0,00 0,50 1,00 1,50 2,00 2,50 3,00 λLT I-Steel beams … beams under tension: Lateral torsional...

I-Steel beams under tension:

Lateral torsional buckling, behaviour and design

João Tomás Mello e Silva

Thesis to obtain the Master of Science Degree in

Civil Engineering

Examination Committee

Chairperson: Professor Doutor Fernando Manuel Fernandes Simões Supervisor: Professor Doutor Dinar Reis Zamith Camotim

Supervisor: Professor Doutor Nicolas Boissonnade Member of the Committee: Professor Doutor Luís Manuel Calado de Oliveira Martins

Member of the Committee: Professor Doutor Pedro Manuel de Castro Borges Dinis

October 2013

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“You must be the change you wish to see in the world.”

Mahatma Gandhi

iii

ACKNOWLEDGMENTS

I would like to thank all the people who contributed in some way to the work described in this thesis.

First and foremost, I thank my academic and scientific supervisor, Professor Dinar Camotim for being an

outstanding supervisor and an excellent professor. His constant encouragement, support and invaluable

suggestions made it possible to carry out the work presented in this dissertation successfully. I would like also

to acknowledge all the opportunities given to me during the last year, which have broaden my personal and

professional horizons considerably. Lastly, I would like to thank him for sharing with me his revolutionary

and perfectionist vision of the professional and academic/research work.

Second, I would like to thank my supervisor, Professor Nicolas Boissonnade, for his constant support and for

always pushing me to the limits to make this dissertation a better work. I also would like to express my

gratitude for receiving me so well in Switzerland and for always making me feel like it was my home.

I would like also to thank Professor Pedro Borges Dinis for his full availability and for all the constructive

advices given during the first part of this dissertation.

I would like to express my deep gratitude and respect to my friend Joanna Nseir, for supporting me during last

year and for the time devoted and constant contributions given to improve the quality of this dissertation.

I would like to express my sincere appreciation, first to my friends, namely Joana and Maria João, as well as to

my “Suisse family”, for their constant support in all my struggles and frustrations, as well as encouraging me

in my decisions in my new life in Switzerland. Even from the distance, each one gave me force to overcome

all kinds of obstacles, supported me to succeed in every new challenge and made me feel that they were always

right there next to me.

I would like to thank my family, especially my mother, father, sister and grandfather, for always believing in

me, for their continuous love and their supports in my decisions. Without them I could not have made it here.

v

ABSTRACT

This dissertation reports the results of an analytical, numerical and experimental investigation dealing with

hot-rolled I-section steel members acted by a combination of major-axis bending and axial tension (“beams

subjected to tension”), which is relatively rare in practice and, therefore, has received little attention from

researchers in the past. In particular, there are no guidelines for the design against buckling ultimate limit states of

such members (only their cross-section resistance is checked). This means that the axial tension favourable effect

on lateral-torsional buckling/failure is neglected, thus leading to over-conservative designs − indeed, a beam

subjected to axial tension is currently designed against lateral-torsional failure as a “pure beam”. In order to

acquire scientific knowledge and provide design guidance on this topic, the lateral-torsional stability, failure

and design of hot-rolled steel I-beams with fork-type end supports and acted by simple transverse loadings (mostly

applied end moments) and various axial tension values are addressed in this work. After developing and validating

an analytical expression to calculate critical buckling moments of beams under uniform bending and axial tension,

numerical (beam finite element) buckling results are presented for the non-uniform bending cases. Then, two full-

scale tests involving a narrow and a wide flange beams under eccentric tension are described and their results are used

to develop shell and beam finite element models − the latter are subsequently employed to perform a parametric

study aimed at gathering a fairly extensive ultimate strength/moment data bank. Finally, this data bank is used to

assess the merits of a design approach proposed in this work for beams subjected to tension and collapsing in

lateral-torsional modes − this design approach, which consists of slightly modifying the current procedure

prescribed in Eurocode 3 to design beams against lateral-torsional failure, is shown to provide ultimate moment

estimates that correlate very well with the values obtained from the numerical simulations. The predictions of the

proposed design approach are also compared with those of the design procedure included in the ENV version of

Eurocode 3 (but later removed).

Keywords:

Hot-rolled I-section steel beams, Combination of major-axis bending and tension, Lateral-torsional buckling, Failure

governed by lateral-torsional buckling, Design approach

vii

RESUMO

Esta dissertação apresenta os resultados de uma investigação analítica, numérica e experimental sobre vigas

de aço laminadas a quente com secção em I, submetidos a uma combinação de flexão em torno do eixo de

maior inércia e tracção (“vigas traccionadas”), a qual ocorre com pouca frequência na prática e, portanto, tem

recebido pouca atenção da comunidade científico-técnica. Em particular, não existem disposições regulamentares

relativas ao dimensionamento, em relação ao estado limite último de encurvadura lateral, de tais elementos

estruturais (apenas se efectua a verificação de secção). Isto significa que o efeito favorável da tracção no colapso

por encurvadura lateral é desprezado, conduzindo a um dimensionamento demasiado conservativo – de facto, uma

viga submetida a tracção é presentemente dimensionada como uma “viga pura”. Com o objectivo de adquirir

conhecimento científico sobre o comportamento estrutural de vigas traccionadas, bem como contribuir para o

seu dimensionamento eficaz, o presente trabalho aborda a estabilidade lateral (por flexão-torção), a resistência última

e o dimensionamento de vigas metálicas laminadas a quente, com secção em I, simplesmente apoiadas (apoio em

“forquilha”) e submetidas a carregamentos transversais simples (sobretudo momentos de extremidade) e

diferentes níveis de tracção axial. Após desenvolver e validar uma expressão analítica para calcular

momentos críticos em vigas submetidas a flexão uniforme e tracção, apresentam-se resultados numéricos

(elemento finito de viga) relativos a vigas submetidas a flexão não-uniforme. Em seguida, descrevem-se dois

ensaios experimentais, efectuados à escala real e envolvendo duas vigas, uma de banzos estreitos e outra de

banzos largo, submetidas a tracção aplicada de forma excêntrica, cujos resultados obtidos são usados para

desenvolver modelos de elementos finitos de casca e viga – este último modelo é, posteriormente, utilizado para

efectuar um estudo paramétrico destinado a reunir uma considerável base de dados de resistências/momentos

últimos de vigas traccionadas. Finalmente, estes resultados são utilizados para avaliar a qualidade das

estimativas fornecidas por uma metodologia de dimensionamento proposta neste trabalho para vigas

submetidas a tracção e cujo colapso é provocado por encurvadura lateral – mostra-se que esta metodologia de

dimensionamento, a qual consiste numa pequena modificação do procedimento prescrito pela actual versão do

Eurocódigo 3 para calcular a resistência de vigas à encurvadura lateral, fornece estimativas da resistência última

que exibem uma correlação muito boa com os valores obtidos através das simulações numéricas. As

estimativas fornecidas pela metodologia de dimensionamento proposta são também comparadas com as que

resultam da aplicação do procedimento preconizado na versão ENV (Pré-Norma Europeia) do Eurocódigo 3, o

qual não figura na versão actual.

Palavras-chave: Vigas de aço laminadas a quente com secção em I, Combinação de flexão em torno do eixo de maior inércia e

tracção, Estabilidade lateral (por flexão-torção), Colapso provocado encurvadura lateral, Metodologia de

dimensionamento

ix

TABLE OF CONTENTS

ACKNOWLEDGMENTS .................................................................................................................. iii

ABSTRACT ......................................................................................................................................... v

RESUMO ........................................................................................................................................ viiii

TABLE OF CONTENTS ................................................................................................................... ix

List of Figures ............................................................................................................................... xiii

List of Tables ................................................................................................................................. xix

Chapter 1 ........................................................................................................................................... 1

Introduction

1.1. Preliminary remarks ....................................................................................................................... 2 1.2. Motivation and scope of the work ............................................................................................... 3 1.3. Organization of the dissertation .................................................................................................. 4

Chapter 2 ........................................................................................................................................... 7

Lateral Torsional Buckling

2.1 Introduction ......................................................................................................................................... 7

2.2 Beams under uniform bending -‐ analytical solution ............................................................. 8

2.3 Beams under non-‐uniform Bending − numerical results .................................................. 11 2.3.1 Beam finite element model ...................................................................................................................... 12 2.3.2 Validation -‐ comparison with the analytical results ..................................................................... 13 2.3.3 Parametric studies ...................................................................................................................................... 14

2.4 Summary ............................................................................................................................................. 20

x

Chapter 3 ......................................................................................................................................... 23

Ultimate Behaviour and Strength − Experimental Study

3.1 Introduction ....................................................................................................................................... 23

3.2 Specimen characterisation ........................................................................................................... 24 3.2.1. Material tests ................................................................................................................................................ 24 3.2.2 Residual stress measurement ................................................................................................................ 25 3.2.3 Determination of the initial geometrical imperfections ............................................................. 26

3.3 Experimental set-‐up and procedure ......................................................................................... 28 3.4 Initial Measurements -‐ beam characterisation ...................................................................... 32

3.5 Test results ......................................................................................................................................... 36 3.5.1. IPE 200 beam ................................................................................................................................................ 36 3.2.2 HEA 160 beam ............................................................................................................................................... 38

3.5.3 Discussion ....................................................................................................................................................... 40 3.6 Numerical simulation ..................................................................................................................... 41

3.6.1. Modelling issues .......................................................................................................................................... 41 3.6.2 Numerical results ......................................................................................................................................... 45

3.7 Summary ............................................................................................................................................. 49

Chapter 4 ......................................................................................................................................... 51

Ultimate Behaviour and Strength − Numerical Parametric Study

4.1 Beam finite element model ........................................................................................................... 52 4.1.1 Description ..................................................................................................................................................... 52 4.1.2 Validation ........................................................................................................................................................ 54

4.2 Effect of axial tension on the ultimate strength -‐ qualitative aspects ............................ 54 4.3 Parametric study .............................................................................................................................. 55 4.3.1 Scope and procedure .................................................................................................................................. 55 4.3.2 Results .............................................................................................................................................................. 56

4.3 Summary ............................................................................................................................................. 61

xi

Chapter 5 ......................................................................................................................................... 63

Development of a design approach

5.1 Proposed design approach ........................................................................................................... 64 5.2 Assessment of the proposed ultimate strength/moment estimates .............................. 65 5.3 Axial tension beneficial influence ............................................................................................. 70 5.4 Comparison with the design procedure prescribed in EC3-‐ENV-‐1-‐1 ............................. 72

5.5 Summary ............................................................................................................................................. 75

Chapter 6 ......................................................................................................................................... 77

Conclusion and Future Developments

6.1 Concluding Remarks ............................................................................................................ 78

6.2 Future developments .......................................................................................................... 80

References ...................................................................................................................................... 81

Annexes ........................................................................................................................................... 83

Annex 1 Analytical formula to calculate critical buckling moments of beams subjected to uniform major-‐axis bending and axial tension .................................................................... A1.1

Annex 2 Numerical Data: critical moments, ultimate moment values and ultimate

moment estimates .............................................................................................................................. A2.1 A2.1. Proposed ultimate moment estimates and design results -‐ IPE300 beams ................... A2.3 A2.2. Proposed ultimate moment estimates and design results -‐ IPE500 beams ................ A2.19 A2.3. Proposed ultimate moment estimates and design results -‐ HEB300 beams .............. A2.35 A2.4. Proposed ultimate moment estimates and design results -‐ HEB500 beams .............. A2.51

Annex 3 Measured initial geometrical imperfections ............................................................ A3.1

xiii

List of Figures

Figure 1.1 -‐ Beam subjected to uniform major-‐axis bending (My) and tension (N) .................................................. 3

Figure'2.1'–'Beam'deformed'configuration'associated'with' the'occurrence'of'LTB:' (a)'member'and' (b)'

cross>section'views'...............................................................................................................................................................................'8'

Figure' 2.2' –' Lateral>torsional' buckling:' fundamental' and' post>buckling' equilibrium' paths' (Reis' &'

Camotim,'2012)'......................................................................................................................................................................................'8'

Figure' 2.3' –' Beam' subjected' to'major>axis' bending'My' and' axial' tension' Nt:' (a)' general' view' and' (b)'

deformed'configuration'associated'with'the'occurrence'of'lateral'torsional'buckling'..........................................'9'

Figure'2.4'–'Variation'of'the'critical'buckling'moment'increase'Mcr'(Nt)'/Mcr'(0)'with'Nt'(IPE'300'+'L=10'

m)'................................................................................................................................................................................................................'10'

Figure'2.5'−' Linear' longitudinal' stress'distributions'at'an' IPE'300'cross>section' for' (a)'β'<'9.6'and' (b)'

β=9.6'..........................................................................................................................................................................................................'11'

Figure'2.6'–'“Fork'conditions”'at'both'end'supports'...........................................................................................................'13'

Figure'2.7'>'Variation'of'Mcr'(Nt)'/Mcr'(0)'with'Nt:'comparison'between'analytical'and'numerical'results'

(IPE'300'+'L=10'm)'.............................................................................................................................................................................'14'

Figure'2.8:'Variation'of'Mcr'(Nt)'/Mcr'(0)'with'β'for'0.5)m'≤'L'≤'15)m'(IPE'300'beams'+'ψ=0)'..........................'16'

Figure'2.9:'Variation'of'Mcr'(Nt)'/Mcr'(0)'with'β'for'0.5)m'≤'L'≤'15)m'(IPE'500'beams'+'ψ=0.5)'.......................'16'

Figure'2.10:'Variation'of'Mcr'(Nt)'/Mcr'(0)'with'β'for'0.5)m'≤'L'≤'15)m'(HEB'500'beams'+'ψ=.1)'....................'17'

Figure'2.11:'Variation'of'Mcr'(Nt)'/Mcr'(0)'with'β'for'various'bending'moment'diagrams'(HEB'300'beams'+'

L=10'm)'....................................................................................................................................................................................................'18'

Figure'2.12:'Variation'of'Mcr'(Nt)'/Mcr'(0)'with'β'for'various'bending'moment'diagrams'(IPE'300'beams'+'

L=5'm)'.......................................................................................................................................................................................................'18'

Figure'2.13'–'Top'views'of'the'LTB'mode'shapes'of'the'beams'subjected'to'(a)'ψ='−'0.5'and'(b)'ψ='−'1'

diagrams'(β=1)'.....................................................................................................................................................................................'19'

Figure'2.14:'Variation'of'Mcr'(Nt)'/Mcr'(0)'with'β'for'beams'with'HEB>IPE'500>300'cross'sections'(L=15m'

+'ψ=0)'.......................................................................................................................................................................................................'20'

xiv

Figure'3.1'–'Standard'tension'coupon'specimens:'(a)'overview'and'(b)'detail'of'the'rupture'zone'..............'24!

Figure'3.2'–'Tensile'coupon'test'and'axial'extension'measured'by'means'an'extensometer'............................'24!

Figure'3.3'–'Cutting'of'thin'strips'to'measure'the'residual'stresses'.............................................................................'25!

Figure'3.4'–'Measuring'strip'length'(after'cutting),'by'means'of'an'extensometer'...............................................'25!

Figure'3.5'–'Stable'Bench'and'LVDT’s'employed'to'measure'the'beam'initial'geometrical'imperfections'.'26!

Figure'3.6'–'Schematic'representation'of'Step'1'...................................................................................................................'27!




Figure'3.10'–'Schematic'representation'of'Step'5'.................................................................................................................'27!

Figure'3.11:'Experimental'setTup:'(a)'overall'view'and'(b)'detail'of'the'beam'end'supports'...........................'28!

Figure'3.12'–'Detail'of'the'secondary'supporting'system'where'the'hydraulic'jacks'are'mounted'...............'29!

Figure'3.13'–'Web'stiffeners'intended'to'preclude'local'buckling'during'the'HEA'160'beam'test'.......................'29!

Figure'3.14'–'Detailed'view'of'the'beam'end'support'conditions'..................................................................................'30!

Figure'3.15'–'Measuring'device'systems'...................................................................................................................................'31!

Figure'3.16'–'Schematic'representations'of'the'steel'σTε'curves'obtained'for'the'(a)'IPE'200'and'(b)'the'

HEA160'beams'......................................................................................................................................................................................'33!

Figure'3.17'–'Residual'stresses'distribution'measured'at'the'IPE200'and'HEA160'beams'(positive'values'

stand'for'compression)'......................................................................................................................................................................'34!

Figure'3.18'–'Comparison'of'the'residual'stresses'distribution:'measured'(red),'linear'(blue)'and'

parabolic'(green)'.................................................................................................................................................................................'34!

Figure'3.19'–'Initial'geometrical'imperfections'measured'on'the'flanges'(points'B'and'H)'for'the'(a)'IPE'

200'and'(b)'HEA'160'..........................................................................................................................................................................'35!

Figure'3.20'–'Initial'geometrical'imperfections'measured'on'the'web'(point'E)'for'the'(a)'IPE'200'and'(b)'

HEA'160'...................................................................................................................................................................................................'35!

Figure'3.21'–'CrossTsection'points'for'which'initial'displacement'profiles'were'measured:'(a)'IPE'200'and'(b)'

HEA'160'beams'......................................................................................................................................................................................'35!

Figure'3.22'–'Overall'view'of'the'test'setTup'and'initial'(deformed)'configuration'of'the'IPE'200'beam'

specimen'..................................................................................................................................................................................................'36!

Figure'3.23'–'Time'evolution'of'the'axial'forces'recorded'by'the'measuring'devices'of'the'hydraulic'jacks'during'

the'IPE'200'beam'test'...........................................................................................................................................................................'36!

xv

Figure'3.1'–'Standard'tension'coupon'specimens:'(a)'overview'and'(b)'detail'of'the'rupture'zone'..............'24!

Figure'3.2'–'Tensile'coupon'test'and'axial'extension'measured'by'means'an'extensometer'............................'24!

Figure'3.3'–'Cutting'of'thin'strips'to'measure'the'residual'stresses'.............................................................................'25!

Figure'3.4'–'Measuring'strip'length'(after'cutting),'by'means'of'an'extensometer'...............................................'25!

Figure'3.5'–'Stable'Bench'and'LVDT’s'employed'to'measure'the'beam'initial'geometrical'imperfections'.'26!





Figure'3.10'–'Schematic'representation'of'Step'5'.................................................................................................................'27!

Figure'3.11:'Experimental'setTup:'(a)'overall'view'and'(b)'detail'of'the'beam'end'supports'...........................'28!

Figure'3.12'–'Detail'of'the'secondary'supporting'system'where'the'hydraulic'jacks'are'mounted'...............'29!

Figure'3.13'–'Web'stiffeners'intended'to'preclude'local'buckling'during'the'HEA'160'beam'test'.......................'29!

Figure'3.14'–'Detailed'view'of'the'beam'end'support'conditions'..................................................................................'30!

Figure'3.15'–'Measuring'device'systems'...................................................................................................................................'31!

Figure'3.16'–'Schematic'representations'of'the'steel'σTε'curves'obtained'for'the'(a)'IPE'200'and'(b)'the'

HEA160'beams'......................................................................................................................................................................................'33!

Figure'3.17'–'Residual'stresses'distribution'measured'at'the'IPE200'and'HEA160'beams'(positive'values'

stand'for'compression)'......................................................................................................................................................................'34!

Figure'3.18'–'Comparison'of'the'residual'stresses'distribution:'measured'(red),'linear'(blue)'and'

parabolic'(green)'.................................................................................................................................................................................'34!

Figure'3.19'–'Initial'geometrical'imperfections'measured'on'the'flanges'(points'B'and'H)'for'the'(a)'IPE'

200'and'(b)'HEA'160'..........................................................................................................................................................................'35!

Figure'3.20'–'Initial'geometrical'imperfections'measured'on'the'web'(point'E)'for'the'(a)'IPE'200'and'(b)'

HEA'160'...................................................................................................................................................................................................'35!

Figure'3.21'–'CrossTsection'points'for'which'initial'displacement'profiles'were'measured:'(a)'IPE'200'and'(b)'

HEA'160'beams'......................................................................................................................................................................................'35!

Figure'3.22'–'Overall'view'of'the'test'setTup'and'initial'(deformed)'configuration'of'the'IPE'200'beam'

specimen'..................................................................................................................................................................................................'36!

Figure'3.23'–'Time'evolution'of'the'axial'forces'recorded'by'the'measuring'devices'of'the'hydraulic'jacks'during'

the'IPE'200'beam'test'...........................................................................................................................................................................'36!

xvi

Figure'4.1'−'(a)'Longitudinal'residual'stress'pattern'and'(b)'initial'geometrical'imperfections'incorporated'into'

the'beam'GMNIA'−'shapes'and'values'taken'from'the'recent'work'of'Boissonnade'&'Somja'(2012)'.............'52'

Figure'4.2'–'Finite'element'model:'beam'discretisation'and'load'application'.........................................................'53'

Figure'4.3'−'Constitutive'law'adopted'to'model'the'steel'material'behaviour'.........................................................'53'

Figure' 4.4' –' Numerical' beam' equilibrium' path' and' deformed' configuration' at' the' brink' of' the' LTB'

collapse'.....................................................................................................................................................................................................'53'

Figure' 4.5' –' Schematic' representation' of' the' crossPsection' plastic' resistance' decrease' caused' by' the'

presence'of'axial'tension'..................................................................................................................................................................'55'

Figure' 4.6' –' Failure' mode' governed' by' lateralPtorsional' buckling' of' a' member' acted' by' majorPaxis'

bending'and'axial'tension'................................................................................................................................................................'55'

Figure'4.7'–'Deformed'configuration'of'the'midPspan'region'of'a'very'slender'beam,'at'collapse'.................'56'

Figure'4.8'−'Variation'of'Mu/Mpl'with'β'and'the'beam'length'(S460'steel'IPE'300'beams'under'uniform'

bending)'...................................................................................................................................................................................................'57'

Figure'4.9'−'Variation'of'Mu/Mpl'with'β'and'the'beam'length'(S355'steel'IPE'500'beams'under'triangular'

bending'–'ψ=0)'......................................................................................................................................................................................'58'

Figure'4.10'−'Variation'of'Mu/Mpl'with'β'and'the'bending'moment'diagram'(L=15*m'S355'steel'HEB'300'beams)

......................................................................................................................................................................................................................'58'

Figure'4.11'−'Variation'of'Mu/Mpl'with'β'and'the'bending'moment'diagram'(L=5*m'S460'steel'IPE'300'beams)'59'

Figure'4.12'–'Variation'of'Mu/Mpl'with'the'beam'lateralPtorsional'slenderness'λLT'..............................................'61'

Figure'5.1'−'Comparison'between'the'Mu'/Mpl,Rk'(numerical'gross'results)'and'Mb,Rd'/Mpl,Rk'(proposed'design'approach)'values'for'ψ=0'..................................................................................................................................................'66'Figure'5.2'−'Comparison'between'the'Mu'/Mpl,Rk'(numerical)'and'Mb,Rd'/Mpl,Rk'(proposed'design'approach)'values'for'ψ=1'.......................................................................................................................................................................................'67'Figure'5.3'−'Comparison'between'the'Mu-/Mpl,Rk-(numerical)'and'Mb,Rd'/Mpl,Rk'(proposed'design'approach)'values'for'ψ=0.5'...................................................................................................................................................................................'67'Figure'5.4'−'Comparison'between'the'Mu-/Mpl,Rk-(numerical)'and'Mb,Rd'/Mpl,Rk'(proposed'design'approach)'values'for'ψ=0'.......................................................................................................................................................................................'68'Figure'5.5'−'Comparison'between'the'Mu-/Mpl,Rk-(numerical)'and'Mb,Rd'/Mpl,Rk'(proposed'design'approach)'values'for'ψ ='−'0.5'.............................................................................................................................................................................'68'

xvii

Figure'5.6'−'Comparison'between'the'Mu#/Mpl,Rk#(numerical)'and'Mb,Rd'/Mpl,Rk'(proposed'design'approach)'

for'ψ='−'1'.................................................................................................................................................................................................'69'

Figure'5.7'−'Pictorial'representation'of'the'ultimate'moment'predictions'−'L=8.0'm'S355'steel'IPE'500'beam'

(ψ=1)'..........................................................................................................................................................................................................'71'

Figure'5.8'−'Illustration'of'the'effective'moment'concept'on'which'the'EC3JENVJ1J1'provisions'are'based'.........'72'

Figure'5.9'−'Values'of'the'ratio'difference'ΔRP-EC3'plotted'against'the'beam'slenderness'(ψ=1)'.....................'74'

Figure'5.10'−'Values'of'the'ratio'difference'ΔRP-EC3'plotted'against'the'beam'slenderness'(ψ=#−#1)'..............'74'

xix

List of Tables

Table&2.1&–&Critical&bending&loads&using&analytic&and&numerical&approaches&...........................................................&13&

Table&2.2&–&Profiles&and&lengths&used&within&LBA&..................................................................................................................&14&

Table&2.3&–&Moment&distribution&evaluated&in&LBA&...............................................................................................................&15&

Table&2.4&–&Comparison&between&geometric&properties&of&the&different&profile&section&.....................................&19&

Table&3.1&–&Measured&and&nominal&beam&cross5section&dimensions&............................................................................&32!

Table&3.2&–&Steel’s&material&properties&.......................................................................................................................................&32!

Table&3.3&5&Analytical,&numerical&and&experimental&results&concerning&the&two&beams&tested&........................&45!

Table&4.1&–&Load.carrying&capacity&of&HEB&300&beams&for&β =&0&....................................................................................&54&Table&4.2&–&Load.carrying&capacity&of&HEB&300&beams&for&β =&1&....................................................................................&54&

Table&5.1&−&Averages,&standard&deviations&and&maximum/minimum&value&of&the&ratio&RM&...............................&70&

Table&5.2&−&Ultimate&moment&predictions&for&the&L=8.0&m&S355&steel&IPE&500&beam&under&uniform&

bending&.....................................................................................................................................................................................................&70&

Table&5.3&−&Averages,&standard&deviations&and&maximum/minimum&values&of&ΔMb,Rd&........................................&71&

Table&5.4&−&Averages,&standard&deviations&and&maximum/minimum&values&of&ΔRP'EC3&for&(a)&ψ&=&1&and&(b)&

ψ&=&L1&.........................................................................................................................................................................................................&75&

Chapter 1

Introduction

2

1.1 Preliminary remarks

In recent years, the technical and scientific community dealing with steel structures has devoted a

considerable effort to the development of efficient (safe and economical) procedures and formulae

(interaction equations) for the design and safety checking of steel members (i) subjected to different

combinations of internal forces and moments and (ii) susceptible to global buckling phenomena,

namely flexural buckling (members under compression) and/or lateral-torsional buckling (open-section

members under major-axis bending). Indeed, it is well known that the failure of most thin-walled steel

members, such as the I-section beams dealt with in this work, is governed by a combination of

instability and plasticity effects − while the latter are more prevalent in stocky beams, the former

dominate in the more slender members. In the particular case of beams subjected to major-axis bending, their

failure often involves lateral-torsional buckling, a complex three-dimensional global instability phenomenon

involving torsion and minor-axis bending, which is mainly triggered by the low torsional stiffness exhibited

by open-section thin-walled cross-sections. Naturally, the ultimate strength and collapse mechanism of the

aforementioned beams can only be adequately predicted provided that in-depth knowledge about their lateral-

torsional buckling mechanics is acquired. Moreover, it is well known that the beam lateral-torsional buckling

behaviour is affected by the presence of axial forces. Furthermore, the influence of compressive forces on

the lateral-torsional buckling behaviour has been thoroughly investigated, not only because of its practical

relevance (most steel frame members are subjected to major-axis bending and compression), but also because

such forces cause a significant reduction of beam ultimate (bending) strength that needs to be

accounted for. As for the influence of tensile forces on the beam lateral-torsional buckling behaviour,

which has much less practical relevance (members subjected to bending and tension are relatively rare), it has

received little attention from researchers − indeed, due to their beneficial effects, tensile forces are often

“ignored” when assessing the beam resistance against lateral-torsional failure (e.g., in the current version of the

part 1-1 of Eurocode 3 − CEN 2005).

As far as steel members are concerned, the vast majority of available studies deal with I-section members,

by far the most widely used in the steel construction industry. This fact is attested by the very large number

of “fine-tuned” expressions (interaction equations), intended for the design and safety checking of I-

section members, which are present in the current steel design codes. For instance, the current version

of part 1-1 of Eurocode 3 (EC3-1-1 − CEN 2005) contains a plethora of rather elaborate (and also fairly

complex) formulae and equations aimed at the design (cross-section and member checks) of I-section members

with narrow-flange (I type) and wide-flange (H type) cross-sections and members subjected to a large

variety of internal forces and moment diagrams − the interested reader can find the background of most of these

formulae and equations in the ECCS (European Convention for Constructional Steelwork) report stemming

3

from the activity of its Technical Committee on Stability (TC8) and co-authored by Boissonnade et al. (2006).

In the particular case of I-section members subjected to major-axis bending (beams), which are highly

prone to lateral-torsional buckling (unlike beams with closed section, such as RHS beams), it is necessary

either (i) to prevent the occurrence of such buckling phenomenon, by appropriately bracing the beam (i.e.,

restraining the lateral deflections and/or twisting rotations at selected cross-section points along the beam

length), or (ii) to develop efficient (safe and economical) procedures to estimate the beam ultimate strength

associated with a collapse governed by lateral-torsional buckling.

1.2 Motivation and scope of the work

For some load combinations, the members of steel frames and/or trusses members may be subjected to

internal force and moment diagrams that combine major-axis bending (predominant) and axial tension −

such members, which are illustrated in Figure 1.1, are sometimes termed “beams under tension”, a

designation adopted hereafter in this work.

Figure 1.1 - Beam subjected to uniform major-axis bending (My) and tension (Nt)

The fact that the above internal force and moment combination is relatively rare and, moreover, can be

conservatively handled by “ignoring” the axial tension when checking against the member buckling ultimate

limit state (only the cross-section resistance needs to be checked), is most likely the reason why very

little attention has been paid to the development of a genuine design and/or safety checking procedure

aimed at estimating the ultimate strength of beams under tension. Indeed, it is fair to say that, quite

surprisingly, virtually no information can currently be found concerning the structural response and design of

I-section beams members subjected to major-axis bending and tension (i.e., beams under tension), namely on

how the presence of tension affects (improves) the beam lateral-torsional buckling behaviour. Indeed, the

rather complete literature search (including publication in both the English and German languages) carried

out by the author bore no fruits and, moreover, no information was obtained from several world-wide

recognized experts on lateral-torsional buckling that were contacted in the last year. The sole exception

to the above situation are the provisions included in Part 1-1 of the ENV (European Pre-Norm) version of

Eurocode 3 (EC3-ENV-Part 1-1, 1992) and concerning the safety checking of beams under tension

against failures triggered by lateral-torsional buckling. Such provisions, whose existence provided the

motivation for the investigation study reported in this work, are based on an “effective (reduced) bending

4

moment” concept to take into account the beneficial effect stemming from the presence of axial tension

− however, once more, no trace of background information concerning these rather “mysterious” provisions

could be found. Of course, part of the explanation for the “information void” on this problem is due to the

fact that (i) beams under tension occur seldom in practice and (ii) neglecting the tension effects leads to

conservative ultimate strength estimates against lateral-torsional failures. The above design provisions were

later removed from the EN (European Norm) version of Eurocode 3 (EC3-EN-Part 1-1, 2005), allegedly due to

space limitations. Thus, it seems fair to argue that the favourable effect of axial tension on failures

governed by lateral-torsional buckling is currently completely neglected, which naturally leads to over-

conservative designs. Indeed, a beam subjected under tension is currently designed as a “pure beam”,

i.e., only (major-axis) bending is taken into account − the presence of axial tension is felt exclusively through

the cross-section resistance check.

Therefore, the objective of this work is to provide a contribution to the investigation of the behaviour,

collapse and design of I-section beams susceptible to lateral-torsional buckling and subjected to tension, namely

by acquiring information on how conservative are the ultimate strength predictions that neglect the tension

effects. In particular, the works aims at bridging the lack of scientific information and technical guidance

concerning the lateral-torsional stability, behaviour/failure and design of beams under tension. It deals

specifically with (doubly symmetric) hot-rolled steel I-section beams exhibiting “fork-type” end supports and

subjected to simple transverse loadings (mostly applied end moments) and not affected by local buckling

phenomena − beams with compact cross-sections (class 1 or 2 cross-sections, according to the EC3

nomenclature) that can reach its plastic resistance prior to the occurrence of local buckling.

1.3 Organisation of the Dissertation

The dissertation is organised into six chapters, the first of which is the present introductory chapter. In the

following paragraphs, brief descriptions of the contents of the remaining of these chapters are presented.

Chapter 2 is devoted to investigate the influence of axial tension of the beam lateral-torsional

stability/buckling (bifurcation) behaviour. After briefly reviewing the fundamental of lateral-torsional

buckling behaviour, attention is paid to the derivation and validation, through the comparison with

beam finite element results, of an analytical expression that provides critical buckling moments associated with

the lateral-torsional stability of uniformly bent beams subjected to tension. Then, the analytical study

is (numerically) extended to beams subjected to non-uniform bending (mostly stemming from unequal

applied end moments, although uniformly loaded beams are also addressed) − several beam finite element

results concerning the beneficial influence of axial tension on the beam lateral-torsional stability are presented

and discussed in detail.

5

Chapter 3, which is concerned with the experimental investigation, is divided into three distinct parts,

which address: (i) the description and characterisation of the specimens tested (one narrow flange beam and one

wide flange beam, both subjected to eccentric axial tension), including all the preliminary measurements

required to obtain information about the steel material properties (tensile coupon tests), residual stresses and

initial geometrical imperfections; (ii) the performance of two full-scale tests, including the description

of the test set-up and procedure and the presentation of the results obtained; and (iii) the numerical

simulations carried out to develop a shell finite element model that is able to simulate adequately the test

results − this was done by means of the software FINELG (2012) and the resulting shell finite element

model was then used to develop and validate a FINELG beam finite element model, subsequently employed to

perform an extensive parametric study.

Chapter 4 deals with the aforementioned parametric study, carried out in order to assemble a fairly large

ultimate strength/moment data bank, involving more than 2000 numerical simulations concerning beams with

various cross-section shapes, lengths, yield stresses, acting bending moment diagrams and axial

tension levels. Particular attention is paid to the distinction between the beams collapses stemming from

plasticity effects (cross-section resistance) and those governed by lateral-torsional buckling effects −

recall that only the latter are investigated in this work.

Chapter 5 uses the gathered experimental (only two specimens) and numerical (over 2000 beams

analysed) ultimate strength/moment data gathered previously to develop/propose design procedures

for beams subjected to tension − in particular, the work (i) revisits the “effective moment” concept included

in EC3-ENV-Part 1-1 and (ii) investigates the merits of using the beam buckling curves currently available in

EC3-EN-Part 1-1 in combination with slenderness values obtained on the basis of critical buckling moments

that incorporate the beneficial effects of the presence of axial tension − i.e., the latter approach merely

consists of a slight modification of the procedure prescribed in the current Eurocode 3 to design beams

against lateral-torsional failures.

Finally, Chapter 6 briefly describes the content of the dissertation, underlining the main conclusions drawn

from the analytical, experimental and numerical research activity reported, and provides a few suggestions for

future developments/extensions of the work carried out by the author.

Chapter 2

Lateral Torsional Buckling

2.1 Introduction

This chapter addresses the influence of axial tension on the lateral-torsional stability/buckling behaviour of

simply supported (“fork-type” supports − free warping and flexural rotations) doubly-symmetric I-

section beams subjected to major-axis bending − i.e., to assess how the presence of an axial tension Nt

changes/increases the critical buckling moment Mcr. Of course, it is assumed that Nt is such that the beam

cross-section resistance (under bending moment and axial force) is not reached prior to the occurrence of

instability (bifurcation) − otherwise, if Nt is large enough to preclude the occurrence of compressive stresses in

the cross-section, the beam collapse stems exclusively from plasticity effects.

Lateral-torsional buckling (LTB) is a three-dimensional instability phenomenon exhibited by beams

subjected to major-axis bending, which causes transverse (vertical) displacements u, as depicted in Figure 2.1 −

the equilibrium path associated with major-axis bending is termed the “fundamental (or pre-buckling)

equilibrium path”, as shown in Figure 2.2. The LTB instability, occurring at a bifurcation point,

involves a combination of minor-axis bending (transverse/horizontal displacements v − see Figure 2.1)

and torsion (angles of twist φ − see Figure 2.1) − the equilibrium path following the

instability/bifurcation is termed the “post-buckling equilibrium path”, as shown in Figure 2.2. The

intersection between the above two equilibrium paths occurs at “the critical buckling moment Mcr” (caused

by the transverse loading).

8

Figure 2.1 – Beam deformed configuration associated with the occurrence of LTB: (a) member and (b) cross-section views

Figure 2.2 – Lateral-torsional buckling: fundamental and post-buckling equilibrium paths (Reis & Camotim, 2012)

This chapter begins by presenting the analytical derivation of an expression providing critical buckling

moments of simply supported I-section beams subjected to uniform major-axis bending and axial tension.

