same curve would not normally be used for a wide range of sections. Curves are given in Figure 7.2 for values of α equal to 0.50, 0.75 and 1.00. Definitive design curves would consider smaller intervals of a, and because it is possible to group curves this would not lead to a large increase in the number of curves needed.

Θ / ©

0.95 - , • ~ d, = 175 mm — d, = 100 mm - d. = 50 mm For Fe E 355

increase Θ^/®* ̂ y: a.

0.6

0.7

0.8

1.0

200 250 300 350 400 450 500 550 600

Steel section (IPE)

Figure 7.3 - Relationship between Θ^/Θα and steel section. Example for IPE sections.

Figure 7.3 gives values of the adjustment factor ©av/©a as a function of the steel section for IPE sections. This adjustment factor takes into account the influence of composite action on the steel section. Values of © a v /© a depend on the slab reinforcement lever arm (dg), the proportion ofthe web depth in compression (a) and the yield strength ofthe steel (fy). Curves are given for Fe E 235 steel. To allow for the use of Fe E 355 steel, values of ©av/©a c a n De increased by an amount which is independent ofthe section and given for each curve on the figure. For example when α = 0.6 values of © a v /© a can be increased by 0.05, 0.04 or 0.03 depending on the reinforcement lever arm, as noted on the respective curves. In deriving the curves in Figure 7.3 it was assumed that S500 reinforcing steel is used. Separate curves are needed for different families of sections.

165

beam load capacity, so an exact knowledge of the length which influences inelastic rotation is not

required.

7.3.2 Semi­compact sections

It is shown in chapters 4 and 6 that to enable calculation of load capacity for beams with semi-compact

sections it is necessary to calculate either the peak resistance moment Qsíiri2}¿) at which buckling

occurs, or the post-buckling resistance moment which corresponds to an available rotation capacity of

the composite section (©av) of 2.0. © a v and the resistance moment of the section are dependent not

only on the section properties but also on the arrangement of spans and loads.

As explained in chapter 6, two distinct cases can be identified. When the arrangements of spans and

loads give an elastic moment ratio of 1.3 the shape ofthe curve showing required rotation capacity as a

function of moment redistribution is such that, as required rotation capacity increases from 0.0 to 2.0,

the gain in moment redistribution leads to an increase in beam load capacity which is balanced by the

loss in load capacity due to the decrease in section resistance moment. For an elastic moment ratio

greater than 1.3 there is a greater increase in moment redistribution, and for an elastic moment ratio less

than 1.3 there is less increase in redistribution. The consequences of this are that:

- When span and load arrangements give an elastic moment ratio less than 1.3 ultimate load may be

assumed to be reached when the section buckles, so the resistance moment ofthe section is taken as

the peak resistance moment (Μ^χ 1 ) which is given by equation 4.11, 4.13, 4.15 (LRFD), or 4.2

(EC4). The value of © a v which corresponds to the rotation at which buckling occurs is derived from

equation 4.21 and given by:

Θ =2 «ν ■**

M . Λ

Mp l '

— 1 (7.1)

J

©

M,

M,

av

•max

Pi

available rotation capacity ofthe composite section

peak resistance moment

plastic resistance moment

- When span and load arrangements give an elastic moment ratio equal to or exceeding 1.3 ultimate

load may be assumed to be reached when the rotation capacity equals 2.0. The resistance moment

which corresponds to this value of © a v is the post-buckling value M21, given by equation 4.17.

A summary ofthe values of resistance moment and available rotation capacity to be used for different

cases with semi-compact sections is given in Table 7.3.

Table 7.3 - Values of resistance moment and Θ^ to be used for different cases with semi­compact

sections.

Elastic moment ratio

Resistance moment

©av

Mel < 1-3

■^max

f^max'.Mpf)

[Equation 7.1]

Pel * 1.3

M2 '

2.0

In conclusion, using the procedure described above © a v and resistance moment can be determined for a

beam with a semi-compact support section. Because these are the same two parameters which define

166

X¿1 : largest value of λ for which buckling is inelastic

L L J : unrestrained length of beam in hogging

i z : radius of gyration about minor-axis

- Flange local buckling,

M^ivy-tiv-M^) λ - λ ρ 1

ν λ =ι- λ ρΐ^ (4.13)

λ = · (4.14)

The elastic resistance moment again allows for a residual stress of 69 N/mm^ in the flanges. - Web local buckling,

M _ ' = M ^ ' - ( M ^ · - ! ^ · ) ^ λ = ι " λ

Ρ ι ; (4.15)

λ = · (4.16)

d : depth of web. No allowance is made for residual stresses in the steel section when calculating the elastic resistance moment for this failure mode.

The resistance moment ofthe section is taken as the lowest ofthe three values for these failure modes, i.e. it is determined by the most critical of lateral torsional buckling, flange local buckling or web local buckling.

Kubo and Galambos

Kubo and Galambos [33] showed that resistance moments calculated using the LRFD model agree well with peak moments measured in tests. They also considered results from three-point bending tests to show that a linear falling branch on the moment vs. rotation curve may be used to represent post-buckling behaviour. The form of this linear falling branch is given by:

M' Mpi' M θ''

«d'

max

M ' = M p l ' M , ' M p l '

•0.1 ' θ 1 M ^ eel' MP1·

(4.17)

support moment plastic resistance moment, calculated using a stress-block model peak resistance moment, calculated using LRFD model rotation at the support rotation at the support which corresponds to the attainment of Mpi', assuming elastic

rigidity.

A typical moment vs. rotation curve predicted by this model is shown in Figure 4.3.

167

behaviour of a beam with a plastic or compact support section, the same design method can be used for beams with any of these three types of section.

7.4 REQUIRED ROTATION CAPACITY

The graphical representation of required rotation capacity (© req) as a function of moment redistribution (Δ) is considered in detail in chapter 6. The use of © r e q vs. Δ curves is an effective way of allowing for all the parameters which affect the rotation capacity required by a beam to achieve a given moment redistribution. These parameters are: - Elastic moment ratio (μ^) and span type (external or internal). These two parameters affect the

basic form of ©req vs. Δ curves. - Plastic moment ratio (|Xpi), which affects values of moment redistribution but not the form of

®req v s · ^ c u r v e s -- Degree of shear connection and construction method (propped or unpropped). These two parameters

may necessitate modifications to the value of moment redistribution which is given by a © r e q vs. Δ curve. The way in which they are taken into account is discussed in section 7.5.

To identify the appropriate © req vs. Δ curve for a given example, the distribution of "uncracked elastic" moments must firstly be determined. This distribution shows whether a mechanism would form first in an external or internal span, and gives the elastic moment ratio. Knowledge ofthe span type and the elastic moment ratio allows the moment redistribution which corresponds to a required rotation capacity of 1.0 to be found from Figure 7.5. This figure shows moment redistribution as a function of elastic moment ratio, and is basically the same as Figure 6.4 except that specific Compcal results are not presented. Two curves are shown, one for external spans and the other for internal spans, for beams with a plastic moment ratio of 0.57. Curves are given for this value of plastic moment ratio so that they agree with the choice of axis used for the curves shown in Figure 7.6. Any value of plastic moment ratio could have been used to establish these curves provided the two figures are in agreement. Curves shown in Figure 7.5 can be used for a beam with any plastic moment ratio. As stated above, both curves in Figure 7.5 relate to a required rotation capacity of 1.0, but any value of rotation capacity could have been chosen since the purpose of Figure 7.5 is merely to fix both co-ordinates of a point on Figure 7.6. The required rotation capacity value of this point is not important, provided that it corresponds to the correct value of moment redistribution.

Knowing the value of moment redistribution which corresponds to a required rotation capacity of 1.0 for a plastic moment ratio of 0.57 enables the appropriate curve to be chosen from Figure 7.6 a) for external spans, or 7.6 b) for internal spans. These figures are derived from Figures 6.3 and 6.5 respectively, and the form ofthe curves is fully described in section 6.3.1. So that the curves shown in Figure 7.6 are applicable to beams with any value of plastic moment ratio, required rotation capacity is not simply given as a function of moment redistribution, rather required rotation capacity is shown as a function of:

Δ - ( θ . 5 7 - μ ρ Ι ) Δ ' (7.2)

Mpi Δ*

moment redistribution plastic moment ratio constant given in Table 7.4 as a function of μ&\. The derivation of this constant is explained below.

The choice of abscissa comes from the fact that curves shown in Figure 7.6, although being ©ren vs. Δ curves for a plastic moment ratio of 0.57, may be used for beams with any value of plastic moment ratio. This is possible because changes in plastic moment ratio merely lead to a series of parallel curves

168

on a ©req vs. Δ diagram, as discussed in chapter 6. These parallel curves can all be represented by the same curve if the abscissa is revised to allow for the differences in moment redistribution between them.

Moment redistribution (Δ) at ©^=1.0 [%]

50

40 -

30 -

20 -

10

External span Internal span

V/-0.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Elastic moment ratio (μ£ΐ)

Figure 7.5- Moment redistribution vs. elastic moment ratio for required rotation capacity of 1.0.

Firstly it is necessary to calculate the rate of change in moment redistribution with plastic moment ratio, given by the parameter Δ*. Knowing this rate of change, differences in moment redistribution can be calculated for given differences in plastic moment ratio. To establish values of Δ* for various arrangements of spans and loads, the difference in moment redistribution between curves for plastic moment ratios of 0.5 and 0.6 was calculated using equation 6.5. Plastic moment ratios of 0.5 and 0.6 were chosen since they are representative of values likely to be found in practice. Values of Δ* were then calculated by dividing the difference in moment redistribution by the change in plastic moment ratio, i.e. 0.6 - 0.5. It was found that Δ* varies as a function ofthe elastic moment ratio (Hgi) ofthe beam. Results are presented in Table 7.4, which gives Δ* for various values of elastic moment ratio. Linear interpolation is possible to calculate Δ* for other arrangements of spans and loads. It should be noted that values of Δ* given in Table 7.4 are only valid when the elastic moment ratio exceeds the plastic moment ratio, so that redistribution is away from the support.

Having calculated Δ* for different arrangements of spans and loads, the value of plastic moment ratio can be calculated for a given beam from a consideration of span and support resistance moments. Knowing Δ* and the plastic moment ratio, allowable moment redistribution can be calculated from the value of Δ-(0.57-μρΐ)Δ* derived from Figure 7.6.

Table 7.4 - Values ofA*as a function of elastic moment ratio.

H\ Δ*

1.8

65

1.6

69

1.4

74

1.2

82

1.0

92

0.8

108

0.6

133

169

Required rotation capacity (θ„ ς)

10 20 30 40 50 60 70 Δ-(0.57-μρ[)Δ* [%]

a)

Required rotation capacity ( θ ^ )

0 10 20 30 40 50 60 70 Δ-(0.57-μρΙ)Δ* [%]

b) Figure 7.6 - Required rotation capacity as a function o/A-(0.57-Upi )Δ* a) external span, b) internal

span.

Curves shown in Figure 7.6 represent the results of specific calculations using Compcal. These differ from definitive design curves which would be based on a large number of simulations, adopting small load steps, using the procedure described in section 6.3.3 to group the curves. Definitive curves would not contain the irregularities evident in the curves shown for specific cases. However, definitive curves

170

Annex 10

Document 3198-3-3 (LABEIN)

(50 pages)

"Technical report n° 4 : Numerical simulations of class 2 & 3 limit and class 3 & 4 limit"

171

FINITE ELEMENT MODEL

INTRODUCTION

These series of simulations have been carried out with the aim of providing information about class limitations in order to verify the EC3 2&3 and 3&4 class limits. 2&3 class limit has been obtained only for extreme values of b/tf and d/tw, although 3&4 class limit has been verified for full range of b/tf and d/tw values. Both limits have been obtained for S235 and S460 steel grades.

MODELLING

Cross sections

* TPEA500, HEA200, HEAA300, HEA280, IPE300, IPEA400, and HEA450 with flange and web thickness modified * Span: 6 m

Meshing 616 elements and 1913 nodes.

Element type

S8R-Abaqus (parabolic 8-node shell element)

Load application * Constraints: Vertical displacement of the central section upper flange nodes linked together

Boundary conditions * Vertical supports: both ends

* Lateral restrains: both ends, central section, and L ^ in accordance with the specified in CM66

LOAD

RESTRAINS

172

Analysis conditions

* h-tf= cte for full range of values

* tf/tw= 1.5 for the central values of b/tf and d/tw

* Fillet radius modelled by means of the following overthicknesses in each case:

ΓΡΕΑ500

L l = R = 21mm

L2= 4/5 R + tfaverage/2 = 26.805 mm tfaverage= 20.01 mm

HEA200

Ll= 4/5 R + twaverage/2 = 16.26 mm twaverage= 3.72 mm

L2=R = 18 mm

HEAA300

Ll= 4/5 R + twaverage/2 = 25.35 mm

L2= R= 27 mm

HEA280

Ll= 4/5 R + twaverage/2 = 23,2 mm

L2= R= 24 mm

IPE300

Ll= R= 15 mm

L2= 4/5 R + tfaverage/2 = 17,35 mm

IPEA400

L1=R=21 mm

L2= 4/5 R + tfaverage/2 = 22,8 mm

173

HEA450 Ll= 4/5 R + twaverige/2 = 27,35 mm L2= R= 27 mm

These values have been taking into account in order to evaluate Mel and Mpi Mel,Mpl = φ (Ll,L2,Section)

. L1 „ 4—1

L2

RESULTS

Linear analysis

The evaluation of each point in the figures has been performed by searching the tw and tf values which give a critical buckling moment equal to the elastic moment of the section. The calculation of the critical buckling loads has been done assuming a linear behaviour.

The class 3&4 limits for S235 and S460 steels obtained in the simulation are showed in

figures 1 to 3.

The simulation results show that central zone of the class 3&4 limit could be adjusted to a straight line, which brings into line with the EC3 classification method. In this area the EC3

174

limit between class 3 and 4 seems to be on the safe side comparing to the simulation results. A safety coefficient of approximately 2 would be obtained. The web-flange combined buckling mode obtained in the simulation for the central zone of the class 3&4 limit is showed in Fig. 13.

As far as the extreme values of b/tf and d/tw are concerned, the EC3 limit between class 3 and 4 seems to be on the safe side too. There is a quite wide band of values considered class 4 by the EC3 , which would be considered as class 3 according to the simulation results.

On the other hand for high values of d/tw the class 3&4 EC3 limit the results of the simulation show that a local web buckling appears and, therefore, the critical buckling load becomes independent on the flange thickness tf. The local web buckling is showed in Fig. 14. As a consequence of this, an increase on tf does not affect the critical load while the elastic moment rises leading to a lower buckling factor (M^/M^).

For example in the case L4_D26 (IPEA500, tf=28.6 mm, tw=2.6 mm):

Mel= 661390.2 N-m

Vzel= Pe/2= 220500 N M

σ= —*d= 202 MPa Normal stress

V τ= ——= 192 MPa Shear tress

A web

Assuming the following elastic critical buckling stresses as a reference, corresponding to a

rectangular plates with all edges simply supported :

a=b= 0.439 re; i= 0.0026 τα, E= 205000 MPa; v=0.285;

i/b=l; K=21.1; rcrit=(K*E*(l/b)')/(l-v')= 60 MP» i/b=l; K=7.75

C= 202 MPa= 1.2*a

T= 192 MPa= 3.2*1,.

175

The web local buckling seems to be produced by the shear stresses rather than by normal stresses. Then a symmetrical web buckling (see fig. 14) can be expected. This is the situation in the case of profiles with a very small value of the web thickness tw and with a very high value of the ratio tf/tw.

To sum up, the EC3 classification method can be considered on the safe side for all of b/tf and d/tw values.

Non-Linear Analysis

The evaluation of each point in the figures has been performed by searching the tw and tf

values which give a maximum moment equal to the elastic moment of the section for the 3&4

class limit, and equal to the plastic moment for 2&3 class limit.

The stress-strain curves used to characterize the material behaviour for the non-linear analysis

are shown on the figures 6 and 7.

The class 2&3 and class 3&4 limits for S235 and S460 steels obtained in the simulation are

showed in figures 9 to 12.

The results seem to present the same tendency that those obtained from the linear analysis, although additional analysis would be required to deduce the conclusions.

176

LINEAR ANALYSIS

Class 3&4 Limit (S235.S460)

20 40 60 80 100

alprv d/tw-eps

120 140 160

IPEA500_460

HEA200_460

♦ IPEA500_235

o HEA200_235

* Limit 460

ώ Limit 235

— : · EC3

°—- EC3

Figure 1

LINEAR ANALYSIS

Class 3&4 Limit (S235)

■-1 00

20 40 60 80

0.5 d/tw

100 120 140 160

­■ IPEA500_235

­a HEA200.235

­♦ Limit 3&4

­o EC3.235

·* EC3_235

Figure 2

LINEAR ANALYSIS

Class 3&4 Limit (S460)

VO

70 ­r

60 ­­

50

40 ­ ­

b/tf

30 ­­

20 +

10 ­­

0 10 20 30 40

1

50

0.5 d/tw

60 70 80

IPEA500_460

HEA200_460

­♦ Limit 3&4

EC3_460

­* EC3_460

90 100

Figure 3

HEA 200

CLASS 3 AND 4 LIMIT LINEAR ANALYSIS EXTREME VALUES

Point la 2a 3a 4a 5a lb 2b 3b 4b 5b

tw 8

5.5 3.72 2,68

2 8

5,5 3,72 2,68

2

tf 2,25 2,3 2,3 2,5 2,7

3,35 3,35 3,8 4,1

4,85

f y (MPa) alphad/tw 235 235 235 235 235 460 460 460 460 460

8,85 12,88 19,04 26,39

35,325 8,79

12,78 18,84 26,1

34,78

b/tf 88,88 86,95 86,95

80 74,07 59,7 59,7

52,63 48,78 41,23

Load Factor 1,05 1,04

0,977 1,03 1,02 1,06

0,971 0,99

0,943 1,04

h-tf Buckling Mode 144 Antimetrical 144 Symmetrical 144 Symmetrical 144 Antimetrical 144 Antimetrical 144 Symmetrical 144 Antimetrical 144 Antimetrical 144 Antimetrical 144 Antimetrical

Acronym

L1_D23 L2_D23 L3 D25

L1_D335 L2.D38 L3 D41

oo o IPEA500

Point 6a 7a 8a 9a

10a 11a 12a 6b 7b 8b 9b

10b l i b

tf 28,6

20,01 14,5

10 8 5 3

28,6 20,01

14,5 10 8 5

tw 2,6 2,3 2,1 1,9

1,75 1,65 1,75 3,45 3,1

2,85 2,6 2,5

2,85

fy (MPa) ι 235 235 235 235 235 235 235 460 460 460 460 460 460

alpha-d/tw 87,28

100,54 111,42 124,34 135,57 144,69

137 65,78 74,59 82,1

90,86 94,9

83,77

b/tf 7

10 13,8

20 25 40

66,66 7

10 13,8

20 25 40

Load Factor 1,04 1,03 1.06 1,07 1,01

0,978 0,955

1,05 1,05 1,07 1,04

0,988 0,962

h-tf Buckling Mode 482,5 Antimetrical 482,5 Antimetrical 482,5 Antimetrical 482,5 Antimetrical 482,5 Antimetrical 482,5 Antimetrical 482,5 Antimetrical 482,5 Antimetrical 482,5 Antimetrical 482,5 Antimetrical 482,5 Antimetrical 482,5 Antimetrical 482,5 Antimetrical

Acronym L4_D26 L5_D23 NL6 D21

L4.D345 L5_D31 L6 D285

Figure ·

CLASS 3 AND 4 LIMIT LINEAR ANALYSIS CENTRAL VALUES

oo

SECCIÓN IPEA500 HEA200 HEAA300 HEA280 IPE300 IPEA400 HEA450 IPEA500 HEA200 HEAA300 HEA280 IPE300 IPEA400 HEA450

fy (MPa) 235 235 235 235 235 235 235 460 460 460 460 460 460 460

tf(mm) 2.75 2.7 4.25 4.0 2.15 2.25 4.5 4.75 4.25 6.8 6.5 3.5 3.75 6.75

tw (mm) 1.8 1.8 2.83 2.66 1.43 1.5 3.0 3.16 2.68 4.53 4.33 2.33 2.5 4.5

tf/tw 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5

b/tf 72.32 74.07 70.58 70.0 69.76 80.0 66.66 42.1 47.05 44.11 43.07 42.85 48.0 44.44

0.5-d/tw 133.26 39.25 47.39 47.55 100.4 127.48 69.08 75.59 26.05 29.32 28.92 61.33 76.25 45.8

Load Factor 0.95 0.98 1.02 0.97 1.09 1.08 1.05 0.94 0.98 1.09 1.08 1.09 1.02 0.99

Buckling Mode Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical Symmetrical Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical

Figure 5

IN^IN­UINCMK /MNMLTOIÖ

ty= 235 ΜΡα

400η

350 ­

300·

250 ­

β (ΜΡα) 200 ­

150 ­

100 ·

50 ­

0<

­ /

ι ■

ι

0.05 αϊ 0.15 Strain

02 025 0.3

Figure 6 182

NON­UNEAR ANALYSIS

600 τ

500­­

400

Stress (ΜΡα) 300

200

100 +

0

0,05

fy= 460 ΜΡα

-«—■-

0,1 0,15

Strain

02

7

0,25 0,3

183

CLASS 2 AND 3 LIMIT CLASS 3 AND 4 LIMIT

NON-LINEAR ANALYSIS

IPEA500

Point 6a 7a 8a 6a 7a 8ά 6b 7b 8b

6'b 7'b 8'b

4̂ .

HEA200

Point l a 2a 3a lá 2á 3á l b 2b 3b l ' b 2'b 3'b

tf (mm) 28,6

20,01 14,5 28,6

20,01 14,5 28,6

20,01 14,5 28,6

20,01 14,5

tf(mm) 1,7 1,8

2 2,6

3 2,8

2,95 3,6 4,3 4,2 4,4 4,8

tw (mm) 3.4

2,55 2

3,65 2,7 2,1

3,575 2,875

2,45 3,85

3 2,57

tw (mm) 5,5

3,72 2,68

5,5 3,72 2,68

5,5 3,72 2,68

5,5 3,72 2,68

fy (MPa) 235 235 235 235 235 235 460 460 460 460 460 460

f y (MPa) 235 235 235 235 235 235 460 460 460 460 460 460

Mel ( tm) 68,3 49,2 36,6 68,6 49,3 36,6

134,2 96,8 72,4

134,7 97,1 72,7

Mel ( tm) 2,1

2,03 2,05 2,71 2,82 2,59

5,8 6,33 7,05 7,43 7,37 7,71

Mpi ( tm) 73,9 52,4 38,6 74,3 52,6 38,8

145,2 103,5 76.9

146.2 103,8 77,3

Mpi ( tm) 2,36

2,2 2,21

3 3,05 2,77 6,38 6,83 7,52 8,14 7,94 8,21

Mmax (t-m) 68,3 49,9 36,4 73,9 52,9 38,9

134,7 96,3 72,3

146,5 103,1 77,5

Mmax (tm) 2,18 2,08 2,03

3 3,09 2,75 5,93 6,35 7,05 8,15 7,91 8,2

alpha-d/tw . 66,75

90,68 117

62,17 85,64

111,42 63,48 80,43 95,51 58,94 77,08 91,05

alpha-d/tw 12,93 19,11 26,49 12,85 18,95 26,34 12,82 18,87 26,06 12,71 18,76 25,97

b/tf 7 10

13,8 7 10

13,8 7 10

13,8 7 10

13,8

b/tf 117,64 111,11

100 76,92 66,66 71,42 67,79 55,55 46,51 47,62 45,45 41,66

Buckling Mode Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical

Buckling Mode Antimetrical Symmetrical Antimetrical Symmetrical Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical Antimetrical