Next, the analytical expression obtained is then used to validate an ABAQUS (Simulia Inc. 2008) beam finite

element model, subsequently employed to perform an extensive parametric study aimed at assessing the effect

of axial tension on the critical moment of simply supported doubly symmetric I-section beams acted by several

non-uniform bending diagrams, namely those caused by unequal applied end moments and uniformly

distributed loads.

2.2 Beams under uniform bending − analytical solution

As mentioned before, this present section addresses the derivation of an analytical expression that provides

critical buckling moments associated with the lateral-torsional buckling/stability (bifurcation) of simply

supported I-section beams subjected to uniform major-axis bending and axial tension − see Figure 2.3. The first

step consists of establishing the differential equilibrium equations governing the behaviour under consideration.

Following the classical monographs by Chen & Atsuta (1977) and Trahair (1993), concerning the LTB

behaviour of beam-columns (i.e., members under uniform major-axis bending and axial compression), it is

9

Figure 2.3 – Beam subjected to major-axis bending My and axial tension Nt: (a) general view and (b) deformed configuration associated with the occurrence of lateral torsional buckling

possible to derive differential equilibrium equations that ensure adjacent equilibrium for members subjected to

major-axis bending and axial tension and deemed to remain undeformed up to bifurcation (i.e., the pre-

buckling deformations are neglected) − they read (note the change in sign of the Nt terms)

where v and φ are the minor-axis bending displacements and torsional rotations, respectively (see Figure 2.1)

The detailed analytical derivation of these equations is presented in Annex 1, included at the end of this

dissertation.

For a simply supported beam, the solution of the eigenvalue problem defined by equations (2.1)-(2.2) is

provided by the sinusoidal eigenfunctions

which define the beam critical mode shape and correspond to critical buckling moments given by the expression

In this expression, (i) Mcr (0) denotes the critical buckling moment of the “pure beam” (member under uniform bending only), and (ii) Pcr,z and Pcr,φ are given by

EIz vIV + Nt vʹ′ʹ′+My φ ʹ′ʹ′= 0

EIw φ IV − (GIt + Nt r02)φ ʹ′ʹ′ + My vʹ′ʹ′= 0

(2.1)

(2.2)

v(x) = A1 sin (π/L x)

φ(x) = A2 sin (π/L x)

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)

10

and their values correspond to the symmetric of the minor-axis flexural and torsion buckling loads of the “pure

column” (member under uniform compression only) − once again, the steps involved in the determination of

Eqs.(2,3)-(2.7) are presented and explained in detail in Annex 1, included at the end of this dissertation.

Eq. (2.5) confirms (and quantifies, for the particular case under consideration) the beneficial effect of tension

on the member lateral-torsional buckling moment − i.e., the additional bending and torsional stiffness values,

stemming from the presence of Nt, lead to a Mcr increase. In order to illustrate the results provided by the

derived analytical expression, Figure 2.4 plots the critical bucking moment increase [Mcr (Nt) /Mcr (0)] against the

ratio β=Nt /My 1 for an IPE 300 beam with length L=10 m. It is observed that the critical moment increase

grows exponentially with the applied tension level − for β larger than 9.6, lateral-torsional buckling no

longer occurs, as the whole beam is under tension.

β = Νt/Μy

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5

Mcr

(N

t) /M

cr (

0)

0.0

5.0

10.0

15.0

20.0

25.0

30.0

Figure 2.4 – Variation of the critical buckling moment increase Mcr (Nt) /Mcr (0) with Nt (IPE 300 + L=10 m)

The Mcr (Nt) /Mcr (0) vs. β curve eventually tends to infinity as β approaches a “limit value” (9.6 in this

particular case) because such limit value corresponds to the absence of compressive stresses in any beam

cross-section. In order to illustrate this statement, Figure 2.5 shows the linear longitudinal stress distributions

of an IPE 300 cross-section subjected to axial tension levels corresponding to (i) β <9.6 and (ii) β=9.6

(limit value). However, note that a loading strategy involving and Nt value established a priori, which

corresponds to applying a transverse loading to a “pre-tensioned beam”, never precludes the occurrence of

lateral-torsional buckling (for any Nt) − indeed, compressive stresses will always eventually occur in the beam.

1 Note that the parameter β has dimension L−1, i.e., these results are valid only for the IPE 300 beam analysed. My stands for the applied major-axis

end bending moment, herein designated simply as M. It is worth mentioning that β, which relates the applied Nt and M values, corresponds to the inverse

of the eccentricity exhibited by tensile axial force Nt causing the moment M.

11

(a)

(b)

Figure 2.5 − Linear longitudinal stress distributions at an IPE 300 cross-section for (a) β < 9.6 and (b) β=9.6

Finally, one last word to mention the similarity between Eq. (2.5) and the classical expression to

evaluate the critical buckling moment of doubly symmetric members subjected to uniform major-axis

bending and axial compression (beam-columns), which reads (e.g., Trahair, 1993)

where Pcr,z and Pcr,φ are now the minor-axis flexural and torsion column buckling. Note that the sole difference

between Eqs. (2.5) and (2.8) are the signs of the terms involving the axial compression − the negative signs

reflect the fact that axial compression lowers the critical buckling moment value.

2.3 Beams under non-uniform bending − numerical results

This section begins by presenting an ABAQUS beam finite element model developed to perform linear buckling

analyses, which will be subsequently used to investigate the influence on axial tension on the LTB behaviour

of beams subjected to non-uniform bending. Then, after using the analytical expression derived in the

previous sub-section to validate the above finite element model, this model is used to perform a parametric

(2.8)

12

study aimed at assessing the beneficial influence of axial tension on the LTB behaviour of doubly symmetric I-

section beams with different geometries (cross-section dimensions and length) and subjected to non-uniform

bending caused by either unequal end moments or uniformly distributed transverse loads.

2.3.1 Beam finite element model

The beams are discretised by means of ABAQUS three-dimensional bar finite elements (BFE). Previous studies

(e.g., Mendonça, 2004) showed that, among the various elements of this type available in the ABAQUS library,

element B31OS (B-“beam”, 3-spatial, 1-linear interpolation functions and OS-“open section”) is the most

suitable for the purpose of this work. The formulation of this BFE is based on Timoshenko’s bar theory (i.e.,

includes the shear deformation) and it exhibits seven degrees of freedom per node, corresponding to

three translations, three rotations and warping.

In order to perform a buckling (linear stability) analysis, it is necessary to input a data file containing (i) a

reference load, which is linked to a load parameter λ (λcr is its sought lowest bifurcation value), and

(ii) one command to initiate the solution of the buckling eigenvalue problem (determination of the

critical buckling load and corresponding buckling mode shape, i.e., the lowest eigenvalue and its associated

eigenvector). In the particular case of the lateral-torsional buckling of beams subjected to major-axis bending,

the code provides critical buckling moment values (Mcr), which correspond to the critical load parameter (λcr)

multiplied by the maximum moments associated with the reference transverse loading − in this work, they may

consist of (i) applied end moments (Mcr=λcr MR,max) or (ii) uniformly distributed loads (Mcr=λcr pR,max L2/8).

Concerning the loading application, the combination of major axis bending (M) and axial tension (Nt)

can be made in many different ways. The two most common ones are either (i) begin by applying a fixed Nt

value and then gradually increase the bending moment diagram until buckling takes place, thus determining

the (unknown) Mcr value associated with the presence of the (known) Nt, or (ii) select reference bending

moment diagram and axial tension value, and then increase them simultaneously while keeping the

relation between them constant − this corresponds to a“proportional loading strategy” and leads to a

pair of (initially unknown) M and Nt values that are associated with the occurrence of lateral-torsional buckling.

The latter load application strategy was employed in this work and the parameter β defines the constant ratio

between the axial tension and the reference maximum bending moment (β=NR,t /MR,max, where it should be

noted that β has dimension L−1 and, therefore, is“beam-dependent”) − buckling occurs for Ncr,t=λcr NR,t and

Mcr,N=λcr MR,max, which define the critical pair (Ncr,t, Mcr,N (Ncr,t)). It is still worth pointing out that, according to

this load application strategy and depending on the β value, it may happen that Mcr,N=∞ − it suffices that β

corresponds to a bending moment and axial tension combination causing no compression in the whole beam.

Concerning the boundary conditions, all the beams analysed are simply supported, i.e., exhibit “fork-type”

end supports, as illustrated in Figure 2.6: the two end sections have (i) prevented transverse displacements,

13

(ii) free flexural rotations, (iii) prevented torsional rotations and (iv) free warping − as for the

longitudinal displacement, it is prevented at one end section and free at the other.

Figure 2.6 – “Fork conditions” at both end supports

2.3.2 Validation − comparison with the analytical results

Before carrying out the numerical investigation, it is convenient to begin by validating the developed ABAQUS

BFE model. This is done in two steps, both including a comparison between the analytical and numerical results

concerning beams subjected to uniform bending (ψ=1). The first step consists of analysing two beams

subjected to pure uniform bending: (i) a 10 m long IPE 300 beam and (ii) a 5 m long HEB 300 beam.

Table 2.1 shows the critical bending moment values of these two beams, obtained analytically and

numerically − the two pairs of values are very similar, as the differences between then do not exceed 2%.

Table 2.1 – Comparison between the analytical and numerical (ABAQUS BFE) critical bending moment values

Mcr [kNm] Analytical Numerical (ABAQUS BFE) Δ (%)

IPE 300 (L=10.0 m) 48.6 47.8 1.6

HEB 300 (L=5.0 m) 1433.26 1401.5 2.2

The second step consists of comparing the analytical and numerical results concerning the LTB behaviour

of a 10 m long IPE 300 beam subjected to uniform bending and different levels of axial tension. Figure

2.7 compares results concerning the variation of the critical bucking moment percentage increase Mcr (Nt)

/Mcr (0) with the ratio Nt /M − the solid line stands for the analytical values, obtained with Eq. (2.5)

and already shown in Figure 2.4, and the small circles correspond to the numerical values determined

by means of the ABAQUS BFE analyses. It is clear that there is a virtually perfect agreement between the

analytical and numerical values, which means that the BFE model may be deemed validated.

14

β = Νt/Μ

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5

Mcr

(N

t) /M

cr (

0)

0.0

5.0

10.0

15.0

20.0

25.0

30.0

Numerical

Analytical

Figure 2.7 - Variation of Mcr (Nt) /Mcr (0) with Nt: comparison between analytical and numerical results (IPE 300 + L=10 m)

2.3.3 Parametric studies

The parametric studies, carried out by means of the developed (and validated) ABAQUS BFE model,

concern I-section beams with the geometries (cross-section dimensions and lengths) defined in Table 2.2.

Table 2.2 – Geometries of the I-section beam analysed in the parametric studies

Length (m) IPE 300 IPE 500 HEB 300 HEB 500

0.5 P 1 P P

2 P P P P 3.5 P P P P

5 P P P P 8 P P P P

10 P P P P 15 P P P P

20 P P P 25 P P

15

For each beam, various levels of axial tension are considered, defined by means of the (dimensional)

parameter β=Nt /M − it takes the values 0; 0.5; 1.0; 1.5; 2.0: 2.5; 3.0. Moreover, several beam transverse

loadings are addressed − they are indicated in Table 2.3. A total of about 1200 LTB results were obtained.

Table 2.3 – Moment distribution evaluated in Lateral Buckling Analysis (LBA)

ψ=1 (uniform bending)

ψ=0.5

ψ=0(triangular bending)

ψ = − 0.5

ψ= − 1 (doubly triangular bending)

Uniformly distributed load p

(applied along the beam shear centre axis)

Next, a representative sample of the critical buckling moments obtained in this work is presented and

discussed − the full set of LTB results can be found (in tabular form) in Annexes 2.1 to 2.4, located at the end

of the dissertation. The main aim of this presentation/discussion is to assess the individual influences of the

various parameters involved.

Figures 2.8 to 2.10 concern (i) IPE 300 beams under ψ=0 bending moment diagrams, (ii) IPE 500

beams under ψ=0.5 bending moment diagrams and (iii) HEB 500 beams under ψ= − 0.5 bending moment

diagrams, respectively. All of them provide the variations of the critical moment ratio Mcr (Nt) /Mcr (0) with β for

various beam lengths, ranging from L=0.5 m to L=25 m. The observation of the LTB results presented in these

three figures make it possible to conclude that, naturally, the relevance of the (beneficial) tension effect

on the LTB behaviour grows with the beam length, i.e., as the beam becomes more prone to this instability

phenomenon. This assertion can easily be confirmed by looking at the slopes of the Mcr (Nt) /Mcr (0) vs. β

curves corresponding to the different beam lengths. Quantitatively speaking, the variation of the beneficial effect

due to axial tension with the beam length is different for the three cases addressed in Figures 2.8 to 2.10 −

in particular, the acting bending moment diagram shape seems to play an important role. The influences of the

various parameters involved in the beam LTB behaviour will be assessed individually next.

16

β = Νt/Μ

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mcr

(N

t) /M

cr (

0)

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6L = 0.5 mL = 1 mL = 2mL = 3.5 mL = 5 mL = 8 mL = 10 mL = 15 m

Figure 2.8: Variation of Mcr (Nt) /Mcr (0) with β for 0.5 m ≤ L ≤ 15 m (IPE 300 beams + ψ=0)

β = Νt/Μ

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mcr

(N

t) /M

cr (

0)

1.01.52.02.53.03.54.04.55.05.56.06.57.0

L = 1 mL = 2 mL = 3.5 mL = 5 mL = 8 mL = 10 mL = 15 mL = 20 m

Figure 2.9: Variation of Mcr (Nt) /Mcr (0) with β for 1 m ≤ L ≤ 20 m (IPE 500 beams + ψ=0.5)

17

β = Νt/Μ0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mcr

(N

t) /M

cr (

0)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0L = 2 mL = 3.5 mL = 5 mL = 8 mL = 10 mL = 15 mL = 20 mL = 25 m

Figure 2.10: Variation of Mcr (Nt) /Mcr (0) with β for 2 m ≤ L ≤ 25 m (HEB 500 beams + ψ=-1)

In order to assess the influence of the bending moment diagram shape, Figures 2.11 (L=10 m HEB 300

beams) and 2.12 (L=5 m IPE 300 beams) provide Mcr (Nt) /Mcr (0) vs. β curves for all the bending moment

diagrams considered in this work. The observation of these numerical results prompts the following

remarks:

(i) The key factor is the shape of the amount of compression exhibited by the axial stress distributions acting

on the flanges along the whole beam length, which stems from the combination of the uniform axial

tension Nt with the varying bending moments. This combination leads to tensile effects that are (i1) highest

for the (triangular) ψ=0 moment diagram, corresponding to the “least compressed” flange pair, and (i2)

lowest for the (uniform) ψ=1 uniform diagram, obviously corresponding to the “most compressed”

flange pair. The same reasoning explains why, between the two moment distributions associated with

flanges part in compression and part in tension due to bending (ψ= − 0.5 and ψ=0.5), the benefits of axial

tension are more pronounced for the beam acted by the ψ= − 0.5 diagram. Figure 2.13, providing the top

views of the LTB mode shapes concerning the (i1) ψ= − 0.5 and (i2) ψ= − 1 bending moment diagrams,

for β=1, clearly shows that the axial tension reinforces the restraint effect of the shorter and less bent

beam right side on its longer and more bent left counterpart − due to symmetry, such restraint effect

is absent from the beam subjected to the ψ= − 1 bending moment diagram.

18

(ii) It is still worth noting that, somewhat surprisingly, the curves associated with the uniformly distributed

load fall in between those concerning the ψ= − 0.5 and ψ=0.5 bending moment diagrams.

β = Νt/Μ

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mcr

(N

t) /M

cr (

0)

1.0

1.5

2.0

2.5

3.0

3.5ψ = 1ψ = 0.5ψ = 0ψ = -0.5ψ = -1Dist. Load

Figure 2.11: Variation of Mcr (Nt) /Mcr (0) with β for various bending moment diagrams (HEB 300 beams + L=10 m)

β = Νt/Μ

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mcr

(N

t) /M

cr (

0)

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6ψ = 1ψ = 0.5ψ = 0ψ = -0.5ψ = -1Dist. Load

Figure 2.12: Variation of Mcr (Nt) /Mcr (0) with β for various bending moment diagrams (IPE 300 beams + L=5 m)

19

(a)

(b)

Figure 2.13 – Top views of the LTB mode shapes of the beams subjected to (a) ψ= − 0.5 and (b) ψ= − 1 diagrams (β=1)

Regarding the influence of the cross-section shape, Figure 2.14 concerns L=15 m beams subjected

to triangular bending moment diagrams (ψ=0) and exhibiting the four cross-sections considered in this

work. The observation of the corresponding Mcr (Nt) /Mcr (0) vs. β curves indicate that the benefits of axial

tension become more pronounced as the cross-section height increases (IPE 500 vs. IPE 300 + HEB 500 vs.

HEB 300). This height increase is also associated with an increase of the web height-to-flange width ratio

(h/b), which naturally leads to a growth of the ratio between the major and minor-axis moments of inertia

(Iy /Iz), a well known “measure” of the beam susceptibility to LTB − Table 2.4 shows the values of these

ratios for the four profiles considered in this work. Finally, it is interesting to notice that, depending on the

cross-section height, the benefits of axial tension may be larger for the IPE profile (IPE 300 vs. HEB 300) or

for the HEB one (IPE 500 vs. HEB 500) − however, the differences are only meaningful for β >1.0 (see

Figure 2.14). Although an extended parametric study would be required to obtain a solid explanation for this

“benefit switch”, it is probably related to the fact that the same web height increase (67%) corresponds to (i) a

25% h/b increase and a 64% Iy /Iz increase, for the IPE profiles, and (ii) a 70% h/b increase and a 167% Iy /Iz

increase, for the HEB profiles − the susceptibility to LTB grows much more for the HEB profiles than for the

IPE ones.

Table 2.4 – Comparison between geometric properties of the different profile section

IPE 300 IPE 500 ΗΕΒ 300 ΗΕΒ 500

h / b 2 2.5 1 1.7 Iy / Iz 14 23 3 8

20

β = Νt/Μ

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mcr

(N

t) /M

cr (

0)

1.2

1.6

2.0

2.4

2.8

3.2

3.6

4.0

4.4IPE 300IPE 500HEB 300HEB 500

Figure 2.14: Variation of Mcr (Nt) /Mcr (0) with β for beams with HEB-IPE 500-300 cross sections (L=15m + ψ=0)

Finally, it is worth mentioning that a preliminary investigation was carried out on the possibility of

finding a relation between the values of Mcr (Nt) concerning beams subjected to non-uniform and uniform

bending − similar to the coefficient C1 adopted in EC3 to relate the critical moments of beams under non-

uniform and uniform pure bending. Although this preliminary investigation bore no fruits (the high dispersion

of the values found precluded the immediate development of a simple expression for a coefficient C1 that takes

into account axial tension), the author is convinced that further research may lead to the sought expression.

2.4 Summary

This chapter reported an analytical and numerical (BFE) investigation concerning the beneficial effect of axial

tension on the lateral-torsional buckling behaviour of I-section steel beams. After deriving an analytical

expression that provides critical buckling moments of beam subjected to uniform bending and tension,

numerical results were presented for beams with various geometries (cross-section dimensions and length)

and subjected to several non-uniform bending moment diagrams. The results obtained and gathered in this

chapter will be used later in the development of a design procedure against the lateral-torsional failure

of I-section beams subjected to axial tension − see Chapter 5.

1.0

21

Out of the various conclusions drawn from the research work reported in this chapter, the following ones

deserve to be specially mentioned:

(i) It was shown analytically, for the case of uniform bending, that axial tension has a beneficial effect on the beam

LTB behaviour, i.e., leads to a Mcr increase.

(ii) The above beneficial effect becomes more relevant as beams are more susceptible to LTB − for instance,

the Mcr increase grows with the beam length or with the ratio between the cross-section major and minor-

axis moments of inertia.

(iii) Moreover, it was possible to assess how the axial tension beneficial effects vary with the bending moment

diagram shape. It was concluded that the highest effects occur for ψ=0 (triangular diagram), which

corresponds to the “least compressed” flange pair. Conversely, the lowest axial tension benefits

occur for ψ=1, corresponding to the “most compressed” flange pair.

(iv) Concerning the cross-section geometry, it is clear that increasing the web height leads to considerably

higher axial tension beneficial effects. However, it became also clear that the increased axial tension

benefits are more pronounced for the HEB (wide flange) profiles than for the IPE (narrow flange) ones.

Chapter 3

Behaviour and Strength − Experimental Study

3.1 Introduction

This chapter reports a limited experimental investigation (only two full-scale tests are involved), concerning the

behaviour and strength of beams subjected to tension − the tests were carried out at the Structures Laboratory of

École d'Ingénieur et d'Architecture de Fribourg, from the Haute École Suisse Occidentale. Besides acquiring

in-depth knowledge about the structural response under consideration, this experimental study aims at gathering

information intended to develop accurate and reliable numerical (finite element) models, which will be

subsequently employed to (i) carry out parametric studies to assemble a fairly large ultimate strength data bank,

in Chapter 4 − the final goal is to assess the merits of the design methodology developed in Chapter 5.

Prior to the performance of each test, standard preliminary measurements were carried out with the

aim of characterising the specimens: (i) measurements to define the member geometry, (ii) tensile coupon

tests, to obtain the steel material properties, (ii) residual stresses measurements and (iii) determination of

the initial geometrical imperfections. Then, after describing the experimental set-up and procedure, the chapter

presents and discusses the test results obtained. Finally, these results are used to validate shell and beam finite

element models developed to simulate the structural response of thin-walled members under major-axis

bending and tension − the validation is made through the comparison between the test results reported

and the corresponding numerical simulations.

24

3.2 Specimen characterisation

3.2.1. Material tests

Regarding the material tests, (i) the real cross-section geometric properties were defined by carefully measuring

the specimen dimensions and (ii) tensile coupon tests were carried out, in order to assess the steel material

behaviour. Initially, several measurements were taken, in order to determine the specimen full length and its

cross section dimensions, namely the web and flange width and thickness − several measurements were

taken along the specimen length, in order to assess the longitudinal variation of the cross-section

dimensions, and a high accuracy digital calliper was employed to perform this task. Then, coupon tests were

carried out to obtain the steel stress-strain curve (constitutive law) at both the web and flanges, following the

provisions for uniaxial tensile prescribed in EN 10002-1 (CEN, 2001) − an extensometer was used to measure the

axial extensions and the coupon specimens were tested up to failure/rupture, as illustrated in Figures 3.1 and

3.2.

(a) (b)

Figure 3.1 – Standard tension coupon specimens: (a) overview and (b) detail of the rupture zone

Figure 3.2 – Tensile coupon test and axial extension measured by means of an extensometer

The performance of a tensile coupon tests consisted of a three-step protocol, which included (i) an

initial loading procedure up to the plastic range, (ii) a full unloading procedure and (iii) a new reloading

procedure until failure/rupture occurred. This protocol was followed to enable a more accurate

25

estimation of the steel profile Young’s modulus, on the basis of Hooke’s law. The steel σ–ε curves

obtained were characterised by four parameters, namely the Young’s modulus E, yield stress fy, failure

stress fu and axial extension at failure εu.

3.2.2 Residual stress measurement

Concerning the residual stress measurement, it was based on a destructive method termed “sectioning

method” and briefly described next. The beam segment used to measure the residual stresses was cut into thin

strips along the cross section mid-line, as shown in Figure 3.3. Prior to cutting, the strips were marked on the

cross section, together with sets of two point (circular) marks located near the beam segment ends (well

apart), intended for the measurement of each strip initial and final length, by means of an extensometer,

as depicted in Figure 3.4. After recording all the strip initial and final lengths, the residual longitudinal stresses

were estimated through the simple relation

Figure 3.3 – Cutting of thin strips to measure the residual stresses

Figure 3.4 – Measuring strip length (after cutting), by means of an extensometer

(3.1)

26

By following the above procedure, which (i) involves measuring the length changes of strips covering the

entire cross section mid-line, (ii) is based on the stress relief experienced by each strip after being cut and (iii)

uses Hooke’s law, it is possible to use Hooke’s law to obtain a reasonable estimate/measurement the cross-

section longitudinal normal residual stress distribution.

3.2.3 Determination of the initial geometrical imperfections

The specimen initial geometrical imperfections (global and local) were measured by resorting to a set of

linear variable displacement transducers (LVDT’s) and a stable bench, which are displayed in Figure 3.5 and

were specifically designed to perform this task. At 10 cm intervals along the specimen length, (i) vertical

displacements at three upper flange points (web-flange corner and flange free ends) and (ii) lateral

displacements at three web points (mid-height and web-flange corners) were recorded. It is worth noting

that, in view of the relative lengths of the stable bench (1.20 m) and specimen (4.0 m), four beam

segments were measured separately and an overlapping was purposely considered to check and ensure the

accuracy of the measurement procedure. After performing this large number of measurements, they were

rigorously treated computationally, thus leading to a quick and reliable determination of the specimen real

initial configuration.

Figure 3.5 – Stable Bench and LVDT’s employed to measure the beam initial geometrical imperfections

Between the data collection (displacement measurement), on the specimen, and the plot of the

corresponding beam initial configuration, a 7-step procedure had to be carried out − each step is described

next, together with the associated simplifying assumptions:

27

Step 1: The initial measurements collected from the

sensors are processed and lead to a plot providing the

initial positions of the bench plus floor (accounting for

their out-of-flatness) − see Figure 3.6. These

reference positions that must be considered when the

subsequent measurements are made on the four

specimen segments.

Figure 3.6 – Schematic representation of Step 1

Step 2: This step consists of making sure that the future

measurements are adequately collected. It is necessary to

match the slopes associated with the sensors belonging to

two adjacent beam segments: the measurements of (i)

the last four sensors of one segment and (ii) the first four

sensors of the adjacent segment must account for the

different reference positions of the bench plus floor

located below each beam segment − see Figure 3.7.


Step 3: In this step, the four measurement series (one per

beam segment) are put together, thus enabling the

calculation of the beam chord position − see Figure

3.8.


Step 4: After knowing the beam chord position, all

measured displacements are related to the horizontal

axis depicted in Figure 3.9.


Step 5: It is assumed that the beam ends (reference

points) share the same position with respect to the z-axis.


Step 6: In this step, the displacement measurements of the points corresponding to the beam segment overlaps are

replaced by their averages, so that a smooth deformed configuration is obtained, which incorporates the beam

(i) initial geometrical imperfections and (ii) deformed configuration caused by the self-weight.

Step 7: In order to isolate the beam initial geometrical imperfections, it suffices to subtract the deformed

configuration caused by the self-weight from the total one obtained in Step 6, thus making it possible to

visualise the beam initial (deformed) configuration.

28

3.3 Experimental set-up and procedure

Figures 3.11(a)-(b) provide an overall picture of the experimental set-up and a detailed view of the beam end

support conditions, which combine (i) “fork-type conditions”, with respect to major and minor-axis bending,

with (ii) full warping restraint of the end cross-sections (the beam “extends” beyond the cross-sections

where the end supports are deemed materialised). The two beams tested had length L=4.00 m (due to

space constraints, the effective beam “free length” was L=3.36 m) and were linked at both ends (symmetrically)

to rigid secondary systems conceived to ensure a smooth application of eccentric tension (minor-axis

eccentricity causing major-axis bending). The tests involved (i) an IPE 200 beam loaded with a 0.25 m

eccentricity and (ii) an HEA 160 beam loaded with a 0.5 m eccentricity.

(a)

(b)

Figure 3.11: Experimental set-up: (a) overall view and (b) detail of the beam end supports

29

Moreover, the eccentricity values were selected so that the associated LT slenderness and axial tension

level allow for the assessment of the influence of axial tension on the LTB behaviour, i.e., do not lead

to experimental failures merely stemming from exceeding the cross-section resistance. Therefore, the

performance of the tests was preceded by preliminary numerical simulations that yielded the following results:

(i) λLT=0.91 and Nu/Npl=0.24, for the IPE 200, and (ii) λLT=0.66 and Nu/Npl=0.16, for the HEA 160 − these λLT

values include already the beneficial effect stemming from the presence of axial tension.

When performing a test, the first steps consisted of (i) welding vertical rigid profiles (HEB 200) to the

specimen ends, thus preventing warping and making it possible to apply the eccentric tensile loads, (ii)

positioning the specimen in between two pairs of end support cylindrical hinges, one resting on the supporting

cross-bar and the other leaning vertically against a short RHS cantilever, ensuring that the symmetry with

respect to the mid cross-section is retained kept, i.e., that the outstand segments, extending beyond each

supporting hinge, are equal, and (iii) placing the hydraulic jacks, which are mounted on secondary structural

systems in such a way that the required axial tension eccentricity is guaranteed − see Figure 3.12.

Figure 3.12 – Detail of the secondary supporting system where the hydraulic jacks are mounted

In addition, (i) a rigid hollow member (RHS 200 x 100 x 12.5) was assembled on the top of the vertical

HEB 200 profile, to ensure a smooth and uniform load transmission between the two hydraulic jacks, and

(ii) stiffeners were attached to the web of the HEB 200, to preclude the occurrence of local buckling during

the performance of the second test (HEA 160 beam) − the one with the higher eccentricity (see Figure 3.13).

Figure 3.13 – Web stiffeners intended to preclude local buckling during the HEA 160 beam test

30

Regarding the beam end support conditions, shown in detail in Figure 3.14, (i) the two cylindrical hinges

ensure free axial displacements and major-axis flexural rotations, while preventing the vertical displacements,

and (ii) a system of welded plates, which provide point supports for the specimen flanges, ensuring free minor-

axis flexural rotations, while preventing the lateral/horizontal displacements. Furthermore, the secondary

system, welded to the HEB 200 profile, together with the support devices described above, ensure full end

section torsional twist and warping.

Figure 3.14 – Detailed view of the beam end support conditions

Another aspect concerns the location of the measuring devices (i) on the hydraulic jacks and (ii) at the

mid-span and end cross-sections. In view of the expected specimen three-dimensional behaviour, a complex

displacement transducer system was devised to enable the measurement of two pairs of mid-span cross-

section transverse displacements (two vertical and two lateral). Figures 3.15(a)-(b) make it possible to visualise

the displacement transducer system, which adopts (i) two LVDTs (Linear Variable Differential Transformers)

to measure the vertical displacements – TK 100 (range of measurement: 0-100 mm) and (ii) a system of

pulleys to record the lateral displacement (including two displacement transducer plungers − WA 200,

with range of measurement 0–200 mm). Moreover, inclinometers (KB–10EB) were attached to the vertical

rigid profiles welded to the specimen ends, in order to measure the major-axis flexural rotations at the supports,

as illustrated in Figure 3.15(c). The real forces applied by the jacks were monitored by means of two load

cells (C6A force transducers) located near each jack, as depicted in Figure 3.15(d).

During the performance of the tests, the above measurement devices recorded values at 0.5 s intervals,

thus providing a fairly continuous displacement/rotation output. Concerning the load application, a two-stage

strategy was adopted, involving (i) large load increments in the elastic range and (ii) much smaller load

increments after the (anticipated) onset of yielding, which was detected by closely monitoring the tensile axial

load level provided by loads cells (also recorded at 0.5 s intervals). Finally, it is worth noticing that the

specimens were tested up to failure, which means that experimental ultimate strength values were obtained.

31

(a) Mid-span vertical displacements (TK 100)

(b) Mid-span lateral displacements (WA 200)

(c) End support inclinations (one inclinometer

KB–10EB and two TT50 LVDT’s)

(d) Loads applied by the hydraulic jacks (load cells C6A)

Figure 3.15 – Measuring device systems

32

3.4 Initial measurements − beam characterisation

As mentioned before, each specimen tested was initially characterised, by measuring its (i) cross-section

dimensions, (ii) material properties (Young’s modulus, yield stress, failure stress and extension at failure), (iii)

longitudinal normal residual stress distribution, and (iv) global and local initial geometrical imperfections. Table

3.1 presents the beam measured cross-section dimensions (each value stands for the average of measurements

taken at three different cross-sections located along the beam length), which are compared with the

corresponding nominal values, i.e., those appearing in standard catalogue. It is observed that, with one

exception, the tested specimens exhibit dimensions larger than the nominal ones − the exception are the

flange thickness values, which are below the nominal ones.

Table 3.1 – Measured and nominal beam cross-section dimensions

IPE 200 (measured)

IPE 200 (nominal) Δ (%) HEA 160

(measured) HEA 160 (nominal) Δ (%)

b [mm] 101.4 100 + 1.4 162.3 160 + 1.4 h [mm] 203.3 200 + 1.7 153.8 152 + 1.2 tf [mm] 8.1 8.5 - 4.7 8.8 9 - 2.2 tw [mm] 6.1 5.6 + 8.9 6.7 6 + 11.7 r [mm] − 12 − 15

Concerning the steel material behaviour, it was characterised by means of σ-ε curves defined by four

representative properties, namely the Young’s modulus (E), yield stress (fy), failure stress (fu) and extension at

failure (εu) − their measures values are given in Table 3.2. Additionally, Figures 3.16 (a)-(b) depict

schematic representations σ-ε curves obtained from the coupon tests taken from the two beams analysed.

(a)

33

(b)

Figure 3.16 – Schematic representations of the steel σ-ε curves obtained for the (a) IPE 200 and (b) the HEA160 beams

Table 3.2 – Steel’s material properties

IPE200 HEA160 E [GPa] 208 212 fy [MPa] 320 307 fu [MPa] 446 449

εu 30.57 31.2

As far as the residual stresses are concerned, Figure 3.17 illustrates the measured residual stress

distributions, which are next analysed and compared with the (theoretical) triangular and parabolic distributions

recently proposed by Boissonnade & Somja (2012). First of all, it is readily noticed that the measured

residual stresses are not self-equilibrated in both cross sections (the flanges exhibit only compressive

stresses in both cases) − this can only be attributed to measurement inaccuracies, most likely due to the

high sensitivity of the device employed to make the measurements (an extensometer). Moreover, the magnitude

of the measured residual stresses also differs considerably from the linear and parabolic ones proposed

by Boissonnade & Somja (2012) − the comparisons are shown in Figure 3.18.

In view of what was mentioned in the previous paragraph, which reflects the poor quality of the obtained

residual stress measurements, the data collected was deemed not valid/reliable and, therefore, the finite element

model developed to simulate the experimental (see Section 3.6) includes the residual stress distributions

proposed by Boissonnade & Somja (2012).

34

Figure 3.17 – Residual stresses distribution measured at the IPE200 and HEA160 beams (positive values stand for compression)

Figure 3.18 – Comparison of the residual stresses distribution: measured (red), linear (blue) and parabolic (green)

Finally, Figures 3.19(a)-(b) and 3.20(a)-(b) display a sample of the measured initial geometrical imperfection

profiles − the complete set of measured initial geometrical imperfection data is presented in Annex 3.

These four profiles concern the initial vertical and horizontal displacements measured along the longitudinal

lines passing through the cross-section mid-flange and mid-web points: points B, E and H indicated in Figure

3.21(a)-(b) − note that these displacement measurements are associated with both local and global initial

deformations. It is worth pointing out that the measured initial geometrical imperfections are included in

the numerical simulations presented further ahead in this chapter.

35

Axial position of measurement [mm]

0 1000 2000 3000 4000

Ver

tica

l dis

pla

cem

ent

[mm

]

0

123456789

1011

Top FlangeBottom Flange


0 1000 2000 3000 4000

Ver

tica

l dis

pla

cem

ent

[mm

]

0

1

2

3

4

5

6Top FlangeBottom Flange

(a) (b)

Figure 3.19 – Initial geometrical imperfections measured on the flanges (points B and H) for the (a) IPE 200 and (b) HEA 160


0 1000 2000 3000 4000

Late

ral d

ispla

cem

ent

[mm

]

0

1

2

3


0 1000 2000 3000 4000

Late

ral dis

pla

cem

ent

[mm

]

0

2

4

6

8

10

12

14

16

18

(a) (b)

Figure 3.20 – Initial geometrical imperfections measured on the web (point E) for the (a) IPE 200 and (b) HEA 160

(a) (b)

Figure 3.21 – Cross-section points for which initial displacement profiles were measured: (a) IPE 200 and (b) HEA 160 beams

36

3.5 Test results

3.5.1 IPE 200 beam

The first test concerns an IPE 200 beam subjected to a tensile axial force applied with a 250 mm minor-axis

eccentricity (uniform major-axis bending moment diagram − ψ=1) − Figure 3.22 provides an overall view of

the test set-up and shows the beam initial (deformed) configuration (prior to testing). Figure 3.23 shows the

time evolution of the axial loads recorded by the measurement devices of the hydraulic jacks located at

each end of the beam. One readily observes the virtual coincidence between the curves concerning the

readings of the two hydraulic jacks, thus confirming that the applied bending moment diagram is, indeed,

uniform. It is worth noting that the experimental failure load is 270 kN, a value that corresponds to an applied

bending moment diagram equal to 67.25 kNm.