Acronym < NL4_D34

NL5.D255 NL6_D2

NL4.D365 NL5.D27 NL6.D21

NL4.D357 NL5_D287 NL6.D245 NL4.D385

NL5_D3 NL6_D257

Acronym < NL1_D17 NL2.D23 NL3.D2

NL1_D26 NL2.D3

NL3_D28 NL1_D295 NL2_D36 NL3_D43 NLLD42 NL2.D44 NL3.D48

2lass Lirr 3&4 3&4 3&4 2&3 2&3 2&3 3&4 3&4 3&4 2&3 2&3 2&3

wlass Lirr 3&4 3&4 3&4 2&3 2&3 2&3 3&4 3&4 3&4 2&3 2&3 2&3

Figure 8

CLASS 2 A N D 3 LIMIT

CLASS 3 A N D 4 LIMIT

EXTREME VALUES

Non­Linear Analysis

120

100 ­ ­

oo

80

b/tf 60

40

­■ IPEA500_el_235

o IPEA500_pl_235

­ ♦ — IPEA500_el_460

­o IPEA500_pl_460

~A HEA200_pl_460

­ * HEA200_el_460

­ · HEA200_pl_235

­ ° HEA200_el 235

20 ­ ­

o—rj=*==i5

20 40 60

d/tw

80 100 120

Figure 9

120 ­τ

CLASS 2 AND 3 LIMIT

CLASS 3 AND 4 LIMIT

Non-Linear Analysis (fy= 235 MPa)

oo

100 ­ ­

80 ­ ­

b/tf 60

­■ IPEA500_el_235

­a IPEA500_pl_235

­♦ HEA200_el_235

­o HEA200_pl_235

40 ­ ­

20 ­ ­

0 20 40 60

0.5-d/fw

80 100 120

Figure 10

oo -J

70 -r

60 --

50

40

b/tf

30

CLASS 2 AND 3 LIMIT CLASS 3 AND 4 LIMIT

Non-Linear Analysis (fy= 460 MPa)

-■ IPEA500_el_460

-° IPEA500_pl_460

HEA200_el_460

-o HEA200_pl_460

20 --

10 --

0

10 20 30 40 50

0.5-d/tw

60 70 80 90 100

Figure 11

> 3 Cl Χ

00

D; o

00 oo

b/tf

90 ­r

80 ­

70

60

50

40

30 ­

20 ­

10

0

CLASS 2 AND 3 LIMIT

CLASS 3 AND 4 LIMIT

EXTREME VALUES

Non­Linear Analysis

20 40 60 80

d/tw

100

­■ IPEA500_el_235

­° IPEA500_pl_235

­♦ IPEA500_el_460

­o IPEA500_pl_460

-* HEA200_pl_460

­Δ HEA200_el_460

­ · ' HEA200_pl_235

­° HEA200_el 235

120 140 160

Ensidesa/Labein Figure 12

0 0 VO

'T)

Η·

«etnei iiïffgSiia^ïïïï!!»»· '·?,. ­ ­'ynAia^as«Kîaswp

TOSSUSI

2 1

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o TI 00.

ABACUS

VO

\ 3

¡PLACEMENT MflGH IF IC AT I OH FACTOR = EIGEHMODE 3 EIGENVALUE = 1 . 0 4 ABAQl/S VERSION! 5 . 4 - 1 DATE! 2 S - M A Y - 9 5 T I M E ! 1 6 ¡ Í 8 ¡ 5 6

STEP 1 INCREMENT 1

ABACUS

VO

»LACEMENT MAGNIFICATION FACTOR =

EIGENMODE 2 EIGEHVALVE = 1 . 0 3

ABAQVS VERSION! 5 . 4 - 1 DATE! 2 $ - M A Y - 3 5 TIME! 0 9 i 1 0 : 5 0

STEP 1 INCREMENT 1

ÄBAQUS1

VO U l

¡PLACEMENT MAGNIFICATION FACTOR =

EIGEHMODE 3 EIGENVALUE = 1 . 0 S

ABAQUS VERSION! 5 . 4 ­ 1 DATE! 2 6 ­ M A Ï ­ 9 S

STEP 1 INCREMENT 1

Τ I ME !

PÜ

ABACUS

t

¡PLACEMENT MAGNIFICATION FACTOR = 2 0 0 .

EIGENMODE 2 EIGENVALUE = 1 . 0 4 ABAQUS VERSION! 5 . 4 - 1 DATE! 2 6 - M A V - 9 5 TIME! 1 3 : 0 1 ¡ 2 9 STEP 1 INCREMENT 1

VO

¡MENT MAGNIFICATION FACTOR =

Î E J T M O D B 1 EIGENVALUE = 1 . 0 4

ABAQUS VERSION! 5 . 4 ­ 1 DATE! 2 2 - Μ Α Ϋ - 9 5 T I M E ! 13 ! 1 0 ! 4 9

STEP 1 INCREMENT 1

ABAQUS

ABAQUS!

VO ON

ίΝΤ MAGNIFICATION FACTOR = 2 5 0 ,

¡NMODE 1 EIGENVALUE = 1 . 0 5

ABAQUS VERSION! 5 . 4 ­ 1 DATE ! 2 3 ­ Μ Λ Υ ­ 9 5 TIME! 1 1 ; 0 9 ¡ 2 3

STEP 1 INCREMENT 1

Klaifc Ι Ε § Ε Ι ^

80

<v­

3

M O M E N Τ

t η

m

60

40

20

0 . 0

&* ^;"-|Ví-jf» O t * · v±íl-^.lsií¡£íí^ í.vf.

1 ■ -UL.

o-

' *, +

-H- *+H + *

L 4 _ D 2 6

N L 4 _ D 3 4

NL4_D3 6 5

MEL D2 6

-A MEL_D3 4

-Φ MPL D3 6 5

1 . 0 1 . 5

ANGLE ( d e g )

* ^ Ì 5 ­1 ΐ;.'

- O

Tik

2 . 0 2 . 5

150

VO

oo

M O M E N Τ

t n

m

100

50

Χ­ι 1

0"— 1 1

¿Λ

o—

—κ I I

—o I ' ι

A Í J »

— e >

L4_D345

NL4_D3 57

NL4 D3 85

MEL_D3 4 5

MEL_D3 5 7

MPL D3 85

2 . 3

ANGLE ( d e g )

s

0 . o 0 . 5 1 .0 1 .5 ANGLE ( d e g )

2 .0 2 .5

^ ^ » ^ y r ^ ^ ^ w K » ^ ^ . . . * * ^ ^ ^ üuaacüPc«iu0oa»K»oflcaaccx»xj5aiiaQflaaui0Stt^ —ttmø&a*emaaammoa

M O M E Ν Τ

η *

1 0 0

80

60

m 4 0

20

Χ Κ L 5 _ D 3 1

-1 h NL5 D2 87

O O NL5_D3

I■■ - . ! MEL_D31

Δ Δ MEL_D2 8 7

<S> e> MPL D3

ANGLE ( d e g )

40

K) o

M O M E N Τ t η m

30

20

10

0. 0.0 0. 5

X-

13-

-X NL6_D2 -+ NL6 D21 -θ MEL_D2 , < MPL D21

1.0 1.5 ANGLE (deg)

2 .0 2 .5

K) o SJ

M O M E N Τ

t η

m

80

60

40

20

1

>^ ­Χ L6_D2 85

­+ NL6 D2 4 5

Θ ö NL6_D2 57

li i ! MEL_D28 5

Δ Δ MEL_D2 4 5

O <0 MPL D2 5 7

I

2 . 3

ANGLE (deg)

s

M o M E N Τ t η * m

Δ-

X L1_D2 3 NLl_Dl7

O NL1_D2 6 ti MEL_.D2 3 A MEL_D17 O MPL D2 6

4. 6 ANGLE (deg)

8. 10.

M O M E N Τ ΐ η Λ m

L l _ D 3 3 5 NL1_D2 95 NL1_D42 MEL_jD3 35 MEL_D2 95 HPL D42

4 . 6 . ANGLE ( d e g )

■*- " ■A—■* —-*- -—* ­ ' ­*[­ ­~τ — r r - v ·

.....>» ̂ ..,..—..... __. ■ ­

- - — — — - — " - - r n r i

3 . -dfe

A-)-. ■R—. -Θ- — i . r v -

M O M E Ν Τ

η *

m

¿r

1 .

■frH-4-

2 .

Χ Χ L 2 _ D 2 3

Η h NL2 Dl 8

θ θ N L 2 _ D 3

i, . MEL_D2 3

Δ Δ MEL_D18

O O MPL D3

4 . 6

ANGLE ( d e g )

8 . IO

8. <r=-

o

M 0 M E N Τ

t η

m

6.

7«tî

v­.'S­i,

^-φ-Λί

X L2_D38 NL2 D3 6

O NL2_D44 : ι MEL_D3 8 Δ MEL_D3 6 O MPL D44

4. 6 ANGLE (deg)

10

2 . 8

K)

3

1 0 . ANGLE ( d e g )

# ■

0

O 00

M 0 M E N Τ

η

m

¿r

6.

0 .

Χ L 3 _ D 4 1

-f NL3 D43

-Ό N L 3 _ D 4 8 : ι MEL_D41

-Δ MEL_D4 3 -O MPL D4 8

4 . 6

ANGLE ( d e g )

1 0

I

Beete

5

2

5 1

r

to K)

K) Ul

to

to

to 5\

to ι — >

^1

IO oo

to VO

to to o

Annex 11

(24 pages)

Exploitation of Labein numerical simulations (ProfilARBED) presented in Document 3198-3-3 (Annex 10)

221

ANNEX 11 Exploitation of Labein numerical simulations (ProfilARBED) presented in Document 3198-3-3 (Annex 10)

1. Tables exploiting numerical simulations about borders between class 3 & 4 cross­sections (Mei):

In following 4 tables, different calculations have been carried out about the cases simulated for evaluation of borders between class 3 & 4 cross-sections submitted to My (see Annex 10; document 3198-3-3). The details of formulas used in those tables are presented hereafter :

1.1 Values of geometrical and statical characteristics (table 1) :

- ¿PARE = h - tf - 2L-2 ,

- ocd/tw (LABEIN) = 0,5(h - 2tf )/tw,

- d/tw (PARE) = dpARE / tw = (h - tf - 2L2)/tw,

- c = b/2,

- Li and L2 = length of overthickness (a) (see Figure 1),

tf

ψ

+ 2a

­ ­

/

' ν

Ν

s /

ia K J ^

'

\

a ν

a .

" T '

1 ι

Ί 5

- - — - -7 i

L2

U + 2a a

h

Figure 1 : Modelization of root fillet

- R = radius of root filet,

a =

overthickness of finite elements in the web and in the flanges for modelization of web-flange connections with root fillets (see figure 1) (Ref. 19) :

-(tf+tw-2.L2-4L1)-J(tf + tw-2L2-4.L1)2-8. -2.L1.tf + l,5.tf.tw-twl2 + 2.tfJv + t w . R + R 2 / ^ ^ ]

sectional area of the cross-section :

tf A = 2btf + ( h - 2 t f )tw +4L1a+2a(2L1 - t w ) + 4a(L2 — -—a)

shear area for the load parallel to the web

Avz = (h-tf)tw

222

radius of gyration about minor axis z-z

iz = V

I Z/A

- moment of inertia according to the minor axis z-z :

il6L3a + t f b 3 + ( t w + 2 a ) 3 Ì L 2 - a - ^ ì ì 3

Iz - * ­6 " ^ + ̂ ( h - t f - 2 L 2 ) ,

- moment of inertia according to the major axis y-y :

I y = ^ ( h - t f - 2 L 2 ) 3+ I ( . w + 2 a ) ( L 2 - í t - a )

+ ÍL1(t f + 2a)3 + L1(tf+2a)(h-tf)2

- elastic section modulus according to the major axis y-y :

2IV

WeLy­γ.

- plastic section modulus according to the major axis y-y :

WpLy =2aL1(h + a) + tfb(h-tf) + 2aL1(h-2tf - a )

+ (2L2 - t f -2a)(tw +2a)-(2h-3tf - 2 a - 2 L 2 )

+ ^ t w ( h - t f - 2 L 2 ) 2

1.2 Values about bending resistance of cross­section (table 2):

- elastic bending moment resistance about major axis y-y :

Mei.y = Wei.y fy (= maximum bending moment applied to the cross-section = Ms<j),

- plastic bending moment resistance about major axis y-y :

Mpl.y = Wpl.y fy ,

- concentrated load related to Meiy for the 3-point bending beam :

Pei = 4Mei.y / L (with L = beam span = 6m),

- shear force related to Pei :

Vei = Pel / 2 (= maximum shear force applied to the cross-section = Vsd),

223

1.3 Values about elastic critical stresses on isolated flange (table 2):

- maximum normal stress in flange :

Œmax = fy >

- maximum critical slendemess of isolated flange (for a rectangular plate with one simply supported edge) :

π ζ Ε

tf V 1 2 ( l - v z ) a m a x

, with ko = 0,43,

- critical normal stress in isolated flange (for a rectangular plate with one simply supported edge) :

π 2 Ε f t ^ 2

1 2 ( 1 - v z ) -ï- ,withko = 0,43,

*v.C

1.4 Values about elastic critical stresses on isolated web (table 3) :

- maximum normal stress in web :

( h - t f - 2 L 2 )

^max — *y

maximum shear stress in web

Vel -max t w ( h - t f - 2 L 2 ) '

maximum crital slendemess of isolated web (for a rectangular plate with two simply supported edges) :

(

= minimum Lw

π ζ Ε π E kT

12(1 - ν 2 ) o m a x Ì l 2 ( l - v 2 ) x m a x

.with 1^ = 2 3 , 9 , ^ = 9,34

and d = h - t f - 2 L 2 ,

critical normal stress in web (for a rectangular plate with two simply supported edges) :

. 2 C / . \2 Gcrit - k α

π E 2 f-T-1 ,withk{J = 23,9(a = l>2 /3) , 12(1 - v z ) V d J 12a ­v*y

critical shear stress in web (for a rectangular plate with two simply supported edges) :

π 2 Ε ^crit - ^τ

­2c / j . \ 2

12(1-V z ) ■3L| , with kt = 9,34 (<x=l) ,

k d

224

1.5 Values about shear buckling of web according to post­critical method (see Eurocode 3) (table 3) :

- web slendemess :

d ( h - t f - 2 L 2 ) if = ; — < 69 , then no shear buckling occurs in web.

,we t w J Ç - web slendemess for post-critical method :

Xy, = ' w ,— , with kT = 5,34 and d = h - tf - 2L2-37,4 ε ^/kT

- simple post-critical shear strength of the web :

fy zba=~7x , ΐ ίλνν^Ο,δ,

f _ _ = -^ ( ΐ ,5 -0 ,625λ ν ) , if 0,8 < λ* £1,2,

fv 0 9 = -j=^- , i fXw>l ,2 ,

V3 Aw - design shear buckling resistance of the web :

Vba.Rd = t w ( h - t f - 2 L 2 ) - ^ - , with TMI = 1,0, ΎΜ1

- design plastic moment resistance of a cross-section consisting of the flanges only : f

Mf.Rd = b tf (h - tf )—— , withYMO = 1,0, ΎΜΟ

- reduced design shear buckling resistance of the web allowing for bending moment interaction :

i f M S d / M f -Rd > 1,0 and, i f V S d / V b a . R d > 0 , 5 , then, VM-ba_Rd * 0

withyM 1 = 1,0, _ vba.Rd I li MSd~Mf\Rd 1 - JU lJsa +1 M p l y - M f . R d J

1.6 Values about bending and shear resistance of the cross-section (table 3):

- plastic shear resistance of cross-section (for shear parallel to web) :

Vpi.z.Rd= VZ%m with 7M0 = 1,0,

225

- reduced design plastic resistance moment Mv.Rd allowing for the shear force

i f Vz.Sd/ vpl .z .Rd> 0 ' 5

then Myjtø Φ 0

wpi.y -2

Vz.Sd

A- U

' Vpl.z.Rd

•1 lvz

4t w ΎΜ0

,withyM O=l ,0 ,

1.7 Values about lateral-torsional buckling (table 4) :

- CM66 rules for lateral-torsional buckling (Ref. 18) :

In the case of the member in bending which contains at one of its ends the plastic hinge allowing for redistribution of bending moments, the conditions of lateral restraints given by §5.21 of Additif 80 - DPU Ρ 22-701 (CM66) should be satisfied in the neighbourhood of the plastified section to avoid the lateral-torsional buckling (LTB).

If the moment varies linearly along a member with the length Lj which is free to buckle laterally, the conditions are as follows :

^ ·<35ε

h_< < (60-40 ψ) ε

if

if

0,625 < ψ < 1

-1 < ψ < 0,625

where : Lj is the length of the member or of the portion of that member where the linear distribution of bending moment is applied,

ψ is the ratio of bending moments at both ends of the member or of the portion of that member (-1 < ψ < 1),

iz is the radius of gyration about minor axis (Λ/1 Ζ / A ), with : Iz - moment of inertia about minor axis of the cross-section,

A - total area of the cross-section.

In tables, we have :

. in h/ize , L = 3 m because lateral restraints were introducced at least at both supports and at concentrated load position,

. if 60ίζε > 3 m , then no need of supplementary lateral restraints,

3 m

ψ = 0

226

if 60ίζε < 3 m, then the length for which more lateral restraints should be introduced is ( L L R ) ^ = 3- 60ίζε

h å J» 4

ι ; L 6 0 i z £ L (klOreal A

y ( C M 6 6 ) - ' ΏΠΧ] (LLR) = maximum length between lateral restraints for ψ (CM66) according to CM66

rales, (if 60ίζε < 3 m). (LLR)max s h o u l d t*5 - (LLR)real t 0 a v o i d L T B P rob lems ·

Eurocode 3 rules for lateral-torsional buckling (Ref. 1) :

. elastic critical moment of cross-section for lateral-torsional buckling :

Mcr (see Ref. 1, Annex F) with either LLR = 3 m or LLR = (LLR) ^ (if 60ΐζε < 3 m).

. non-dimensional slendemess for LTB : Mply

λ LT = —í—i- , (see Ref. 1, 5.5.2(5)), should be < 0,40 if sufficient lateral restraints are V Mcr

ensured.

. design buckling resistance moment of member in bending :

Mb R d = X L T pLy , (see Ref. 1 (5.48)), with YMI = 1,0. Y Ml

1.8 Comments on results in tables :

(1) The very conservative approach of elastic buckling theory of separate part of cross-sections, highlights critical parts of different simulated cases (see in table 2 : ratios of maximum applied stresses ( σ ^ and/or xma3[) and elastic critical stresses (σ,̂ , and/or Tcrit)) :

- high ( σ ^ / acrit ) (> 2) values on isolated flange for S235 simulations n° "IL" to " 13L" and for S460 simulations n° "IH" to "13H",

- high (cmn I acrit) and (xmax / τ^) values (> 1,15) on isolated web for S235 simulations n° "11L" to "19L" and for S460 simulations n° "14H" to "18H".

(2) Per definition the classification of cross-sections (as presented in Eurocode 3) is determined by local buckling only induced by normal stresses (σ) in part(s) of cross-sections. The resistance of cross -sections to shear forces has to be checked further: shear resistance Vp^, shear buckling V^^if too high web slendemess meaning d/(t,£) > 69 : see table 3), interaction with bending moment resistance,...

(3) Finite elements numerical simulations do not separate each phenomenon and take into account all effects interacting together. According to the comment of clause (2), the following

227

simulations should be rejected to evaluate the border between class 3 and class 4 cross-sections because shear effects show a big influence or a clear predominancy on the failure mode :

- two S235 simulations (n° "18L" and "19L") do not fulfill requirements of Eurocode 3 about shear resistance of the cross-sections (see (VSd/VpLRd) ratios in table 3),

- S235 simulations n° "11L" to "19L" and S460 simulations n° "13H" to "18H" do not fulfil requirements of Eurocode 3 about shear buckling resistance Vte Rd interacting with bending moment (see (VsJVm» Rd) ratios in table 3).

For high values of d/tw the results of the simulation show that a local web buckling appears and, therefore, the critical buckling load becomes independent on the flange thickness tf. As a consequence of this, an increase on tf does not affect the critical load while the elastic moment rises leading to a lower buckling factor (Mcrit / Mei).

The web local buckling seems to be produced by the shear stresses rather than by normal stresses. Then a symmetrical web buckling can be expected. This is the situation in the case of profiles with a very small value of the web thickness tw and with a very high value of the ratio tfAw

(4) Against lateral-torsional buckling (LTB), lateral restraints have been introduced at both supports and at concentrated load position (mid-span). For several simulations supplementary lateral restraints were introduced because of requirements of CM66 rales (Ref. 18) : cases where 60ίζε is lower than 3 m (see table 4). In spite of those conditions, several numerical simulations are sensitive to LTB and should be checked with LTB problems according to :

- CM66 (Ref. 18) : (LLR)max < (LLR)real in table 4, for S235 simulation n° "10L" and, for S460 simulations n° "IH", "2H" and "10H",

- Eurocode 3 (Ref. 1) : λυτ > 0,40 in table 4, for S235 simulations n° "16L" and "17L", for S460 simulations n° "7H", "10H" and "12H".

i f 6 0 U > 3 m :

Β + if 60U < 3 m

y ι Lateral restraints £—Xr

A Β 3 m ¥ 3m

X X X X x-Β

eoi^

3 m

6 0 ^

3 m

228

The following simulations should be rejected according to Eurocode 3 rules because the applied bending moment MSd (= Md) is greater than the allowed bending resistance considering the effect of lateral-torsional buckling M,^ (see (MSd/MbRd) ratios in table 4):

- for S235 simulations n° "16L" and "17L", - for S460 simulations n° "7H" to "10H" and "12H".

But if more lateral restraints were introduced along the simulated beams LTB should not be anymore a predominant failure mode (λ^. < 0,40) and the results could be improved: local buckling - which interacts with LTB conditions - should be related to higher load level (Md) with same values of ((c/tf), (d/tj). In fact because of sensitivity to LTB, present results are conservative and better results (greater values of ((c/tf), (d/tj)) could be expected if better provisions were taken against LTB.

(5) All the simulated points for the borders between class 3 & 4 cross-sections are plotted in c i d i