Figure 3.22 – Overall view of the test set-up and initial (deformed) configuration of the IPE 200 beam specimen

Time [s]

0 100 200 300 400 500

Appl

ied

Tens

ion

[kN

]

0

30

60

90

120

150

180

210

240

270

300

Jacks (Side 1)Jacks (Side 2)

Figure 3.23 – Time evolution of the axial forces recorded by the measuring devices of the hydraulic jacks during the IPE 200 beam test

37

Figures 3.24 and 3.25 provided several views of the beam deformed configuration at the brink of collapse

− these views provide clear experimental evidence of the three-dimensional nature of the beam collapse

mechanism, which combines minor-axis (lateral) flexural and torsional deformations.

Figure 3.24 – IPE 200 beam deformed configuration at the brink of collapse − overall view

Figure 3.25 – IPE 200 beam deformed configuration at the brink of collapse − detailed views of the (a) mid-span and (b) end regions

Figures 3.26(a)-(b3) display the recorded IPE 200 beam equilibrium paths, which plot the applied tensile

force versus (i) the end cross-section flexural rotation θy (measured twice, by means of either two LVDTs or an

inclinometer), and (ii) the mid-span (ii1) vertical displacement v, (ii2) lateral displacement u and (iii3)

torsional rotation θx.

38

End section flexural rotation θy [°]

-10 -5 0 5 10

Applie

d T

ensi

on [

kN]

0

50

100

150

200

250

300

LVDT_Support2LVDT_Support1Incl._Support2Incl._Support1

Mid-span vertical displacement v [mm]

0 5 10 15 20 25 30 35 40

Applie

d T

ensi

on [

kN

]

0

50

100

150

200

250

(a) (b1)

Mid-span lateral displacement u [mm]

0 5 10 15 20

Applie

d T

ensi

on [

kN]

0

50

100

150

200

250

Torsional rotation θx [°]

0 2 4 6 8 10 12 14 16 18 20

Applie

d T

ensi

on [

kN]

0

50

100

150

200

250

(b2) (b3)

Figure 3.26 – IPE 200 beam equilibrium paths relating the applied tensile force with (a) the end cross-section flexural rotation θy and

(b) the mid-span (b1) vertical displacement v, (b2) lateral displacement u and (b3) torsional rotation θx

3.5.2 HEA 160 beam

The second test involves an HEA 160 beam and is similar to the first one − however, the tensile axial force

is now applied with a 500 mm minor-axis eccentricity − Figure 3.27 provides an overall view of the test set-up

and shows a view of the beam deformed configuration at the onset of collapse. As for the IPE 200 beam, the

curves concerning the readings of the two hydraulic jacks, depicted in Figure 3.28, are virtually coincident,

39

which means that the applied bending moment diagram is again uniform. The experimental failure load now

reads 145 kN, a value corresponding to an applied bending moment diagram of 72.50 kNm.

Figure 3.27 – HEA 160 beam deformed configuration at the brink of collapse − overall view

Time [s]

0 600 1200 1800 2400 3000 3600

Appl

ied

Tens

ion

[kN

]

0

40

80

120

160

Jacks (Side 1)Jacks (Side 2)

Figure 3.28 – Time evolution of the axial forces recorded by the measuring devices of the hydraulic jacks during the HEA 160 beam

As before, Figures 3.29(a)-(b3) display the recorded IPE 200 beam equilibrium paths, which plot the applied

tensile force versus (i) the end cross-section flexural rotation θy (measured once more by means of two LVDTs

and an inclinometer), and (ii) the mid-span (ii1) vertical displacement v, (ii2) lateral displacement u

and (ii3) torsional rotation θx.

40


-10 -5 0 5 10

Applie

d T

ensi

on [

kN]

0

20

40

60

80

100

120

140

LVDT_Support2LVDT_Support1Incl._Support2Incl._Support1


0 5 10 15 20 25 30 35 40 45 50

Applie

d T

ensi

on [

kN]

0

20

40

60

80

100

120

140

(a) (b1)


-0.5 0.0 0.5 1.0 1.5

Applied T

ensi

on [

kN

]

0

20

40

60

80

100

120

140


0.0 0.5 1.0 1.5 2.0

Applied T

ensio

n [

kN

]

0

20

40

60

80

100

120

140

(b2) (b3)

Figure 3.29 – HEA 160 beam equilibrium paths relating the applied tensile force with (a) the end cross-section flexural rotation θy

and (b) the mid-span (b1) vertical displacement v, (b2) lateral displacement u and (b3) torsional rotation θx

3.5.3 Discussion

The observation of the beam equilibrium paths presented in the previous two sub-sections prompts

the following remarks:

41

(i) In most of the equilibrium paths, the beam are in the elastic regime − in terms of applied load, the

elastic behaviour extends up to 90% of the failure load. However, the elastic regime is responsible for

only a very small fraction (about 2%) of the displacements reached at the onset of collapse.

(ii) The IPE 200 beam exhibited non-negligible lateral displacements and torsional rotations, thus providing

clear experimental evidence concerning the occurrence of a collapse mechanism governed by

LTB. On the other hand, the HEA 160 beam collapse mechanism was characterized by smaller

lateral displacements and rotations, which provides experimental evidence of the “exhaustion” of the

mid-span plastic capacity, but still some signs of LTB.

(iii) The lateral displacements and rotations exhibited by the IPE 200 beam were higher than those recorded

for the HEA 160 beam, which is just a logical consequence of the fact that the IPE beams (narrow

flanges) are more prone to the occurrence of LTB.

(iv) The horizontal plateaus exhibited by the equilibrium paths (of both beams) concerning the mid-span

lateral displacements and torsional rotations provide clear indication that that the collapse is triggered by

the beam central region − this is just logical, since LTB governs the beam failures.

(v) With the exception of the IPE 200 beam equilibrium path concerning the end section flexural rotation,

all the remaining equilibrium paths exhibit an ascending slope, a feature that may be misleading, in

the sense that it appears to indicate that beam is able to withstand a larger applied load. Indeed, these

equilibrium path end slopes are due to erroneous measurements occurring at the onset of collapse, due to a

considerable decrease in accuracy of the displacement transducers measuring three-dimensional

deformed configurations.

3.6 Numerical simulations

3.6.1 Modelling issues

The experimental results are now employed to develop, calibrate and validate a shell finite element

model able to handle realistic material constitutive laws, end support conditions, load application

procedures, initial geometrical imperfections and residual stresses. This is done using the non-linear

FEM software FINELG (2012), which was originally developed by Ville de Goyet (1989), at the University of

Liège, and has been continuously updated by several researchers at that University and also at the Greisch

Design Office. In the context of this dissertation, this software is used mainly to perform elastic buckling,

elastic-plastic first-order and elastic-plastic second-order analyses.

In order to shed some light on the capabilities of the FINELG shell finite element model, the next few

lines are devoted to describing some important modelling features exhibited by this model (Boissonnade &

Somja, 2012). The first issue concerns the fact that the real hot-rolled beam cross-section cannot be

42

modelled by merely considering an assembly of three plates/walls, due to the existence of the rounded

web-flange corner areas. In order to model adequately these areas, the FINELG model places an additional node

within the web height and located at the exact vertical position of the radius zone centroid, as depicted in

Figure 3.30. Besides being linked to the web elements, this node also bears an additional beam finite element,

oriented in the longitudinal (x) direction and having a cross-section area equal to the difference between the

radius zones and the overlapped area − see Figure 3.30. The presence of this beam element, which is assumed

to exhibit the same constitutive law as the various wall shell finite elements, makes it possible to achieve

nearly exact cross-sectional properties with the developed model.

Figure 3.30 – FINELG finite element modelling of the web-flange corner areas (Boissonnade & Somja, 2012)

Since the “nominal beam” (member with length 3.36 m and subjected to major-axis bending and axial

tension) end support conditions are fairly complex, due to the flexural rotation and warping restraint provided by

the two 0.3 m overhang segments attached to the vertical rigid profiles (see Figure 3.12), it was decided to

attempt to simulate the beam “real end support conditions”. This was done by modelling the entire

experimental set-up mainly by means of fine meshes of 4-node shell elements based on Kirchhoff’s bending

theory, thus ensuring that the influence of the beam “surroundings” is adequately taken into account. The only

exception concerns the rectangular hollow section segments, which are depicted in blue in Figure 3.32 and

were modelled by means of 3D beam finite elements, in order to facilitate the application of the end nodal

forces.

At this stage, it is worth mentioning that the parametric study addressed in the next chapter involves

exclusively simply supported beams with “fork-type” end supports: a combination of (i) prevented flexural

displacements and torsional rotation with (ii) free axial extension, warping and flexural rotations. Two aspects

deserve to be specially mentioned concerning the modeling of these end support conditions. The first one deals

with the handling of the end cross-section in-plane local supports, in order to preclude the occurrence of local

buckling stemming from the (concentrated) reactive forces − the arrangement adopted is depicted in Figure

3.31 and consists of preventing the local displacements normal to the wall thickness along the whole cross-

section contour. The second aspect concerns the allowance for longitudinal displacements ensuring that the end

43

cross-section exhibits free axial extension, flexural rotations and warping. As illustrated in Figure 3.31, this

was achieved by allowing four (adequately selected) cross-section nodes to have free longitudinal

displacements, while restraining the remaining ones to guarantee linear variations along all three wall mid-lines

− for symmetry reasons, the flange tips were selected as the four nodes exhibiting free longitudinal

displacements.

Figure 3.31 – FINELG modelling of the “fork-type” end support conditions (Boissonnade & Somja, 2012)

Finally, Figures 3.32 to 3.34 concern the IPE200 beam test and provide (i) an overall view of the

experimental set-up discretisation, (ii) the shape of the initial geometrical imperfections included in the analysis

and (iii) the load application system adopted in the analysis. As for Figures 3.35 and 3.36, they concern the

HEA 160 beam and provide (i) a detailed view of the web stiffeners added to the vertical rigid element (to

prevent local buckling) and (ii) the shape of the initial geometrical imperfections included in the numerical

simulation.

Figure 3.32 – Overall view of the experimental set-up discretisation using the developed shell finite element model (IPE 200 beam)

44

Figure 3.33 – Measured initial geometrical imperfections included in the shell finite element analysis (IPE 200 beam)

Figure 3.34 – Load introduction adopted in the shell finite element analysis

Figure 3.35 – Web stiffeners added to the vertical rigid element to prevent local buckling (HEB 200 beam)

45

Figure 3.36 – Measured initial geometrical imperfections included in the shell finite element analysis (HEA 160 beam)

3.6.2 Numerical results

The developed shell finite element model was employed to perform elastic buckling and elastic-plastic

geometrically non-linear analyses of the two beams tested. Table 3.3 shows a comparison between the

experimental and numerical ultimate moments (Mu) obtained − moreover, this table also provides the (i)

analytical cross-section plastic moments Mpl (under pure bending), (ii) numerical (FINELG) critical moments

Mcr, calculated for the “real experimental set-up conditions” modelled (Figure 3.37 displays half of the

IPE200 beam critical lateral-torsional buckling mode shape) and (iii) beam lateral-torsional slenderness

values λLT=(Mpl,Rk /Mcr)0.5, calculated on the basis of the presented Mpl and Mcr values. It is observed that

there is a quite good correlation between the experimental and numerical and experimental ultimate

moments − indeed, the numerical values either underestimate by 6% (IPE 200 beam) or overestimate by

2% (HEA 160 beam) their experimental counterparts.

Table 3.3 - Analytical, numerical and experimental results concerning the two beams tested

Numerical Experimental

Mpl [kNm] λLT [-] Mcr [kNm] Mu [kNm] Mu [kNm]

IPE 200 70.6 0.90 86.5 63.0 67.3

HEA 160 75.3 0.66 179.0 76.9 75

46

Figure 3.37 – Half of the critical lateral-torsional buckling mode provided by FINELG for the IPE200 beam

As for Figures 3.38 and 3.39, they provide the numerical (FINELG) IPE200 and the HEA160 beam

deformed configurations at collapse − note the qualitative and quantitative similarity with their experimental

counterparts, shown previously in Figures 3.24 and 3.27. Moreover, Figure 3.40 shows the amount

yielding taking place at the collapse of the IPE 200 beam − note the heavy spread of plasticity clearly

visible along the flanges.

Figure 3.38 – IPE200 beam deformed configuration at collapse obtained with FINELG

47

Figure 3.39 – HEA160 beam deformed configuration at collapse obtained with FINELG

Figure 3.40 − Amount yielding taking place at the collapse of the IPE 200 beam

Lastly, Figures 3.41(a)-(b3) and 3.42(a)-(b3) show comparisons between the experimental and

numerical equilibrium paths relating the applied load to the (i) end cross section flexural rotation θy

(measured twice, using two LVDTs and an inclinometer) and (ii) mid-span (ii1) vertical displacement

v, (ii2) lateral displacement u and (ii3) torsional rotation θx − note that the experimental equilibrium

paths had already been shown in Figures 3.26 and 3.29. At first glance it becomes clear that, with one

exception, there is a virtually perfect coincidence in the elastic regime, beyond which the numerical model

becomes a bit stiffer and, therefore, underestimates the experimentally measured displacements (v and u) and

rotations (θy and θx).

The exception concerns the mid-span lateral displacement of the IPE 200 beam, whose

experimental equilibrium path shows a very pronounced displacement reversal taking place during the

test − such displacement reversal is also visible in the corresponding numerical equilibrium path, but

to a much lesser extent. A possible (and quite reasonable) explanation for this behaviour and

discrepancy stems from the fact that the beam collapse occurred in a direction opposite to that of the measured

48

initial geometrical imperfections (lateral displacements) − however, it should be noted that also in this

case the numerical simulation follows the qualitative trend recorded during the performance of the

experimental test.


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Figure 3.41 – Experimental and numerical equilibrium paths relating the applied load/tension with the (a) end cross section

flexural rotation and (b) mid-span (b1) vertical displacement, (b2) lateral displacement and (b3) torsional rotation (IPE 200 beam)

49


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Figure 3.42 – Experimental and numerical equilibrium paths relating the applied load/tension with the (a) end cross section

flexural rotation and (b) mid-span (b1) vertical displacement, (b2) lateral displacement and (b3) torsional rotation (HEA 160 beam)

At this stage, it is still worth mentioning that some discrepancies between the numerical and experimental

equilibrium paths may stem from the three-dimensional nature of the beam deformed configurations, which is

probably the source of erroneous measurements. Indeed, it was concluded that the accuracy of the transducer

measurements decreases considerably when the point under consideration experiences various displacement

components. Although some corrections were made to overcome this situation, on the basis of geometrical

considerations, they were found to become gradually less effective as the beam deformation increases,

rendering almost inevitable the underestimation of the measured displacements and rotations.

50

In view of the fairly good agreement observed between the experimental results obtained and the

corresponding numerical simulations, it seems fair to conclude that the shell finite element model

developed provides reasonably accurate results and, therefore, can be employed to validate the beam finite

element model adopted to perform the parametric study addressed in the next chapter.

3.7 Summary

This chapter presented an experimental investigation comprising two beams tested under axial

tension applied with a minor-axis eccentricity, thus leading to a uniform major-axis bending moment diagram.

After describing the beam material and geometrical characterisation, experimental set-up and experimental

measurements, the test results were presented and discussed. Both beams were tested up to failure and

it was observed that their collapses were governed by lateral-torsional buckling, which was clearly more

pronounced for the first test (IPE 200 beam). The experimental were then used to calibrate and validate a

shell finite element model developed in the code FINELG − a fairly good agreement was found between the

numerical and experimental results (equilibrium paths and ultimate moments). The above shell finite element

model will be used to validate a FINELG beam finite element model, subsequently used to perform the

parametric study addressed in the next chapter.

Chapter 4

Ultimate Behaviour and Strength − Numerical

Parametric Study

The shell finite element model just developed is now employed to validate a beam finite element model, which

is subsequently used to perform a numerical parametric study comprising geometrically and materially non-

linear analyses of about 2000 simply supported beams subjected to major-axis bending and axial tension, and

containing initial geometrical imperfections and residual stresses − this type of structural analysis is often

identified by the acronym GMNIA. Specifically, this chapter includes (i) the description and validation of a

FINELG beam finite element model; (ii) the performance of the aforementioned parametric study, aimed at

obtaining a beam ultimate strength/moment data bank, and (iii) the analysis of this ultimate strength/moment

data bank, in order assess the influence of the axial tension on the lateral-torsional buckling behaviour

and collapse of the beams under consideration.

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4.1 Beam Finite Element Model

4.1.1 Description

The FINELG beam finite element employed to perform the GMNIA is based on Vlasov’s theory for open-

section thin-walled members, reported in Vlasov (1961), and has seven degrees of freedom per node: three

displacements, three rotations and warping. The beams are discretised into unequal-length 28 beam elements

− finer meshes are considered at the beam end section and mid-span regions (an overall view of the

beam discretisation can be observed in Figure 4.2). Moreover, longitudinal residual stresses and initial

geometrical imperfections are incorporated into the analyses. The formed exhibit the parabolic pattern

depicted in Figure 4.1(a), with the values given as percentages of the steel yield stress, and the latter are

sinusoidal and consist of a combination of minor-axis flexure and torsion, as shown in Figure 4.1(b) − these

shapes and values were taken from the recent work of Boissonnade & Somja (2012).

(a) (b)

Figure 4.1 − (a) Longitudinal residual stress pattern and (b) initial geometrical imperfections incorporated into the beam GMNIA −

shapes and values taken from Boissonnade & Somja (2012)

As mentioned earlier, all the beams analysed are simply supported, i.e., exhibit “fork-type” end supports that

combine (i) prevented flexural displacements and torsional rotation with (ii) free axial extension, warping and

flexural rotations. Additionally, in order to preclude the occurrence of a beam rigid-body axial translation, the

axial displacement was prevented at the mid-span cross-section. Concerning the load application, axial forces

were imposed at the end cross-section nodes − such forces are statically equivalent to the particular combination

of major-axis bending moment and axial tension considered (recall that no in-span transverse loads were

considered in this parametric study) − see Figure 4.2. Since the 28 finite element mesh is refined near

the supports and at mid-span, it is possible to (i) ensure a smooth introduction of the applied loads and (ii)

capture the continuous spread of plasticity occurring at the onset of the beam LTB collapse. The steel

53

material behaviour was modelled as depicted in Figure 4.3 and corresponds to the usual elastic-perfectly

plastic constitutive lay with marginal strain-hardening taking place for very large strains.

Figure 4.2 – Finite element model: beam discretisation and load application

Figure 4.3 − Constitutive law adopted to model the steel material behaviour

The beam load-carrying capacities were determined by means GMNIA, employing a standard arc-length

numerical technique (Memon & Su, 2003). Figure 4.4 shows the output of each of the analyses performed,

namely a schematic representation of the beam equilibrium path relating the applied force (F) with the mid-

span vertical displacement (δ), and the beam deformed configuration at the brink of the LTB collapse.

Figure 4.4 – Beam numerical F-δ equilibrium path and deformed configuration at the brink of the LTB collapse

!

!!

F

δ

54

4.1.2 Validation

In order to validate the above beam finite element model, Tables 4.1 and 4.2 provide load-carrying

capacities of simply supported HEB 300 beams, with yield stress fy=355 MPa and subjected to either pure

uniform bending (β =0) or uniform bending combined with axial tension (β =1), obtained with (i) the

beam finite element (BFE) model described in the previous sub-section and (ii) the shell finite element

(SFE) model developed and validated in Chapter 3, through the comparison with the experimental results.

Table 4.1 – Load-carrying capacity of HEB 300 beams for β = 0

L λLT Mu (BFE) [kNm] Mu (SFE) [kNm]

5 0.69 583.5 553.5 5.42% 10 1.09 443.2 421.9 5.06% 15 1.37 339.5 325.3 4.35% 25 1.80 203.4 198.7 2.39%

Table 4.2 – Load-carrying capacity of HEB 300 beams for β = 1

L λLT Mu (BFE) [kNm] Mu (SFE) [kNm]

5 0.63 643.4 609.3 5.60% 10 0.98 473.4 454.1 4.18% 15 1.21 439.7 424.2 3.53% 20 1.49 327.8 320.6 2.19%

The observation of the ultimate moments given in the two tables clearly shows that there is a quite good

correlation between the BFE and SFE values − indeed, the differences never reach 6% and decrease as the

beam length increases. Moreover, it is also noticed that the SFE values are always the lowest ones, which is just

a logical consequence of the fact that they are influenced by local deformation effects (not captured by the

BFE analyses) that invariably lower the beam load-carrying capacity − naturally, these local deformation

effects become less relevant as the beam length increases. In view of the similarity between the BFE and

SFE values presented in Tables 4.1 and 4.2, it seems fair to consider the beam finite element model validated,

which means that it can be adequately used to perform the parametric study addressed later in this chapter.

4.2 Effect of axial tension on the ultimate strength − qualitative aspects

Concerning the influence of axial tension on the beam load-carrying capacity, it may be either (i) beneficial,

if the beam collapse is governed by lateral-torsional buckling (critical moment increase), or (ii) detrimental, if

the beam collapse is governed by plasticity effects (cross-section plastic resistance decrease − see Figure 4.5).

This means that the presence of axial tension (i) decreases the beam load-carrying capacity of stocky beams and

55

(ii) increases their slender beam counterparts. The present dissertation is mainly concerned with the

first situation, i.e., with the beneficial influence of axial tension on the beam ultimate moments associated

with failure modes governed by lateral-torsional buckling − Figure 4.6 depicts such a failure mode.

Figure 4.5 – Schematic representation of the cross-section plastic resistance decrease caused by the presence of axial tension

Figure 4.6 – Failure mode governed by lateral-torsional buckling of a member acted by major-axis bending and axial tension

4.3 Parametric study

4.3.1 Scope

The parametric study carried out comprise beams exhibiting several slenderness values, stemming from (i)

eight span lengths (between 0.5 and 25 m), (ii) two yield stresses (fy=355; 460 MPa − the steel material

behaviour modelled is depicted in Figure 4.3. and (iii) four cross-section shapes (IPE 300, IPE 500, HEB 300,

HEB 500). The beams are subjected to (i) five bending moment diagrams (ψ=1; 0.5, 0, − 0.5, − 1 − all caused

by applied end moments) and (ii) six axial tension levels, corresponding to β=Nt /My ratios equal to 0; 0.5; 0.75;

1.0; 1.5; 2.0 − a total of over 2000 numerical simulations are carried out. As mentioned earlier, the beams

contain (i) longitudinal normal residual stresses and (ii) global sinusoidal initial geometrical imperfections with

the patterns depicted in Figures 4.1(a)-(b).

56

4.3.2 Results

Before presenting the ultimate strengths/moments obtained from the parametric study carried out, it is

important to stress again the fact that this dissertation focuses on beams whose collapse is governed

by lateral-torsional buckling. Therefore, the ultimate strengths/moments concerning collapses stemming from

plasticity effects (cross-section plastic resistance) are only briefly commented and will not be included in the

ultimate strength/moment data bank used to develop design rules, in Chapter 5. It is still worth mentioning

that, in the most stocky beams, the cross-section plastic resistance is sometimes exceeded, which is due to the

inclusion of the (small) strain-hardening in the steel material behaviour.

Another feature that deserves to be specially mentioned concerns the most slender beams and consists of

the fact that the collapse occurs at extremely high deformation levels (e.g., torsional rotations close to 90°)

and, therefore, is associated with very large ultimate moments − in order to illustrate this statement, Figure 4.7

depicts the deformed configuration of the mid-span region of a very slender beam, at collapse. Indeed,

for this high deformation/rotation levels, the beam major-axis bending resistance is “activated”, thus

rendering the beam capable of withstanding ultimate loadings much larger than expected. Since such high

deformation/rotation levels are unacceptable for practical purposes, it was decided to consider as “ultimate

strength/moment”, for these beams, the value corresponding to a torsional rotation of about 15°.

Figure 4.7 – Deformed configuration of the mid-span region of a very slender beam, at collapse

The results presented and discussed next constitute a representative fraction of those obtained from the

parametric study carried out − the full set of results are given, in tabular form, at the end of this dissertation (in

Annex 2). They make it possible to assess the influence of the axial tension on the ultimate strength of the

beam, for different lengths and moment distributions.

Figures 4.8 and 4.9 concern the influence of the axial tension level on the ultimate strength of IPE 300,

IPE 500 and HEB 300 beams made of S355 and S460 steel, exhibiting various lengths, comprised between

L=0.5 m and L=15 m, and subjected to several bending moment diagrams, all stemming from applied end

moments. Both figures provide the variation of the ultimate moment Mu, normalized with respect to the cross-

57

section plastic bending resistance Mpl (calculated for pure bending on the basis of fy), with the loading ratio

β=Nt /My − the values between parentheses, given above or below each point (beam analysed) provide the Mu

percentage increase due to axial tension: [Mu (β) − Mu (0)] /Mu (0). While Figures 4.8 and 4.9 focus on the

combined effect of β and the beam length, Figures 4.10 and 4.11 address the joint influence of β and the

bending moment diagram. It is worth noting that the negative (red) and underlined positive (blue) values in

Figures 4.8 to 4.11 correspond to beams whose collapse is governed by the cross-section resistance, which

naturally decreases with β − all the remaining (positive/green) values are associated with collapses governed by

LTB. It is worth noting that the underlined values concern beams whose collapse becomes governed by the

cross-section due to the axial tension level − for lower or null axial tension levels, the collapse is

governed by LTB. Note also that, after the descending curve (corresponding to the plastic moment reduction

caused by the axial tension) intersects a particular Mu /Mpl vs. β curve, for a given axial tension level, they

become coincident for higher axial tension levels (β values). This means that, for some axial load levels, the

same descending curve point applies to several curves. In such cases, the various Mu /Mpl values (either

negative or underlined) are displayed in “column form” (i.e., one above the other) − naturally, in each

“column” the values are ordered according to the corresponding Mu /Mpl vs. β curves, i.e., in ascending order

“top down”. For instance, in Figure 4.10, the three values associated with β=2.0 concern the curves

corresponding to the ψ= − 1 (top value), ψ= − 0.5 (intermediate value) and ψ= 0 (bottom value) bending

moment diagrams.

Figure 4.8 − Variation of Mu/Mpl with β and the beam length (S460 steel IPE 300 beams under uniform bending)

58

Figure 4.9 − Variation of Mu/Mpl with β and the beam length (S355 steel IPE 500 beams under triangular bending – ψ=0)

Figure 4.10 − Variation of Mu/Mpl with β and the bending moment diagram (L=15 m S355 steel HEB 300 beams)

59

Figure 4.11 − Variation of Mu/Mpl with β and the bending moment diagram (L=5 m S460 steel IPE 300 beams)

The observation of the numerical results displayed in these figures prompts the following remarks:

(i) First of all, as mentioned earlier, the influence of axial tension is completely different in the stocky and

slender beams, due to the fact that their collapses are governed by plasticity and instability effects,

respectively. In the former (e.g., the L=0.5; 1.0 m beams in Figure 4.8 and the L=1 to 3.5 m beams in

Figure 4.9), axial tension leads to an ultimate moment decrease, stemming exclusively from the drop

in cross-section resistance. In the latter (e.g., the L=8; 10; 15 m beams in Figure 4.8 and the L=15; 20

m beams in Figure 4.9), axial tension leads to an ultimate moment increase, which grows with β and

stems from the improved lateral-torsional buckling resistance.

(ii) In Figure 4.8, the comparison between the Mu /Mpl vs. β curves concerning the (ii1) L=8; 10; 15 m and (ii2)

L=3.5; 5 m beams show different trends, even if all these curves have positive slopes throughout the whole

β range considered. While in the former group Mu /Mpl grows with β at an always increasing rate (upward

curvature), which becomes percentage-wise more relevant as L increases, the latter group exhibit points

of inflexion, i.e., the curvature changes from upward to downward at a given β value that seems to

increase with L. These different trends reflect the contradicting influence of axial tension on the

lateral-torsional buckling and cross-section resistances: the latter becomes progressively more relevant

as β increases and L decreases. This assertion is fully confirmed by looking at the Mu /Mpl vs. β curve

concerning the L=2 m beam, which exhibits very little growth and ends up merging with their L=0.5; 1.0 m

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beam counterparts for β=2.0 − it would start descending for larger β values, whenever collapse would

become governed by plasticity in the beam mid-span region.

(iii) Naturally, the Mu /Mpl percentage growth with β is considerably larger for the longer (more slender)

beams − e.g., in Figure 4.8, for L=15 m and β=2.0, Mu /Mpl increases by almost 84% (for L=5 m this same

increase is just about 27%).

(iv) The results presented in Figure 4.9 show the same qualitative trends exhibited by those displayed

in Figure 4.8. However, it should be noticed that the different moment distribution leads to (iv1)

higher Mu /Mpl growths, which may exceed 200% for the 20 m beam, and also (iv2) larger Mu /Mpl

drops for most of the beams with length below 5 m.

(v) Concerning the influence of the bending moment diagram shape on the axial tension benefit, it should

be mentioned that the lateral-torsional slenderness (λLT) of the beams included in Figure 4.10 varies

between 0.5 and 1.1, while those included in Figure 4.11 exhibit, in the majority of the cases, λLT values

larger than 1.0. This fact explains why, regardless of the moment distribution, the Mu /Mpl percentage

growths are always larger in Figure 4.11.

(vi) In Figure 4.10, dealing with the L=15 m S355 steel HEB 300 beams, the first important observation is

that only the ψ=1 and ψ=0.5 (marginally) curves (i.e., those leading to more relevant lateral-torsional

buckling effects) are not limited by the descending curve associated with the mid-span cross-section full

yielding up to β=2.0 − indeed, the ψ=0 and ψ= − 0.5 curves merge into this curve at lower (decreasing) β

values and following an “almost horizontal” segment. Finally the ψ= − 1 curve decreases monotonically

with ψ, thus meaning that the beam collapse is always governed by the mid-span cross-section

resistance. Quantitatively speaking, the largest Mu /Mpl percentage increases occur for the beams acted

by the ψ=0.5 bending moment diagram − they slightly exceed their ψ=0 and ψ=1 diagram counterparts

(in this order).

(vii) The results presented in Figure 4.11 are qualitatively similar to those shown in Figure 4.10.

However, they differ considerably in quantitative terms, as already explained in item (v). Indeed, the

descending curve associated with the decreasing cross-section resistance only limits the Mu /Mpl vs. β

curves associated with the moment distributions less prone to LTB at high axial tension levels. The curves

concerning the moment distributions more prone to LTB, namely ψ=1 and ψ=0.5, show a significant Mu

/Mpl growth with β and involve exclusively collapses governed by LTB.

Finally, Figure 4.12 shows the variation of Mu /Mpl with the lateral-torsional slenderness λLT=(Mpl,Rk /Mcr)0.5,

where Mcr is calculated taking into account the axial tension, for various combinations of beam length, cross-

section shape and steel grade. This figure provides clear evidence that the net effect of the presence of an

increasing axial tension is to move the Mu /Mpl vs. λLT “beam points” (i) to the left (lateral-torsional slenderness

decrease) and (ii) upwards (ultimate moment increase), thus reflecting the double influence of Nt. Moreover, it

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can also be observed in this figure that the whole set of points, corresponding to various beams and β values

(including β=0), remain nicely “aligned” along a “design-like” strength curve. The design approach for beams

subjected to axial tension that is proposed in the next chapter takes advantage of this feature.

Figure 4.12 – Variation of Mu/Mpl with the beam lateral-torsional slenderness λLT

4.3 Summary

The results of a parametric study comprising about 2000 numerical simulations, concerning beams (i) subjected

to major-axis bending and axial tension, and (ii) exhibiting collapse modes governed by either lateral-torsional

buckling or plasticity effects, were presented and discussed in this chapter. It was shown that:

(i) In the slender beams, whose collapse is governed by lateral-torsional buckling (not plasticity effects), the

presence of axial tension causes a load-carrying capacity growth that increases with the axial

tension level (provided that such growth is not “interrupted” by the exhaustion of the mid-span cross-

section resistance). Although the above load-carrying capacity growth is non-linear and depends

on several parameters, it may said, generally, that larger growths occur for (i1) longer (more slender)

beams and (i2) moment distributions more prone to lateral torsional buckling.

(ii) In the stocky beams, whose collapse is governed by plasticity effects, the presence of axial tension

naturally causes a load-carrying capacity drop that increases with the axial tension level. Such beams do

not constitute the primary focus of this dissertation, which is mainly concerned with beams whose

collapse is governed by lateral-torsional buckling (with and without axial tension).

(iii) The influence of axial tension on the ultimate strength of beams exhibiting LTB-based collapses

is two-fold: (iii1) reduces the beam vulnerability to lateral-torsional buckling, i.e., decreases its lateral-

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torsional slenderness λLT, and (iii2) increases the beam load-carrying capacity, i.e., leads to higher

ultimate strengths/moments.

(iv) The normalised ultimate strengths/moments plotted against λLT are aligned along a “design-like” strength

curve, which indicates that a design approach may be successfully sought − this will be done in Chapter 5,

taking advantage of the extensive ultimate strength/moment data bank gathered in this chapter (note that

only those values concerning beams exhibiting collapses governed by lateral-torsional buckling, both with

and without axial tension, are considered).

Chapter 5

Development of a Design Approach

As mentioned earlier, Part 1-1 of Eurocode 3 (EC3-1-1 − CEN 2005) currently lacks design guidance for

beams susceptible to lateral-torsional buckling that are subjected to axial tension − moreover, this topic has

been very seldom been addressed in the literature. This means that the provisions of the current EC3-1-1

completely neglect the beneficial influence of axial tension on the beam ultimate strength associated

with collapses governed by lateral-torsional buckling, thus leading to overly conservative designs − indeed, as

far as this type of collapse is concerned, the beams is treated as if they were subjected to pure bending.

This chapter presents the development and assesses the merits of a design approach aimed at providing

efficient (safe and economic) predictions of the ultimate strength of beams subjected to major-axis bending and

axial tension whose collapse is governed by lateral-torsional buckling. The proposed approach is based on

the use of beam buckling/strength curves currently prescribed by EC3-1-1 in combination with slenderness

values calculated on the basis of critical buckling moments that account for the beneficial effect of axial tension.

In order to assess the merits of this approach, its estimates are compared with the numerical ultimate strength

data gathered in Chapter 4 − moreover, the benefits of incorporating axial tension in the proposed design

approach are quantified (in percentage terms). Finally, the chapter closes with (i) a comparison between the

design procedure proposed in this dissertation and that included in the previous version of EC3-1-1 (EC3-

ENV-1-1, 1992), which no longer appears in the current EC3-1-1, and (ii) the presentation and discussion of a

couple of illustrative examples.

64

5.1 Proposed design approach

Following the currently available design guidance, a beam subjected to axial tension is designed against

lateral-torsional instability ultimate limit states as “pure beam” (i.e., only major-axis bending is taken into

account), and the (detrimental) influence of axial tension is only felt through the cross-section resistance. The

aim of the design approach proposed in this dissertation is to change the above situation, by incorporating the

axial tension effects in the ultimate moment prediction prescribed by EC31-1 for compact hot-rolled

steel beams (the so-called “special method”).

The proposed design approach is based on the current EC3-1-1 methodology, which stipulates that the

ultimate moment of (compact) beams subjected to axial tension (MRd) is the least of two values: (i) the

cross-section reduced plastic moment (MN,Rk) and (ii) the beam bending resistance against a failure governed by

lateral-torsional buckling. While the former is determined through classical strength of materials concepts,

the latter is obtained by means of a procedure based of the use of “beam strength curves” − Mb,Rd. This

procedure involves the following steps (the EC3-1-1 nomenclature is adopted):

(i) Determine the beam lateral-torsional slenderness λLT=(Mpl,Rk /Mcr)0.5, where Mpl,Rk is the cross-section

plastic moment (bending resistance) and Mcr is the beam critical buckling moment, which obviously

depends on the acting major-axis bending moment diagram.

(ii) On the basis of λLT, use the appropriate buckling curve (depends on the cross-section geometry and

fabrication process − curve b for all the profiles considered in this work) to obtain the reduction factor χLT.

(iii) Further modify/increase the reduction factor obtained in the previous step, by means of the relation

χLT.mod=χLT /f, where the parameter f ≤ 1.0 depends on the bending moment diagram and beam slenderness

λLT − it supposedly reflects the influence of the spread of plasticity taking place prior to the beam collapse.

(iv) Evaluate the beam bending resistance against lateral-torsional buckling failure, which is termed Mb,Rd

and given by Mb,Rd=χLT.mod × Mpl,Rk.

At this stage, it is worth recalling that this dissertation is exclusively concerned with instability limit states

and, therefore, the cross-section resistance safety check falls outside the scope of this dissertation.

However, the interested reader may found detailed information in EC3-1-1 (section 6.2.9), which includes a set

of formulae and interaction equations aimed at estimating the plastic resistance of I cross-sections subjected to

major-axis bending and axial force.