graphs — = f presented in chapter 2. tf£ [t^ej

229

Points

1L

2L

3L

4L

5L

6L

7L

8L

9L

10L

11L

12L

13L

14L

15L

16L

17L

18L

19L

1H

2H

3H

4H

5H

6H

7H

8H

9H

10H

11H

12H

13H

14H

15H

16H

17H

18H

Simulations

LABEIN

1a

2a

3a

4a

5a

14a

15a

16a

19a

17a

18a

13a

12a

11a

10a

9a

8a

7a

6a

1b

2b

3b

4b

13b

15b

14b

5b

18b

16b

12b

17b

11b

10b

9b

8b

7b

6b

h

mm

146,3

146,3

146,3

146,5

146,7

146,7

280,3

261,0

423,5

291,5

387,3

485,3

485,5

487,5

490,5

492,5

497,0

502,5

511,1

147,4

147,4

147,8

148,1

148,3

263,5

282,8

148,9

425,8

292,8

487,3

388,8

487,5

490,5

492,5

497,0

502,5

511,1

b

mm

200,0

200,0

200,0

200,0

200,0

200,0

300,0

280,0

300,0

150,0

180,0

198,9

200,0

200,0

200,0

200,0

200,1

200,1

200,2

200,0

200,0

200,0

200,0

200,0

280,0

299,9

200,0

300,0

150,0

200,0

180,0

200,0

200,0

200,0

200,1

200,1

200,2

tw

mm

8

5,5

3,72

2,68

2

1,8

2,83

2,66

3

1,43

1,5

1,8

1,75

1,65

1,75

1,9

2,1

2,3

2,6

8

5,5

3,72

2,68

2,68

4,33

4,53

2

4,5

2,33

3,16

2,5

2,85

2,5

2,6

2,85

3,1

3,45

tf

mm

2,25

2,3

2,3

2,5

2,7

2,7

4,25

4

4,5

2,15

2,25

2,75

3

5

8

10

14,5

20

28,6

3,35

3,35

3,8

4,1

4,25

6,5

6,8

4,85

6,75

3,5

4,75

3,75

5

8

10

14,5

20

28,6

fy

N/mm2

235

235

235

235

235

235

235

235

235

235

235

235

235

235

235

235

235

235

235

460

460

460

460

460

460

460

460

460

460

460

460

460

460

460

460

460

460

Q PARE

mm

108,0

108,0

108,0

108,0

108,0

108,0

222,0

209,0

365,0

254,6

339,4

428,9

428,9

428,9

428,9

428,9

428,9

428,9

428,9

108,0

108,0

108,0

108,0

108,0

209,0

222,0

108,0

365,0

254,6

428,9

339,4

428,9

428,9

428,9

428,9

428,9

428,9

Geometrical and statical characteristics

oc.d/tw

LABEIN

8,85

12,88

19,04

26,39

35,325

39,25

47,39

47,55

69,08

100,4

127,48

133,26

137

144,69

135,57

124,34

111,42

100,54

87,28

8,79

12,78

18,84

26,1

26,05

28,92

29,32

34,78

45,8

61,33

75,59

76,25

83,77

94,9

90,86

82,1

74,59

65,78

d/tw

PARE

13,5

19,6

29,0

40,3

54

60,0

78,4

78,6

121,7

178,0

226,3

238,3

245,1

259,9

245,1

225,7

204,2

186,5

165,0

13,5

19,6

29,0

40,3

40,3

48,3

49,0

54,0

81,1

109,3

135,7

135,8

150,5

171,6

165,0

150,5

138,4

124,3

b/t,

88,88

86,95

86,95

80

74,07

74,07

70,58

70

66,66

69,76

80

72,32

66,66

40

25

20

13,8

10

7

59,7

59,7

52,63

48,78

47,05

43,07

44,11

41,23

44,44

42,85

42,1

48

40

25

20

13,8

10

7

c/t,

44,44

43,475

43,475

40

37,035

37,035

35,29

35

33,33

34,88

40

36,16

33,33

20

12,5

10

6,9

5

3,5

29,85

29,85

26,315

24,39

23,525

21,535

22,055

20,615

22,22

21,425

21,05

24

20

12,5

10

6,9

5

3,5

h­t,

mm

144

144

144

144

144

144

276

257

419

289,3

385

482,5

482,5

482,5

482,5

482,5

482,5

482,5

482,5

144

144

144

144

144

257

276

144

419

289,3

482,5

385

482,5

482,5

482,5

482,5

482,5

482,5

L1

mm

16,26

16,26

16,26

16,26

16,26

16,26

25,35

23,2

27,35

15

21

21

21

21

21

21

21

21

21

16,26

16,26

16,26

16,26

16,26

23,2

25,35

16,26

27,35

15

21

21

21

21

21

21

21

21

L2

mm

18

18

18

18

18

18

27

24

27

17,35

22,8

26,805

26,805

26,805

26,805

26,805

26,805

26,805

26,805

18

18

18

18

18

24

27

18

27

17,35

26,805

22,8

26,805

26,805

26.805

26,805

26,805

26,805

R

mm

18

18

18

18

18

18

27

24

27

15

21

21

21

21

21

21

21

21

21

18

18

18

18

18

24

27

18

27

15

21

21

21

21

21

21

21

21

a

mm

2,00

1,85

1,75

1,71

1,68

1,67

2,40

2,07

2,18

1,10

1,56

1,43

1,44

1,50

1,61

1,69

1,90

2,21

2,84

2,24

2,02

1,94

1,88

1,89

2,40

2,75

1,89

2,40

1,19

1,53

1,66

1,53

1,65

1,75

2,02

2,41

3,20

A

mm2

2382

2031

1767

1695

1676

1646

4009

3452

4603

1251

1767

2330

2413

3172

4431

5311

7231

9562

13203

2847

2470

2388

2355

2416

5324

6062

2559

6612

1922

3796

2701

3749

4796

5655

7606

9970

13652

Avz

mm2

1152

792

536

386

288

259

781

684

1257

414

578

869

844

796

844

917

1013

1110

1255

1152

792

536

386

386

1113

1250

288

1886

674

1525

963

1375

1206

1255

1375

1496

1665

¡z mm

35,7

39,0

41,8

44,5

46,5

46,9

69,3

65,3

66,5

31,2

35,5

39,5

40,9

46,0

49,2

50,2

51,8

52,9

53,9

39,8

42,6

46,2

48,3

48,5

66,9

71.2

50,3

67,9

32,1

41,0

36,9

42,3

47,3

48,6

50,5

51,8

53,0

Iz mm

4

3030970

3090686

3087413

3353644

3619168

3618956

19224813

14705099

20365039

1219072

2225990

3641353

4034914

6704281

10706938

13375811

19410220

26775771

38319435

4501190

4493100

5089891

5488809

5685978

23853364

30708940

6485447

30502474

1979040

6370726

3686982

6706116

10708762

13378055

19414007

26781814

38329810

'y mm

4

8212322

7648122

7168626

7315441

7556015

7501376

65572507

48936883

164052909

20230518

50640612

100969364

106738256

152766767

224132129

272579745

380643198

512757224

719910395

10616626

9917302

10384802

10730277

11049072

75087642

98664498

12130752

233791818

30634673

160825393

75698582

163877004

231326443

279451766

388391628

521543132

730173958

We,.y

mm3

11230

10455

9799Î

9987C

10301

10226

46795

37499

77474

13882

26154

41615

43970

62673

91389

110692

153176

204078

28171C

14410

13460

140521

144901

149061

569921

69776!

16299:

109825

20925:

660131

389441

672311

94322

113483

156294

207575

285726

- T - „ I _ l _ ι 1 · . · ! . . . _ . _ ft /■Μ — ­ . ­ . Λ — ­

Points

1L

2L

3L

4L

5L

6L

7L

8L

9L

10L

11L

12L

13L

14L

15L

16L

17L

18L

19L

1H

2H

3H

4H

5H

6H

7H

8H

9H

10H

11H

12H

13H

14H

15H

16H

17H

18H

Simulations

LABEIN

1a

2a

3a

4a

5a

14a

15a

16a

19a

17a

18a

13a

12a

11a

10a

9a

8a

7a

6a

1b

2b

3b

4b

13b

15b

14b

5b

18b

16b

12b

17b

11b

10b

9b

8b

7b

6b

Bending resistance of cross­section

Mel.y = M S d

N.m

26392

24570

23030

23469

24208

24033

109970

88124

182066

32624

61462

97796

103331

147283

214765

260127

359964

479584

662019

66286

61920

64642

66657

68568

262166

320974

74977

505199

96256

303662

179145

309265

433884

522022

718954

954846

1314342

Mp|.y

N.m

30275

27407

25112

25164

25671

25410

116549

93389

194973

35349

66517

106997

112390

156777

226747

274612

380717

510117

712581

74636

68156

69710

71078

73114

280663

343450

79470

544895

105176

335732

195783

338785

464272

556909

766727

1022436

1421927

Pel

N

17594

16380

15353

15646

16139

16022

73313

58749

121377

21750

40975

65197

68887

98189

143176

173418

239976

319723

441346

44191

41280

43094

44438

45712

174777

213982

49985

336799

64171

202441

119430

206177

289256

348014

479303

636564

876228

Ve.=Vs d

N

8797

8190

7677

7823

8069

8011

36657

29375

60689

10875

20487

32599

34444

49094

71588

86709

119988

159861

220673

22095

20640

21547

22219

22856

87389

106991

24992

168400

32085

101221

59715

103088

144628

174007

239651

318282

438114

Elastic critical stresses on isolated flange

Gmax

N/mm2

235,0

235,0

235,0

235,0

235,0

235,0

235,0

235,0

235,0

235,0

235,0

235,0

235,0

235,0

235,0

235,0

235,0

235,0

235,0

460,0

460,0

460,0

460,0

460,0

460,0

460,0

460,0

460,0

460,0

460,0

460,0

460,0

460,0

460,0

460,0

460,0

460,0

C/t f

max

18,6

18,6

18,6

18,6

18,6

18,6

18,6

18,6

18,6

18,6

18,6

18,6

18,6

18,6

18,6

18,6

18,6

18,6

18,6

13,3

13,3

13,3

13,3

13,3

13,3

13,3

13,3

13,3

13,3

13,3

13,3

13,3

13,3

13,3

13,3

13,3

13,3

σο Μ

N/mm2

41

43

43

51

60

60

66

67

73

67

51

62

73

204

522

816

1714

3265

6662

92

92

118

137

147

176

168

192

165

178

184

142

204

522

816

1714

3265

6662

Œmax'Œcrlt

<■ 1,0

5,69

5,44

5,44

4,61

3,95

3,95

3,59

3,53

3,20

3,50

4,61

3,76

3,20

1,15

0,45

0,29

0,14

0,07

0,04

5,02

5,02

3,90

3,35

3,12

2,61

2,74

2,40

2,78

2,59

2,50

3,25

2,25

0,88

0,56

0,27

0,14

0,07

Elast c critical stresses on isolated web

σΓΤ13Χ

N/mm2

173,5

173,5

173,5

173,2

173,0

173,0

186,2

188,2

202,5

205,3

206,0

207,7

207,6

206,7

205,5

204,6

202,8

200,6

197,2

337,2

337,2

336,1

335,4

335,1

364,9

361,1

333,8

394,4

400,0

404,9

401,6

404,7

402,2

400,6

397,0

392,6

386,0

^max

N/mm2

10

14

19

27

37

41

58

53

55

30

40

42

46

69

95

106

133

162

198

26

35

54

77

79

97

106

116

103

54

75

70

84

135

156

196

239

296

d/tw

max

161,7

161,7

161,7

161,8

161,9

161,9

156,1

155,3

149,7

148,7

148,4

147,8

147,8

148,1

136,3

129,1

115,4

104,6

94,6

116,0

116,0

116,2

116,3

116,3

111,5

112,1

116,6

107,3

106,5

105,8

106,3

105,9

106,2

106,4

95,1

86,1

77,4

°crlt

N/mm2

24890

11764

5382

2793

1556

1260

737

735

306

143

89

80

76

67

76

89

109

130

167

24890

11764

5382

2793

2793

1947

1889

1556

689

380

246

246

200

154

167

200

237

294

xcr l t

N/mm2

9727

4597

2103

1092

608

492

288

287

120

56

35

31

30

26

30

35

43

51

65

9727

4597

2103

1092

1092

761

738

608

269

148

96

96

78

60

65

78

93

115

σ/σ0Γ|,

S 1,0

0,01

0,01

0,03

0,06

0,11

0,14

0,25

0,26

0,66

1,43

2,32

2,60

2,75

3,06

2,72

2,30

1,86

1,54

1,18

0,01

0,03

0,06

0,12

0,12

0,19

0,19

0,21

0,57

1,05

1,64

1,63

2,02

2,61

2,40

1,98

1,66

1,32

t / t c r l t

£ 1,0

0,00

0,00

0,01

0,02

0,06

0,08

0,20

0,18

0,46

0,53

1,16

1,35

1,55

2,64

3,23

3,06

3,13

3,18

3,04

0,00

0,01

0,03

0,07

0,07

0,13

0,14

0,19

0,38

0,36

0,78

0,73

1,08

2,24

2,40

2,50

2,58

2,58

Table 2: numerical simulations about borders between Class 3 and 4 cross­sections (2/4)

Points

1L

2L

3L

4L

5L

6L

7L

8L

9L

10L

11L

12L

13L

14L

15L

16L

17L

18L

19L

1H

2H

3H

4H

5H

6H

7H

8H

9H

10H

11H

12H

13H

14H

15H

16H

17H

18H

Simulations

LABEIN

1a

2a

3a

4a

5a

14a

15a

16a

19a

17a

18a

13a

12a

11a

10a

9a

8a

7a

6a

1b

2b

3b

4b

13b

15b

14b

5b

18b

16b

12b

17b

11b

10b

9b

8b

7b

6b

Shear buckling ­ post critical method

d/tw.£

<. 69

13,5

19,6

29,0

40,3

54,0

60,0

78,4

78,6

121,7

178,0

226,3

238,3

245,1

259,9

245,1

225,7

204,2

186,5

165,0

18,9

27,5

40,6

56,4

56,4

67,5

68,6

75,6

113,5

152,9

189,9

189,9

210,5

240,0

230,8

210,5

193,6

173,9

Ay,

0,156

0,227

0,336

0,466

0,625

0,694

0,908

0,909

1,408

2,060

2,618

2,757

2,836

3,008

2,836

2,612

2,363

2,158

1,909

0,219

0,318

0,470

0,652

0,652

0,781

0,793

0,874

1,313

1,769

2,197

2,198

2,436

2,777

2,670

2,436

2,240

2,012

tba

N/mm2

135,7

135,7

135,7

135,7

135,7

135,7

126,5

126,4

86,7

59,3

46,6

44,3

43,1

40,6

43,1

46,8

51,7

56,6

64,0

265,6

265,6

265,6

265,6

265,6

265,6

265,6

253,3

182,0

135,1

108,8

108,8

98,1

86,1

89,5

98,1

106,7

118,8

Vba.Rd

N

117225

80592

54510

39270

29306

26376

79505

70284

94981

21581

23745

34193

32320

28732

32320

38098

46540

55827

71341

229462

157755

106700

76870

76870

240343

267084

54706

298994

80158

147439

92282

119930

92282

99812

119930

141893

175742

Vsd/Vba.Rd

0,08

0,10

0,14

0,20

0,28

0,30

0,46

0,42

0,64

0,50

0,86

0,95

1,07

1,71

2,21

2,28

2,58

2,86

3,09

0,10

0,13

0,20

0,29

0,30

0,36

0,40

0,46

0,56

0,40

0,69

0,65

0,86

1,57

1,74

2,00

2,24

2,49

Mf.Rd

N.m

15226

15565

15565

16920

18273

18273

82687

67642

132914

21923

36642

62014

68026

113388

181420

226775

328988

454004

649225

44380

44380

50341

54316

56293

215126

258954

64242

390259

69854

210826

119543

221950

355120

443900

643977

888688

1270824

MSd/M,.Rd

1,73

1,58

1,48

1,39

1,32

1,32

1,33

1,30

1,37

1,49

1,68

1,58

1,52

1,30

1,18

1,15

1,09

1,06

1,02

1,49

1,40

1,28

1,23

1,22

1,22

1,24

1,17

1,29

1,38

1,44

1,50

1,39

1,22

1,18

1,12

1,07

1,03

VM.ba.Rd

N

*

*

*

*

*

*

*

*

69148

15651

16756

24829

23462

21086

24469

29531

38009

48504

67536

•

*

*

*

*

*

*

•

225241

*

111074

67696

90106

70487

77635

97374

121381

162016

Vsd/Vwi.ba.Rd

<■ 1,0

•

•

*

*

*

*

*

*

0,88

0,69

1,22

1,31

1,47

2,33

2,93

2,94

3,16

3,30

3,27

•

*

•

»

*

•

*

•

0,75

*

0,91

0,88

1,14

2,05

2,24

2,46

2,62

2,70

Bending and shear resistance of the cross­secti«

VpLz.Rd

N

156300

107456

72680

52361

39075

35168

105975

92752

170546

56130

78354

117836

114563

108016

114563

124382

137475

150568

170207

305949

210340

142266

102493

102493

295541

332051

76487

500753

179020

404932

255622

365207

320357

333172

365207

397243

442093

Vz.sd/Vp|.z.Rd

S 1,0

0,06

0,08

0,11

0,15

0,21

0,23

0,35

0,32

0,36

0,19

0,26

0,28

0,30

0,45

0,62

0,70

0,87

1,06

1,30

0,07

0,10

0,15

0,22

0,22

0,30

0,32

0,33

0,34

0,18

0,25

0,23

0,28

0,45

0,52

0,66

0,80

0,99

Mv.Rd

N.m

220769

264367

359302

474776

655933

553808

742889

972435

1331224

Mz.sd/Mv.i

< 1,0 •

*

*

•

*

*

*

*

*

«

•

*

*

*

0,97

0,98

1,00

1,01

1,01

*

*

*

*

•

•

*

*

*

*

•

*

•

•

0,94

0,97

0,98

0,99

Points

1L 2L 3L 4L 5L 6L 7L 8L 9L 10L 11L 12L 13L 14L 15L 16L 17L 18L 19L

1H 2H 3H 4H 5H 6H 7H 8H 9H 10H 11H 12H 13H 14H 15H 16H 17H 18H

Simulations

LABEIN 1a 2a 3a 4a 5a 14a 15a 16a 19a 17a 18a 13a 12a 11a 10a 9a 8a 7a 6a

1b 2b 3b 4b 13b 15b 14b 5b 18b 16b 12b 17b 11b 10b 9b 8b 7b 6b

Lateral-torsional bucklinq resistance L/(iz.e)

<, 60 84,1 76,9 71,8 67,4 64,6 64,0 43,3 46,0 45,1 96,1 84,5 75,9 73,4 65,3 61,0 59,8 57,9 56,7 55,7

105,6 98,4 90,9 86,9 86,5 62,7 59,0 83,4 61,8 130,8 102,5 113,6 99,2 88,8 86,3 83,1 81,0 79,2

60.ίζ.ε £ 3 m 2,14 2,34 2,51 2,67 2,79 2,81 4,16 3,92 3,99 1,87 2,13 2,37 2,45 2,76 2,95 3,01 3,11 3,18 3,23

1,71 1,83 1,98 2,07 2,08 2,87 3,05

2,16 2,91 1,38 1,76 1,58 1,81 2,03 2,09 2,17 2,22 2,27

(LLR)real [m] If 60.lz.e < 3 m

0,86 0,66 0,49 0,33 0,21 0,19

» • *

1,13 0,87 0,63 0,55 0,24 0,05

* * * *

1,29 1.17 1,02 0,93 0,92 0,13

* 0,84 0,09 1,62 1,24 1,42 1,19 0,97 0,91 0,83 0,78 0,73

ψ (CM 66) (-1 £ ψ * 1 )

0,71 0,78 0,84 0,89 0,93 0,94

* * *

0,62 0,71 0,79 0,82 0,92 0,98

* * M

» 0,57 0,61 0,66 0,69 0,69 0,96

* 0,72 0,97 0,46 0,59 0,53 0,60 0,68 0,70 0,72 0,74 0,76

(l_LR)max [m] 2(LLR)real

1,25 1,37 1,46 1,56 1,63 1,64

• • *

1,09 1,24 1,38 1,43 1,61 1,72

* * • •

0,99 1,07 1,15 1,21 1.21 1.67

* 1,26 2,91 1,38 1,76 1,58 1,81 2,03 2,09 2,17 2,22 2,27

Cf

1,168 1,124 1,092 1,062 1,040 1,035 1,879 1,879 1,879 1,233 1,170 1,118 1,103 1,045 1,009 1,879 1,879 1,879 1,879

1,273 1,243 1,207 1,185 1,182 1,024 1,879 1,163 1,016 1,363 1,261 1,302 1,247 1,195 1,181 1,161 1,148 1,137

Mer N.m

731390 1197421 2082837 4853235 12549872 16084277 1157538 825617 1854109 355004 1371523 5162381 7450702

60047687 2121193598

1437996 2150833 3113810 4885289

530438 616720 886928 1129880 1193557

388625061 1872531 1601525

1767966900 308635

2602360 958543

2975411 6767278 9479818 16309971

L 25638174 41741787

λ LT

S 0,40 0,19 0,14 0,11 0,07 0,04 0,04 0,31 0,33 0,31 0,30 0,21 0,14 0,12 0,05 0,01 0,43 0,41 0,39 0,37

0,35 0,32 0,27 0,24 0,24 0,03 0,41 0,22 0,02 0,56 0,34 0,43 0,32 0,25 0,23 0,21 0,19 0,18

Φ LT

0,61 0,61

* * . * * • • •

0,61 * *

0,69 *

0,62 * * * w

* *

XLT

1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 0,95 0,95 1,00 1,00

1,00 1,00 1,00 1,00 1,00 1,00 0,95 1,00 1,00 0,91 1,00 0,94 1,00 1,00 1,00 1,00 1,00 1,00

Mb.Rd N.m

26392 24570 23030 23469 24208 24033 109970 88124 182066 32624 61462 97796 103331 147283 214765 246079 342101 479584 662019

66286 61920 64642 66657 68568

262166 304623 74977

505199 87119

303662 169127 309265 433884 522022 718954 954846 1314342

MSd/Mbi

S 1,0 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,06 1,05 1,00 1,00

1,00 1,00 1,00 1,00 1,00 1,00 1,05 1.00 1,00 1,10 1,00 1,06 1,00 1,00 1,00 1,00 1,00 1,00

Table 4: numerical simulations about borders between Class 3 and 4 cross-sections (4/4)

2. Graphs exploiting numerical simulatoins about borders between class 3 & 4 cross­sections (Me¡) and borders between class 2 & 3 (Mpi) :

2.1 EC3 borders and simulated borders (linear analysis) between class 3 & 4 cross­sections

(Mei) :

c (1) The evaluation of each point in the graphs — = f

tfE has been performed by searching the

tw and tf values which give a critical buckling moment equal to the elastic moment of the section (Mei)· The calculation of the critical buckling loads has been done assuming a linear behaviour.

(2) The class 3 & 4 limits for S235 and S460 steels obtained in the simulations are showed in Figure 1 (lines respectively called "S235 - simulation border" and "S460 - simulation border with EC3 epsilon factor").

|235 The factor is issued from present Eurocode 3 mies : ε = ni , with η = 2.

Figure 1 also includes other results :

- the simulations numbers (la to 19a and lb to 18b) and the points numbers defined in tables (IL to 19L and IH to 18H), respectively for S235 and S460 steel grades;

- the simulations governed by shear buckling failure mode (see chapter 1.2 : comments on results in tables), have been excluded (see specific tines between concerned points in Figure 1); because of these excluded simulations, new limits have been proposed (see specific lines in Figure 1);

- a new border for class 3 & 4 cross-sections is proposed;

- present mies of Eurocode 3 are provided;

- the safety reserve between present mies and new proposal is highlighted by hatching; for flange slendemess, a safety coefficient of 1,7 to 2,3 can be obtained; for web slendemess, a safety coefficient of 1,3 can be obtained.

(3) In the upper graphs of Figures 2 to 8 the results shown in Figure 1 are presented with characteristic values of (c/Οϊε) ; d/(tv\£)) for standard I or Η hot-rolled profiles and for both steel grades S235 and S460 : respectively, IPE, IPEA, IPEO, HEAA, HEA, HEB and HEM. For those standard profiles, the web slendemess (d/(twe)) is not determinant (see Figure 2 with IPE profiles for the worst cases of slender web) but the flange slendemess (c/tø)) is more relevant (see Figure 5 with HEAA profiles for the worst cases of slender flanges).

2.2 EC3 borders and simulated borders (non­linear analysis) between class 2 & 3 cross­sections (Mpi) :

f d ï

^ we

y

has been performed by searching the c

(1) The evaluation of each point in the graphs — = f tfe

tw and tf values which give a maximum moment equal to the plastic moment (Mpi) for class 2 & 3 limit.

(2) The class 2 & 3 limits for S235 and S460 steel grades obtained in the simulations are showed in lower graphs of Figures 2 to 8, with characteristic values (c/(tfjE) ; d/(tv^)) for standard I or Η hot-roUed profiles : respectively, IPE, IPEA, IPEO, HEAA, HEA, HEB and HEM. Those lower graphs of Figures 2 to 8 also includes other results :

234

- the simulations governed by shear buckling failure mode have been excluded (see specific lines between concerned points with high d/(tw£) web slendemess and low c/tø) flange slendemess);

- a new border for class 2 & 3 cross-sections is proposed;

- present rules of Eurocode 3 are provided;

- the safety zone between present rules and new proposal is highlighted by hatching; for flange slendemess a safety coefficient of 1,8 to 32 can be obtained; for web slendemess a safety coefficient of 1,5 could be probably obtained.

Although additional analysis would be required for class 2 & 3 limits in order to draw final conclusions.

(3) As mentioned in chapter 2.1 (class (3)), the flange slendemess c/tø) is more relevant for standard profiles given in Figures 2 to 8. Therefore these simulations results are interesting even if they are not complete.

2.3 Influence οίε factor :

The figure 9 shows the results of simulations (linear analysis) about borders between class 3 & 4 cross-sections for S235 and S460 steel grades. In graphs (c/tø) ; d/(twE)) both simulated curves for S235 and S460 steel grades (where ε factor according to Eurocode 3 :

[235 ε = J , with η = 2) should fit together if ε factor expressed correctly the dependence of

local buckling in function of the yield strength fy. But ε factor does not seem to be correct : for instance, the same formula with η = 1,8 should be preferable to present η = 2, for cases of linear analysis with high c/tø) and low d/(tw£).

235

3. Summary of results :

(1) On the basis of these numerical simulations presented in details in Annex 10 (see working document 3198-3-3) and exploited in present document, Eurocode 3 present rules for classification of steel cross-sections submitted to bending about major axis yy (My), are shown to be too conservative for all values of flange slendemess (c/tø)) and web slendemess (d/(tv\Æ)), in cases of borders between class 3 & 4 cross-sections (Mel is reached) and borders between class 2 & 3 cross-sections (Mpi is reached).

At present state, following improved mies could be proposed :

Present EC 3 mies

class 2

web

d/tw<

flange

cAf<

New proposals

class 2

web

d/tw^

flange

c/tf<

83 ε 11 ε 124 ε 35 ε -6,2t

w

I + I

_ /

I - 1

fy

r NMy

Present EC 3 rules

class 3

web

d/tw^

124 ε

flange

C/tf<

15 ε

New proposals

class 3

web

oVtw^

165 ε

flange

c/tf<

max (25 ε; 35 ε -8tw

(2) But more developments should be necessary to reach general and safe conclusion and to define precisely new improved limits.

236

^1

Linear Analysis - Border Class 3&4 elastic cross-sections (= M.IRd is reached) |

tf.E

45 τ

ÅL 2.U

2A.