The proposed design approach consists of merely incorporating the axial tension beneficial effects into the

above procedure. This is done exclusively through the value of the critical buckling moment used to determine

the beam slenderness, while keeping all the remaining steps unchanged − in particular, Mpl,Rk still remains the

cross-section pure bending plastic resistance (i.e., does not account for the presence of axial tension). In other

65

words, Mcr≡Mcr (0) is replaced by Mcr (Nt,Ed), where Nt,Ed is the acting axial tension. This leads to a λLT decrease

and, therefore, also to larger χLT.mod and Mb,Rd values. It is worth noting that the calculation of Mcr (Nt,Ed) must be

done by means of a numerical beam buckling analysis (e.g., using beam finite elements) − in the future, the

authors plan to develop analytical expressions and/or other design aids that will render the performance of

this task easier and more straightforward for the practitioners.

5.2 Assessment of the proposed ultimate strength/moment estimates

The assessment of the quality of the ultimate strength/moment estimates provided by the proposed

modification of the current EC3-1-1 design rules is based on the results of the numerical simulations

addressed in Chapter 4 and reported in Annex 2. These results consist of, for each combination of beam

geometry (cross-section and length), steel grade, bending moment diagram and β value, the beam (i) critical

moment Mcr (accounting for the axial tension), (ii) plastic moment Mpl,Rk, (iii) reduced (by the axial tension)

plastic bending resistance MN,Rk, (iv) numerical ultimate moment Mu, (v) lateral-torsional slenderness λLT (based

on Mcr(Nt) and Mpl,Rk), (vi) reduction factor χLT.mod (obtained with the EC3-1-1 curve b), (vii) predicted ultimate

moment Mb,Rd (for a collapse governed by lateral-torsional buckling) and (viii) numerical-to-estimated moment

ratio RM=Mu /Mu.est, where Mu.est is the lower between MN,Rk and Mb,Rd − whenever Mu.est=MN,Rk, the value of RM

reflects the cross-section over-strength due to the small strain-hardening included in the steel constitutive law

considered in this work.

Before comparing the obtained numerical and estimated ultimate moments, it should be pointed out again

that this comparison concerns exclusively the beams whose collapse does not correspond to exhausting the

beam mid-span cross-section resistance, i.e., beams failing in lateral-torsional modes occurring prior to the

attainment of MN,Rk − the proposed design approach only deals with the latter.

The numerical ultimate moments Mu, normalised with respect to the cross-section “pure” (not reduced)

plastic moment Mpl (Mu /Mpl≡χLT), are plotted against the beam “modified” (by accounting for the influence of

axial tension Nt on Mcr) lateral-torsional slenderness λLT. Also plotted is the EC3-1-1 design curve b, making

it possible to compare the numerical Mu /Mpl values with their predictions provided by the proposed design

approach. Figure 5.1, concerning the results obtained for beams subjected to a bending moment diagram

defined by ψ=0, provides an illustrative example of the Mu /Mpl values corresponding to the ultimate moments

effectively calculated through the FINELG beam finite element analyses. Although it is clear that there is a very

good agreement between the numerical ultimate moments and the EC3-1-1 curve b, it is impossible not to

notice the few striking exceptions to the above general rule, practically all of them concerning the very slender

beams: an L=25 m HEB 500 beams (both the S355 and S460 steel grades). The explanation for these

discrepancies lies in the fact that, as already mentioned in Chapter 4, for most loadings (i.e., whenever the

66

mid-span cross-section resistance does not govern) these beams collapse at extremely high deformation levels

(e.g., torsional rotations above 90°), which correspond to ultimate moments that are clearly underestimated by

the design curve. If the ultimate moments are linked to “acceptable deformation levels” (instead of the actual

equilibrium path limit points) their values drop considerably and end up much closer to the design curve. For

instance, if the (quite logical) torsional rotation limit of 15° is adopted as a beam ultimate limit state, none

of the Mu /Mpl values associated with the above very slender beams in Figure 5.1 exceeds the design

curve (EC3-1-1 curve b) by more than 13% − currently, the underestimation can be as high as 35% (for the

L=25 m S355 HEB 500 beam and β =1).

λLT

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mu

/ M

pl

[-]

0.00.10.20.30.40.50.60.70.80.91.01.11.2

β = 0β = 0.5β = 0.75β = 1β = 1.5β = 2EC3-1-1 curve b ( ψ = 0)

Figure 5.1 − Comparison between the Mu /Mpl,Rk (numerical gross results) and Mb,Rd /Mpl,Rk (proposed design approach) values

for ψ=0

In order to exclude all the numerical moments associated with “unacceptably high” deformation levels, it

was decided to limit the beam torsional rotation to 15°, which means that the associated applied moment is

hereafter termed “ultimate moment”, even if it does not correspond to the equilibrium path limit point. Then, the

ultimate moments provided in Figures 5.2 to 5.6, for the various beams analysed subjected bending

moment diagrams defined by ψ=1, ψ=0.5, ψ=0, ψ= − 0.5 and ψ= − 1, respectively, are in accordance with

the above criterion − in particular, the comparison between Figures 5.4 and 5.1 makes it possible to assess its

implications. Moreover, the observation of the results displayed in these five figures prompts the following

remarks:

67

λLT

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mu

/ M

pl

[-]

0.00.10.20.30.40.50.60.70.80.91.01.11.2


Figure 5.2 − Comparison between the Mu /Mpl,Rk (numerical) and Mb,Rd /Mpl,Rk (proposed design approach) values for ψ=1

λLT

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mu

/ M

pl

[-]

0.00.10.20.30.40.50.60.70.80.91.01.11.2

β = 0β = 0.5β = 0.75β = 1β = 1.5β = 2EC3-1-1 curve b ( ψ = 0.5)

Figure 5.3 − Comparison between the Mu /Mpl,Rk (numerical) and Mb,Rd /Mpl,Rk (proposed design approach) values for ψ=0.5

68

λLT

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mu

/ M

pl

[-]

0.00.10.20.30.40.50.60.70.80.91.01.11.2


Figure 5.4 − Comparison between the Mu /Mpl,Rk (numerical) and Mb,Rd /Mpl,Rk (proposed design approach) values for ψ=0

λLT

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mu

/ M

pl

[-]

0.00.10.20.30.40.50.60.70.80.91.01.11.2

β = 0β = 0.5β = 0.75β = 1β = 1.5β = 2EC3-1-1 curve b ( ψ = - 0.5)

Figure 5.5 − Comparison between the Mu /Mpl,Rk (numerical) and Mb,Rd /Mpl,Rk (proposed design approach) values for ψ = − 0.5

69

λLT

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mu

/ M

pl

[-]

0.00.10.20.30.40.50.60.70.80.91.01.11.2

β = 0β = 0.5β = 0.75β = 1β = 1.5β = 2EC3-1-1 curve b ( ψ = -1)

Figure 5.6 − Comparison between the Mu /Mpl,Rk (numerical) and Mb,Rd /Mpl,Rk (proposed design approach) for ψ= − 1

(i) First of all, it is worth noting that the length of the EC3-1-1 design curve b horizontal plateau depends on

the bending moment diagram acting on the beam − indeed, this plateau length increases gradually

from 0.4 (ψ=1) to 0.55 (ψ=0.5), 0.70 (ψ=0), 0.75 (ψ=− 0.5) and 0.80 (ψ= − 1).

(ii) Then, it is impossible not to notice the remarkable closeness between the numerical ultimate moments and

their predictions provided by the proposed design approach. Indeed, in the five figures the numerical

values are very nicely aligned slightly above the design curve.

(iii) It is also clearly noticeable that, as one travels from Figure 5.1 to Figure 5.5, there is a clear trend

of the numerical results to shift to the left and upwards, i.e., towards the plastic plateau. Additionally, the

number of simulations decreases as one travels from ψ = 1 to ψ = − 1, since the moment distribution

change renders the beam less prone to LTB and, therefore, the collapse becomes gradually more often

governed by the cross section plastic resistance.

(iv) Table 5.1 provides the averages, standard deviations and maximum/minimum values of the ratio

RM=Mu /Mu.est corresponding to Figures 5.1 to 5.5, for the various axial tension levels. These indicators

reflect the excellent quality of the ultimate strength/moment estimates − indeed, the overwhelming

majority of them are safe and extremely accurate. It is still worth noticing that the least accurate

estimations (higher average and standard deviation) concern β=1.

70

Table 5.1 − Averages, standard deviations and maximum/minimum value of the ratio RM

Average St. Dev. Max Min

β =0 1.03 0.04 1.12 0.92

β =0.5 1.04 0.03 1.11 0.93

β =0.75 1.05 0.03 1.15 0.95

β =1 1.06 0.04 1.13 0.97

β =1.5 1.05 0.04 1.16 0.92

β =2 1.05 0.03 1.16 0.98

In view of what was mentioned above, it seems fair to conclude that the proposed design approach for

beams subjected to major-axis bending and axial tension provides excellent estimates of all the numerical

ultimate moments obtained in this work (associated with lateral-torsional collapse modes) and, therefore, can

be considered as a very promising candidate for inclusion in a future version of Eurocode 3 − of course,

additional parametric studies, involving other loadings (particularly transverse loads) and reliability assessments

studies are required before this goal can be actually achieved. The only foreseeable hurdle for designers is the

lack of an easy and user-friendly way to calculate critical buckling moment in the presence of axial tension − as

mentioned earlier, it is planned to work on the removal of this hurdle through the development of analytical

expressions and/or other design aids to calculate Mcr (Nt,Ed).

5.3 Axial tension beneficial influence

In order to assess the beneficial influence of the presence of axial tension on the beam ultimate

strength/moment, let us begin by considering, as an illustrative example, the L=8.0 m S355 steel IPE 500 beam

subjected to uniform bending and six axial tension levels (β=NEd /MEd). Table 5.2 shows the corresponding λLT,

χLT.mod and Mb,Rd values, and also the Mb,Rd percentage increases with respect to the “pure bending” value

(ΔMb,Rd). Figure 5.7 provides a pictorial representation of the various Mb,Rd and ΔMb,Rd values − it is very clear

that how an increase in axial leading causes a slenderness drop and the corresponding ultimate moment

increase.

Table 5.2 − Ultimate moment predictions for the L=8.0 m S355 steel IPE 500 beam under uniform bending

β λLT χLT.mod Mb.Rd [kNm] ΔMb.Rd [kNm]

0 1.683 0.357 278.1 − 0.5 1.575 0.396 308.9 11.1%

0.75 1.509 0.423 329.9 18.6%

1 1.464 0.443 345.1 24.1%

1.5 1.350 0.498 387.8 39.4%

2 1.231 0.562 437.8 57.4%

71

Figure 5.7 − Pictorial representation of the ultimate moment predictions − L=8.0 m S355 steel IPE 500 beam (ψ=1)

Finally, Table 5.3 provides the averages, standard deviations and maximum/minimum values of the

percentage ultimate moment increases (ΔMb,Rd) due to axial tension corresponding to the beam ultimate

moments included in Figures 5.1 to 5.5. It is observed that all the above axial tension benefit indicators increase

with β, with the sole exception of the minimum value − it remains constant and very small, because it

always corresponds to a slenderness located very close to the end of the design curve plateau.

Table 5.3 − Averages, standard deviations and maximum/minimum values of ΔMb,Rd

β Average St. Dev. Max Min

β =0.5 14.7% 8.9% 43% 0.01%

β =0.75 24.5% 14.3% 64% 0.01%

β =1 34.0% 19.5% 91% 0.01%

β =1.5 53.7% 30.4% 135% 0.01%

β =2 79.9% 46.4% 202% 0.01%

72

5.4 Comparison with the design procedure prescribed in EC3-ENV-1-1

This section compares the design approach proposed in this work with the provisions prescribed by

EC3-ENV-1-1 (1992) for the safety checking of beams subjected to major-axis bending and axial tension,

and failing in lateral-torsional modes. Such provisions were based on the concept of effective moment

(Meff,Ed) − the influence of axial tension was taken into account by decreasing the magnitude of the

applied major-axis bending moment, before comparing it with the beam resistance against lateral-torsional

buckling (Mb,Rd). Figure 5.7 illustrates this concept for the case of a doubly symmetric I-section beam acted

by a bending moment diagram with maximum value My,Ed and axial tension Nt,Ed.

Figure 5.8 − Illustration of the effective moment concept on which the EC3-ENV-1-1 provisions are based

The value of Meff,Ed is obtained from the expression

with Wcomp denoting the cross-section elastic modulus concerning the most compressed fibre and σcomp,Ed is

calculated by means of the expression

where the vectorial reduction factor ψvec takes the value 1.0 or 0.8 depending on whether the applied bending

moment and axial tension stem from the same or distinct actions − in the latter case, only 80% of the beneficial

effect is taken into account. If ψvec =1.0, σcomp,Ed is the maximum compressive stress acting on the

Meff,Ed = Wcomp σcomp,Ed (5.1)

σcomp,Ed = MEd /Wcomp,Ed − ψvec NEd /A (5.2)

73

cross-section subjected to the highest bending moment. It is still worth noting that σcomp,Ed may be higher than

fy in class 1 or 2 cross-sections acted by bending moments exceeding Mel,Rk and small axial tension values.

Finally, the safety checking of the beams subjected to major-axis bending and axial tension merely

consisted of comparing Meff,Ed with the beam LTB resistance Mb,Rd, calculated as prescribed by EC3-ENV-1-1.

In order to compare the ultimate moments provided by the design approach proposed in this dissertation

and the design methodology prescribed in EC3-ENV-1-1, two illustrative examples are first presented, both

concerning IPE beams. In order to have a meaningful comparison, it is assumed that ψvec =1.0, i.e., that My,Ed

and Nt,Ed stem from the same action − otherwise, the calculation of Mcr should be based on only 80% of the

acting axial tension. Moreover, the value of Mb,Rd is calculated according to EC3-1-1 and not EC3-ENV-1-1

(the two values are not identical). Since axial tension benefits are captured differently in the two design

procedures (one increases the bending resistance and the other reduces the applied moment), it is necessary to

define a criterion for their comparison. The following one is adopted here: for a beam subjected to a bending

moment diagram with maximum value My,Ed and axial tension Nt,Ed, related by a given β value, and with LTB

resistance Mb,Rd (0) (without considering the axial tension beneficial influence), (i) the benefit of the

proposed approach is measured by the ratio RP=[Mb,Rd (Ntu,Ed) − Mb,Rd (0)]/Mb,Rd (0) and (ii) that associated with the

EC3-ENV-1-1 methodology by the ratio REC3=[Mb,Rd (0) − Meff,Ed (Ntu,Ed)]/Mb,Rd (0), where the calculation of

Meff,Ed (Ntu,Ed) is based on Mb,Rd (0) − in both cases, Ntu,Ed denotes the value of the axial tension at the beam lateral-

torsional collapse, calculated on the basis of the proposed design approach. Then, in order to assess the strength

increases, stemming from using the proposed design approach and the EC3-ENV-1-1 methodology, the

two aforementioned ratios are compared for all the beams analysed that collapse in modes governed by LTB −

the percentage difference between these two ratios, termed ΔRP-EC3, will be used to quantify this comparison.

The first illustrative example concerns a L=8 m S460 IPE 300 beam under uniform bending (ψ=1) and

subjected to a loading strategy corresponding to β=1. The corresponding design values are the following:

(i) Mb,Rd (Ntu,Ed)=85.9 kNm, (ii) Ntu,Ed=85.9 kN, (iii) Mb,Rd (0)=57.0 kNm and (iv) Meff,Rd (Ntu,Ed)=51.1 kNm.

They correspond to RP=0.51%, REC3=0.10 and, thus, ΔRP-EC3=41%.

The second illustrative example concerns a L=15 m S460 IPE 500 beam under a bending moment diagram

defined by ψ= − 1) and subjected to a loading strategy corresponding to β=0.5. The corresponding design

values are the following: (i) Mb,Rd (Ntu,Ed)=393.3 kNm, (ii) Ntu,Ed=196.7 kN, (iii) Mb,Rd (0)=335.9 kNm and

(iv) Meff,Rd (Ntu,Ed)=308.0 kNm. They correspond to RP=0.17, REC3=0.08 and, thus, ΔRP-EC3=9%.

Although the two illustrative examples indicate that the proposed design approach leads to higher benefits

stemming from the presence of axial tension that the methodology prescribed in EC3-ENV-1-1, such an

assertion can only be general after checking it against a much larger of beams. Therefore, the above comparison

is extended to 400 beams, exhibiting (i) various lengths (comprised between 0.5 and 25 m), (ii) two yield

stresses (fy=355; 460 MPa), (iii) four cross-section shapes (IPE 300, IPE 500, HEB 300, HEB 500), (iv) two

74

bending moment diagrams (ψ=1 and ψ= − 1), and (v) five axial tension levels, corresponding to β=Nt /My

ratios equal to 0.5; 0.75; 1.0; 1.5; 2.0 – this is a sizeable fraction of the parametric study carried out in Chapter 4.

Since all these beams collapse in modes governed lateral-torsional buckling, the corresponding ultimate

moments fall outside the design curve b plastic plateau, i.e., λLT >0.4 (ψ=1) and λLT >0.8 (ψ= − 1).

Figures 5.9 (ψ=1) and 5.10 (ψ= − 1) plot the ΔRP-EC values against the beam lateral-torsional slenderness λLT.

Moreover, Tables 5.4(a)-(b) provide the ΔRP-EC3 averages, standard deviations and maximum/minimum values

for the total number of beams considered. After observing these results, the following remarks are appropriate:

Figure 5.9 − Values of the ratio difference ΔRP-EC3 plotted against the beam slenderness (ψ=1)

Figure 5.10 − Values of the ratio difference ΔRP-EC3 plotted against the beam slenderness (ψ= − 1)

75

Table 5.4 − Averages, standard deviations and maximum/minimum values of ΔRP-EC3 for (a) ψ=1 and (b) ψ= − 1

Average St. Dev. Max Min Beam Number esults.

ψ = 1 23% 45% 195% − 42% 270

(a)

Average St. Dev. Max Min Beam Number

ψ = −1 10% 27% 102% − 37% 108

(b)

(i) First of all, it is readily observed the huge scatter of the ΔRP-EC3 values, particularly for ψ=1 − indeed, the

maximum and minimum values are 237% (ψ=1) and 139% (ψ= − 1) apart, even if the average

values (45% and 10%) are relatively small.

(ii) Then, it is also clear that the proposed design approach generally leads higher ultimate strength/moment

prediction increases due to the presence of axial tension. Moreover, the increases associated with

the uniformly bent beams (ψ=1) are naturally considerably larger than those concerning the

beams acted by ψ= − 1 bending moment diagrams − this is because the former are much more

prone to LTB, which means that “feel more intensely” the axial tension benefits.

(iii) However, in spite of what was mentioned in the previous two items, it is also noticeable that a

distinction must be made between the beams with low slenderness values (close to the design curve b

plastic plateau and, generally speaking, below 1.0) and those with λLT values above 1.0. For the vast

majority of the former beams, the use of the design methodology prescribed in EC3-ENV-1-1 leads to

higher axial tension benefits. Conversely, the proposed design approach ensures higher axial tension

benefits for virtually all the beams associated with λLT >1.0.

5.5 Summary

This chapter presented the development of a design approach aimed at predicting the ultimate strength/moment

of beams subjected to major-axis bending and axial tension whose collapse is governed by lateral-torsional

buckling. This approach consists of a slight modification of the current EC3-1-1 design rules for beams prone to

lateral-torsional buckling and, in particular, used its design curves to obtain the ultimate moment estimates.

The modification consists of calculating the beam slenderness on the basis of a critical moment value that

accounts for the beneficial effect of axial tension.

76

Out of the various conclusions drawn from the research work reported in this chapter, the following ones

deserve to be specially mentioned:

(i) Neglecting the influence of axial tension on the LTB behaviour of failure of beam subjected to major-axis

bending may lead to a considerable underestimation of their load-carrying capacity. This underestimation

is more pronounced for the beams more prone to lateral-torsional buckling (and, obviously, subjected to

higher axial tension levels).

(ii) The proposed design approach was shown to provide ultimate strength/moment estimates that correlate quite

well with the numerical values obtained from the parametric study performed in Chapter 4. Indeed, the

overwhelming majority of the predictions are safe and rather accurate.

(iii) The application of the proposed design approach is quite straightforward. The only difficulty concerns the

determination of critical moments of beams subjected to axial tension − this difficulty should be overcome by

developing easy-to-use formulae to calculate (more or less approximately) these critical moments.

(iv) The comparison between the ultimate moment estimates provided by the proposed design approach and the

design methodology prescribed in EC3-ENV-1-1 showed that, generally speaking, the former leads to

higher axial tension benefits than the latter. However, a closer observation of the results obtained made it

possible to conclude that the above assertion is mostly true for beams with λLT >1.0. For the lesser

slender beams, higher axial tension benefits can be achieved using the EC3-ENV-1-1 methodology.

Chapter 6

Conclusion and Future Developments

This dissertation reported the results of an analytical, numerical and experimental investigation on the

lateral-torsional stability, failure and design of hot-rolled steel I-section beams with fork-type end

supports and acted by simple transverse loadings (mostly applied end moments) and various axial

tension values. Initially, the derivation and validation of an analytical expression providing critical

buckling moments of uniformly bent beams subjected to tension was presented. Then, this analytical

finding was followed by a numerical study on the beneficial influence of axial tension on beams under

non-uniform bending, namely caused by unequal applied end moments − several beam finite element

results were presented and discussed in some detail. Next, the dissertation addressed the performance of two

experimental tests, carried out at the University of Fribourg and aimed at determining the behaviour

and ultimate strength of a narrow and a wide flange beams subjected to eccentric axial tension. These

results were also used to develop and validate FINELG beam and shell finite element models that were

subsequently employed to perform an extensive parametric study that (i) involved more than 2000 numerical

simulations, concerning beams with various cross-section shapes, lengths, yield stresses, bending moment

diagrams and axial tension levels, and (ii) was carried out to gather a fairly large ultimate strength/moment

data bank. Finally, these data were used to assess the merits of a design approach proposed for beams subjected

to tension and collapsing in modes governed by lateral-torsional buckling − this design approach

consists of slightly modifying the current procedure prescribed in Eurocode 3 to design beams against

lateral-torsional failures (through the incorporation of the axial tension influence on the critical buckling

moment that is used to evaluate the beam slenderness). The predictions of the proposed design approach

were also compared with those of the design procedure included in the ENV version of Eurocode 3, which

was later removed and is absent from the current version.

78

6.1 Concluding Remarks

The most relevant findings and conclusions of the research work carried out in this dissertation are the following:

(i) An analytical expression to calculate critical moments of doubly symmetric I-section beams subjected to

uniform bending and axial tension was developed and validated by means of a comparison with beam

finite element results. This expression made it possible to acquire in-depth knowledge about the beneficial

influence of axial tension on the beam lateral-torsional buckling behaviour, namely by increasing its

critical buckling moment (Mcr).

(ii) In order to assess the influence of the cross-section shape, bending moment diagram and loading

characteristics, and at the same time gather critical buckling moment data to be used subsequently in the

development of a design approach, a fairly wide numerical (ABAQUS beam finite element) parametric

study was carried out. Its results made it possible to conclude that the Mcr increase due to axial tension is

more pronounced for (ii1) slender beams, (ii2) cross-sections with narrow flanges (higher ratio between the

major and minor-axis moments of inertia) and (ii3) triangular moment distributions (ψ=0).

(iii) In order to obtain a better feel concerning the mechanics of the lateral-torsional collapse of beams

subjected to major-axis bending and axial tension, as well as to assemble experimental results to be used in

the development of a FINELG shell finite element model, two full scale tests were performed. They

involved beams subjected to eccentric tension and provided clear experimental evidence of the occurrence

of collapse modes governed by lateral-torsional buckling. Moreover, the beam specimens were fully

characterised prior to testing, namely by (iii1) carrying out tensile coupon tests to obtain the steel material

behaviour (stress-strain law) and (iii2) carefully measuring the beam initial geometrical imperfections.

(iv) The measurements and results gathered from the above two tests were then used to develop and validate a

FINELG shell finite element model. After the validation procedure, which required modelling the whole

test set-up to obtain an acceptable correlation between the numerical and experimental results, the shell

finite element model was used to develop and validate an accurate FINELG beam finite element model.

(v) The above beam finite element model was then employed to perform an extensive parametric study,

comprising about 2000 numerical simulations and aimed at gathering ultimate strength/moment data

concerning beams subjected to major-axis bending and axial tension and exhibiting collapse modes

stemming from either lateral-torsional buckling or plasticity effects (cross-section resistance). It was

shown clearly that, as expected, the presence axial tension either increases or reduces the ultimate

strength/moment, depending on whether failure is due to LTB or plasticity effects. It is worth noting

that the focus of this dissertation was beams failing in lateral-torsional collapse modes.

(vi) By plotting the extensive ultimate strength/moment data, normalised with respect to the cross-section

plastic resistance, against the beam lateral-torsional slenderness λLT, it became very clear that ultimate the

79

numerical Mu /Mpl values exhibited a typical “design curve” alignment, thus suggesting the development

of a design/strength curve to estimate them. Moreover, it was found that the axial tension benefits stem

from (vi1) decreasing the beam vulnerability to lateral-torsional buckling (critical buckling moment

increase that reduces λLT) and, therefore, (vi2) increasing the beam load-carrying capacity associated with

lateral-torsional collapses.

(vii) A design approach for beams subjected to major-axis bending (only end applied moments were dealt

with) and axial tension that collapse in lateral-torsional modes was developed and the quality of its

estimates was assessed by means of the comparison with the ultimate strength/moment data obtained

previously − a very good correlation was found for the overwhelming majority of the beams considered

(moreover, practically all the predictions are on the safe side). The proposed design approach consists of a

slight modification of the procedure currently prescribed in EC3-1-1 to determine the beam resistance

against lateral-torsional failures − the modification consists of determining λLT on the basis of a critical

buckling moments that account for the beneficial effect of axial tension.

(viii) The application of the proposed design approach is rather simple and straightforward. The only difficulty

resides in the determination of the critical moment of a beam subjected to major-axis bending and axial

tension. In order to overcome this hurdle, easy-to-use formulae to calculate (more or less approximately) such

critical moments are required.

(ix) It was clearly shown that neglecting the beneficial effect stemming from the presence of axial tension may

lead to highly over-conservative designs, particularly in the beams most prone to lateral-torsional buckling.

(x) Finally, a comparison between the ultimate moment estimates provided by the proposed design approach

and the design methodology prescribed in EC3-ENV-1-1 showed that, generally speaking, the former

leads to higher axial tension benefits (which were confirmed by the numerical results). However, it was

also found that the predictions provided by the EC3-ENV-1-1 methodology lead to higher axial tension

benefits for less slender beams (λLT <1.0), even if most of these predictions are most likely probably

unsafe (this fact was not checked in this work). Therefore, it seems fair to say that the proposed design

approach looks quite promising, at least in the context of the beams analysed in this dissertation.

Finally, it is still worth mentioning that the work reported in this dissertation originated two papers presented in

international conferences and published in the respective proceedings − moreover, a paper intended to

be submitted for publication in a peer-reviewed international journal is currently under preparation. The

references of the above two publications are the following:

(i) Tomás J, Nseir J, Camotim D, Boissonnade N (2013). “Stability, failure and design of I-section beams

subjected to tension”, USB Key Drive Proceedings of Structural Stability Research Council Annual

Stability Conference (St. Louis, 16-20/4).

80

(ii) Tomás J, Nseir J, Camotim D, Boissonnade N (2013). “Stability, failure and design of I-section steel

beams subjected to tension”, Research and Applications in Structural Engineering, Mechanics and

Computation (SEMC-2013 − Cape Town, 2-4/9), A. Zingoni (ed.), Taylor & Francis (London), 465-466.

(full paper in CD-ROM Proceedings − 1303-1308).

6.2 Future Developments

This dissertation constitutes a first contribution towards understanding the structural behaviour and

providing design guidance concerning members subjected to major-axis bending and torsion. Although it is fair

to say that the output of the dissertation is clearly positive and successful, it is also important to recognise that

much remains to be done before this topic can be deemed mastered. Without claiming to be exhaustive, several

extensions of the work presented in this dissertation that require future research are listed below:

(i) Consideration of additional transverse loadings, namely those involving transverse loadings − in this case,

it is essential to consider the location of the point of application of the transverse loads (with respect to the

cross-section shear centre).

(ii) Consideration of additional beam end support conditions, namely those involving warping fixity.

(iii) Consideration of slender beams, namely class 3 and class 4 beams. This issue is particularly relevant, in

view of the current trend of the steel construction industry to use higher-grade steels exhibiting very large

yield stress values.

(iv) Assessment of whether the design approach proposed in this dissertation can be successfully applied to the

beams described in the previous three items − naturally, such an assessment requires also the performance

of a reasonable number of experimental tests. If successful, this task may lead to the proposal of design

provisions for beams under major-axis bending and torsion to be included in a future Eurocode 3 version.

(v) Extension of the work carried out in this dissertation, as well as all the topics listed above, to singly-

symmetric beams, namely welded beams with unequal flanges.

(vi) Extension of the work carried out in this dissertation, as well as of all the research concerning the

topics listed above, to beams subjected to bi-axial bending and torsion.

References

Boissonnade N, Greiner R, Jaspart J-P, Lindner J (2006). Rules for Member Stability in EN 1993-1-1:

Background Documentation and Design Guidelines (ECSS Technical Committee 8 − Stability), ECCS

Publication Nº 119, Brussels.

Boissonnade N, Somja H (2012). “Influence of imperfections in FEM modeling of lateral-torsional buckling”,

USB Key Drive Proceedings of SSRC Annual Stability Conference (Grapevine. 17-21/4).

Reis A & Camotim D, (2012). Estabilidade e Dimensionamento de Estruturas, Orion Press, Lisbon.

(Portuguese)

CEN (Comité Européen de Normalisation) (1992). Eurocode 3: Design of Steel Structures − Part 1.1: General

Rules and Rules for Buildings (ENV 1993-1-1). Brussels.

CEN (Comité Européen de Normalisation) (2005). Eurocode 3: Design of Steel Structures − Part 1.1: General

Rules and Rules for Buildings (EN 1993-1-1), Brussels.

Chen W-F, Atsuta T (1977). Theory of Beam-Columns − Space Behavior and Design (vol. 2), McGraw-Hill.

New York.

Culver C (1966). “Exact solution of the biaxial bending equations”, Journal of the Structural Division (ASCE),

92(2), 63-83.

Falgoust MK (2004). On the validity of the Wagner Hypothesis in Thin-Walled Open-Profile

Members, M.A.Sc. Thesis in Civil Engineering, University of Pittsburgh.

FINELG (Non-Linear Finite Element Analysis Program) (2012). User’s Manual (vrs. 9.3), ArGEnCo

Department, University of Liège and Greisch Info S.A.. Liège.

Fukumoto Y, Galambos T (1966). “Inelastic lateral-torsional buckling of beam-columns”, Journal of Structural

Division (ASCE), 92(2), 41-55.

Memon, B-A, Su X-Z. (2003). “Arc-length technique for nonlinear finite element analysis”, Journal of

Zhejiang University − Science, Department of Structural Engineering, Tongji University, Shanghai.

Mendonça P. (2006). Dimensionamento de Colunas-Viga através das Equações de Interacção do Eurocódigo 3,

M.A.Sc. Dissertation (Structural Engineering), Instituto Superior Técnico, Technical University of Lisbon.

(Portuguese)

Simulia Inc. (2008). Abaqus Standard (vrs. 6.7-5).

Tededge N, Alpsten G, Tall L, (1973). “Residual stress measurement by the sectioning method”, Proceedings of

the Society of Experimental Stress Analysis”, 30, nº1.

Trahair NS (1993). Flexural Torsional Buckling of Structures, E&FN Spon (Chapman & Hall), London.

82

Ville de Goyet V (1989). L’Analyse Statique Non Linéaire par la Méthode des Élements Finis des Structures

Spatiales Formées de Poutres à Section Symétrique. Ph.D. Thesis in Applied Science, University of Liège.

(French)

Vlasov VZ (1961), Thin-walled Elastic Beams, English translation by the National Science Foundation and

Department of Commerce, Washington D.C.

Annexes Annex 1 Analytical formula to calculate critical buckling moments of beams

subjected to uniform major-‐axis bending and axial tension .............................. A1.1

Annex 2 Numerical data: critical moments, ultimate moment values and

ultimate moment estimates ........................................................................................... A2.1

A2.1. Proposed ultimate moment estimates and design results -‐ IPE300 beams ................... A2.3 A2.2. Proposed ultimate moment estimates and design results -‐ IPE500 beams ................ A2.19 A2.3. Proposed ultimate moment estimates and design results -‐ HEB300 beams .............. A2.35 A2.4. Proposed ultimate moment estimates and design results -‐ HEB500 beams .............. A2.51

Annex 3 Measured initial geometrical imperfections ............................................................ A3.1

A1-1

Annex 1: Analytical Formula to Calculate Critical Buckling Moments of Beams

Subjected to Uniform Major-Axis Bending and Axial Tension

This annex concerns the establishment and analytical solution of the differential equilibrium

equations governing the lateral-torsional buckling behaviour of doubly symmetric beams subjected

to uniform major-axis uniform bending and axial tension. Figure A1.1 depicts the beam deformed

configuration at an equilibrium state adjacent to the fundamental equilibrium path − also shown are

(i) the coordinate axes at the beam undeformed (x-y-z − corresponding to displacements w-v-u) and

deformed (ζ-η-ξ) configurations and (ii) the applied/external forces and moments. In the undeformed

and deformed configuration, the y-z and ζ-η axes coincide with the cross-section centroidal principal

axes. As for the ξ axis, it is tangent to the beam deformed longitudinal axis ζ.

Figure A1.1 – Beam undeformed and deformed (adjacent equilibrium state) configurations, including the corresponding

coordinate axes and the applied forces and moments

In the case of a beam under uniform major-axis bending (My) and axial tension (Nt), the

applied forces and moments at the deformed configuration shown in Figure A1.1, expressed

in terms of the undeformed coordinate axes, read

𝐹!𝐹!𝐹!

=−𝑁!00

(A.1)

𝑀!𝑀!𝑀!

=0𝑀!0

+0 𝑢 −𝑣−𝑢 0 𝑥𝑣 −𝑥 0

−𝑁!00

=0

𝑀! + 𝑢𝑁!−𝑣𝑁!

(A.2)

By using the rotation matrix 𝑅 , associated with the hypothesis of “small displacements and

moderate rotations” and defined by Chen & Atsuta (1977)

A1-2

𝜁𝜂𝜉= 𝑅

𝑥𝑦𝑧

(A.3)

𝑅 =cos(𝜁 𝑥) cos(𝜁 𝑦) cos(𝜁 𝑧)cos(𝜂 𝑥) cos(𝜂 𝑦) cos(𝜂 𝑧)cos(𝜉 𝑥) cos(𝜉 𝑦) cos(𝜉 𝑧)

=1 𝑣′ 𝑢′−𝑣′ 1 𝜙−𝑢′ −𝜙 1

(A.4)

it is possible to rewrite the applied/external moments in terms of the deformed coordinate axes, as

𝑀!𝑀!𝑀! !"#

=1 𝑣′ 𝑢′−𝑣′ 1 𝜙−𝑢′ −𝜙 1

−𝑢𝑁!𝑀! + 𝑢𝑁!−𝑣𝑁!

(A.5)

In the equation system (A.5), it is worth noting that the three equations concern the bending

moment and bi-moment equilibrium in the beam deformed configuration and are written with respect

to the deformed location of the origin of the cross-section coordinate system (𝜂, 𝜉) − note that, since

only doubly symmetric beams are dealt with in this work, this origin coincides with both the cross-

section centroid and shear/torsion centre.

The next step in the establishment of the beam differential equilibrium equations consists of

developing generalised elastic stress-strain relationships, based on the assumption of a linear

longitudinal extension variation over the whole cross section. Following the work of Chen & Atsuta

(1977) and using the deformed configuration coordinate system (𝜁 , 𝜂 , 𝜉 ), the longitudinal

extensions can be related to the cross-section generalised deformations, namely the axial extension

ε!, the two principal flexural curvatures 𝜒! and 𝜒! , and the “twistature” 𝜃′′! (adopting the

neologism coined by Trahair, 1993), by means of the expression

𝜀 = 𝜀! + 𝜒!𝜉 − 𝜒!𝜂 − 𝜁𝜃!!! = 𝜀! + 𝜃!!!𝜉 − 𝜃!!!𝜂 − 𝜁𝜃!!! (A.6)

where in which 𝜃! , 𝜃! and 𝜃! are the cross-section rotations about the coordinate axes ζ – η – ξ.

Taking into account that the longitudinal normal stresses are obtained from the axial extensions

through the multiplication by the material (steel) Young’s modulus, the cross-section stress

resultants (internal forces and moments) stemming from these normal stresses are given by

Axial Force 𝑁! = 𝜎 𝑑𝐴𝑁! = 𝐸𝐴𝜀! (A.7)

(A.8)

(A.9)

Bending Moments 𝑀! = 𝜎 𝜉 𝑑𝐴𝑀! = 𝐸𝐼!𝜒!

𝑀! = − 𝜎 𝜂 𝑑𝐴𝑀! = − 𝐸𝐼!𝜒!

A1-3

Bi-moment 𝑀𝜁 = 𝜎𝜁 𝑑𝐴𝑀𝜁 = −𝐸𝐼!𝜃′′!

(A.10)

where A, Iy, Iz and Iw are the cross-sectional area, major and minor moments of inertia and

warping constant. In other words, the longitudinal normal stresses acting on the are given by

𝜎 = 𝑁!𝐴+𝑀!

𝐼!𝜉 −

𝑀!

𝐼!𝜂 +

𝑀𝜁𝐼!𝜁 (A.11)

Concerning the torsional/twisting moment equilibrium, it reads (Chen & Atsuta, 1977)

𝑀! = 𝑇! + 𝑇!" = −𝐸𝐼!𝜃!!!! + 𝐺𝐼!𝜃′! (A.12)

where the first and second terms in the right hand side stand for the warping and Saint-Venant

torsional moments − while the former is caused by the shear stress resultant stemming from the

longitudinal variation of the bi-moment (the so-called bi-shear shear stresses constant across the wall

thickness), the latter is due to the Saint-Venant shear stresses (linear across the wall thickness)

stemming from the longitudinal variation of the cross-section torsional rotation (Trahair, 1993).

Moreover, it is still necessary to take into account a second-order effect associated with the

longitudinal normal stresses caused by the axial load and bi-moment, which is designated in the

literature as “Wagner effect”. According to Falgoust (2004), Wagner’s hypothesis states that when a

thin-walled open-section member is subjected to longitudinal normal stresses and begins to twist,

these normal stresses become inclined with respect to the deformed cross-section plane and, therefore,

produce a torsional moment about the member deformed longitudinal axis tangent − see Figure A1.2.

The contribution of the normal stress acting at a given fibre is proportional to its distance to the cross-

section shear centre (coincident with the centroid in doubly symmetric beams) − its value is given

through the multiplication by

Figure A1.2 − Wagner effect due to inclined longitudinal normal stresses (Chen & Atsuta, 1977)

A1-4

𝑠𝜃′! = 𝜂 − 𝜂! ! + 𝜉 − 𝜉! !𝜃′! (A.13)

where s is the distance to the cross section to the shear centre and 𝜃′! is the twist (derivative

of the torsional rotation) − see Figure A1.2. Then, the total contribution is given by 𝐾𝜃′!

𝑇! = 𝜎𝑠!𝜃′! 𝑑𝐴 = 𝐾𝜃′! (A.14)

where 𝐾 is the “Wagner coefficient” − for doubly symmetric section and when the normal stresses

due to axial loading are the most relevant ones (recall that no applied bi-moments are considered in this

work), (A.14) may be simplified to (Chen & Atsuta, 1977)

𝐾 = 𝜎𝑠! 𝑑𝐴𝐾𝜃′! ≈𝑁!𝐴

𝜂 − 𝜂! ! + 𝜉 − 𝜉! ! 𝑑𝐴 = 𝑁! 𝑟!! (A.15)

where 𝑟! is the cross-section polar radius of gyration.

Then, the total equilibrium reads

𝑀! = −𝐸𝐼!𝜃!!!! + 𝐺𝐼! + 𝐾 𝜃′! (A.16)

Before equating the external/applied and internal axial force, major and minor-axis bending

moments and torsional/twisting moment (i.e., establishing the equilibrium equations governing the

“tensioned beam” lateral-torsional buckling behaviour), it is necessary to express the latter in the

undeformed beam coordinate system (x-y-z), corresponding to displacements (w-v-u) − see Figure A1.1.

Under the assumption of the small displacements and moderate rotations, the rotation first derivatives

can be written as

𝜃!!𝜃!!𝜃!!

=1 𝑣′ 𝑢′−𝑣′ 1 𝜙−𝑢′ −𝜙 1

−𝑢′′𝑣′′𝜙′

=−𝑢!! + 𝑣!𝑣!! + 𝑢′𝜙′𝑣!𝑢!! + 𝑣!! + 𝜙𝜙′𝑢!𝑢!! − 𝜙𝑣!! + 𝜙′

(A.17)

Then, the corresponding torsional rotation second and third derivatives (i.e., the “twistature” and

its first derivative) read

𝜃!!! = 𝑢!! ! + 𝑢!𝑢!!! − 𝜙′𝑣!! − 𝜙𝑣!!! + 𝜙′′ (A.18)

𝜃!!!! ≈ −𝜙′′𝑣!!! − 𝜙𝑣!!! + 𝜙′′′ (A.19)

where the fourth derivative was neglected in (A.19). Finally, the axial strain is given by

𝜀! = 𝑤! + !!𝑣! ! + !

!𝑢! ! (A.20)

A1-5

Using the above relationships, the internal axial force, bending moments and torsional/twisting

moment are expressed as (see equations (A.7)-(A.10) )

−𝑁!𝑀!𝑀! !"#

= 𝐸𝐴 0 0 00 𝐼! 0 00 0 𝐼! 0

𝑤! + !!𝑣! ! + !

!𝑢! !′

−𝑢!! + 𝑣!𝑣!! + 𝑢′𝜙′𝑣!𝑢!! + 𝑣!! + 𝜙𝜙′

(A.21)

𝑀! = −𝐸𝐼! 𝑢!𝑢!! − 𝜙𝑣!! + 𝜙 + 𝐺𝐼! + 𝐾 −𝜙!!!!!! − 𝜙𝑣!!! + 𝜙!!! −

−𝐸𝐼!( 𝑢!! ! + 𝑢!𝑢!!! − 𝜙′𝑣!! − 𝜙𝑣!!! + 𝜙′′)

(A.22)

Taking into account that the external forces are given by (see equation (A.5))

−𝑁!𝑀!𝑀!𝑀! !"#

=

1 0 0 00 1 𝜙 −𝑣′0 −𝜙 1 −𝑢′0 𝑣′ 𝑢′ 1

−𝑁!𝑀! + 𝑢𝑁!−𝑣𝑁!−𝑢𝑁!

(A.23)

and equating the external and internal axial forces, bending moments and torsional/twisting

moments, one is led to (note that, since only adjacent equilibrium is sought, all the non-linear terms

are neglected)

𝐸𝐴 0 00 𝐼! 00 0 𝐼!

𝑤!

−𝑢!!𝑣!!

=1 0 0 00 1 𝜙 −𝑣′0 −𝜙 1 −𝑢′

−𝑁!

𝑀! + 𝑢𝑁!−𝑣𝑁!

(A.24)

−𝐸𝐼! 𝐺𝐼! + 𝐾𝜃!!!!𝜃!!

= 0 𝑣′ 𝑢′ 1

−𝑁!𝑀! + 𝑢𝑁!−𝑣𝑁!−𝑢𝑁!

(A.25)

or, in explicit form

where it should be noted that equation (A.26) is trivially satisfied. In order to render the solution

of the above equation system easier, it is convenient to rewrite it as the fourth-order differential

equation system (Chen & Atsuta, 1977)

𝐸𝐴𝑤! = −𝑁!

−𝐸𝐼!𝑢!! = 𝑀! + 𝑢𝑁! + 𝑣𝜙𝑁! ⟺ 𝐸𝐼!𝑢!! +𝑀! + 𝑁!𝑢 = 0

𝐸𝐼!𝑣!! = −𝜙𝑀! − 𝑢𝜙𝑁! − 𝑣𝑁! ⟺ 𝐸𝐼!𝑢!! +𝑀! + 𝑁!𝑢 = 0

−𝐸𝐼!𝜙!!!! + 𝐺𝐼! + 𝐾 𝜙!! = 𝑣!𝑀! ⟺ 𝐸𝐼!𝜙!!!! − 𝐺𝐼! + 𝐾 𝜙!! + 𝑣

!𝑀! = 0

(A.26)

(A.27)

(A.28)

(A.29)

A1-6

Since (i) equation (A.30) concerns the major-axis bending moment equilibrium along the tensioned

beam fundamental equilibrium path and (ii) the pre-buckling deformation is neglected (in this case,

its consideration would lead to a beam stiffening effect − neglecting such effect is conservative, in

the sense that higher critical moments are obtained), this equation can be disregarded and the

critical buckling moment is provided by the solution of the eigenvalue problem defined by equations

(A.31) and (A.32). Making My≡M and integrating these two equations twice leads to the following

system of second-order differential equations.

whose complete solution consists of two parts, namely a particular solution and a complementary

solution. Since the terms on the right hand sides of these equations are linear functions of x, their

particular solutions 𝑣! and 𝜙! must satisfy the relations

which means that they are given by

On the other hand, the complementary solutions 𝑣! and 𝜙𝑝 are determined by solving the

homogeneous system of second-order differential equilibrium equation system

In the most general case, the solution of the above system is of the form 𝑣 = 𝐴!𝑒!" and

𝜙 = 𝐴!𝑒!", and its incorporation into equations (A.39) and (A.40), together with the assemblage of

the terms 𝑒!", leads to the algebraic equation system

𝐸𝐼!𝑢!" + 𝑁!𝑢!! = 0

𝐸𝐼!𝑣!" + 𝑁!𝑣!! +𝑀!𝜙′′ = 0

𝐸𝐼!𝜙!" − 𝐺𝐼! + 𝑁! 𝑟!! 𝜙!! +𝑀!𝑣!! = 0

(A.30)

(A.31)

(A.32)

𝐸𝐼!𝑣!! − 𝑁!𝑣 +𝑀 𝜙 = 𝐶!𝑥 + 𝐶!

𝐸𝐼!𝜙!! − 𝐺𝐼! + 𝑁!𝑟!! 𝜙 +𝑀𝑣 = 𝐶!𝑥 + 𝐶!

(A.33)

(A.34)

−𝑁𝑡𝑣 +𝑀 𝜙 = 𝐶!𝑥 + 𝐶!

− 𝐺𝐼! +𝑁𝑡𝑟!! 𝜙 +𝑀𝑣 = 𝐶!𝑥 + 𝐶!

(A.35)

(A.36)

𝑣! = 𝐶!𝑥 + 𝐶! 𝐺𝐼! +𝑁𝑡𝑟!! + 𝐶!𝑥 + 𝐶! 𝑀

𝑀! −𝑁𝑡 𝐺𝐼! +𝑁𝑡𝑟!!

𝜙! =𝐶!𝑥 + 𝐶! 𝑀 𝑁𝑡 + 𝐶!𝑥 + 𝐶!𝑁𝑡 𝑀 𝑁𝑡 ! + 𝑟!! − 𝐺𝐼!

(A.37)

(A.38)

𝐸𝐼!𝑣!! −𝑁𝑡𝑣 +𝑀 𝜙 = 0

𝐸𝐼!𝜙!! − 𝐺𝐼! +𝑁𝑡𝑟!! 𝜙 +𝑀𝑣 = 0

(A.39)

(A.40)

A1-7

whose non-trivial solutions are obtained by solving the characteristic equation

It can be shown (Culver, 1966) that the four solutions of (A.42) are of the form 𝐷 = ±𝑖𝜆! and

𝐷 = ±𝜆!, where λ1 and λ2 must be evaluated numerically. Then, the complementary solutions

v! and ϕ! are given by

Finally, the complete solution of the original system reads

where the eight integrations constants are determined from the beam end supporting conditions,

concerning the lateral displacement and torsional rotation values and second derivatives at the beam

end cross-sections, through the equations

EI!D! −𝑁𝑡 𝑀𝑀 EI!D! − GI! +𝑁𝑡r!!

v𝜙 = 0

0 (A.41)

𝐸𝐼!𝐷! − 𝑁! 𝐸𝐼!𝐷! − 𝐺𝐼! + 𝑁!𝑟!! − 𝑀 ! = 0 (A.42)

𝑣! = −𝑀𝐸𝐼!

1−𝜆!

! − 𝜆!! 𝐵! 𝑠𝑖𝑛 𝜆! 𝑥 + 𝐵! 𝑐𝑜𝑠 𝜆! 𝑥

+1

𝜆!! − 𝜆!

! 𝐵! 𝑠𝑖𝑛ℎ 𝜆! 𝑥 + 𝐵! 𝑐𝑜𝑠ℎ 𝜆! 𝑥

𝜙! = 𝐵!𝑠𝑖𝑛 (𝜆!𝑥) + 𝐵! 𝑐𝑜𝑠 (𝜆!𝑥) + 𝐵! 𝑠𝑖𝑛ℎ (𝜆!𝑥) + 𝐵! 𝑐𝑜𝑠ℎ (𝜆!𝑥)

𝜆!! =

𝑁𝑡𝐸𝐼!

(A.43)

(A.44)

(A.45)

𝑣 = 𝑣! + 𝑣! (A.46)

𝜙 = 𝜙! + 𝜙! (A.47)

𝑣 0 = 0 ⇔ −𝑀𝐸𝐼!

𝐵!−𝜆!

! − 𝜆!! +

𝐵!𝜆!

! − 𝜆!! +

𝐶! 𝐺𝐼! + 𝑁!𝑟!! + 𝐶! 𝑀𝑀2 − 𝑁! 𝐺𝐼𝑡 + 𝑁!𝑟02

= 0 (A.48)

𝑣!! 0 = 0⇔ −𝑀𝐸𝐼!

−𝜆!!

−𝜆!! − 𝜆!

! 𝐵! +𝜆!

!

𝜆!! − 𝜆!

! 𝐵! = 0 (A.49)

𝜙 0 = 0⇔ 𝐵! + 𝐵! +𝐶! 𝑀 𝑁! + 𝐶!

𝑁! 𝑀 𝑁! 2 + 𝑟02 − 𝐺𝐼𝑡= 0 (A.50)

𝜙!! 0 = 0⇔ −𝜆!!𝐵! + 𝜆!

!𝐵! = 0 (A.51)

(

A1-8

From the above first four equations ((A.48) to (A.51)), it can be readily concluded that

𝐵! = 𝐵! = 𝐶! = 𝐶! = 0. From the last four equations ((A. 52) to (A.55)), it can also be easily

concluded that C1=C3=0. As for B5 and B7, they can be either (i) null, which corresponds to the trivial

solution, or (ii) arbitrary, provided that the following conditions hold,

Finally, since sinh ! !! L = sin !

! L , the expressions providing of the two eigenmodes

of the problem, 𝑢 and 𝜙, read

Introducing now the above eigenmode expressions in equations (A.33) and (A.34), they become

and may be rewritten as

where

𝑣 𝐿 = −𝑀𝐸𝐼!

𝐵! 𝑠𝑖𝑛 𝜆! 𝐿−𝜆!

! − 𝜆!! +

𝐵! 𝑠𝑖𝑛ℎ 𝜆! 𝐿𝜆!

! − 𝜆!! +

𝐶!𝐿 𝐺𝐼! + 𝑁!𝑟!! + 𝐶!𝐿 𝑀𝑀2 − 𝑁! 𝐺𝐼𝑡 + 𝑁!𝑟02

= 0 A.52)

𝑣′′ 𝐿 = −𝑀𝐸𝐼!

−𝜆!! 𝐵! 𝑠𝑖𝑛 𝜆! 𝐿−𝜆!

! − 𝜆!! + 𝜆!

! 𝐵! 𝑠𝑖𝑛ℎ 𝜆! 𝐿𝜆!

! − 𝜆!! = 0 (A.53)

𝜙 𝐿 = 𝐵! 𝑠𝑖𝑛 𝜆!𝐿 + 𝐵! 𝑠𝑖𝑛ℎ 𝜆!𝐿 +𝐶!𝐿 𝑀 𝑁! + 𝐶!𝐿

𝑁! 𝑀 𝑁! 2 + 𝑟02 − 𝐺𝐼𝑡= 0 (A.54)

𝜙′′ 𝐿 = −𝜆!!𝐵! 𝑠𝑖𝑛 𝜆!𝐿 + 𝜆!

!𝐵! 𝑠𝑖𝑛ℎ 𝜆!𝐿 = 0 (A.55)

𝑠𝑖𝑛ℎ 𝜆! 𝐿 = 0 ⇒ 𝜆! =𝑖 𝜋𝐿

(A.56)

𝑠𝑖𝑛 𝜆! 𝐿 = 0 ⇒ 𝜆! =𝜋𝐿

(A.57)

𝑣 𝑥 = 𝐵! 𝑠𝑖𝑛𝜋𝐿 𝑥 (A.58)

𝜙 𝑥 = 𝐵! 𝑠𝑖𝑛𝜋𝐿 𝑥 (A.59)

𝐸𝐼! 𝐵!𝜋𝐿

!𝑠𝑖𝑛

𝜋𝐿 𝑥 −𝑁𝑡 −𝐵!

𝜋𝐿

!𝑠𝑖𝑛

𝜋𝐿 𝑥 +𝑀 − 𝐵!

𝜋𝐿

!𝑠𝑖𝑛

𝜋𝐿 𝑥 = 0 (A.60)

𝐸𝐼! 𝐵!𝜋𝐿

!𝑠𝑖𝑛

𝜋𝐿 𝑥 − 𝐺𝐼! + 𝑁!𝑟!! − 𝐵!

𝜋𝐿

!𝑠𝑖𝑛

𝜋𝐿 𝑥 +𝑀 −𝐵!

𝜋𝐿

!𝑠𝑖𝑛

𝜋𝐿 𝑥 = 0 (A.61)

𝑃!",! +𝑁𝑡 𝑀𝑀 𝑃𝑐𝑟,𝜙 +𝑁𝑡 𝑟!!

𝐵!𝐵!

= 00 (A.62)

A1-9

are the symmetric of the (i) minor-axis flexural and (ii) torsional buckling loads of this member

when subjected to uniform compression only (i.e., treated as a “pure column”). Since B5 and B7 are

both non-null, the solution of the eingenvalue problem defined by equation (A.62) correspond

to the readily obtainable root of the characteristic equation

Taking into account that the solution of equation (A.65) for Nt=0 (i.e., for the case of a

simply supported “pure beam” under uniform bending) is given by

equation (A.65) can be cast in the form

which leads to

The above expression provides the critical buckling moment of a beam subjected to uniform

bending and an axial tension Nt. It is clearly demonstrated that the axial tension leads to a critical

moment increase, thus reducing the beam susceptibility to lateral-torsional buckling.

𝑃!",! = 𝜋!𝐸𝐼!𝐿!

𝑃!",! = 𝜋!𝐸𝐼!𝐿!

+ 𝐺𝐼!1𝑟!!

(A.63)

(A.64)

𝑃!",! + 𝑁! 𝑃!",! + 𝑁! 𝑟!! −𝑀! = 0 (A.65)

𝑀!"(0) = 𝑟! 𝑃!",! ∙ 𝑃!",! =𝜋𝐿∙ 𝐸𝐼! ∙ 𝐺𝐼! ∙ 1 +

𝜋!𝐸𝐼!𝐿!𝐺𝐼!

(A.66)

1 +𝑁!𝑃!",!

1 +𝑁!𝑃!",!

=𝑀!

𝑀!"(0) ! (A.67)

𝑀!"(𝑁!) = 1 +𝑁!𝑃!",!

1 +𝑁!𝑃!",!

𝑀!"(0) (A.68)

A2 -‐ 1

Annex 2: Numerical Data: Critical Moments, Ultimate Moment Values

and Ultimate Moment Estimates

This Annex presents the (i) numerical critical buckling moments and ultimate moments obtained from

the parametric studies reported in Chapters 2 and 4, respectively, and (ii) the ultimate moment

estimates provided by the design approach proposed in Chapter 5. The numerical data are included in

four distinct sections, each of them concerning beams exhibiting the same cross-section: IPE 300

(section A2.1), IPE 500 (section A2.2) HEA 300 (section A2.3) and HEA 500 (section A2.4).

In the above sections, the information is provided in different tables, one per combination of

beam geometry (cross-section shape and length), steel yield stress and bending moment diagram. In

each table and for each β value considered in this work, the results displayed are the beam (i)

critical moment Mcr (accounting for the axial tension) (ii) plastic moment Mpl,Rk, (iii) reduced (by the

axial tension) plastic bending resistance MN,Rk, (iv) numerical ultimate moment Mu, (v) lateral-torsional

slenderness λLT (based on Mcr(Nt) and Mpl,Rk), (vi) reduction factor χLT.mod (obtained with the EC3-1-1

curve b), (vii) ultimate moment prediction Mb,Rd (provided by the EC3-1-1 curve b) and (viii)

numerical-to-estimated ultimate moment ratio RM=Mu /Mu.est, where Mu.est is the lower between MN,Rk

and Mb,Rd. When Mu.est=MN,Rk (mostly shorter beams), it should be noted that some RM values exceed 1.0

(up to 15%) − this somewhat surprising fact is due to the small strain-hardening exhibited by the

employed steel constitutive law model, which leads to some degree of cross-section over-strength.

A2 -‐ 2

A2.1. Proposed ultimate moment estimates and design results - IPE300 beams

A2.3

β Mcr [kNm] Mpl,Rk [kNm] MN,Rk [kNm] Mu [kNm] λLT XLT Mb,Rd [kNm] Mu,est [kNm] RM

0 6612,8 223,2 223,2 244,2 0,184 1,000 223,2 223,2 1,090,5 7067,0 223,2 222,0 242,9 0,178 1,000 223,2 222,0 1,08

0,75 7235,8 223,2 220,2 241,0 0,176 1,000 223,2 220,2 1,071 7582,8 223,2 218,4 239,1 0,172 1,000 223,2 218,4 1,09

1,5 8172,2 223,2 213,1 233,1 0,165 1,000 223,2 213,1 1,102 8850,1 223,2 206,3 225,7 0,159 1,000 223,2 206,3 1,09


0 8705,8 223,2 223,2 224,3 0,160 1,000 223,2 223,2 1,010,5 9497,9 223,2 222,0 228,8 0,153 1,000 223,2 222,0 1,03

0,75 9876,5 223,2 220,2 230,2 0,150 1,000 223,2 220,2 1,051 10431,9 223,2 218,4 235,7 0,146 1,000 223,2 218,4 1,08

1,5 11543,9 223,2 213,1 236,5 0,139 1,000 223,2 213,1 1,112 12880,6 223,2 206,3 237,2 0,132 1,000 223,2 206,3 1,15


0 11953,3 223,2 223,2 202,5 0,137 1,000 223,2 223,2 0,910,5 13358,3 223,2 222,0 204,5 0,129 1,000 223,2 222,0 0,92

0,75 12345,8 223,2 220,2 208,9 0,134 1,000 223,2 220,2 0,951 15059,5 223,2 218,4 216,6 0,122 1,000 223,2 218,4 0,99

1,5 17130,8 223,2 213,1 219,5 0,114 1,000 223,2 213,1 1,032 19657,0 223,2 206,3 223,7 0,107 1,000 223,2 206,3 1,08


0 15755,0 223,2 223,2 197,5 0,119 1,000 223,2 223,2 0,880,5 17699,5 223,2 222,0 203,9 0,112 1,000 223,2 222,0 0,92

0,75 18563,8 223,2 220,2 206,7 0,110 1,000 223,2 220,2 0,941 20004,9 223,2 218,4 207,3 0,106 1,000 223,2 218,4 0,95

1,5 22738,7 223,2 213,1 221,2 0,099 1,000 223,2 213,1 1,042 25971,4 223,2 206,3 228,1 0,093 1,000 223,2 206,3 1,11


0 15755,0 223,2 223,2 197,5 0,119 1,000 223,2 223,2 0,880,5 17699,5 223,2 222,0 203,9 0,112 1,000 223,2 222,0 0,92

0,75 18563,8 223,2 220,2 206,7 0,110 1,000 223,2 220,2 0,941 20004,9 223,2 218,4 207,3 0,106 1,000 223,2 218,4 0,95

1,5 22738,7 223,2 213,1 221,2 0,099 1,000 223,2 213,1 1,042 25971,4 223,2 206,3 228,1 0,093 1,000 223,2 206,3 1,11

IPE300 / L = 0.5 m / fy = 355 MPa

ψ = -‐0.5

ψ = 0

ψ = 1

ψ = 0.5

ψ = -‐1


A2.4


0 6612,8 289,2 289,2 317,0 0,209 1,000 289,2 289,2 1,100,5 7067,0 289,2 287,6 315,3 0,202 1,000 289,2 287,6 1,11

0,75 7235,8 289,2 285,3 312,8 0,200 1,000 289,2 285,3 1,081 7582,8 289,2 283,1 310,3 0,195 1,000 289,2 283,1 1,09

1,5 8172,2 289,2 276,1 302,6 0,188 1,000 289,2 276,1 1,102 8850,1 289,2 267,3 293,0 0,181 1,000 289,2 267,3 1,10


0 8705,8 289,2 289,2 289,8 0,182 1,000 289,2 289,2 1,000,5 9497,9 289,2 287,6 285,3 0,174 1,000 289,2 287,6 0,99

0,75 9876,5 289,2 285,3 288,4 0,171 1,000 289,2 285,3 1,011 10431,9 289,2 283,1 289,8 0,166 1,000 289,2 283,1 1,02

1,5 11543,9 289,2 276,1 286,3 0,158 1,000 289,2 276,1 1,042 12880,6 289,2 267,3 282,3 0,150 1,000 289,2 267,3 1,06


0 11953,3 289,2 289,2 267,3 0,156 1,000 289,2 289,2 0,920,5 13358,3 289,2 287,6 265,4 0,147 1,000 289,2 287,6 0,92

0,75 12345,8 289,2 285,3 262,5 0,153 1,000 289,2 285,3 0,921 15059,5 289,2 283,1 277,3 0,139 1,000 289,2 283,1 0,98

1,5 17130,8 289,2 276,1 281,9 0,130 1,000 289,2 276,1 1,022 19657,0 289,2 267,3 275,5 0,121 1,000 289,2 267,3 1,03


0 15755,0 289,2 289,2 263,6 0,135 1,000 289,2 289,2 0,910,5 17699,5 289,2 287,6 266,4 0,128 1,000 289,2 287,6 0,93

0,75 18563,8 289,2 285,3 273,6 0,125 1,000 289,2 285,3 0,961 20004,9 289,2 283,1 277,4 0,120 1,000 289,2 283,1 0,98

1,5 22738,7 289,2 276,1 278,3 0,113 1,000 289,2 276,1 1,012 25971,4 289,2 267,3 276,6 0,106 1,000 289,2 267,3 1,03


0 15443,0 289,2 289,2 280,6 0,137 1,000 289,2 289,2 0,970,5 16967,6 289,2 287,6 290,5 0,131 1,000 289,2 287,6 1,01

0,75 1737,0 289,2 285,3 294,5 0,408 1,000 289,2 285,3 1,031 18731,3 289,2 283,1 288,6 0,124 1,000 289,2 283,1 1,02

1,5 20777,1 289,2 276,1 285,4 0,118 1,000 289,2 276,1 1,032 23153,1 289,2 267,3 274,4 0,112 1,000 289,2 267,3 1,03

ψ = -‐1

ψ = 1

ψ = 0.5

ψ = 0

ψ = -‐0.5

IPE300 / L = 0.5 m / fy = 460 MPa


A2.5


0 1798,2 223,2 223,2 211,6 0,352 1,000 223,2 223,2 0,950,5 1865,8 223,2 222,0 213,9 0,346 1,000 223,2 222,0 0,960,75 1922,7 223,2 220,2 216,9 0,341 1,000 223,2 220,2 0,98

1 2065,3 223,2 218,4 219,6 0,329 1,000 223,2 218,4 1,011,5 2230,0 223,2 213,1 221,5 0,316 1,000 223,2 213,1 1,042 2422,1 223,2 206,3 222,0 0,304 1,000 223,2 206,3 1,08


0 2373,5 223,2 223,2 240,0 0,307 1,000 223,2 223,2 1,080,5 2592,7 223,2 222,0 242,5 0,293 1,000 223,2 222,0 1,09

0,75 2454,7 223,2 220,2 241,0 0,302 1,000 223,2 220,2 1,091 2854,5 223,2 218,4 237,3 0,280 1,000 223,2 218,4 1,09

1,5 3172,0 223,2 213,1 236,4 0,265 1,000 223,2 213,1 1,112 3563,8 223,2 206,3 229,7 0,250 1,000 223,2 206,3 1,11


0 3312,5 223,2 223,2 235,7 0,260 1,000 223,2 223,2 1,060,5 3720,3 223,2 222,0 239,6 0,245 1,000 223,2 222,0 1,08

0,75 3976,2 223,2 220,2 242,9 0,237 1,000 223,2 220,2 1,101 4227,3 223,2 218,4 246,4 0,230 1,000 223,2 218,4 1,13

1,5 4867,8 223,2 213,1 245,4 0,214 1,000 223,2 213,1 1,152 5690,4 223,2 206,3 237,5 0,198 1,000 223,2 206,3 1,15


0 4585,6 223,2 223,2 203,3 0,221 1,000 223,2 223,2 0,910,5 5206,6 223,2 222,0 205,8 0,207 1,000 223,2 222,0 0,93

0,75 5476,6 223,2 220,2 206,8 0,202 1,000 223,2 220,2 0,941 5966,8 223,2 218,4 207,8 0,193 1,000 223,2 218,4 0,95

1,5 6907,7 223,2 213,1 208,3 0,180 1,000 223,2 213,1 0,982 8085,1 223,2 206,3 208,5 0,166 1,000 223,2 206,3 1,01


0 4710,3 223,2 223,2 213,1 0,218 1,000 223,2 223,2 0,950,5 5212,2 223,2 222,0 211,6 0,207 1,000 223,2 222,0 0,95

0,75 5342,9 223,2 220,2 212,5 0,204 1,000 223,2 220,2 0,971 5808,9 223,2 218,4 206,5 0,196 1,000 223,2 218,4 0,95

1,5 6526,5 223,2 213,1 207,6 0,185 1,000 223,2 213,1 0,972 7400,3 223,2 206,3 204,1 0,174 1,000 223,2 206,3 0,99

IPE300 / L = 1 m / fy = 355 MPa

ψ = 0

ψ = 1

ψ = 0.5

ψ = -‐0.5

ψ = -‐ 1


A2.6


0 1798,2 289,2 289,2 301,3 0,401 1,000 289,2 289,2 1,040,5 1922,7 289,2 287,6 299,6 0,388 1,000 289,2 287,6 1,03

0,75 1865,8 289,2 285,3 297,3 0,394 1,000 289,2 285,3 1,051 2065,3 289,2 283,1 294,9 0,374 1,000 289,2 283,1 1,04

1,5 2230,0 289,2 276,1 287,6 0,360 1,000 289,2 276,1 1,042 2422,1 289,2 267,3 278,5 0,346 1,000 289,2 267,3 1,05


0 2373,5 289,2 289,2 304,8 0,349 1,000 289,2 289,2 1,050,5 2592,7 289,2 287,6 309,6 0,334 1,000 289,2 287,6 1,08

0,75 2454,7 289,2 285,3 310,0 0,343 1,000 289,2 285,3 1,091 2854,5 289,2 283,1 307,3 0,318 1,000 289,2 283,1 1,09

1,5 3172,0 289,2 276,1 305,1 0,302 1,000 289,2 276,1 1,102 3563,8 289,2 267,3 297,3 0,285 1,000 289,2 267,3 1,11


0 3312,5 289,2 289,2 286,2 0,295 1,000 289,2 289,2 0,990,5 3720,3 289,2 287,6 291,8 0,279 1,000 289,2 287,6 1,01

0,75 3976,2 289,2 285,3 293,5 0,270 1,000 289,2 285,3 1,031 4227,3 289,2 283,1 294,7 0,262 1,000 289,2 283,1 1,04

1,5 4867,8 289,2 276,1 292,7 0,244 1,000 289,2 276,1 1,062 5690,4 289,2 267,3 288,2 0,225 1,000 289,2 267,3 1,08


0 4585,6 289,2 289,2 270,7 0,251 1,000 289,2 289,2 0,940,5 5206,6 289,2 287,6 264,6 0,236 1,000 289,2 287,6 0,92

0,75 5476,6 289,2 285,3 249,1 0,230 1,000 289,2 285,3 0,871 5966,8 289,2 283,1 250,8 0,220 1,000 289,2 283,1 0,89

1,5 6907,7 289,2 276,1 251,0 0,205 1,000 289,2 276,1 0,912 8085,1 289,2 267,3 251,5 0,189 1,000 289,2 267,3 0,94


0 4710,3 289,2 289,2 291,5 0,248 1,000 289,2 289,2 1,010,5 5212,2 289,2 287,6 265,3 0,236 1,000 289,2 287,6 0,92

0,75 5342,9 289,2 285,3 268,2 0,233 1,000 289,2 285,3 0,941 5808,9 289,2 283,1 242,9 0,223 1,000 289,2 283,1 0,86

1,5 6526,5 289,2 276,1 259,3 0,211 1,000 289,2 276,1 0,942 7400,3 289,2 267,3 252,9 0,198 1,000 289,2 267,3 0,95

IPE300 / L = 1 m / fy = 460 MPaψ = 1

ψ = 0.5

ψ = 0

ψ = -‐0.5

ψ = -‐ 1


A2.7


0 494,9 223,2 223,2 187,6 0,672 0,884 197,2 197,2 0,950,5 529,7 223,2 222,0 197,7 0,649 0,894 199,6 199,6 0,99

0,75 535,1 223,2 220,2 201,3 0,646 0,896 200,0 200,0 1,011 569,7 223,2 218,4 203,6 0,626 0,905 202,0 202,0 1,01

1,5 616,0 223,2 213,1 206,2 0,602 0,916 204,5 204,5 1,012 670,2 223,2 206,3 206,4 0,577 0,927 207,0 206,3 1,00


0 653,5 223,2 223,2 228,3 0,584 0,988 220,4 220,4 1,040,5 714,9 223,2 222,0 231,3 0,559 1,000 223,2 222,0 1,04

0,75 759,3 223,2 220,2 231,2 0,542 1,000 223,2 220,2 1,051 788,4 223,2 218,4 231,7 0,532 1,000 223,2 218,4 1,06

1,5 878,0 223,2 213,1 232,5 0,504 1,000 223,2 213,1 1,092 989,4 223,2 206,3 224,6 0,475 1,000 223,2 206,3 1,09


0 914,8 223,2 223,2 238,2 0,494 1,000 223,2 223,2 1,070,5 1029,5 223,2 222,0 243,8 0,466 1,000 223,2 222,0 1,10

0,75 1081,8 223,2 220,2 243,1 0,454 1,000 223,2 220,2 1,101 1172,7 223,2 218,4 245,6 0,436 1,000 223,2 218,4 1,12

1,5 1355,1 223,2 213,1 244,9 0,406 1,000 223,2 213,1 1,152 1592,3 223,2 206,3 235,8 0,374 1,000 223,2 206,3 1,14


0 1281,5 223,2 223,2 236,6 0,417 1,000 223,2 223,2 1,060,5 1457,7 223,2 222,0 225,6 0,391 1,000 223,2 222,0 1,02

0,75 1580,3 223,2 220,2 230,2 0,376 1,000 223,2 220,2 1,051 1674,5 223,2 218,4 224,3 0,365 1,000 223,2 218,4 1,03

1,5 1945,6 223,2 213,1 222,5 0,339 1,000 223,2 213,1 1,042 2291,5 223,2 206,3 229,7 0,312 1,000 223,2 206,3 1,11


0 1339,5 223,2 223,2 242,2 0,408 1,000 223,2 223,2 1,090,5 1480,7 223,2 222,0 248,2 0,388 1,000 223,2 222,0 1,12

0,75 1532,6 223,2 220,2 252,0 0,382 1,000 223,2 220,2 1,141 1650,1 223,2 218,4 243,2 0,368 1,000 223,2 218,4 1,11

1,5 1856,6 223,2 213,1 234,1 0,347 1,000 223,2 213,1 1,102 2113,2 223,2 206,3 227,0 0,325 1,000 223,2 206,3 1,10

IPE300 / L = 2 m / fy = 355 MPaψ = 1

ψ = 0.5

ψ = 0

ψ = -‐0.5

ψ = -‐1


A2.8


0 494,9 289,2 289,2 263,2 0,764 0,836 241,9 241,9 1,090,5 529,7 289,2 287,6 266,4 0,739 0,850 245,8 245,8 1,08

0,75 535,1 289,2 285,3 266,9 0,735 0,852 246,3 246,3 1,081 569,7 289,2 283,1 269,7 0,712 0,863 249,7 249,7 1,08

1,5 616,0 289,2 276,1 272,1 0,685 0,877 253,6 253,6 1,072 670,2 289,2 267,3 276,1 0,657 0,891 257,6 257,6 1,07


0 653,5 289,2 289,2 287,8 0,665 0,952 275,2 275,2 1,050,5 714,9 289,2 287,6 297,2 0,636 0,965 279,1 279,1 1,06

0,75 759,3 289,2 285,3 298,0 0,617 0,974 281,6 281,6 1,061 788,4 289,2 283,1 297,3 0,606 0,979 283,0 283,0 1,05

1,5 878,0 289,2 276,1 296,7 0,574 1,000 289,2 276,1 1,072 989,4 289,2 267,3 289,8 0,541 1,000 289,2 267,3 1,08