3L

■S 235 - Simulation border *S 460 - Simulation border with EC3 epsilon factor -Simulations excluded by shear buckling failure mode Proposed limits of simulations Proposal of new border class 3&4 EC3 limits border class 3&4 EC3 limits border class 2&3 EC3 limits border class 1&2 Simulations numbers related to points of enclosed tables ML

J2<L·

Figure 1

EC3 borders and simulated borders (linear analysis) between Class 3&4 versus IPE profiles \

c

4 5 - r

B S 235 - Simulation border "S 460 - Simulation border with EC3 epsilon factor (n = 2) - Simulations excluded by shear buckling failure mode bv ¡

lula Proposed limits of simulations Proposal of new border class 3&4 EC3 limits border class 3&4

— E C 3 limits border class 2&3 -EC3 limits border class 1&2

A IPE profiles - S 235 Π IPE profiles - S 460

100 120 140 160 180 200 220 240 260

EC3 borders and simulated borders (non-linear analysis) between Class 2&3 versus IPE profiles |

A S 235 - Simulation border

"S 460 - Simulation border with EC3 epsilon factor

- Simulations excluded by shear buckling failure mode

Proposal of new border class 2&3

— EC3 limits border class 3&4

EC3 limits border class 2&3

• EC3 limits border class 1&2

A IPE profiles - S 235

Π IPE profiles - S 460

Border meaning that MpLRd is reached

OH—ι—ι—ι—ι—ι—ι—π—H—* 0 20 40 60 80 100 120 140

Figure 2

238

EC3 borders and simulated borders (linear analysis) between Class 3&4 versus IPE A profiles]

45-r

•S 235 - Simulation border "S 460 - Simulation border with EC3 epsilon factor (n = 2) - Simulations excluded bv shear buckling failure mode • Proposed limits of simulations Proposal of new border class 3&4 EC3 limits border class 3&4

- -EC3 limits border class 2&3 EC3 limits border class 1&2

A IPE A profiles-S 235 D IPE A profiles-S 460

100 120 140 160 180 200 220 240 260

EC3 borders and simulated borders (non-linear analysis) between Class 2&3 versus IPE A profiles |

A S 235 - Simulation border "S 460 - Simulation border with EC3 epsilon factor -Simulations excluded by shear buckling failure mode Proposal of new border class 2&3 EC3 limits border class 3&4 EC3 limits border class 2&3 EC3 limits border class 1&2

A IPE A profiles-S 235 D IPE A profiles-S 460

Border meaning that MpLRd is reached

100 120 140

Figure 3 180 260

239

EC3 borders and simulated borders (linear analysis) between Class 3&4 versus IPE O profiles I

■S 235 - Simulation border S 460 - Simulation border with EC3 epsilon factor in = 2) Simulations excluded by shear buckling failure moae

- - Proposed limits of simulations - -Proposal of new border class 3&4 • — EC3 limits border class 3&4

— EC3 limits border class 2&3 -EC3 limits border class 1&2

A IPE O profiles - S 235 D IPE O profiles-S 460

60 80 100 120 140 160 180 200 220 240 260

EC3 borders and simulated borders (non-linear analysis) between Class 2&3 versus IPE O profiles!

A S 235 - Simulation border

"S 460 - Simulation border with EC3 epsilon factor

-Simulations excluded by shear buckling failure mode

Proposal of new border class 2&3

EC3 limits border class 3&4

EC3 limits border class 2&3

EC3 limits border class 1&2

A IPE O profiles - S 235

Π IPE O profiles-S 460

Border meaning that MpLRd is reached

60 80 100 120 140 160

Figure 4

180 200 220 240 260

240

EC3 borders and simulated borders (linear analysis) between Class 3&4 versus HE AA profiles I

c

t^~S 235 - Simulation border S 460 - Simulation border with EC3 epsilon factor (n = 2) Simulations excluded by shear buckling failure mode by!

nula Proposed limits of simulations Proposal of new border class 3&4 EC3 limits border class 3&4

— E C 3 limits border class 2&3 -EC3 limits border class 1&2

A HE AA profiles - S 235 D HE AA profiles - S 460

80 100 120 140 160 180 200 220 240 260

EC3 borders and simulated borders (non-linear analysis) between Class 2&3 versus HE AA profiles!

A S 235 - Simulation border

"S 460 - Simulation border with EC3 epsilon factor

- Simulations excluded by shear buckling failure mode

Proposal of new border class 2&3

■ EC3 limits border class 3&4

EC3 limits border class 2&3

EC3 limits border class 1&2

A HE AA profiles - S 235

Π HE AA profiles - S 460

Border meaning that MpLRd is reached

20 40 60 100 120 140 160

Figure 5

180 200 220 240 260

241

EC3 borders and simulated borders (linear analysis) between Class 3&4 versus HE Β profiles] S 235 - Sjmulation border S 460 - Simulation border with EC3 epsilon factor (n = 2) Simulations excluded by shear buckling failure mode by; Proposed limits of simulations Proposal of new border class 3&4

• —EC3 limits border class 3&4 - -EC3 limits border class 2&3

-EC3 limits border class 1&2 A HE Β profiles - S 235 Π HE Β profiles - S 460

20 40 80 100 120 140 160 180 200 220 240 260

EC3 borders and simulated borders (non-linear analysis) between Class 2&3 versus HE Β profiles |

A S 235 - Simulation border •S 460 - Simulation border with EC3 epsilon factor -Simulations excluded by shear buckling failure mode Proposal of new border class 2&3 EC3 limits border class 3&4 EC3 limits border class 2&3 EC3 limits border class 1&2

A HE Β profiles - S 235 Π HE Β profiles-S 460

Border meaning that MpLRd is reached

20 40 100 120 140 Figure 7

160 180 200 220 240 260

242

EC3 borders and simulated borders (linear analysis) between Class 3&4 versus HE M profiles!

c

tfX

A~~—S 235 - Simulation border S 460 - Simulation border with EC3 epsilon factor (n = 2) Simulations excluded by shear buckling failure mode by¡

nula Proposed limits of simulations Proposal of new border class 3&4 EC3 limits border class 3&4

- EC3 limits border class 2&3 -EC3 limits border class 1&2

A HE M profiles-S 235 D HE M profiles-S 460

20 40 80 100 120 140 160 180 200 220 240 260

EC3 borders and simulated borders (non-linear analysis) between Class 2&3 versus HE M profiles \

A S 235 - Simulation border

"S 460 - Simulation border with EC3 epsilon factor

- Simulations excluded by shear buckling failure mode

Proposal of new border class 2&3

EC3 limits border class 3&4

EC3 limits border class 2&3

EC3 limits border class 1&2

A HE M profiles-S 235

Π HE M profiles - S 460

Border meaning that MpLRd is reached

20 40 60 80 100 120 140 160

Figure 8

180 200 220 240 260

243

SJ

Linear analysis - Border Class 3&4 elastic cross-sections (= MtlRd is reached) : influence of ε factor I

—A—S 235 - Simulation border —■—S 460 - Simulation border with EC3 epsilon factor (n = 2) HI—S 460 - Simulation border with η = 1,5 HI—S 460 - Simulation border with η = 1,8

• -D—S 460 - Simulation border with η = 3 Proposal of new border class 3&4 EC3 limits border class 3&4 EC3 limits border class 2&3 EC3 limits border class 1&2

} ε = ρ

235

460

120 140 160

Figure 9

240 260 280

Annex 12

Document 3198-2-10 (CTICM)

(13 pages)

"Some numerical tests for checking the influence of yield strength on limiting b/t ratios"

245

ECSC Project n°3198

IMPROVED CLASSIFICATION OF CROSS-SECTIONS

WORKING DOCUMENT N° ..^ÄF.zÅ.T^O

SOME NUMERICAL TESTS FOR CHECKING

THE INFLUENCE OF YIELD STRENGTH ON LIMITING ΒΛ RATIOS

CONTENT

Page

1

2

3

1.

2.

3.

4.

GENERAL

STRESS­STRAIN BEHAVIOUR OF STEELS

WEB

3.1 Numerical model

3.2 Results

FLANGE

4.1 Numerical model

4.2 Results

ANNEX : fable of numerical simulations 13

1. GENERAL

According to Eurocode 3 (Table 5.3.1), the influence of yield strength on limiting b/t ratios of

section elements is generally taken into account through the parameter

fy in N/mm2

Except for circular hollow sections, the limiting b/t ratios are proportional In ε.

The following study is a tentative to check the relevancy of the choice of this proportionality

in ε for high strength steels. For S420 and S460 steels, these criteria seem to be too severe

because on one hand the plastic plateau is shorter than for S235 and the strain hardening

may increase the rotation capacity in appreciable proportions, and on the other hand,

weakest residual stresses may lead to Initial equivalent imperfections smaller than for S235.

This study deals with webs in pure bending and flanges in pure compression. It Is based on

numerical simulations with the ANSYS program.

246

simply supported

simply supported

Figure la : web Figure lb : flange

In Table 5.3.1 (Sheet 1) of Eurocode 3, the limiting b/t ratio between Class 1 and Class 2 is :

b/t = 72 ε for a web in pure bending

b/t = 10 ε for a flange in pure compression (rolled profiles)

We chose to study plate behaviour just at these limits.

In order to initiate the local plate buckling, this study takes into account an initial out-of-plane imperfection. The shape of this imperfection is deduced from the first p late elastic buckling mode with a magnitude w 0 .

The effect of residual stresses is also investigated.

2. STRESS-STRAIN BEHAVIOUR OF STEELS

The two extreme grades S235 and S460 are studied. The material characteristics for each of them are described below in figure 2 and Table 1 and their resulting σ-ε curve are plotted in figure 3.

Steel grade S235 S460

fv (MPa) 235 460

f„ (MPa) 360 552

Ev

0.001119 0.002190

Est

0.02238 0.02190

Eu

0.04024 0.04380

ε,»/^ 20 10

E/Ert

30 50

Table I

For both steel grades : E = 210000 MPa

247

600

500

400

300

200

100

0

rj (MPa)

I

I

S460

S235

0.01 0.02 0.03 0.04 0.05 0.06 Figure 3

Note : the yield criteria is isotropic (based on von Mises equivalent stress)

3. WEB

3.1 Numerical Model

In order to optimize the meshing and the run time, it has been taken into account that, for a long simply supported plate subject to pure bending, the buckling mode has a "free" half-wave length (that is to say giving the smallest critical bending stress - see figure 4) equal to 2/3 of the width of the plate.

28

26

24

22

23.9

\ v \

0.4 0.

rr = "

6 ;

2/3

\ \

m

0.8 1.0

Π - π

I umi

m =

1.2

>er

*—',

2

of fl

s .

alf-\

** "-̂ ̂

vav

► < ^

es

m = 3

| 1.4 1.6 1.8

4/3 = 2*2/3

σ„ = k.-π E

12(l-v ¿)

V

lb,

So, and with symmetry considerations more, the studied model is the one shown in figure 6.

Dimensions

The plate dimensions are:

Thickness :

. a/2 = 334 mm ­ b = 1000 mm

. Class 1 / Class 2 limit :

(a = 2/3)

t = 1000/72=13.89 mm for S235

t = 13.89/^235/460 = 19.43 mm for S460

Meshing

The meshing is : 16x16 = 256 finite elements (4 nodes shell elements ­ Element n°43 in ANSYS).

248

Initial plate imperfection

An initial plate imperfection is introduced according to the first buckling mode (without initial stresses) : the first buckling mode of the plate is firstly determined by ANSYS and the node coordinates are then modified.

Residual stresses

It is not possible to directly input initial stresses with ANSYS program. So the influence of residual stresses has been taken into account through stress-strain curves. The diagram of residual stresses has a linear variation as shown in Figure 5, with two cases :

- σ 0 = 235 / 2= 117,5 MPa - σ 0 = 460 / 2 = 230 MPa

(with grades S235 and S460) (with steel grade S460)

Residual stresses

Figure 5

The plate is divided into 16 bands. For each band, the stress-strain curve is adap ted in such a way that the sum of the residual stresses and the bending stresses reaches the yield strength for the same imposed displacement. This led to define 8 new curves.

Support conditions

The support conditions are described on figure 6 (restraint degrees of freedom are boxed).

Loading

The bending stress diagram is introduced by imposing DX displacements at the nodes of the extreme left section of the plate. The imposed DX displacements are linear through the height of the plate. In the extreme horizontal fibers of the plate, the DX displacement increases gradually from 0 up to 20 mm.