0 914,8 289,2 289,2 290,7 0,562 1,000 289,2 289,2 1,010,5 1029,5 289,2 287,6 271,6 0,530 1,000 289,2 287,6 0,94

0,75 1081,8 289,2 285,3 293,2 0,517 1,000 289,2 285,3 1,031 1172,7 289,2 283,1 314,6 0,497 1,000 289,2 283,1 1,11

1,5 1355,1 289,2 276,1 313,5 0,462 1,000 289,2 276,1 1,142 1592,3 289,2 267,3 308,6 0,426 1,000 289,2 267,3 1,15


0 1281,5 289,2 289,2 305,5 0,475 1,000 289,2 289,2 1,060,5 1457,7 289,2 287,6 309,2 0,445 1,000 289,2 287,6 1,07

0,75 1580,3 289,2 285,3 311,2 0,428 1,000 289,2 285,3 1,091 1674,5 289,2 283,1 314,2 0,416 1,000 289,2 283,1 1,11

1,5 1945,6 289,2 276,1 319,9 0,386 1,000 289,2 276,1 1,162 2291,5 289,2 267,3 313,1 0,355 1,000 289,2 267,3 1,17


0 1339,5 289,2 289,2 294,0 0,465 1,000 289,2 289,2 1,020,5 1480,7 289,2 287,6 298,0 0,442 1,000 289,2 287,6 1,04

0,75 1532,6 289,2 285,3 299,6 0,434 1,000 289,2 285,3 1,051 1650,1 289,2 283,1 300,1 0,419 1,000 289,2 283,1 1,06

1,5 1856,6 289,2 276,1 300,7 0,395 1,000 289,2 276,1 1,092 2113,2 289,2 267,3 296,2 0,370 1,000 289,2 267,3 1,11

IPE300 / L = 2 m / fy = 460 MPaψ = 1

ψ = 0.5

ψ = 0

ψ = -‐0.5

ψ = -‐ 1


A2.9


0 192,2 223,2 223,2 108,7 1,433 0,457 102,0 102,0 1,070,5 206,4 223,2 222,0 113,0 1,405 0,470 104,9 104,9 1,08

0,75 216,2 223,2 220,2 116,9 1,382 0,482 107,5 107,5 1,091 222,7 223,2 218,4 118,3 1,374 0,486 108,4 108,4 1,09

1,5 241,6 223,2 213,1 120,2 1,363 0,491 109,6 109,6 1,102 263,8 223,2 206,3 121,4 1,356 0,494 110,4 110,4 1,10


0 253,7 223,2 223,2 181,5 0,938 0,791 176,6 176,6 1,030,5 278,6 223,2 222,0 190,9 0,895 0,820 183,0 183,0 1,04

0,75 292,3 223,2 220,2 200,3 0,874 0,834 186,1 186,1 1,081 308,5 223,2 218,4 204,0 0,851 0,849 189,4 189,4 1,08

1,5 344,9 223,2 213,1 209,3 0,804 0,877 195,7 195,7 1,072 390,2 223,2 206,3 210,2 0,756 0,905 201,9 201,9 1,04


0 354,2 223,2 223,2 219,8 0,794 0,957 213,5 213,5 1,030,5 400,0 223,2 222,0 229,5 0,747 0,985 219,9 219,9 1,04

0,75 433,7 223,2 220,2 230,8 0,717 1,000 223,2 220,2 1,051 456,7 223,2 218,4 230,9 0,699 1,000 223,2 218,4 1,06

1,5 528,3 223,2 213,1 232,6 0,650 1,000 223,2 213,1 1,092 620,5 223,2 206,3 236,6 0,600 1,000 223,2 206,3 1,15


0 494,7 223,2 223,2 233,7 0,672 1,000 223,2 223,2 1,050,5 562,3 223,2 222,0 237,5 0,630 1,000 223,2 222,0 1,07

0,75 592,2 223,2 220,2 239,6 0,614 1,000 223,2 220,2 1,091 644,0 223,2 218,4 241,9 0,589 1,000 223,2 218,4 1,11

1,5 745,0 223,2 213,1 246,8 0,547 1,000 223,2 213,1 1,162 873,1 223,2 206,3 255,5 0,506 1,000 223,2 206,3 1,24


0 522,6 223,2 223,2 228,8 0,653 1,000 223,2 223,2 1,030,5 574,2 223,2 222,0 231,0 0,623 1,000 223,2 222,0 1,04

0,75 597,4 223,2 220,2 235,4 0,611 1,000 223,2 220,2 1,071 636,3 223,2 218,4 226,2 0,592 1,000 223,2 218,4 1,04

1,5 712,1 223,2 213,1 231,8 0,560 1,000 223,2 213,1 1,092 806,6 223,2 206,3 218,0 0,526 1,000 223,2 206,3 1,06

IPE300 / L = 3.5 m / fy = 355 MPa

ψ = -‐0.5

ψ = -‐ 1

ψ = 1

ψ = 0.5

ψ = 0


A2.10


0 192,2 289,2 289,2 162,0 1,227 0,564 163,1 163,1 0,990,5 206,4 289,2 287,6 173,8 1,184 0,589 170,2 170,2 1,02

0,75 216,2 289,2 285,3 179,4 1,157 0,605 174,9 174,9 1,031 222,7 289,2 283,1 192,1 1,140 0,615 177,8 177,8 1,08

1,5 241,6 289,2 276,1 211,6 1,094 0,642 185,7 185,7 1,142 263,8 289,2 267,3 225,1 1,047 0,671 194,0 194,0 1,16


0 253,7 289,2 289,2 203,3 1,068 0,701 202,6 202,6 1,000,5 278,6 289,2 287,6 213,4 1,019 0,735 212,6 212,6 1,00

0,75 292,3 289,2 285,3 224,9 0,995 0,752 217,5 217,5 1,031 308,5 289,2 283,1 230,3 0,968 0,771 222,8 222,8 1,03

1,5 344,9 289,2 276,1 234,8 0,916 0,806 233,2 233,2 1,012 390,2 289,2 267,3 247,2 0,861 0,842 243,5 243,5 1,02


0 354,2 289,2 289,2 261,8 0,904 0,881 254,7 254,7 1,030,5 400,0 289,2 287,6 284,6 0,850 0,919 265,8 265,8 1,07

0,75 433,7 289,2 285,3 292,1 0,817 0,942 272,4 272,4 1,071 456,7 289,2 283,1 295,6 0,796 0,956 276,3 276,3 1,07

1,5 528,3 289,2 276,1 295,4 0,740 0,989 286,1 276,1 1,072 620,5 289,2 267,3 298,6 0,683 1,000 289,2 267,3 1,12


0 494,7 289,2 289,2 298,8 0,765 1,000 289,2 289,2 1,030,5 562,3 289,2 287,6 303,0 0,717 1,000 289,2 287,6 1,05

0,75 592,2 289,2 285,3 304,2 0,699 1,000 289,2 285,3 1,071 644,0 289,2 283,1 306,2 0,670 1,000 289,2 283,1 1,08

1,5 745,0 289,2 276,1 300,8 0,623 1,000 289,2 276,1 1,092 873,1 289,2 267,3 298,0 0,576 1,000 289,2 267,3 1,11


0 522,6 289,2 289,7 292,9 0,744 1,000 289,2 289,2 1,010,5 574,2 289,2 294,4 294,9 0,710 1,000 289,2 289,2 1,02

0,75 597,4 289,2 297,8 293,5 0,696 1,000 289,2 289,2 1,011 636,3 289,2 301,9 297,8 0,674 1,000 289,2 289,2 1,03

1,5 712,1 289,2 309,2 295,5 0,637 1,000 289,2 289,2 1,022 806,6 289,2 309,4 295,7 0,599 1,000 289,2 289,2 1,02

IPE300 / L = 3.5 m / fy = 460 MPaψ = 1

ψ = 0.5

ψ = -‐0.5

ψ = -‐ 1

ψ = 0


A2.11


0 113,3 223,2 223,2 110,7 1,404 0,471 105,1 105,1 1,050,5 122,4 223,2 222,0 120,1 1,351 0,497 111,0 111,0 1,080,75 125,4 223,2 220,2 122,9 1,334 0,506 112,9 112,9 1,09

1 132,8 223,2 218,4 127,3 1,296 0,526 117,3 117,3 1,081,5 144,9 223,2 213,1 135,6 1,241 0,556 124,1 124,1 1,092 159,1 223,2 206,3 144,0 1,184 0,588 131,3 131,3 1,10


0 149,4 223,2 223,2 135,0 1,222 0,594 132,5 132,5 1,020,5 165,3 223,2 222,0 149,9 1,162 0,635 141,6 141,6 1,06

0,75 173,9 223,2 220,2 156,6 1,133 0,655 146,2 146,2 1,071 184,3 223,2 218,4 158,0 1,100 0,678 151,2 151,2 1,04

1,5 207,5 223,2 213,1 168,9 1,037 0,722 161,2 161,2 1,052 236,2 223,2 206,3 174,7 0,972 0,768 171,4 171,4 1,02


0 207,8 223,2 223,2 177,6 1,036 0,775 173,1 173,1 1,030,5 236,2 223,2 222,0 198,6 0,972 0,828 184,7 184,7 1,07

0,75 255,4 223,2 220,2 205,4 0,935 0,857 191,2 191,2 1,071 271,1 223,2 218,4 210,7 0,907 0,878 195,9 195,9 1,08

1,5 314,4 223,2 213,1 213,7 0,843 0,924 206,3 206,3 1,042 369,1 223,2 206,3 215,4 0,778 0,967 215,8 206,3 1,04


0 288,3 223,2 223,2 216,1 0,880 0,923 205,9 205,9 1,050,5 328,2 223,2 222,0 224,9 0,825 0,963 214,8 214,8 1,05

0,75 352,1 223,2 220,2 226,6 0,796 0,982 219,1 219,1 1,031 375,5 223,2 218,4 227,8 0,771 0,998 222,7 218,4 1,04

1,5 432,9 223,2 213,1 230,5 0,718 1,000 223,2 213,1 1,082 504,8 223,2 206,3 216,7 0,665 1,000 223,2 206,3 1,05


0 307,2 223,2 223,2 227,9 0,852 0,982 219,1 219,1 1,040,5 336,2 223,2 222,0 232,8 0,815 1,000 223,2 222,0 1,05

0,75 344,8 223,2 220,2 233,1 0,804 1,000 223,2 220,2 1,061 371,0 223,2 218,4 233,7 0,776 1,000 223,2 218,4 1,07

1,5 413,5 223,2 213,1 235,2 0,735 1,000 223,2 213,1 1,102 466,4 223,2 206,3 238,2 0,692 1,000 223,2 206,3 1,15

IPE300 / L = 5 m / fy = 355 MPaψ = 1

ψ = 0.5

ψ = 0

ψ = -‐1

ψ = -‐0.5


A2.12


0 113,3 289,2 289,2 108,8 1,598 0,388 112,1 112,1 0,970,5 122,4 289,2 287,6 116,7 1,537 0,412 119,0 119,0 0,98

0,75 125,4 289,2 285,3 120,5 1,518 0,419 121,3 121,3 0,991 132,8 289,2 283,1 129,0 1,476 0,438 126,6 126,6 1,02

1,5 144,9 289,2 276,1 142,2 1,413 0,467 135,0 135,0 1,052 159,1 289,2 267,3 161,7 1,348 0,499 144,2 144,2 1,12


0 149,4 289,2 289,2 142,8 1,391 0,487 141,0 141,0 1,010,5 165,3 289,2 287,6 145,4 1,323 0,529 152,9 152,9 0,95

0,75 173,9 289,2 285,3 172,9 1,289 0,550 158,9 158,9 1,091 184,3 289,2 283,1 187,2 1,253 0,573 165,8 165,8 1,13

1,5 207,5 289,2 276,1 201,1 1,181 0,622 179,8 179,8 1,122 236,2 289,2 267,3 221,8 1,107 0,673 194,7 194,7 1,14


0 207,8 289,2 289,2 194,0 1,180 0,658 190,2 190,2 1,020,5 236,2 289,2 287,6 216,1 1,106 0,718 207,6 207,6 1,04

0,75 255,4 289,2 285,3 226,3 1,064 0,753 217,7 217,7 1,041 271,1 289,2 283,1 243,0 1,033 0,778 225,1 225,1 1,08

1,5 314,4 289,2 276,1 256,8 0,959 0,838 242,3 242,3 1,062 369,1 289,2 267,3 267,1 0,885 0,894 258,6 258,6 1,03


0 288,3 289,2 289,2 226,0 1,002 0,824 238,3 238,3 0,950,5 328,2 289,2 287,6 255,4 0,939 0,876 253,5 253,5 1,01

0,75 352,1 289,2 285,3 261,8 0,906 0,902 260,9 260,9 1,001 375,5 289,2 283,1 260,5 0,878 0,924 267,3 267,3 0,97

1,5 432,9 289,2 276,1 274,7 0,817 0,968 279,8 276,1 0,992 504,8 289,2 267,3 268,1 0,757 1,000 289,2 267,3 1,00


0 307,2 289,2 289,2 246,9 0,970 0,883 255,4 255,4 0,970,5 336,2 289,2 287,6 267,8 0,927 0,921 266,3 266,3 1,01

0,75 344,8 289,2 285,3 285,0 0,916 0,931 269,2 269,2 1,061 371,0 289,2 283,1 285,7 0,883 0,958 277,0 277,0 1,03

1,5 413,5 289,2 276,1 278,9 0,836 0,994 287,4 276,1 1,012 466,4 289,2 267,3 268,5 0,787 1,000 289,2 267,3 1,00

IPE300 / L = 5 m / fy = 460 MPaψ = 1

ψ = 0.5

ψ = 0

ψ = -‐1

ψ = -‐0.5


A2.13


0 61,7 223,2 223,2 59,0 1,902 0,291 65,0 65,0 0,910,5 67,8 223,2 222,0 64,5 1,815 0,315 70,4 70,4 0,920,75 68,9 223,2 220,2 65,5 1,800 0,320 71,3 71,3 0,92

1 74,8 223,2 218,4 72,8 1,727 0,342 76,3 76,3 0,951,5 83,0 223,2 213,1 85,7 1,640 0,372 83,0 83,0 1,032 92,6 223,2 206,3 104,9 1,552 0,405 90,5 90,5 1,16


0 81,2 223,2 223,2 86,2 1,657 0,366 81,6 81,6 1,060,5 91,8 223,2 222,0 92,3 1,559 0,403 89,9 89,9 1,030,75 96,1 223,2 220,2 101,6 1,524 0,417 93,1 93,1 1,09

1 104,5 223,2 218,4 109,4 1,462 0,448 100,0 100,0 1,091,5 119,8 223,2 213,1 118,5 1,365 0,503 112,3 112,3 1,062 138,7 223,2 206,3 131,7 1,268 0,563 125,7 125,7 1,05


0 112,1 223,2 223,2 112,2 1,411 0,485 108,2 108,2 1,040,5 130,1 223,2 222,0 122,1 1,310 0,556 124,2 124,2 0,980,75 139,8 223,2 220,2 130,3 1,264 0,591 132,0 132,0 0,99

1 151,6 223,2 218,4 146,7 1,213 0,631 140,8 140,8 1,041,5 177,4 223,2 213,1 165,6 1,122 0,705 157,4 157,4 1,052 208,9 223,2 206,3 186,5 1,034 0,778 173,6 173,6 1,07


0 153,4 223,2 223,2 149,3 1,206 0,647 144,5 144,5 1,030,5 176,7 223,2 222,0 172,7 1,124 0,718 160,3 160,3 1,08

0,75 188,2 223,2 220,2 182,4 1,089 0,749 167,1 167,1 1,091 203,0 223,2 218,4 191,4 1,049 0,784 174,9 174,9 1,09

1,5 233,9 223,2 213,1 205,2 0,977 0,845 188,6 188,6 1,092 271,9 223,2 206,3 213,0 0,906 0,902 201,4 201,4 1,06


0 165,7 223,2 223,2 159,2 1,161 0,706 157,6 157,6 1,010,5 181,3 223,2 222,0 175,4 1,110 0,754 168,3 168,3 1,04

0,75 187,3 223,2 220,2 182,0 1,091 0,771 172,1 172,1 1,061 199,9 223,2 218,4 192,1 1,057 0,804 179,4 179,4 1,07

1,5 222,6 223,2 213,1 203,0 1,001 0,855 190,8 190,8 1,062 250,7 223,2 206,3 217,5 0,943 0,907 202,4 202,4 1,07

IPE300 / L = 8 m / fy = 355 MPaψ = 1

ψ = 0.5

ψ = 0

ψ = -‐ 1

ψ = -‐0.5


A2.14


0 61,7 289,2 289,2 62,0 2,165 0,210 60,7 60,7 1,020,5 67,8 289,2 287,6 67,8 2,066 0,240 69,4 69,4 0,980,75 68,9 289,2 285,3 68,8 2,048 0,256 74,1 74,1 0,93

1 74,8 289,2 283,1 76,5 1,966 0,275 79,6 79,6 0,961,5 83,0 289,2 276,1 90,1 1,867 0,301 86,9 86,9 1,042 92,6 289,2 267,3 110,2 1,767 0,340 98,3 98,3 1,12


0 81,2 289,2 289,2 85,5 1,887 0,295 85,4 85,4 1,000,5 91,8 289,2 287,6 96,5 1,775 0,327 94,6 94,6 1,020,75 96,1 289,2 285,3 103,5 1,735 0,340 98,2 98,2 1,05

1 104,5 289,2 283,1 112,1 1,664 0,364 105,1 105,1 1,071,5 119,8 289,2 276,1 120,4 1,553 0,405 117,1 117,1 1,032 138,7 289,2 267,3 127,6 1,444 0,458 132,4 132,4 0,96


0 112,1 289,2 289,2 112,9 1,607 0,384 111,1 111,1 1,020,5 130,1 289,2 287,6 131,6 1,491 0,434 125,5 125,5 1,050,75 139,8 289,2 285,3 137,5 1,438 0,467 135,1 135,1 1,02

1 151,6 289,2 283,1 150,3 1,381 0,505 146,1 146,1 1,031,5 177,4 289,2 276,1 167,8 1,277 0,581 168,1 168,1 1,002 208,9 289,2 267,3 191,6 1,177 0,660 190,9 190,9 1,00


0 153,4 289,2 289,2 153,5 1,373 0,515 149,0 149,0 1,030,5 176,7 289,2 287,6 177,2 1,279 0,587 169,8 169,8 1,04

0,75 188,2 289,2 285,3 187,0 1,239 0,620 179,2 179,2 1,041 203,0 289,2 283,1 194,0 1,194 0,658 190,3 190,3 1,02

1,5 233,9 289,2 276,1 206,6 1,112 0,729 210,7 210,7 0,982 271,9 289,2 267,3 214,0 1,031 0,798 230,9 230,9 0,93


0 165,7 289,2 289,2 166,6 1,321 0,564 163,0 163,0 1,020,5 181,3 289,2 287,6 184,7 1,263 0,613 177,4 177,4 1,04

0,75 187,3 289,2 285,3 199,4 1,242 0,631 182,6 182,6 1,091 199,9 289,2 283,1 206,6 1,203 0,667 193,0 193,0 1,07

1,5 222,6 289,2 276,1 222,0 1,140 0,726 209,8 209,8 1,062 250,7 289,2 267,3 241,9 1,074 0,787 227,7 227,7 1,06

ψ = 1

ψ = 0.5

ψ = 0

ψ =-‐ 0.5

ψ = -‐1

IPE300 / L = 8 m / fy = 460 MPa


A2.15


0 47,8 223,2 223,2 48,1 2,161 0,234 52,1 52,1 0,920,5 52,9 223,2 222,0 53,9 2,054 0,255 57,0 57,0 0,950,75 55,0 223,2 220,2 59,3 2,015 0,264 58,9 58,9 1,01

1 59,3 223,2 218,4 63,4 1,940 0,282 62,9 62,9 1,011,5 66,7 223,2 213,1 69,9 1,830 0,311 69,4 69,4 1,012 75,4 223,2 206,3 81,9 1,721 0,344 76,8 76,8 1,07


0 62,6 223,2 223,2 69,5 1,888 0,295 65,8 65,8 1,060,5 72,0 223,2 222,0 77,7 1,760 0,332 74,0 74,0 1,050,75 78,6 223,2 220,2 82,4 1,685 0,356 79,5 79,5 1,04

1 83,3 223,2 218,4 88,3 1,637 0,373 83,3 83,3 1,061,5 96,9 223,2 213,1 101,9 1,518 0,420 93,7 93,7 1,092 113,5 223,2 206,3 113,9 1,402 0,481 107,4 107,4 1,06


0 86,0 223,2 223,2 87,8 1,611 0,383 85,4 85,4 1,030,5 101,6 223,2 222,0 95,5 1,482 0,439 98,1 98,1 0,970,75 110,9 223,2 220,2 111,4 1,419 0,480 107,1 107,1 1,04

1 119,8 223,2 218,4 119,9 1,365 0,517 115,3 115,3 1,041,5 141,3 223,2 213,1 137,0 1,257 0,596 133,1 133,1 1,032 166,9 223,2 206,3 159,8 1,157 0,677 151,0 151,0 1,06


0 116,9 223,2 223,2 115,5 1,382 0,509 113,6 113,6 1,020,5 136,2 223,2 222,0 128,3 1,280 0,587 130,9 130,9 0,980,75 144,4 223,2 220,2 139,8 1,243 0,617 137,6 137,6 1,02

1 157,4 223,2 218,4 152,4 1,191 0,660 147,4 147,4 1,031,5 182,0 223,2 213,1 171,5 1,107 0,732 163,4 163,4 1,052 211,8 223,2 206,3 192,8 1,027 0,803 179,1 179,1 1,08


0 126,9 223,2 223,2 130,9 1,326 0,560 124,9 124,9 1,050,5 139,4 223,2 222,0 145,4 1,265 0,611 136,4 136,4 1,07

0,75 148,9 223,2 220,2 153,7 1,224 0,648 144,6 144,6 1,061 154,3 223,2 218,4 162,7 1,203 0,667 149,0 149,0 1,09

1,5 172,4 223,2 213,1 174,2 1,138 0,728 162,4 162,4 1,072 194,8 223,2 206,3 182,8 1,070 0,791 176,5 176,5 1,04

IPE300 / L = 10 m / fy = 355 MPaψ = 1

ψ = 0.5

ψ = 0

ψ = -‐0.5

ψ = -‐1


A2.16


0 47,8 289,2 289,2 52,0 2,460 0,185 53,6 53,6 0,970,5 52,9 289,2 287,6 56,8 2,338 0,203 58,7 58,7 0,970,75 55,0 289,2 285,3 57,7 2,293 0,210 60,8 60,8 0,95

1 59,3 289,2 283,1 64,1 2,208 0,225 65,0 65,0 0,991,5 66,7 289,2 276,1 75,5 2,083 0,249 72,0 72,0 1,052 75,4 289,2 267,3 92,5 1,959 0,277 80,1 80,1 1,15


0 62,6 289,2 289,2 65,2 2,149 0,236 68,2 68,2 0,960,5 72,0 289,2 287,6 79,3 2,004 0,266 77,0 77,0 1,030,75 78,6 289,2 285,3 85,5 1,918 0,287 83,0 83,0 1,03

1 83,3 289,2 283,1 90,1 1,863 0,301 87,2 87,2 1,031,5 96,9 289,2 276,1 104,6 1,728 0,342 98,9 98,9 1,062 113,5 289,2 267,3 115,1 1,596 0,388 112,3 112,3 1,02


0 86,0 289,2 289,2 88,8 1,834 0,310 89,6 89,6 0,990,5 101,6 289,2 287,6 107,5 1,687 0,355 102,8 102,8 1,050,75 110,9 289,2 285,3 119,4 1,615 0,381 110,2 110,2 1,08

1 119,8 289,2 283,1 124,2 1,553 0,405 117,1 117,1 1,061,5 141,3 289,2 276,1 142,3 1,431 0,472 136,5 136,5 1,042 166,9 289,2 267,3 165,6 1,316 0,552 159,5 159,5 1,04


0 116,9 289,2 289,2 122,4 1,573 0,397 114,9 114,9 1,070,5 136,2 289,2 287,6 136,1 1,457 0,456 132,0 132,0 1,03

0,75 144,4 289,2 285,3 143,7 1,415 0,485 140,3 140,3 1,021 157,4 289,2 283,1 157,2 1,356 0,528 152,8 152,8 1,03

1,5 182,0 289,2 276,1 176,7 1,261 0,602 174,2 174,2 1,012 211,8 289,2 267,3 192,2 1,169 0,679 196,5 196,5 0,98


0 126,9 289,2 289,2 131,5 1,509 0,423 122,4 122,4 1,070,5 139,4 289,2 287,6 145,8 1,440 0,471 136,1 136,1 1,07

0,75 148,9 289,2 285,3 155,7 1,394 0,506 146,2 146,2 1,061 154,3 289,2 283,1 166,6 1,369 0,525 151,8 151,8 1,10

1,5 172,4 289,2 276,1 177,2 1,295 0,586 169,3 169,3 1,052 194,8 289,2 267,3 199,2 1,218 0,653 188,8 188,8 1,05

ψ = 1IPE300 / L = 10 m / fy = 460 MPa

ψ = 0.5

ψ = 0

ψ = -‐1

ψ = -‐0.5


A2.17


0 30,4 223,2 223,2 36,3 2,709 0,155 34,7 34,7 1,050,5 35,2 223,2 222,0 40,0 2,519 0,178 39,6 39,6 1,010,75 38,6 223,2 220,2 45,0 2,403 0,193 43,1 43,1 1,04

1 40,8 223,2 218,4 48,7 2,338 0,203 45,3 45,3 1,071,5 47,5 223,2 213,1 55,7 2,167 0,232 51,8 51,8 1,072 55,4 223,2 206,3 62,6 2,007 0,266 59,3 59,3 1,06


0 40,0 223,2 223,2 45,9 2,363 0,199 44,4 44,4 1,030,5 48,2 223,2 222,0 56,8 2,153 0,235 52,5 52,5 1,080,75 53,9 223,2 220,2 62,1 2,035 0,259 57,9 57,9 1,07

1 58,1 223,2 218,4 67,9 1,960 0,277 61,7 61,7 1,101,5 69,9 223,2 213,1 74,3 1,786 0,324 72,2 72,2 1,032 84,1 223,2 206,3 88,1 1,629 0,376 83,9 83,9 1,05


0 54,5 223,2 223,2 59,9 2,024 0,262 58,4 58,4 1,030,5 67,4 223,2 222,0 72,9 1,820 0,314 70,0 70,0 1,040,75 73,5 223,2 220,2 78,7 1,742 0,337 75,3 75,3 1,04

1 81,9 223,2 218,4 86,8 1,651 0,368 82,1 82,1 1,061,5 98,1 223,2 213,1 95,4 1,508 0,424 94,6 94,6 1,012 117,0 223,2 206,3 121,0 1,381 0,505 112,7 112,7 1,07


0 73,2 223,2 223,2 78,0 1,746 0,336 75,0 75,0 1,040,5 88,0 223,2 222,0 89,2 1,592 0,390 87,0 87,0 1,030,75 93,8 223,2 220,2 96,2 1,542 0,410 91,4 91,4 1,05

1 103,6 223,2 218,4 104,2 1,468 0,449 100,3 100,3 1,041,5 121,2 223,2 213,1 120,9 1,357 0,527 117,6 117,6 1,032 142,1 223,2 206,3 146,1 1,253 0,609 135,8 135,8 1,08


0 80,0 223,2 223,2 80,6 1,670 0,361 80,6 80,6 1,000,5 89,3 223,2 222,0 90,5 1,581 0,394 88,0 88,0 1,030,75 95,4 223,2 220,2 98,7 1,530 0,415 92,6 92,6 1,07

1 100,4 223,2 218,4 103,1 1,491 0,435 97,1 97,1 1,061,5 113,8 223,2 213,1 115,1 1,400 0,500 111,7 111,7 1,032 130,2 223,2 206,3 137,5 1,309 0,574 128,0 128,0 1,07

IPE300 / L = 15 m / fy = 355 MPaψ = 1

ψ = 0.5

ψ = -‐0.5

ψ = 0

ψ = -‐1


A2.18


0 30,4 289,2 289,2 36,0 3,084 0,122 35,4 35,4 1,020,5 35,2 289,2 287,6 39,6 2,867 0,140 40,5 40,5 0,980,75 38,6 289,2 285,3 46,6 2,736 0,153 44,1 44,1 1,06

1 40,8 289,2 283,1 44,4 2,661 0,161 46,4 46,4 0,961,5 47,5 289,2 276,1 61,2 2,467 0,184 53,3 53,3 1,152 55,4 289,2 267,3 69,8 2,284 0,212 61,2 61,2 1,14


0 40,0 289,2 289,2 47,2 2,690 0,157 45,5 45,5 1,040,5 48,2 289,2 287,6 52,3 2,451 0,187 53,9 53,9 0,970,75 53,9 289,2 285,3 58,4 2,316 0,206 59,7 59,7 0,98

1 58,1 289,2 283,1 65,2 2,231 0,221 63,8 63,8 1,021,5 69,9 289,2 276,1 76,5 2,033 0,260 75,1 75,1 1,022 84,1 289,2 267,3 94,8 1,855 0,304 87,9 87,9 1,08


0 54,5 289,2 289,2 63,6 2,304 0,208 60,3 60,3 1,060,5 67,4 289,2 287,6 74,8 2,071 0,251 72,7 72,7 1,030,75 73,5 289,2 285,3 83,0 1,983 0,271 78,4 78,4 1,06

1 81,9 289,2 283,1 90,0 1,879 0,297 85,9 85,9 1,051,5 98,1 289,2 276,1 101,9 1,717 0,346 99,9 99,9 1,022 117,0 289,2 267,3 122,3 1,572 0,397 114,9 114,9 1,06


0 73,2 289,2 289,2 85,0 1,988 0,270 78,1 78,1 1,090,5 88,0 289,2 287,6 95,4 1,813 0,316 91,3 91,3 1,040,75 93,8 289,2 285,3 99,1 1,755 0,333 96,3 96,3 1,03

1 103,6 289,2 283,1 110,4 1,671 0,361 104,4 104,4 1,061,5 121,2 289,2 276,1 122,7 1,545 0,409 118,1 118,1 1,042 142,1 289,2 267,3 142,8 1,426 0,477 138,0 138,0 1,03


0 80,0 289,2 289,2 87,5 1,901 0,291 84,3 84,3 1,040,5 89,3 289,2 287,6 97,5 1,799 0,320 92,5 92,5 1,050,75 95,4 289,2 285,3 103,1 1,741 0,338 97,6 97,6 1,06

1 100,4 289,2 283,1 108,3 1,697 0,352 101,8 101,8 1,061,5 113,8 289,2 276,1 123,1 1,594 0,389 112,5 112,5 1,092 130,2 289,2 267,3 137,9 1,490 0,435 125,9 125,9 1,09

ψ = 0.5

ψ = -‐ 0.5

ψ = 0.5

ψ = -‐ 1

ψ = 1IPE300 / L = 15 m / fy = 460 MPa


A2.19


0 10449,6 779,2 779,2 719,6 0,273 1,000 779,2 779,2 0,920,5 11668,7 779,2 769,0 765,1 0,258 1,000 779,2 769,0 0,99

0,75 12755,4 779,2 755,2 779,2 0,247 1,000 779,2 755,2 1,031 13191,8 779,2 741,3 791,7 0,243 1,000 779,2 741,3 1,07

1,5 15140,9 779,2 702,6 782,5 0,227 1,000 779,2 702,6 1,112 17706,8 779,2 659,2 761,7 0,210 1,000 779,2 659,2 1,16


0 13785,0 779,2 779,2 756,6 0,238 1,000 779,2 779,2 0,970,5 15953,6 779,2 769,0 782,4 0,221 1,000 779,2 769,0 1,02

0,75 16996,4 779,2 755,2 798,2 0,214 1,000 779,2 755,2 1,061 18865,3 779,2 741,3 790,8 0,203 1,000 779,2 741,3 1,07

1,5 22932,2 779,2 702,6 783,7 0,184 1,000 779,2 702,6 1,122 28871,6 779,2 659,2 756,4 0,164 1,000 779,2 659,2 1,15


0 19168,1 779,2 779,2 776,7 0,202 1,000 779,2 779,2 1,000,5 23197,0 779,2 769,0 817,0 0,183 1,000 779,2 769,0 1,06

0,75 2489,6 779,2 755,2 823,2 0,559 1,000 779,2 755,2 1,091 28921,1 779,2 741,3 818,9 0,164 1,000 779,2 741,3 1,10

1,5 37294,6 779,2 702,6 790,9 0,145 1,000 779,2 702,6 1,132 49675,0 779,2 659,2 763,0 0,125 1,000 779,2 659,2 1,16


0 26230,6 779,2 779,2 782,2 0,172 1,000 779,2 779,2 1,000,5 32135,6 779,2 769,0 813,3 0,156 1,000 779,2 769,0 1,06

0,75 37324,5 779,2 755,2 823,9 0,144 1,000 779,2 755,2 1,091 40232,8 779,2 741,3 805,7 0,139 1,000 779,2 741,3 1,09

1,5 51524,3 779,2 702,6 796,7 0,123 1,000 779,2 702,6 1,132 67315,2 779,2 659,2 757,1 0,108 1,000 779,2 659,2 1,15


0 26634,7 779,2 779,2 823,7 0,171 1,000 779,2 779,2 1,060,5 31272,2 779,2 769,0 830,7 0,158 1,000 779,2 769,0 1,08

0,75 3562,8 779,2 755,2 802,6 0,468 1,000 779,2 755,2 1,061 37376,1 779,2 741,3 797,3 0,144 1,000 779,2 741,3 1,08

1,5 45588,6 779,2 702,6 759,8 0,131 1,000 779,2 702,6 1,082 56817,2 779,2 659,2 741,6 0,117 1,000 779,2 659,2 1,12

IPE500 / L = 1 m / fy = 355 MPaψ = 1

ψ = 0.5

ψ = 0

ψ = -‐0.5

ψ = -‐ 1


A2.20


0 10449,6 1009,7 1009,7 964,1 0,311 1,000 1009,7 1009,7 0,950,5 11668,7 1009,7 996,5 972,3 0,294 1,000 1009,7 996,5 0,98

0,75 12755,4 1009,7 978,5 982,6 0,281 1,000 1009,7 978,5 1,001 13191,8 1009,7 960,5 1009,6 0,277 1,000 1009,7 960,5 1,05

1,5 15140,9 1009,7 910,3 934,0 0,258 1,000 1009,7 910,3 1,032 17706,8 1009,7 854,2 888,0 0,239 1,000 1009,7 854,2 1,04


0 13785,0 1009,7 1009,7 1038,3 0,271 1,000 1009,7 1009,7 1,030,5 15953,6 1009,7 996,5 1061,0 0,252 1,000 1009,7 996,5 1,060,75 16996,4 1009,7 978,5 1064,6 0,244 1,000 1009,7 978,5 1,091 18865,3 1009,7 960,5 1044,8 0,231 1,000 1009,7 960,5 1,091,5 22932,2 1009,7 910,3 991,2 0,210 1,000 1009,7 910,3 1,092 28871,6 1009,7 854,2 949,5 0,187 1,000 1009,7 854,2 1,11


0 19168,1 1009,7 1009,7 905,3 0,230 1,000 1009,7 1009,7 0,900,5 23197,0 1009,7 996,5 931,9 0,209 1,000 1009,7 996,5 0,94

0,75 2489,6 1009,7 978,5 950,7 0,637 1,000 1009,7 978,5 0,971 28921,1 1009,7 960,5 965,2 0,187 1,000 1009,7 960,5 1,00

1,5 37294,6 1009,7 910,3 965,9 0,165 1,000 1009,7 910,3 1,062 49675,0 1009,7 854,2 956,9 0,143 1,000 1009,7 854,2 1,12


0 26230,6 1009,7 1009,7 912,5 0,196 1,000 1009,7 1009,7 0,900,5 32135,6 1009,7 996,5 928,1 0,177 1,000 1009,7 996,5 0,93

0,75 37324,5 1009,7 978,5 933,8 0,164 1,000 1009,7 978,5 0,951 40232,8 1009,7 960,5 911,2 0,158 1,000 1009,7 960,5 0,95

1,5 51524,3 1009,7 910,3 948,5 0,140 1,000 1009,7 910,3 1,042 67315,2 1009,7 854,2 941,8 0,122 1,000 1009,7 854,2 1,10


0 26634,7 1009,7 1009,7 987,0 0,195 1,000 1009,7 1009,7 0,980,5 31272,2 1009,7 996,5 995,7 0,180 1,000 1009,7 996,5 1,00

0,75 3562,8 1009,7 978,5 1030,9 0,532 1,000 1009,7 978,5 1,051 37376,1 1009,7 960,5 987,0 0,164 1,000 1009,7 960,5 1,03

1,5 45588,6 1009,7 910,3 894,5 0,149 1,000 1009,7 910,3 0,982 56817,2 1009,7 854,2 871,9 0,133 1,000 1009,7 854,2 1,02

IPE500 / L = 1 m / fy = 460 MPaψ = 1

ψ = 0.5

ψ = 0

ψ = -‐0.5

ψ = -‐ 1


A2.21


0 2793,8 779,2 779,2 697,1 0,528 1,000 779,2 779,2 0,890,5 3124,1 779,2 769,0 704,5 0,499 1,000 779,2 769,0 0,92

0,75 3389,7 779,2 755,2 714,7 0,479 1,000 779,2 755,2 0,951 3540,0 779,2 741,3 729,1 0,469 1,000 779,2 741,3 0,98

1,5 4079,1 779,2 702,6 731,1 0,437 1,000 779,2 702,6 1,042 4804,2 779,2 659,2 732,9 0,403 1,000 779,2 659,2 1,11


0 3689,3 779,2 779,2 731,9 0,460 1,000 779,2 779,2 0,940,5 4279,5 779,2 769,0 770,9 0,427 1,000 779,2 769,0 1,00

0,75 4617,4 779,2 755,2 808,8 0,411 1,000 779,2 755,2 1,071 5082,2 779,2 741,3 811,6 0,392 1,000 779,2 741,3 1,09

1,5 6231,4 779,2 702,6 787,9 0,354 1,000 779,2 702,6 1,122 7991,9 779,2 659,2 751,3 0,312 1,000 779,2 659,2 1,14


0 5164,8 779,2 779,2 798,9 0,388 1,000 779,2 779,2 1,030,5 6284,5 779,2 769,0 810,6 0,352 1,000 779,2 769,0 1,05

0,75 7173,4 779,2 755,2 829,9 0,330 1,000 779,2 755,2 1,101 7917,7 779,2 741,3 833,9 0,314 1,000 779,2 741,3 1,12

1,5 10429,3 779,2 702,6 812,8 0,273 1,000 779,2 702,6 1,162 14503,2 779,2 659,2 760,4 0,232 1,000 779,2 659,2 1,15


0 7225,0 779,2 779,2 791,7 0,328 1,000 779,2 779,2 1,020,5 8934,8 779,2 769,0 808,1 0,295 1,000 779,2 769,0 1,05

0,75 22718,2 779,2 755,2 823,4 0,185 1,000 779,2 755,2 1,091 11359,1 779,2 741,3 806,2 0,262 1,000 779,2 741,3 1,09

1,5 14960,0 779,2 702,6 740,0 0,228 1,000 779,2 702,6 1,052 20599,9 779,2 659,2 732,3 0,194 1,000 779,2 659,2 1,11


0 7511,6 779,2 779,2 815,4 0,322 1,000 779,2 779,2 1,050,5 8862,6 779,2 769,0 830,7 0,297 1,000 779,2 769,0 1,08

0,75 9026,0 779,2 755,2 846,7 0,294 1,000 779,2 755,2 1,121 10698,7 779,2 741,3 810,7 0,270 1,000 779,2 741,3 1,09

1,5 13307,2 779,2 702,6 799,8 0,242 1,000 779,2 702,6 1,142 17217,9 779,2 659,2 752,3 0,213 1,000 779,2 659,2 1,14

IPE500 / L = 2 m / fy = 355 MPaψ = 1

ψ = 0

ψ = 0.5

ψ = -‐0.5

ψ = -‐ 1


A2.22


0 2793,8 1009,7 1009,7 915,4 0,601 0,917 925,5 925,5 0,990,5 3124,1 1009,7 996,5 922,3 0,569 1,000 1009,7 996,5 0,93

0,75 3389,7 1009,7 978,5 936,6 0,546 1,000 1009,7 978,5 0,961 3540,0 1009,7 960,5 1015,4 0,534 1,000 1009,7 960,5 1,06

1,5 4079,1 1009,7 910,3 932,3 0,498 1,000 1009,7 910,3 1,022 4804,2 1009,7 854,2 885,8 0,458 1,000 1009,7 854,2 1,04


0 3689,3 1009,7 1009,7 948,1 0,523 1,000 1009,7 1009,7 0,940,5 4279,5 1009,7 996,5 995,5 0,486 1,000 1009,7 996,5 1,000,75 4617,4 1009,7 978,5 1013,4 0,468 1,000 1009,7 978,5 1,041 5082,2 1009,7 960,5 1025,2 0,446 1,000 1009,7 960,5 1,071,5 6231,4 1009,7 910,3 952,9 0,403 1,000 1009,7 910,3 1,052 7991,9 1009,7 854,2 900,4 0,355 1,000 1009,7 854,2 1,05


0 5164,8 1009,7 1009,7 1018,1 0,442 1,000 1009,7 1009,7 1,010,5 6284,5 1009,7 996,5 1039,6 0,401 1,000 1009,7 996,5 1,040,75 7173,4 1009,7 978,5 1060,1 0,375 1,000 1009,7 978,5 1,08

1 7917,7 1009,7 960,5 1042,6 0,357 1,000 1009,7 960,5 1,091,5 10429,3 1009,7 910,3 996,7 0,311 1,000 1009,7 910,3 1,092 14503,2 1009,7 854,2 909,5 0,264 1,000 1009,7 854,2 1,06


0 7225,0 1009,7 1009,7 1029,0 0,374 1,000 1009,7 1009,7 1,020,5 8934,8 1009,7 996,5 1043,6 0,336 1,000 1009,7 996,5 1,050,75 22718,2 1009,7 978,5 1067,8 0,211 1,000 1009,7 978,5 1,09

1 11359,1 1009,7 960,5 1093,9 0,298 1,000 1009,7 960,5 1,141,5 14960,0 1009,7 910,3 915,8 0,260 1,000 1009,7 910,3 1,012 20599,9 1009,7 854,2 912,4 0,221 1,000 1009,7 854,2 1,07


0 7511,6 1009,7 1009,7 1034,0 0,367 1,000 1009,7 1009,7 1,020,5 8862,6 1009,7 996,5 1052,6 0,338 1,000 1009,7 996,5 1,060,75 9026,0 1009,7 978,5 1067,8 0,334 1,000 1009,7 978,5 1,09

1 10698,7 1009,7 960,5 1099,9 0,307 1,000 1009,7 960,5 1,151,5 13307,2 1009,7 910,3 974,8 0,275 1,000 1009,7 910,3 1,072 17217,9 1009,7 854,2 941,4 0,242 1,000 1009,7 854,2 1,10

IPE500 / L = 2 m / fy = 460 MPaψ = 1

ψ = 0

ψ = 0.5

ψ = -‐0.5

ψ = -‐ 1


A2.23


0 1005,2 779,2 779,2 650,9 0,880 0,771 601,2 601,2 1,080,5 1126,9 779,2 769,0 666,0 0,832 0,800 623,0 623,0 1,07

0,75 1180,5 779,2 755,2 678,0 0,812 0,810 631,4 631,4 1,071 1280,4 779,2 741,3 680,8 0,780 0,828 645,2 645,2 1,06

1,5 1480,0 779,2 702,6 695,7 0,726 0,857 667,5 667,5 1,042 1749,9 779,2 659,2 708,1 0,667 0,886 690,2 659,2 1,07


0 1327,4 779,2 779,2 708,6 0,766 0,899 700,5 700,5 1,010,5 1544,7 779,2 769,0 773,4 0,710 0,929 724,1 724,1 1,07

0,75 1763,0 779,2 755,2 775,1 0,665 0,952 741,7 741,7 1,051 1840,8 779,2 741,3 781,0 0,651 0,959 746,9 741,3 1,05

1,5 2266,7 779,2 702,6 785,5 0,586 0,987 768,9 702,6 1,122 2925,4 779,2 659,2 827,7 0,516 1,000 779,2 659,2 1,26


0 1858,3 779,2 779,2 809,2 0,648 1,000 779,2 779,2 1,040,5 2267,9 779,2 769,0 820,0 0,586 1,000 779,2 769,0 1,07

0,75 2576,0 779,2 755,2 822,2 0,550 1,000 779,2 755,2 1,091 2864,4 779,2 741,3 804,4 0,522 1,000 779,2 741,3 1,09

1,5 3783,5 779,2 702,6 788,0 0,454 1,000 779,2 702,6 1,122 5295,3 779,2 659,2 751,0 0,384 1,000 779,2 659,2 1,14


0 2606,1 779,2 779,2 787,1 0,547 1,000 779,2 779,2 1,010,5 3222,5 779,2 769,0 812,7 0,492 1,000 779,2 769,0 1,06

0,75 3731,8 779,2 755,2 828,0 0,457 1,000 779,2 755,2 1,101 4093,4 779,2 741,3 796,7 0,436 1,000 779,2 741,3 1,07

1,5 5399,5 779,2 702,6 731,0 0,380 1,000 779,2 702,6 1,042 7514,7 779,2 659,2 711,1 0,322 1,000 779,2 659,2 1,08


0 2734,2 779,2 779,2 832,5 0,534 1,000 779,2 779,2 1,070,5 3122,2 779,2 769,0 815,8 0,500 1,000 779,2 769,0 1,06

0,75 3481,3 779,2 755,2 791,0 0,473 1,000 779,2 755,2 1,051 3866,9 779,2 741,3 772,0 0,449 1,000 779,2 741,3 1,04

1,5 4811,1 779,2 702,6 758,0 0,402 1,000 779,2 702,6 1,082 6269,3 779,2 659,2 740,7 0,353 1,000 779,2 659,2 1,12

IPE500 / L = 3.5 m / fy = 355 MPaψ = 1

ψ = 0

ψ = 0.5

ψ = -‐0.5

ψ = -‐ 1


A2.24


0 1005,2 1009,7 1009,7 761,4 1,002 0,698 705,1 705,1 1,080,5 1126,9 1009,7 996,5 775,8 0,947 0,732 739,3 739,3 1,05

0,75 1180,5 1009,7 978,5 799,4 0,925 0,745 752,5 752,5 1,061 1280,4 1009,7 960,5 813,7 0,888 0,767 774,5 774,5 1,05

1,5 1480,0 1009,7 910,3 824,5 0,826 0,803 810,5 810,5 1,022 1749,9 1009,7 854,2 838,2 0,760 0,839 847,1 847,1 0,99


0 1327,4 1009,7 1009,7 861,1 0,872 0,835 842,9 842,9 1,020,5 1544,7 1009,7 996,5 966,1 0,809 0,874 882,8 882,8 1,09

0,75 1763,0 1009,7 978,5 996,1 0,757 0,904 913,0 913,0 1,091 1840,8 1009,7 960,5 1002,7 0,741 0,913 922,0 922,0 1,09

1,5 2266,7 1009,7 910,3 981,9 0,667 0,951 959,8 910,3 1,082 2925,4 1009,7 854,2 965,2 0,587 0,986 995,8 854,2 1,13


0 1858,3 1009,7 1009,7 1020,5 0,737 0,991 1000,3 1000,3 1,020,5 2267,9 1009,7 996,5 1042,3 0,667 1,000 1009,7 996,5 1,050,75 2576,0 1009,7 978,5 1044,7 0,626 1,000 1009,7 978,5 1,07

1 2864,4 1009,7 960,5 992,6 0,594 1,000 1009,7 960,5 1,031,5 3783,5 1009,7 910,3 974,1 0,517 1,000 1009,7 910,3 1,072 5295,3 1009,7 854,2 956,0 0,437 1,000 1009,7 854,2 1,12


0 2606,1 1009,7 1009,7 1025,1 0,622 1,000 1009,7 1009,7 1,020,5 3222,5 1009,7 996,5 1056,6 0,560 1,000 1009,7 996,5 1,060,75 3731,8 1009,7 978,5 986,0 0,520 1,000 1009,7 978,5 1,01

1 4093,4 1009,7 960,5 968,4 0,497 1,000 1009,7 960,5 1,011,5 5399,5 1009,7 910,3 955,5 0,432 1,000 1009,7 910,3 1,052 7514,7 1009,7 854,2 903,6 0,367 1,000 1009,7 854,2 1,06


0 2734,2 1009,7 1009,7 922,8 0,608 1,000 1009,7 1009,7 0,910,5 3122,2 1009,7 996,5 961,9 0,569 1,000 1009,7 996,5 0,97

0,75 3481,3 1009,7 978,5 1004,4 0,539 1,000 1009,7 978,5 1,031 3866,9 1009,7 960,5 1032,3 0,511 1,000 1009,7 960,5 1,07

1,5 4811,1 1009,7 910,3 978,7 0,458 1,000 1009,7 910,3 1,082 6269,3 1009,7 854,2 966,5 0,401 1,000 1009,7 854,2 1,13

IPE500 / L = 3.5 m / fy = 460 MPa

ψ = 0

ψ = 0.5

ψ = 1

ψ = -‐0.5

ψ = -‐ 1


A2.25


0 552,7 779,2 779,2 467,7 1,187 0,587 457,1 457,1 1,020,5 622,3 779,2 769,0 491,3 1,119 0,627 488,6 488,6 1,01

0,75 681,7 779,2 755,2 515,7 1,069 0,657 512,2 512,2 1,011 710,0 779,2 741,3 525,6 1,048 0,671 522,5 522,5 1,01

1,5 824,3 779,2 702,6 567,4 0,972 0,717 558,4 558,4 1,022 979,0 779,2 659,2 599,5 0,892 0,765 595,8 595,8 1,01


0 729,6 779,2 779,2 606,6 1,033 0,725 564,9 564,9 1,070,5 853,5 779,2 769,0 653,9 0,956 0,779 607,2 607,2 1,08

0,75 978,3 779,2 755,2 684,2 0,892 0,822 640,2 640,2 1,071 1022,2 779,2 741,3 702,4 0,873 0,834 650,0 650,0 1,08

1,5 1264,6 779,2 702,6 741,9 0,785 0,888 692,1 692,1 1,072 1638,9 779,2 659,2 761,5 0,690 0,940 732,3 659,2 1,16


0 1019,4 779,2 779,2 699,8 0,874 0,902 702,9 702,9 1,000,5 1249,5 779,2 769,0 783,5 0,790 0,959 747,6 747,6 1,05

0,75 1383,4 779,2 755,2 792,0 0,751 0,983 766,0 755,2 1,051 1580,9 779,2 741,3 795,1 0,702 1,000 779,2 741,3 1,07

1,5 2084,3 779,2 702,6 784,4 0,611 1,000 779,2 702,6 1,122 2901,6 779,2 659,2 754,4 0,518 1,000 779,2 659,2 1,14


0 1425,7 779,2 779,2 803,5 0,739 1,000 779,2 779,2 1,030,5 1761,0 779,2 769,0 820,8 0,665 1,000 779,2 769,0 1,07

0,75 2568,5 779,2 755,2 828,0 0,551 1,000 779,2 755,2 1,101 2227,2 779,2 741,3 816,3 0,592 1,000 779,2 741,3 1,10

1,5 2920,6 779,2 702,6 787,1 0,517 1,000 779,2 702,6 1,122 4044,4 779,2 659,2 752,7 0,439 1,000 779,2 659,2 1,14


0 1505,0 779,2 779,2 817,3 0,720 1,000 779,2 779,2 1,050,5 1757,9 779,2 769,0 836,1 0,666 1,000 779,2 769,0 1,09

0,75 1806,7 779,2 755,2 845,1 0,657 1,000 779,2 755,2 1,121 2104,6 779,2 741,3 784,7 0,608 1,000 779,2 741,3 1,06

1,5 2605,7 779,2 702,6 735,5 0,547 1,000 779,2 702,6 1,052 3383,9 779,2 659,2 716,9 0,480 1,000 779,2 659,2 1,09

IPE500 / L = 5 m / fy = 355 MPaψ = 1

ψ = 0.5

ψ = -‐0.5

ψ = 0

ψ = -‐ 1


A2.26


0 552,7 1009,7 1009,7 514,0 1,352 0,497 501,6 501,6 1,020,5 622,3 1009,7 996,5 564,3 1,274 0,538 542,9 542,9 1,04

0,75 681,7 1009,7 978,5 580,5 1,217 0,569 575,0 575,0 1,011 710,0 1009,7 960,5 602,1 1,192 0,584 589,2 589,2 1,02

1,5 824,3 1009,7 910,3 656,5 1,107 0,634 640,6 640,6 1,022 979,0 1009,7 854,2 720,0 1,016 0,690 696,8 696,8 1,03


0 729,6 1009,7 1009,7 607,0 1,176 0,625 630,7 630,7 0,960,5 853,5 1009,7 996,5 710,6 1,088 0,687 693,3 693,3 1,02

0,75 978,3 1009,7 978,5 775,8 1,016 0,737 744,4 744,4 1,041 1022,2 1009,7 960,5 817,3 0,994 0,753 760,0 760,0 1,08

1,5 1264,6 1009,7 910,3 868,8 0,894 0,821 828,9 828,9 1,052 1638,9 1009,7 854,2 879,2 0,785 0,888 896,9 854,2 1,03


0 1019,4 1009,7 1009,7 865,7 0,995 0,809 816,9 816,9 1,060,5 1249,5 1009,7 996,5 914,8 0,899 0,884 892,6 892,6 1,02

0,75 1383,4 1009,7 978,5 959,0 0,854 0,916 925,2 925,2 1,041 1580,9 1009,7 960,5 982,7 0,799 0,953 962,6 960,5 1,02

1,5 2084,3 1009,7 910,3 927,2 0,696 1,000 1009,7 910,3 1,022 2901,6 1009,7 854,2 883,0 0,590 1,000 1009,7 854,2 1,03


0 1425,7 1009,7 1009,7 1093,1 0,842 0,951 960,0 960,0 1,140,5 1761,0 1009,7 996,5 1103,0 0,757 1,000 1009,7 996,5 1,110,75 2568,5 1009,7 978,5 1051,1 0,627 1,000 1009,7 978,5 1,07

1 2227,2 1009,7 960,5 1018,1 0,673 1,000 1009,7 960,5 1,061,5 2920,6 1009,7 910,3 1055,8 0,588 1,000 1009,7 910,3 1,162 4044,4 1009,7 854,2 985,8 0,500 1,000 1009,7 854,2 1,15


0 1505,0 1009,7 1009,7 950,5 0,819 1,000 1009,7 1009,7 0,940,5 1757,9 1009,7 996,5 959,2 0,758 1,000 1009,7 996,5 0,96

0,75 1806,7 1009,7 978,5 1000,9 0,748 1,000 1009,7 978,5 1,021 2104,6 1009,7 960,5 959,3 0,693 1,000 1009,7 960,5 1,00

1,5 2605,7 1009,7 910,3 943,7 0,622 1,000 1009,7 910,3 1,042 3383,9 1009,7 854,2 918,1 0,546 1,000 1009,7 854,2 1,07

IPE500 / L = 5 m / fy = 460 MPaψ = 1

ψ = 0.5

ψ = -‐0.5

ψ = 0

ψ = -‐ 1


A2.27


0 275,2 779,2 779,2 293,4 1,683 0,357 278,1 278,1 1,060,5 314,1 779,2 769,0 309,8 1,575 0,396 308,9 308,9 1,00

0,75 342,2 779,2 755,2 332,0 1,509 0,423 329,9 329,9 1,011 363,4 779,2 741,3 359,5 1,464 0,443 345,1 345,1 1,04

1,5 427,6 779,2 702,6 405,1 1,350 0,498 387,8 387,8 1,042 514,6 779,2 659,2 470,1 1,231 0,562 437,8 437,8 1,07


0 699,0 779,2 779,2 567,7 1,056 0,709 552,6 552,6 1,030,5 866,5 779,2 769,0 640,9 0,948 0,784 611,1 611,1 1,050,75 927,7 779,2 755,2 687,9 0,916 0,806 627,8 627,8 1,10

1 1089,9 779,2 741,3 714,8 0,846 0,852 663,7 663,7 1,081,5 1413,7 779,2 702,6 713,3 0,742 0,912 710,8 702,6 1,022 1929,7 779,2 659,2 726,2 0,635 0,965 752,3 659,2 1,10


0 504,3 779,2 779,2 475,1 1,243 0,607 473,2 473,2 1,000,5 627,7 779,2 769,0 588,9 1,114 0,711 554,2 554,2 1,06

0,75 678,9 779,2 755,2 609,8 1,071 0,747 581,8 581,8 1,051 799,8 779,2 741,3 683,5 0,987 0,816 635,5 635,5 1,08

1,5 1051,2 779,2 702,6 709,1 0,861 0,912 710,4 702,6 1,012 1443,5 779,2 659,2 724,1 0,735 0,992 773,0 659,2 1,10


0 699,0 779,2 779,2 624,3 1,056 0,777 605,7 605,7 1,030,5 866,5 779,2 769,0 735,3 0,948 0,869 676,8 676,8 1,09

0,75 927,7 779,2 755,2 754,8 0,916 0,894 696,8 696,8 1,081 1089,9 779,2 741,3 769,5 0,846 0,948 738,6 738,6 1,04

1,5 1413,7 779,2 702,6 735,1 0,742 1,000 779,2 702,6 1,052 1929,7 779,2 659,2 690,1 0,635 1,000 779,2 659,2 1,05


0 746,2 779,2 779,2 655,4 1,022 0,836 651,5 651,5 1,010,5 865,6 779,2 769,0 740,7 0,949 0,902 703,0 703,0 1,05

0,75 927,7 779,2 755,2 768,1 0,916 0,930 724,7 724,7 1,061 1028,8 779,2 741,3 780,1 0,870 0,968 754,2 741,3 1,05

1,5 1264,0 779,2 702,6 779,8 0,785 1,000 779,2 702,6 1,112 1628,1 779,2 659,2 812,3 0,692 1,000 779,2 659,2 1,23

IPE500 / L = 8 m / fy = 355 MPaψ = 1

ψ = 0.5

ψ = 0

ψ = -‐0.5

ψ = -‐ 1


A2.28


0 275,2 1009,7 1009,7 295,5 1,915 0,288 290,5 290,5 1,020,5 314,1 1009,7 996,5 328,7 1,793 0,322 324,8 324,8 1,01

0,75 342,2 1009,7 978,5 355,0 1,718 0,345 348,5 348,5 1,021 363,4 1009,7 960,5 375,8 1,667 0,362 365,9 365,9 1,03

1,5 427,6 1009,7 910,3 429,4 1,537 0,412 415,9 415,9 1,032 514,6 1009,7 854,2 514,6 1,401 0,472 477,1 477,1 1,08


0 699,0 1009,7 1009,7 614,7 1,202 0,607 613,2 613,2 1,000,5 866,5 1009,7 996,5 693,2 1,079 0,692 699,2 699,2 0,99

0,75 927,7 1009,7 978,5 743,9 1,043 0,718 725,0 725,0 1,031 1089,9 1009,7 960,5 805,7 0,963 0,774 782,0 782,0 1,03

1,5 1413,7 1009,7 910,3 865,3 0,845 0,852 860,2 860,2 1,012 1929,7 1009,7 854,2 954,4 0,723 0,922 931,3 854,2 1,12


0 504,3 1009,7 1009,7 517,9 1,415 0,482 487,1 487,1 1,060,5 627,7 1009,7 996,5 604,1 1,268 0,588 593,4 593,4 1,02

0,75 678,9 1009,7 978,5 650,2 1,220 0,626 631,9 631,9 1,031 799,8 1009,7 960,5 742,9 1,124 0,704 710,4 710,4 1,05

1,5 1051,2 1009,7 910,3 910,4 0,980 0,821 829,2 829,2 1,102 1443,5 1009,7 854,2 962,9 0,836 0,929 937,8 854,2 1,13


0 699,0 1009,7 1009,7 676,2 1,202 0,651 657,5 657,5 1,030,5 866,5 1009,7 996,5 820,5 1,079 0,757 764,1 764,1 1,07

0,75 927,7 1009,7 978,5 853,4 1,043 0,788 795,9 795,9 1,071 1089,9 1009,7 960,5 894,7 0,963 0,857 865,2 865,2 1,03

1,5 1413,7 1009,7 910,3 922,9 0,845 0,948 957,4 910,3 1,012 1929,7 1009,7 854,2 952,9 0,723 1,000 1009,7 854,2 1,12


0 746,2 1009,7 1009,7 689,1 1,163 0,704 710,5 710,5 0,970,5 865,6 1009,7 996,5 802,4 1,080 0,782 789,3 789,3 1,02

0,75 927,7 1009,7 978,5 865,9 1,043 0,816 824,2 824,2 1,051 1028,8 1009,7 960,5 916,5 0,991 0,865 873,2 873,2 1,05

1,5 1264,0 1009,7 910,3 970,9 0,894 0,949 958,2 910,3 1,072 1628,1 1009,7 854,2 996,0 0,788 1,000 1009,7 854,2 1,17

IPE500 / L = 8 m / fy = 460 MPaψ = 1

ψ = 0.5

ψ = 0

ψ = -‐0.5

ψ = -‐ 1


A2.29


0 204,5 779,2 779,2 227,8 1,952 0,279 217,1 217,1 1,050,5 236,3 779,2 769,0 256,8 1,816 0,315 245,4 245,4 1,05

0,75 258,5 779,2 755,2 279,3 1,736 0,339 264,3 264,3 1,061 276,6 779,2 741,3 302,9 1,678 0,358 279,3 279,3 1,08

1,5 329,3 779,2 702,6 349,7 1,538 0,411 320,4 320,4 1,092 400,6 779,2 659,2 396,2 1,395 0,475 370,4 370,4 1,07


0 269,6 779,2 779,2 286,6 1,700 0,351 273,5 273,5 1,050,5 325,6 779,2 769,0 332,0 1,547 0,408 317,6 317,6 1,05

0,75 374,8 779,2 755,2 358,1 1,442 0,459 357,5 357,5 1,001 401,7 779,2 741,3 402,6 1,393 0,487 379,1 379,1 1,06

1,5 509,5 779,2 702,6 467,4 1,237 0,584 455,0 455,0 1,032 672,1 779,2 659,2 573,9 1,077 0,694 541,0 541,0 1,06


0 373,1 779,2 779,2 369,3 1,445 0,463 360,5 360,5 1,020,5 470,8 779,2 769,0 457,8 1,287 0,574 447,1 447,1 1,02

0,75 565,2 779,2 755,2 517,1 1,174 0,662 516,0 516,0 1,001 603,9 779,2 741,3 558,7 1,136 0,693 540,3 540,3 1,03

1,5 793,1 779,2 702,6 691,3 0,991 0,812 632,9 632,9 1,092 1081,3 779,2 659,2 737,4 0,849 0,920 717,0 659,2 1,12


0 513,7 779,2 779,2 494,4 1,232 0,626 487,9 487,9 1,010,5 640,8 779,2 769,0 580,4 1,103 0,737 573,9 573,9 1,01

0,75 756,8 779,2 755,2 639,2 1,015 0,813 633,4 633,4 1,011 805,7 779,2 741,3 698,8 0,983 0,839 654,1 654,1 1,07

1,5 1040,9 779,2 702,6 708,6 0,865 0,934 727,5 702,6 1,012 1411,0 779,2 659,2 715,6 0,743 1,000 779,2 659,2 1,09


0 551,8 779,2 779,2 540,2 1,188 0,681 530,3 530,3 1,020,5 639,9 779,2 769,0 594,7 1,104 0,760 591,9 591,9 1,00

0,75 701,9 779,2 755,2 645,5 1,054 0,807 628,5 628,5 1,031 759,9 779,2 741,3 688,7 1,013 0,845 658,2 658,2 1,05

1,5 932,3 779,2 702,6 739,8 0,914 0,932 726,2 702,6 1,052 1198,0 779,2 659,2 757,9 0,807 1,000 779,2 659,2 1,15

IPE500 / L = 10 m / fy = 355 MPa

ψ = 0.5

ψ = 1

ψ = 0

ψ = -‐0.5

ψ = -‐ 1


A2.30


0 204,5 1009,7 1009,7 225,9 2,222 0,222 224,4 224,4 1,010,5 236,3 1009,7 996,5 289,0 2,067 0,252 254,8 254,8 1,13

0,75 258,5 1009,7 978,5 309,3 1,976 0,273 275,4 275,4 1,121 276,6 1009,7 960,5 325,7 1,910 0,289 291,8 291,8 1,12

1,5 329,3 1009,7 910,3 372,0 1,751 0,334 337,7 337,7 1,102 400,6 1009,7 854,2 449,9 1,588 0,392 395,4 395,4 1,14


0 269,6 1009,7 1009,7 303,6 1,935 0,283 285,5 285,5 1,060,5 325,6 1009,7 996,5 351,4 1,761 0,331 334,6 334,6 1,05

0,75 374,8 1009,7 978,5 373,6 1,641 0,371 375,1 375,1 1,001 401,7 1009,7 960,5 424,7 1,585 0,392 396,2 396,2 1,07

1,5 509,5 1009,7 910,3 557,1 1,408 0,478 482,6 482,6 1,152 672,1 1009,7 854,2 626,7 1,226 0,591 596,9 596,9 1,05


0 373,1 1009,7 1009,7 404,2 1,645 0,370 373,7 373,7 1,080,5 470,8 1009,7 996,5 517,7 1,465 0,450 454,7 454,7 1,14

0,75 565,2 1009,7 978,5 591,6 1,337 0,537 542,1 542,1 1,091 603,9 1009,7 960,5 641,0 1,293 0,569 574,4 574,4 1,12

1,5 793,1 1009,7 910,3 791,9 1,128 0,700 706,4 706,4 1,122 1081,3 1009,7 854,2 915,2 0,966 0,832 840,2 840,2 1,09


0 513,7 1009,7 1009,7 569,9 1,402 0,494 499,2 499,2 1,140,5 640,8 1009,7 996,5 649,2 1,255 0,607 612,6 612,6 1,06

0,75 756,8 1009,7 978,5 743,6 1,155 0,691 697,8 697,8 1,071 805,7 1009,7 960,5 843,1 1,119 0,722 728,9 728,9 1,16

1,5 1040,9 1009,7 910,3 910,2 0,985 0,838 846,3 846,3 1,082 1411,0 1009,7 854,2 967,7 0,846 0,948 956,8 854,2 1,13


0 551,8 1009,7 1009,7 602,8 1,353 0,538 543,1 543,1 1,110,5 639,9 1009,7 996,5 689,1 1,256 0,619 625,3 625,3 1,10

0,75 701,9 1009,7 978,5 739,4 1,199 0,670 676,8 676,8 1,091 759,9 1009,7 960,5 812,8 1,153 0,713 720,4 720,4 1,13

1,5 932,3 1009,7 910,3 947,7 1,041 0,819 826,6 826,6 1,152 1198,0 1009,7 854,2 988,4 0,918 0,929 937,8 854,2 1,16

IPE500 / L = 10 m / fy = 460 MPa

ψ = 0.5

ψ = 1

ψ = 0

ψ = -‐0.5

ψ = -‐ 1


A2.31


0 125,8 779,2 779,2 151,8 2,488 0,181 141,4 141,4 1,070,5 151,0 779,2 769,0 175,8 2,272 0,214 166,5 166,5 1,06

0,75 167,6 779,2 755,2 196,3 2,156 0,234 182,6 182,6 1,071 183,2 779,2 741,3 228,0 2,062 0,253 197,4 197,4 1,15

1,5 225,4 779,2 702,6 263,6 1,859 0,303 235,8 235,8 1,122 282,5 779,2 659,2 323,0 1,661 0,365 284,0 284,0 1,14


0 165,6 779,2 779,2 182,6 2,169 0,232 180,7 180,7 1,010,5 209,5 779,2 769,0 226,4 1,929 0,284 221,6 221,6 1,02

0,75 245,4 779,2 755,2 269,9 1,782 0,325 253,2 253,2 1,071 268,9 779,2 741,3 294,7 1,702 0,350 272,9 272,9 1,08

1,5 351,2 779,2 702,6 365,1 1,489 0,433 337,7 337,7 1,082 470,8 779,2 659,2 451,4 1,287 0,551 429,7 429,7 1,05


0 227,5 779,2 779,2 241,5 1,851 0,305 237,6 237,6 1,020,5 299,5 779,2 769,0 330,0 1,613 0,382 297,6 297,6 1,110,75 346,0 779,2 755,2 369,6 1,501 0,428 333,5 333,5 1,11

1 392,4 779,2 741,3 403,5 1,409 0,486 378,9 378,9 1,061,5 517,2 779,2 702,6 551,6 1,227 0,619 482,7 482,7 1,142 698,8 779,2 659,2 678,6 1,056 0,759 591,7 591,7 1,15


0 309,3 779,2 779,2 331,9 1,587 0,392 305,2 305,2 1,090,5 395,9 779,2 769,0 433,5 1,403 0,494 384,7 384,7 1,13

0,75 464,8 779,2 755,2 492,9 1,295 0,575 448,0 448,0 1,101 501,9 779,2 741,3 523,4 1,246 0,614 478,7 478,7 1,09

1,5 648,4 779,2 702,6 625,1 1,096 0,742 578,3 578,3 1,082 871,6 779,2 659,2 715,8 0,946 0,871 678,6 659,2 1,09


0 335,9 779,2 779,2 366,5 1,523 0,417 325,3 325,3 1,130,5 393,3 779,2 769,0 434,0 1,408 0,495 385,6 385,6 1,13

0,75 436,6 779,2 755,2 479,0 1,336 0,551 429,6 429,6 1,111 470,9 779,2 741,3 501,6 1,286 0,593 462,1 462,1 1,09

1,5 581,0 779,2 702,6 595,5 1,158 0,708 552,1 552,1 1,082 747,8 779,2 659,2 737,2 1,021 0,837 652,3 652,3 1,13

IPE500 / L = 15 m / fy = 355 MPa

ψ = 0.5

ψ = 1

ψ = 0

ψ = -‐0.5

ψ = -‐ 1


A2.32


0 125,8 1009,7 1009,7 151,8 2,833 0,143 144,6 144,6 1,050,5 151,0 1009,7 996,5 182,9 2,586 0,169 170,8 170,8 1,07

0,75 167,6 1009,7 978,5 212,7 2,454 0,186 187,8 187,8 1,131 183,2 1009,7 960,5 228,0 2,348 0,201 203,4 203,4 1,12

1,5 225,4 1009,7 910,3 273,6 2,116 0,242 244,5 244,5 1,122 282,5 1009,7 854,2 323,2 1,891 0,294 297,1 297,1 1,09


0 165,6 1009,7 1009,7 198,3 2,469 0,184 185,8 185,8 1,070,5 209,5 1009,7 996,5 255,3 2,195 0,227 229,2 229,2 1,110,75 245,4 1009,7 978,5 298,9 2,028 0,261 263,3 263,3 1,13

1 268,9 1009,7 960,5 324,1 1,938 0,282 284,9 284,9 1,141,5 351,2 1009,7 910,3 379,6 1,695 0,353 356,0 356,0 1,072 470,8 1009,7 854,2 489,9 1,464 0,447 450,9 450,9 1,09


0 227,5 1009,7 1009,7 250,0 2,107 0,244 246,5 246,5 1,010,5 299,5 1009,7 996,5 327,5 1,836 0,309 312,1 312,1 1,05

0,75 346,0 1009,7 978,5 364,4 1,708 0,348 351,6 351,6 1,041 392,4 1009,7 960,5 422,1 1,604 0,385 389,0 389,0 1,09

1,5 517,2 1009,7 910,3 537,5 1,397 0,494 499,1 499,1 1,082 698,8 1009,7 854,2 687,9 1,202 0,640 645,9 645,9 1,06


0 309,3 1009,7 1009,7 352,4 1,807 0,318 320,6 320,6 1,100,5 395,9 1009,7 996,5 419,8 1,597 0,388 391,7 391,7 1,07

0,75 464,8 1009,7 978,5 473,0 1,474 0,445 449,7 449,7 1,051 501,9 1009,7 960,5 548,6 1,418 0,483 487,6 487,6 1,13

1,5 648,4 1009,7 910,3 698,2 1,248 0,613 618,7 618,7 1,132 871,6 1009,7 854,2 862,2 1,076 0,760 766,9 766,9 1,12


0 335,9 1009,7 1009,7 350,6 1,734 0,340 343,3 343,3 1,020,5 393,3 1009,7 996,5 415,2 1,602 0,386 389,6 389,6 1,07

0,75 436,6 1009,7 978,5 458,3 1,521 0,418 422,5 422,5 1,081 470,9 1009,7 960,5 508,5 1,464 0,453 457,9 457,9 1,11

1,5 581,0 1009,7 910,3 636,8 1,318 0,566 571,5 571,5 1,112 747,8 1009,7 854,2 775,6 1,162 0,705 711,7 711,7 1,09

IPE500 / L = 15 m / fy = 460 MPa

ψ = 0.5

ψ = 1

ψ = 0

ψ = -‐0.5

ψ = -‐ 1


A2.33


0 91,4 779,2 779,2 115,1 2,920 0,135 105,5 105,5 1,090,5 114,3 779,2 769,0 139,0 2,611 0,166 129,6 129,6 1,07

0,75 129,3 779,2 755,2 152,8 2,455 0,186 144,9 144,9 1,061 144,2 779,2 741,3 173,4 2,325 0,205 159,8 159,8 1,09