249

Figure 6

1000

axis of Symmetry

DX, RY

3.2 Results

3.2.1 Influence of initial plate imperfection magnitude

In order to evaluate the influence of the magnitude of the plate imperfection, 4 values of w 0 have been studied for a web with a slendemess b/t = 72 (steel grade S235), without residual stresses :

w 0 / b = 1/10000 w 0 / b = 1/1000 w 0 / b =3/1000 w 0 / b =1/100

The results are plotted in figure 8 where

- M is the moment corresponding to the current

imposed displacement dx

- φ is the rotation at left support

- Mpi is the theoretical plastic moment of the plate (Mp, = fy.tb2/4)

- φρι is the rotation at left support corresponding

M

" w .

t o M , Figure 7

pi

250

o.yo -

υ./υ -

0.60 "

u.ou "

0.40 "

O.JO "

0.20 "

0.10 "

o.oo -

ΑΛ/Μρ I

(5) 1/1000 , — ~ -—QJt

S235 ι I I 1

b/t = 72

ι

~i

_ f n i/ioooo

φ/φ, PI 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00

Figure 8 - Influence of plate Imperfection magnitude

For α perfectly flat plate, the Μ-φ curve should begin to leave the elastic range at M = Mei

(Me| = elastic moment) that is M/Mp i = 0,667 for a rectangular cross-section. Because of the initial p late imperfection which generates plate bending stresses, draws aside the elastic curve before this loading level, especially for w 0 / b = 1/100. The biggest the w 0 / b ratio is, the biggest the plate bending stresses are, the smallest Mm a x is. For hight values of dx or φ, all the curves converge because the influence of the initial plate imperfection decreases more and more. The magnitude of geometrical imperfection is normally between b/100 and b/1000.

Theoretically, the Μ-φ curve is asymptotic to the theoretical value Mp!. The local buckling occurs at a point whose position is governed by the steel grade and the b/t ratio of the plate.

3.2.2 Influence of steel grade

The figure 9 shows the influence of the steel grade on the rotation capaci ty of the plate the w 0 / b ratio of which is fixed to 1/10000 and for b/t = 72. As expected in that case, this rotation capac i ty is much smaller for S460 than for S235 steel grade (without residual stresses).

251

1.00

0.90

0.80

0.70

0.40

0.50

0.40

0.30

0.20

0.10

0.00

Μ/Λ/

/ /

/ /

/ /

/

J y ι

Ipl 1

¡MUU ­

¡460­ ilt = 7i

an— IA

— m —

wo/b = 1/10000

~ÇÙ~

φ/φ pi

0.00 1.00 2.00 3.00 •4.00 5.00 6.00 7.00 8.00

3.2.3 Influence of residual stresses

Figure 9

The influence of residual stresses has been investigated for the steel grade S235 (see Figure

10). Residual stresses seem to have influence if imposed rotation is greater than 4 φρ|. For

steel grade S235, the level of residual stresses is 117,5 MPa.

ι on -r

Π flO

0 70 -

0.00 -

W/Mpl

S235

~^}*L wo/b-1 /10000 _ j ( s ) '

v ) I — C I wo/b-3/1000 l_

\¿J '

j

ÍS 1 1

. ' I T

V \ I X \

1 With raxlriunl <tr»«»« I

~,

φ/φ pi

0.00 2.00 4.00 6.00 8.00 10.00 12.00

Figure 10

14.00 16.00 18.00

252

3.2.4 Test of validation of the linear variation of b/t ratio in ε at the limit Class 1 / Class 2

On the figure 9 are plotted the results of the two following simulations :

- S235 and b/t = 72 - S460 and b/t = 72.ε = 72Λ /235 /460 = 51,5

According Eurocode 3, for both cases, the plate is at the limit Class! / Class2 and should have the same rotation capacity. This is nearly the case as the figure11shows a very g o o d fit up to φ/φρι = 6. For greater values, the strain hardening which occurs earlier for S460 steel grade becomes more and more influent.

Λ η -

n R -

υ. / -

o.o -U.1 -

0. ύ J

\j.¿ -

υ. ι ­

η -ι

Μ/Μ

¡ / / / / / / / / /

Pi

/ / / / / / /

wo/b = 1/10000

, I

I

I

ι ι

S440 - b / i = 51 ι

ι

5

Γ~Ί —Ss1— - b / * - " J

4 5 6

Figure 11

φ/φ 10

pi

The same comparison is made for wo /b = 1/100, and for w o / b = 3/1000 for both cases. The results are plotted on figures 12 and 13.

M/Mpi

0.900

0.BOO

0.700

0.600

0.400

0.300

0.200

0.100

0.000

j / / / / / / J 1 1

wo/b = 3/1000 * . κ ssiau ai sire ¡sses i y / ¿

—(Í — ( t

ì>S4iO-b/ t -51 .5 .

^S23S-b / t"72 —

Φ/Φρ| 0.000 1.000 2.000 3.000 5.000 6.000 7.000 8.000 9.000 10.000

Figure 12

253

M7M 1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Pi

wo/b = 1/100

S4 160 b/t = 51,

S235 b/t = 72

5

"

φ/φ 2 3 4 5 6 7 8 9 10

Figure 13

Pi

3.2.5 Conclusions

This study constitutes α preliminary study of the problem of relevancy of b/t ratios given in EC3 for high strength steels. The influence of the steel grade is taken into account through a proportionnality with the ε parameter. For a very small initial plate imperfection, the figure 9 shows that this assumption seems to be valid as long as the plastic rotation involves moderate influence of strain hardening. For greater initial plate imperfection, the figure 10 shows that for a given loading level (M given), the difference between the Μ-φ curves for S235 and S460 becomes more important. In that case, this comparison is more difficult because the curves are very flat.

4. FLANGE

4.1 Numerical model

Dimensions

The plate dimensions are

Thickness

a/2 = 100 mm - b = 100 mm

Class 1 / Class 2 limit :

t = 100/10 =10 mm forS235 t = 10Λ/235/460 = 13,99 mm for S460

Meshing

The meshing is : 16x16 = 256 finite elements (4 node shell elements - Element n°43 in ANSYS)

254

Infilai plate imperfection

An initial plate imperfection is introduced according to the first buckling mode (without

initial stresses) : the first buckling mode of the plate is firstly determined by ANSYS and the

node coordinates are then modified.

The magni tude of the intial imperfection is : wo /b = 6/1000.

Residual stresses

Τ DX Imposed

DI

100

v\

Í.

100

°o

Residnal stresses

Figure 14

4.2 Results

4.2.1 Influence of residual stresses

Figure 15 and 16 show the influence of residual stresses for steel grades S235 and S460.

Residual stresses reduce the deformation capaci ty in a significative way. For steel grade

S460, it is important to consider that the residual stresses do not exceed 235 / 2 = 117,5 MPa.

Therefore, curve 23 is nearer to the real behaviour than curve 18 and the deformation

capac i t y is better.

N/Npl

I / / / / / / / / \

/ f

S235

| w

I I 1 _|_WHtioutr»lidual!ft.i!·. L

rth reiidual streu·« j

d/dy

Figure 15

255

η οπ-

π RO-

Π ACV

0.00-

Ν / Ν

y /

/

/

/

/

/

/

/

/

ol

j

h h

/ /

/ /

/

/

/

/

I

'ff / f

S460

iMktuol ifreiit-i : 230 MFc

WVttout mtduol t t iMMi

-I

1

/

/ /

[T7~ 7 ^ r

l*4duoJ i h * « « : 117J ΜΓα

•

=-<Í7U =-^7>-

—Gèr-

—lie»—

0.00 1.00 2.00 6.00 7.00 3.00 4.00 5.00

Figure 16

4.2.2 Influence of the expression of ε

Figure Υψshows the curves obtained by calculating ε with the following formula

,235,1/n ε = fv,

with η = 1,2,3,4

d / d y 8.00

These numerical simulations take into account residual stresses with σ 0 =

Simulations have also been made for η = 2, 3 and 4, with σ 0 = 117,5 MPa.

230 MPa.

0.00 -

N/Npl

= Ξ ^ ^p^=r

S460

•rsnc«

^ ^ ΐ τ > ^

> 4

curv»(1

/ * " *

ί̂ il V

~~—~ ■ — — .

—.

.

■

-/Ta K L

?5< «4

d / d y

1.00 2.00 4.00

Figure 17

256

1 00­

0 90­

0 80­

0 70­

0 60­

0 10­

0.00­

M / M p l

I / /

/

/

/

/

/ / /

μ ¡ι ι

li

f S /

■

^*fc_

y

" " " " " ■ ^ ^ ^

Reference curve\\ò)

S460

' ' Residual stresses : 117,5 ΜΡα

;^-_ * — ■ * " ^ _ X^­—

z^z^ — _ - @ - n « 2 —

1 - 4 —

d / d y

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

Figure 1Σ

CONCLUSIONS

All simulations are compared to a reference simulation with steel grade S235 taking into

accoun t residual stresses (simulation 16). On figure Yfy the level of the residual stresses is

assumed to be half the yield strength for the steel grade S460 : 230 MPa. Curve 18 is the

nearest curve to the reference curve : η = 2 seems to be the best value for ε.

If we assumed that the level of residual stresses does not depend on the yield strength, we

must take σ 0 = 117,5 MPa (see Figure 15). Then η = 3 seems to be the best value for

calculat ing the factor ε.

These simulations only allow us to make comparisons, but it is not possible to deduce

general conclusions, for many reasons :

­ residual stresses depend on the shape of the cross­section

­ simulations cou ld be done with more realistic stress­strain curves

­ comparisons have been only with steel grade S460, and not with steel g rade S355.

257

ANNEX

Table of numerical simulations

Steel n b/t w0/b σ 0 (MPa)

WEB 1 2 3 4 5 6 7 8 9 10 11 12 13 14

S235 S235 S235 S235 S235 S235 S460 S460 S460 S460 S460 S460

2 2 2 2 2 2 2 2 2 2 2 2

S460 3 S460 4

72 72 72 72 72 72 72 51,5 51,5 51,5 51,5 51,5 57,5 60,9

1/10000 1/1000 3/1000 1/100 1/10000 3/1000 1/10000 1/10000 3/1000 1/10000 3/1000 3/1000 3/1000 3/1000

0 0 0 0

117.5 117,5 0 0 0 230 230 117.5 117,5 117,5

FLANGE 15 16 17 18 19 20 21 22 23 24

S235 S235 S460 S460 S460 S460 S460 S460 S460 S460

2 2 2 2 1 3 4 2 3 4

10 10 7,15 7,15 5,11 7,99 8,45 7,15 7,99 8,45

6/1000 6/1000 6/1000 6/1000 6/1000 6/1000 6/1000 6/1000 6/1000 6/1000

0 117,5 0 230 230 230 230 117,5 117,5 117,5

258

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European Commission

EUR 18404 — Properties and in-service performance Improved classification of steel and composite cross-sections: new rules for local buckling in Eurocodes 3 and 4

J. B. Schleich, Β. Chabrolin, F. Espiga

Luxembourg: Office for Official Publications of the European Communities

1998 — 258 pp. — 21 χ 29.7 cm

Technical steel research series

ISBN 92-828-4466-8

Price (excluding VAT) in Luxembourg: ECU 43

In each specification detailing the design of structural steel members there are usually rules about the local buckling. These rules are based on the combination of cross-sectional dimensions (slendemess of different parts of profiles, b/t for the web and the flange) and on the yield point; for these combinations a critical level is defined over which local buckling appears (classification of cross-sections). Thus, this classification does not take into account the real stresses of the cross-sections which are rarely equal to the yield point. Besides, for high strength steels (yield point = 460 MPa), these rules have been extrapolated without verification and because of their definition, they discriminate against these steels.

For a designer the usual procedure is to choose a cross-section 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, beam-column).

Therefore, the local buckling rules play an important part in the design of structural steel and composite members.

In this research we propose to evaluate the local buckling problem for all main steel grades (S 235, S 355 and S 460 steels) with a more realistic approach based on test results and numerical simulations. This approach should take into account the existing stresses in members submitted to global buckling (cross-sections loadded by centred and also eccentric compression) and should also take into account the real boundary conditions of the cross-sections (for instance, in a composite cross-section the collaborating concrete slab greatly influences the stability of the steel beam web).

The aim of this research is to improve the classification of steel and composite cross-sections in Eurocodes 3 and 4 by taking a more realistic approach. The practical result of this research consists of new rules of classification of cross-sections which will be introduced in both Eurocodes 3 and 4 with the support of experts. In such a way, the competitiveness of steel and composite (steel-concrete) cross-sections will be improved and these sections will not be evaluated too conservatively as is done presently because of a lack of knowledge in the field of local buckling problems.

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