1,5 183,4 779,2 702,6 202,1 2,061 0,254 197,6 197,6 1,022 236,2 779,2 659,2 259,6 1,816 0,315 245,2 245,2 1,06


0 120,1 779,2 779,2 145,1 2,547 0,174 135,5 135,5 1,070,5 159,9 779,2 769,0 194,7 2,208 0,225 175,2 175,2 1,110,75 192,9 779,2 755,2 233,8 2,010 0,265 206,5 206,5 1,13

1 213,5 779,2 741,3 254,3 1,910 0,289 225,2 225,2 1,131,5 285,8 779,2 702,6 310,9 1,651 0,368 286,7 286,7 1,082 387,0 779,2 659,2 406,5 1,419 0,472 367,5 367,5 1,11


0 164,2 779,2 779,2 180,9 2,179 0,230 179,3 179,3 1,010,5 226,3 779,2 769,0 243,1 1,856 0,304 236,6 236,6 1,03

0,75 276,3 779,2 755,2 295,9 1,679 0,358 279,0 279,0 1,061 302,5 779,2 741,3 318,5 1,605 0,385 299,9 299,9 1,06

1,5 401,3 779,2 702,6 404,4 1,394 0,497 387,2 387,2 1,042 541,1 779,2 659,2 557,3 1,200 0,641 499,8 499,8 1,12


0 221,2 779,2 779,2 231,4 1,877 0,298 232,1 232,1 1,000,5 291,9 779,2 769,0 321,4 1,634 0,374 291,6 291,6 1,10

0,75 324,1 779,2 755,2 354,5 1,551 0,406 316,5 316,5 1,121 375,3 779,2 741,3 404,6 1,441 0,467 364,1 364,1 1,11

1,5 488,0 779,2 702,6 514,9 1,264 0,600 467,5 467,5 1,102 655,3 779,2 659,2 656,6 1,090 0,747 582,2 582,2 1,13


0 241,5 779,2 779,2 262,0 1,796 0,321 249,9 249,9 1,050,5 287,8 779,2 769,0 290,7 1,645 0,370 288,3 288,3 1,01

0,75 312,7 779,2 755,2 321,0 1,579 0,395 307,8 307,8 1,041 350,1 779,2 741,3 366,2 1,492 0,434 338,6 338,6 1,08

1,5 437,6 779,2 702,6 453,8 1,334 0,553 430,6 430,6 1,052 567,6 779,2 659,2 584,9 1,172 0,696 542,2 542,2 1,08

IPE500 / L = 20 m / fy = 355 MPa

ψ = 0

ψ = 0.5

ψ = 1

ψ = -‐0.5

ψ = -‐ 1


A2.34


0 91,4 1009,7 1009,7 111,7 3,324 0,106 107,4 107,4 1,040,5 114,3 1009,7 996,5 132,1 2,972 0,131 132,3 132,3 1,00

0,75 129,3 1009,7 978,5 154,2 2,795 0,147 148,2 148,2 1,041 144,2 1009,7 960,5 163,4 2,646 0,162 163,8 163,8 1,00

1,5 183,4 1009,7 910,3 204,4 2,346 0,202 203,6 203,6 1,002 236,2 1009,7 854,2 259,7 2,068 0,252 254,7 254,7 1,02


0 120,1 1009,7 1009,7 142,7 2,899 0,137 138,5 138,5 1,030,5 159,9 1009,7 996,5 191,4 2,513 0,178 180,0 180,0 1,06

0,75 192,9 1009,7 978,5 229,8 2,288 0,211 213,1 213,1 1,081 213,5 1009,7 960,5 249,6 2,175 0,231 233,1 233,1 1,07

1,5 285,8 1009,7 910,3 314,4 1,880 0,297 300,0 300,0 1,052 387,0 1009,7 854,2 402,1 1,615 0,381 384,7 384,7 1,05


0 164,2 1009,7 1009,7 197,3 2,480 0,183 184,3 184,3 1,070,5 226,3 1009,7 996,5 251,4 2,112 0,243 245,3 245,3 1,02

0,75 276,3 1009,7 978,5 297,3 1,912 0,289 291,6 291,6 1,021 302,5 1009,7 960,5 332,1 1,827 0,312 314,7 314,7 1,06

1,5 401,3 1009,7 910,3 417,8 1,586 0,392 395,9 395,9 1,062 541,1 1009,7 854,2 549,1 1,366 0,516 520,9 520,9 1,05


0 221,2 1009,7 1009,7 261,3 2,137 0,238 240,5 240,5 1,090,5 291,9 1009,7 996,5 308,7 1,860 0,302 305,4 305,4 1,01

0,75 324,1 1009,7 978,5 348,9 1,765 0,330 333,3 333,3 1,051 375,3 1009,7 960,5 381,3 1,640 0,372 375,4 375,4 1,02

1,5 488,0 1009,7 910,3 480,6 1,438 0,469 473,6 473,6 1,012 655,3 1009,7 854,2 652,7 1,241 0,618 624,2 624,2 1,05


0 241,5 1009,7 1009,7 262,0 2,045 0,257 259,7 259,7 1,010,5 287,8 1009,7 996,5 310,7 1,873 0,299 301,8 301,8 1,03

0,75 312,7 1009,7 978,5 351,0 1,797 0,320 323,5 323,5 1,081 350,1 1009,7 960,5 376,2 1,698 0,352 355,1 355,1 1,06

1,5 437,6 1009,7 910,3 455,5 1,519 0,419 423,2 423,2 1,082 567,6 1009,7 854,2 590,4 1,334 0,553 558,6 558,6 1,06

IPE500 / L = 20 m / fy = 460 MPaψ = 1

ψ = 0

ψ = 0.5

ψ = -‐ 1

ψ = -‐0.5

A2.3. Proposed ultimate moment estimates and design results - HEB300 beams

A2.35


0 6504,4 664,0 664,0 657,8 0,320 1,000 664,0 664,0 0,990,5 7033,1 664,0 657,1 665,5 0,307 1,000 664,0 657,1 1,01

0,75 7367,9 664,0 647,6 629,2 0,300 1,000 664,0 647,6 0,971 7653,0 664,0 638,0 661,9 0,295 1,000 664,0 638,0 1,04

1,5 8388,9 664,0 610,4 650,2 0,281 1,000 664,0 610,4 1,072 9275,5 664,0 578,4 619,9 0,268 1,000 664,0 578,4 1,07


0 8582,7 664,0 664,0 714,3 0,278 1,000 664,0 664,0 1,080,5 9516,2 664,0 657,1 725,8 0,264 1,000 664,0 657,1 1,10

0,75 10190,7 664,0 647,6 737,5 0,255 1,000 664,0 647,6 1,141 10667,8 664,0 638,0 710,4 0,249 1,000 664,0 638,0 1,11

1,5 12119,1 664,0 610,4 686,7 0,234 1,000 664,0 610,4 1,132 13994,4 664,0 578,4 646,0 0,218 1,000 664,0 578,4 1,12


0 11954,0 664,0 664,0 715,9 0,236 1,000 664,0 664,0 1,080,5 13697,6 664,0 657,1 722,2 0,220 1,000 664,0 657,1 1,10

0,75 14373,1 664,0 647,6 738,4 0,215 1,000 664,0 647,6 1,141 15954,4 664,0 638,0 706,0 0,204 1,000 664,0 638,0 1,11

1,5 18940,2 664,0 610,4 653,3 0,187 1,000 664,0 610,4 1,072 22970,8 664,0 578,4 601,9 0,170 1,000 664,0 578,4 1,04


0 16456,2 664,0 664,0 662,8 0,201 1,000 664,0 664,0 1,000,5 19151,4 664,0 657,1 699,0 0,186 1,000 664,0 657,1 1,06

0,75 20773,2 664,0 647,6 712,6 0,179 1,000 664,0 647,6 1,101 22586,0 664,0 638,0 718,0 0,171 1,000 664,0 638,0 1,13

1,5 27012,6 664,0 610,4 696,9 0,157 1,000 664,0 610,4 1,142 32775,9 664,0 578,4 644,8 0,142 1,000 664,0 578,4 1,11


0 16836,5 664,0 664,0 656,7 0,199 1,000 664,0 664,0 0,990,5 19099,1 664,0 657,1 667,1 0,186 1,000 664,0 657,1 1,02

0,75 20237,3 664,0 647,6 683,7 0,181 1,000 664,0 647,6 1,061 21868,7 664,0 638,0 652,7 0,174 1,000 664,0 638,0 1,02

1,5 25305,4 664,0 610,4 626,6 0,162 1,000 664,0 610,4 1,032 29631,0 664,0 578,4 599,2 0,150 1,000 664,0 578,4 1,04

HEB300 / L = 2 m / fy = 355 MPaψ = 1

ψ = 0.5

ψ = 0

ψ = -‐ 0.5

ψ = -‐ 1


A2.36


0 6504,4 860,5 860,5 820,7 0,364 1,000 860,5 860,5 0,950,5 7033,1 860,5 851,5 844,0 0,350 1,000 860,5 851,5 0,99

0,75 7367,9 860,5 839,1 857,5 0,342 1,000 860,5 839,1 1,021 7653,0 860,5 826,7 866,3 0,335 1,000 860,5 826,7 1,05

1,5 8388,9 860,5 790,9 889,2 0,320 1,000 860,5 790,9 1,122 9275,5 860,5 749,5 912,3 0,305 1,000 860,5 749,5 1,22


0 8582,7 860,5 860,5 854,2 0,317 1,000 860,5 860,5 0,990,5 9516,2 860,5 851,5 865,1 0,301 1,000 860,5 851,5 1,02

0,75 10190,7 860,5 839,1 870,4 0,291 1,000 860,5 839,1 1,041 10667,8 860,5 826,7 873,6 0,284 1,000 860,5 826,7 1,06

1,5 12119,1 860,5 790,9 831,6 0,266 1,000 860,5 790,9 1,052 13994,4 860,5 749,5 922,3 0,248 1,000 860,5 749,5 1,23


0 11954,0 860,5 860,5 862,6 0,268 1,000 860,5 860,5 1,000,5 13697,6 860,5 851,5 878,0 0,251 1,000 860,5 851,5 1,03

0,75 14373,1 860,5 839,1 897,6 0,245 1,000 860,5 839,1 1,071 15954,4 860,5 826,7 872,7 0,232 1,000 860,5 826,7 1,06

1,5 18940,2 860,5 790,9 844,3 0,213 1,000 860,5 790,9 1,072 22970,8 860,5 749,5 795,9 0,194 1,000 860,5 749,5 1,06


0 16456,2 860,5 860,5 852,7 0,229 1,000 860,5 860,5 0,990,5 19151,4 860,5 851,5 865,0 0,212 1,000 860,5 851,5 1,02

0,75 20773,2 860,5 839,1 874,4 0,204 1,000 860,5 839,1 1,041 22586,0 860,5 826,7 891,4 0,195 1,000 860,5 826,7 1,08

1,5 27012,6 860,5 790,9 882,1 0,178 1,000 860,5 790,9 1,122 32775,9 860,5 749,5 831,6 0,162 1,000 860,5 749,5 1,11


0 16836,5 860,5 860,5 880,7 0,226 1,000 860,5 860,5 1,020,5 19099,1 860,5 851,5 916,9 0,212 1,000 860,5 851,5 1,08

0,75 20237,3 860,5 839,1 926,9 0,206 1,000 860,5 839,1 1,101 21868,7 860,5 826,7 944,9 0,198 1,000 860,5 826,7 1,14

1,5 25305,4 860,5 790,9 896,2 0,184 1,000 860,5 790,9 1,132 29631,0 860,5 749,5 857,5 0,170 1,000 860,5 749,5 1,14

ψ = -‐ 1

ψ = -‐ 0.5

ψ = 0

HEB300 / L = 2 m / fy = 460 MPaψ = 1

ψ = 0.5


A2.37


0 2450,6 664,0 664,0 625,3 0,521 1,000 664,0 664,0 0,940,5 2651,3 664,0 657,1 645,6 0,500 1,000 664,0 657,1 0,98

0,75 2707,2 664,0 647,6 657,7 0,495 1,000 664,0 647,6 1,021 2887,1 664,0 638,0 675,8 0,480 1,000 664,0 638,0 1,06

1,5 3168,0 664,0 610,4 634,7 0,458 1,000 664,0 610,4 1,042 3508,2 664,0 578,4 575,7 0,435 1,000 664,0 578,4 1,00


0 3234,5 664,0 664,0 688,9 0,453 1,000 664,0 664,0 1,040,5 3588,8 664,0 657,1 707,9 0,430 1,000 664,0 657,1 1,08

0,75 3754,1 664,0 647,6 725,1 0,421 1,000 664,0 647,6 1,121 4027,3 664,0 638,0 708,4 0,406 1,000 664,0 638,0 1,11

1,5 4582,8 664,0 610,4 692,6 0,381 1,000 664,0 610,4 1,132 5306,9 664,0 578,4 659,1 0,354 1,000 664,0 578,4 1,14


0 4514,5 664,0 664,0 702,7 0,384 1,000 664,0 664,0 1,060,5 5175,3 664,0 657,1 722,0 0,358 1,000 664,0 657,1 1,10

0,75 5633,3 664,0 647,6 718,9 0,343 1,000 664,0 647,6 1,111 6032,8 664,0 638,0 684,9 0,332 1,000 664,0 638,0 1,07

1,5 7174,1 664,0 610,4 675,4 0,304 1,000 664,0 610,4 1,112 8734,8 664,0 578,4 641,8 0,276 1,000 664,0 578,4 1,11


0 6284,0 664,0 664,0 743,1 0,325 1,000 664,0 664,0 1,120,5 7302,0 664,0 657,1 745,4 0,302 1,000 664,0 657,1 1,13

0,75 8195,0 664,0 647,6 733,4 0,285 1,000 664,0 647,6 1,131 8601,9 664,0 638,0 728,6 0,278 1,000 664,0 638,0 1,14

1,5 10294,5 664,0 610,4 684,8 0,254 1,000 664,0 610,4 1,122 12550,5 664,0 578,4 653,7 0,230 1,000 664,0 578,4 1,13


0 6571,0 664,0 664,0 750,1 0,318 1,000 664,0 664,0 1,130,5 7414,8 664,0 657,1 752,4 0,299 1,000 664,0 657,1 1,15

0,75 7949,2 664,0 647,6 740,3 0,289 1,000 664,0 647,6 1,141 8457,5 664,0 638,0 735,4 0,280 1,000 664,0 638,0 1,15

1,5 9772,2 664,0 610,4 691,3 0,261 1,000 664,0 610,4 1,132 11469,5 664,0 578,4 659,9 0,241 1,000 664,0 578,4 1,14

ψ = -‐ 1

ψ = -‐ 0.5

ψ = 0

ψ = 0.5

HEB300 / L = 3.5 m / fy = 355 MPaψ = 1


A2.38


0 2450,6 860,5 860,5 789,0 0,593 0,920 792,0 792,0 1,000,5 2651,3 860,5 851,5 803,8 0,570 1,000 860,5 851,5 0,94

0,75 2707,2 860,5 839,1 825,4 0,564 1,000 860,5 839,1 0,981 2887,1 860,5 826,7 848,9 0,546 1,000 860,5 826,7 1,03

1,5 3168,0 860,5 790,9 806,4 0,521 1,000 860,5 790,9 1,022 3508,2 860,5 749,5 756,1 0,495 1,000 860,5 749,5 1,01


0 3234,5 860,5 860,5 861,0 0,516 1,000 860,5 860,5 1,000,5 3588,8 860,5 851,5 869,1 0,490 1,000 860,5 851,5 1,02

0,75 3754,1 860,5 839,1 889,4 0,479 1,000 860,5 839,1 1,061 4027,3 860,5 826,7 896,3 0,462 1,000 860,5 826,7 1,08

1,5 4582,8 860,5 790,9 926,8 0,433 1,000 860,5 790,9 1,172 5306,9 860,5 749,5 941,8 0,403 1,000 860,5 749,5 1,26


0 4514,5 860,5 860,5 885,7 0,437 1,000 860,5 860,5 1,030,5 5175,3 860,5 851,5 920,3 0,408 1,000 860,5 851,5 1,08

0,75 5633,3 860,5 839,1 939,5 0,391 1,000 860,5 839,1 1,121 6032,8 860,5 826,7 925,1 0,378 1,000 860,5 826,7 1,12

1,5 7174,1 860,5 790,9 937,8 0,346 1,000 860,5 790,9 1,192 8734,8 860,5 749,5 952,8 0,314 1,000 860,5 749,5 1,27


0 6284,0 860,5 860,5 977,5 0,370 1,000 860,5 860,5 1,140,5 7302,0 860,5 851,5 987,6 0,343 1,000 860,5 851,5 1,16

0,75 8195,0 860,5 839,1 986,5 0,324 1,000 860,5 839,1 1,181 8601,9 860,5 826,7 980,2 0,316 1,000 860,5 826,7 1,19

1,5 10294,5 860,5 790,9 958,1 0,289 1,000 860,5 790,9 1,212 12550,5 860,5 749,5 935,5 0,262 1,000 860,5 749,5 1,25


0 6571,0 860,5 860,5 907,6 0,362 1,000 860,5 860,5 1,050,5 7414,8 860,5 851,5 936,3 0,341 1,000 860,5 851,5 1,10

0,75 7949,2 860,5 839,1 950,8 0,329 1,000 860,5 839,1 1,131 8457,5 860,5 826,7 971,5 0,319 1,000 860,5 826,7 1,18

1,5 9772,2 860,5 790,9 974,8 0,297 1,000 860,5 790,9 1,232 11469,5 860,5 749,5 980,6 0,274 1,000 860,5 749,5 1,31

ψ = -‐ 1

ψ = -‐ 0.5

ψ = 0

ψ = 0.5

HEB300 / L = 3.5 m / fy = 460 MPaψ = 1


A2.39


0 1401,5 664,0 664,0 583,5 0,688 0,875 581,3 581,3 1,000,5 1519,7 664,0 657,1 617,8 0,661 0,889 590,2 590,2 1,05

0,75 1576,3 664,0 647,6 641,8 0,649 0,894 593,9 593,9 1,081 1658,8 664,0 638,0 643,4 0,633 0,902 599,0 599,0 1,07

1,5 1824,6 664,0 610,4 631,5 0,603 0,916 608,0 608,0 1,042 2025,6 664,0 578,4 581,0 0,573 1,000 664,0 578,4 1,00


0 1849,1 664,0 664,0 682,5 0,599 0,981 651,7 651,7 1,050,5 2057,3 664,0 657,1 693,5 0,568 1,000 664,0 657,1 1,06

0,75 2195,1 664,0 647,6 713,9 0,550 1,000 664,0 647,6 1,101 2314,9 664,0 638,0 715,2 0,536 1,000 664,0 638,0 1,12

1,5 2641,2 664,0 610,4 683,6 0,501 1,000 664,0 610,4 1,122 3066,8 664,0 578,4 641,7 0,465 1,000 664,0 578,4 1,11


0 2575,1 664,0 664,0 709,5 0,508 1,000 664,0 664,0 1,070,5 2957,1 664,0 657,1 728,6 0,474 1,000 664,0 657,1 1,11

0,75 3253,3 664,0 647,6 744,0 0,452 1,000 664,0 647,6 1,151 3449,1 664,0 638,0 720,3 0,439 1,000 664,0 638,0 1,13

1,5 4098,4 664,0 610,4 690,1 0,403 1,000 664,0 610,4 1,132 4978,2 664,0 578,4 655,9 0,365 1,000 664,0 578,4 1,13


0 3577,8 664,0 664,0 716,6 0,431 1,000 664,0 664,0 1,080,5 4145,6 664,0 657,1 735,9 0,400 1,000 664,0 657,1 1,12

0,75 4515,4 664,0 647,6 744,7 0,383 1,000 664,0 647,6 1,151 4861,3 664,0 638,0 727,5 0,370 1,000 664,0 638,0 1,14

1,5 5785,7 664,0 610,4 697,0 0,339 1,000 664,0 610,4 1,142 7015,8 664,0 578,4 662,5 0,308 1,000 664,0 578,4 1,15


0 3782,1 664,0 664,0 723,7 0,419 1,000 664,0 664,0 1,090,5 4236,1 664,0 657,1 743,3 0,396 1,000 664,0 657,1 1,13

0,75 4488,9 664,0 647,6 745,5 0,385 1,000 664,0 647,6 1,151 4797,8 664,0 638,0 734,8 0,372 1,000 664,0 638,0 1,15

1,5 5508,5 664,0 610,4 704,0 0,347 1,000 664,0 610,4 1,152 6431,8 664,0 578,4 669,1 0,321 1,000 664,0 578,4 1,16

ψ = -‐ 1

ψ = -‐ 0.5

ψ = 0

ψ = 0.5

HEB300 / L = 5 m / fy = 355 MPaψ = 1


A2.40


0 1401,5 860,5 860,5 721,1 0,784 0,826 710,8 710,8 1,010,5 1519,7 860,5 851,5 780,3 0,752 0,843 725,1 725,1 1,08

0,75 1576,3 860,5 839,1 819,3 0,739 0,850 731,3 731,3 1,121 1658,8 860,5 826,7 823,6 0,720 0,859 739,5 739,5 1,11

1,5 1824,6 860,5 790,9 874,2 0,687 0,876 753,9 753,9 1,162 2025,6 860,5 749,5 821,5 0,652 0,893 768,5 749,5 1,10


0 1849,1 860,5 860,5 872,4 0,682 0,943 811,8 811,8 1,070,5 2057,3 860,5 851,5 885,8 0,647 0,960 826,3 826,3 1,07

0,75 2195,1 860,5 839,1 912,9 0,626 0,970 834,4 834,4 1,091 2314,9 860,5 826,7 887,5 0,610 0,977 840,6 826,7 1,07

1,5 2641,2 860,5 790,9 833,4 0,571 1,000 860,5 790,9 1,052 3066,8 860,5 749,5 811,2 0,530 1,000 860,5 749,5 1,08


0 2575,1 860,5 860,5 852,2 0,578 1,000 860,5 860,5 0,990,5 2957,1 860,5 851,5 864,3 0,539 1,000 860,5 851,5 1,02

0,75 3253,3 860,5 839,1 912,9 0,514 1,000 860,5 839,1 1,091 3449,1 860,5 826,7 941,1 0,499 1,000 860,5 826,7 1,14

1,5 4098,4 860,5 790,9 905,9 0,458 1,000 860,5 790,9 1,152 4978,2 860,5 749,5 868,7 0,416 1,000 860,5 749,5 1,16


0 3577,8 860,5 860,5 898,1 0,490 1,000 860,5 860,5 1,040,5 4145,6 860,5 851,5 872,9 0,456 1,000 860,5 851,5 1,03

0,75 4515,4 860,5 839,1 913,8 0,437 1,000 860,5 839,1 1,091 4861,3 860,5 826,7 950,5 0,421 1,000 860,5 826,7 1,15

1,5 5785,7 860,5 790,9 900,4 0,386 1,000 860,5 790,9 1,142 7015,8 860,5 749,5 860,0 0,350 1,000 860,5 749,5 1,15


0 3782,1 860,5 860,5 904,8 0,477 1,000 860,5 860,5 1,050,5 4236,1 860,5 851,5 879,4 0,451 1,000 860,5 851,5 1,03

0,75 4488,9 860,5 839,1 920,6 0,438 1,000 860,5 839,1 1,101 4797,8 860,5 826,7 957,6 0,423 1,000 860,5 826,7 1,16

1,5 5508,5 860,5 790,9 905,4 0,395 1,000 860,5 790,9 1,142 6431,8 860,5 749,5 863,2 0,366 1,000 860,5 749,5 1,15

ψ = -‐ 1

ψ = -‐ 0.5

ψ = 0

ψ = 0.5

ψ = 1HEB300 / L = 5 m / fy = 460 MPa


A2.41


0 735,6 664,0 664,0 507,1 0,950 0,730 484,8 484,8 1,050,5 805,3 664,0 657,1 532,3 0,908 0,755 501,5 501,5 1,06

0,75 835,1 664,0 647,6 545,4 0,892 0,765 507,9 507,9 1,071 887,5 664,0 638,0 556,6 0,865 0,780 518,3 518,3 1,07

1,5 985,6 664,0 610,4 573,0 0,821 0,806 535,0 535,0 1,072 1104,7 664,0 578,4 580,5 0,775 0,831 551,5 551,5 1,05


0 969,4 664,0 664,0 622,8 0,828 0,863 572,9 572,9 1,090,5 1091,4 664,0 657,1 647,2 0,780 0,891 591,7 591,7 1,09

0,75 1143,8 664,0 647,6 671,7 0,762 0,901 598,6 598,6 1,121 1242,1 664,0 638,0 686,8 0,731 0,918 609,7 609,7 1,13

1,5 1432,4 664,0 610,4 630,3 0,681 0,944 626,9 610,4 1,032 1679,2 664,0 578,4 587,6 0,629 0,968 643,1 578,4 1,02


0 1341,2 664,0 664,0 686,6 0,704 1,000 664,0 664,0 1,030,5 1556,2 664,0 657,1 683,4 0,653 1,000 664,0 657,1 1,04

0,75 1667,5 664,0 647,6 662,7 0,631 1,000 664,0 647,6 1,021 1826,9 664,0 638,0 684,8 0,603 1,000 664,0 638,0 1,07

1,5 2174,3 664,0 610,4 643,0 0,553 1,000 664,0 610,4 1,052 2630,9 664,0 578,4 586,1 0,502 1,000 664,0 578,4 1,01


0 1844,9 664,0 664,0 702,7 0,600 1,000 664,0 664,0 1,060,5 2140,4 664,0 657,1 710,8 0,557 1,000 664,0 657,1 1,08

0,75 2280,7 664,0 647,6 681,5 0,540 1,000 664,0 647,6 1,051 2499,4 664,0 638,0 735,1 0,515 1,000 664,0 638,0 1,15

1,5 2951,0 664,0 610,4 677,3 0,474 1,000 664,0 610,4 1,112 3542,4 664,0 578,4 594,0 0,433 1,000 664,0 578,4 1,03


0 1979,7 664,0 664,0 700,4 0,579 1,000 664,0 664,0 1,050,5 2197,8 664,0 657,1 697,1 0,550 1,000 664,0 657,1 1,06

0,75 2306,5 664,0 647,6 676,0 0,537 1,000 664,0 647,6 1,041 2467,4 664,0 638,0 698,5 0,519 1,000 664,0 638,0 1,09

1,5 2808,3 664,0 610,4 655,8 0,486 1,000 664,0 610,4 1,072 3251,7 664,0 578,4 597,9 0,452 1,000 664,0 578,4 1,03

ψ = -‐ 1

ψ = -‐ 0.5

ψ = 0

ψ = 1HEB300 / L = 8 m / fy = 355 MPa

ψ = 0.5


A2.42


0 735,6 860,5 860,5 607,9 1,082 0,650 559,1 559,1 1,090,5 805,3 860,5 851,5 616,0 1,034 0,679 584,3 584,3 1,05

0,75 835,1 860,5 839,1 635,5 1,015 0,690 594,1 594,1 1,071 887,5 860,5 826,7 638,6 0,985 0,709 610,1 610,1 1,05

1,5 985,6 860,5 790,9 672,3 0,934 0,740 636,3 636,3 1,062 1104,7 860,5 749,5 688,6 0,883 0,770 662,8 662,8 1,04


0 969,4 860,5 860,5 675,4 0,942 0,788 678,4 678,4 1,000,5 1091,4 860,5 851,5 760,5 0,888 0,825 709,5 709,5 1,07

0,75 1143,8 860,5 839,1 784,5 0,867 0,838 721,0 721,0 1,091 1242,1 860,5 826,7 788,4 0,832 0,860 739,9 739,9 1,07

1,5 1432,4 860,5 790,9 829,9 0,775 0,894 769,2 769,2 1,082 1679,2 860,5 749,5 850,1 0,716 0,926 797,1 749,5 1,13


0 1341,2 860,5 860,5 758,0 0,801 0,952 819,3 819,3 0,930,5 1556,2 860,5 851,5 853,4 0,744 0,987 849,3 849,3 1,00

0,75 1667,5 860,5 839,1 916,6 0,718 1,000 860,5 839,1 1,091 1826,9 860,5 826,7 867,4 0,686 1,000 860,5 826,7 1,05

1,5 2174,3 860,5 790,9 837,8 0,629 1,000 860,5 790,9 1,062 2630,9 860,5 749,5 756,4 0,572 1,000 860,5 749,5 1,01


0 1844,9 860,5 860,5 835,9 0,683 1,000 860,5 860,5 0,970,5 2140,4 860,5 851,5 896,1 0,634 1,000 860,5 851,5 1,05

0,75 2280,7 860,5 839,1 914,4 0,614 1,000 860,5 839,1 1,091 2499,4 860,5 826,7 929,0 0,587 1,000 860,5 826,7 1,12

1,5 2951,0 860,5 790,9 890,9 0,540 1,000 860,5 790,9 1,132 3542,4 860,5 749,5 840,4 0,493 1,000 860,5 749,5 1,12


0 1979,7 860,5 860,5 859,6 0,659 1,000 860,5 860,5 1,000,5 2197,8 860,5 851,5 949,9 0,626 1,000 860,5 851,5 1,12

0,75 2306,5 860,5 839,1 919,1 0,611 1,000 860,5 839,1 1,101 2467,4 860,5 826,7 893,4 0,591 1,000 860,5 826,7 1,08

1,5 2808,3 860,5 790,9 816,2 0,554 1,000 860,5 790,9 1,032 3251,7 860,5 749,5 791,8 0,514 1,000 860,5 749,5 1,06

ψ = -‐ 1

ψ = -‐ 0.5

ψ = 0

ψ = 1HEB300 / L = 8 m / fy = 460 MPa

ψ = 0.5


A2.43


0 560,1 664,0 664,0 443,2 1,089 0,645 428,5 428,5 1,030,5 618,8 664,0 657,1 456,8 1,036 0,678 450,0 450,0 1,02

0,75 641,7 664,0 647,6 470,0 1,017 0,689 457,6 457,6 1,031 688,1 664,0 638,0 473,9 0,982 0,710 471,8 471,8 1,00

1,5 771,0 664,0 610,4 522,0 0,928 0,743 493,6 493,6 1,062 871,8 664,0 578,4 581,7 0,873 0,776 515,3 515,3 1,13


0 737,6 664,0 664,0 565,3 0,949 0,784 520,5 520,5 1,090,5 839,9 664,0 657,1 579,6 0,889 0,824 547,0 547,0 1,06

0,75 899,6 664,0 647,6 599,1 0,859 0,843 559,9 559,9 1,071 966,3 664,0 638,0 611,6 0,829 0,862 572,4 572,4 1,07

1,5 1125,3 664,0 610,4 643,9 0,768 0,898 596,2 596,2 1,082 1330,4 664,0 578,4 669,4 0,706 0,931 618,3 578,4 1,16


0 1016,5 664,0 664,0 665,1 0,808 0,948 629,2 629,2 1,060,5 1192,2 664,0 657,1 686,7 0,746 0,985 654,4 654,4 1,05

0,75 1314,9 664,0 647,6 706,6 0,711 1,000 664,0 647,6 1,091 1409,8 664,0 638,0 691,6 0,686 1,000 664,0 638,0 1,08

1,5 1683,6 664,0 610,4 667,0 0,628 1,000 664,0 610,4 1,092 2036,3 664,0 578,4 625,1 0,571 1,000 664,0 578,4 1,08


0 1388,9 664,0 664,0 689,1 0,691 1,000 664,0 664,0 1,040,5 1619,9 664,0 657,1 703,0 0,640 1,000 664,0 657,1 1,07

0,75 1775,0 664,0 647,6 697,0 0,612 1,000 664,0 647,6 1,081 1893,5 664,0 638,0 657,5 0,592 1,000 664,0 638,0 1,03

1,5 2232,3 664,0 610,4 631,6 0,545 1,000 664,0 610,4 1,032 2671,0 664,0 578,4 625,3 0,499 1,000 664,0 578,4 1,08


0 1499,8 664,0 664,0 704,9 0,665 1,000 664,0 664,0 1,060,5 1662,9 664,0 657,1 760,8 0,632 1,000 664,0 657,1 1,16

0,75 1752,1 664,0 647,6 727,0 0,616 1,000 664,0 647,6 1,121 1864,2 664,0 638,0 672,6 0,597 1,000 664,0 638,0 1,05

1,5 2118,5 664,0 610,4 677,2 0,560 1,000 664,0 610,4 1,112 2448,7 664,0 578,4 664,3 0,521 1,000 664,0 578,4 1,15

ψ = -‐ 1

ψ = -‐ 0.5

ψ = 0.5

ψ = 0

ψ = 1HEB300 / L = 10 m / fy = 355 MPa


A2.44


0 560,1 860,5 860,5 509,8 1,239 0,557 479,1 479,1 1,060,5 618,8 860,5 851,5 530,9 1,179 0,591 508,8 508,8 1,04

0,75 641,7 860,5 839,1 575,6 1,158 0,604 519,5 519,5 1,111 688,1 860,5 826,7 561,8 1,118 0,628 540,0 540,0 1,04

1,5 771,0 860,5 790,9 648,8 1,056 0,665 572,3 572,3 1,132 871,8 860,5 749,5 700,6 0,993 0,704 605,5 605,5 1,16


0 737,6 860,5 860,5 601,5 1,080 0,692 595,5 595,5 1,010,5 839,9 860,5 851,5 681,6 1,012 0,740 636,7 636,7 1,07

0,75 899,6 860,5 839,1 716,0 0,978 0,764 657,2 657,2 1,091 966,3 860,5 826,7 769,0 0,944 0,787 677,5 677,5 1,13

1,5 1125,3 860,5 790,9 804,8 0,874 0,833 717,1 717,1 1,122 1330,4 860,5 749,5 834,5 0,804 0,877 754,6 749,5 1,11


0 1016,5 860,5 860,5 850,5 0,920 0,868 747,0 747,0 1,140,5 1192,2 860,5 851,5 851,7 0,850 0,920 791,3 791,3 1,08

0,75 1314,9 860,5 839,1 832,9 0,809 0,947 814,9 814,9 1,021 1409,8 860,5 826,7 823,3 0,781 0,965 830,0 826,7 1,00

1,5 1683,6 860,5 790,9 812,4 0,715 1,000 860,5 790,9 1,032 2036,3 860,5 749,5 752,8 0,650 1,000 860,5 749,5 1,00


0 1388,9 860,5 860,5 862,7 0,787 0,988 849,8 849,8 1,020,5 1619,9 860,5 851,5 864,0 0,729 1,000 860,5 851,5 1,01

0,75 1775,0 860,5 839,1 866,3 0,696 1,000 860,5 839,1 1,031 1893,5 860,5 826,7 828,6 0,674 1,000 860,5 826,7 1,00

1,5 2232,3 860,5 790,9 824,0 0,621 1,000 860,5 790,9 1,042 2671,0 860,5 749,5 748,2 0,568 1,000 860,5 749,5 1,00


0 1499,8 860,5 860,5 871,4 0,757 1,000 860,5 860,5 1,010,5 1662,9 860,5 851,5 872,7 0,719 1,000 860,5 851,5 1,02

0,75 1752,1 860,5 839,1 875,1 0,701 1,000 860,5 839,1 1,041 1864,2 860,5 826,7 836,9 0,679 1,000 860,5 826,7 1,01

1,5 2118,5 860,5 790,9 832,4 0,637 1,000 860,5 790,9 1,052 2448,7 860,5 749,5 755,8 0,593 1,000 860,5 749,5 1,01

ψ = -‐ 1

ψ = -‐ 0.5

ψ = 0.5

ψ = 0

ψ = 1HEB300 / L = 10 m / fy = 460 MPa


A2.45


0 352,5 664,0 664,0 339,5 1,373 0,486 322,9 322,9 1,050,5 400,3 664,0 657,1 394,9 1,288 0,530 352,0 352,0 1,12

0,75 421,5 664,0 647,6 420,1 1,255 0,548 363,9 363,9 1,151 457,3 664,0 638,0 439,7 1,205 0,576 382,7 382,7 1,15

1,5 525,8 664,0 610,4 448,6 1,124 0,624 414,5 414,5 1,082 609,2 664,0 578,4 510,8 1,044 0,673 446,7 446,7 1,14


0 463,5 664,0 664,0 417,0 1,197 0,611 405,5 405,5 1,030,5 546,0 664,0 657,1 501,9 1,103 0,676 448,9 448,9 1,12

0,75 596,2 664,0 647,6 516,3 1,055 0,709 471,1 471,1 1,101 648,1 664,0 638,0 562,7 1,012 0,740 491,3 491,3 1,15

1,5 775,4 664,0 610,4 588,3 0,925 0,800 531,1 531,1 1,112 936,4 664,0 578,4 603,5 0,842 0,854 567,0 567,0 1,06


0 633,8 664,0 664,0 558,3 1,024 0,786 521,9 521,9 1,070,5 768,3 664,0 657,1 661,0 0,930 0,861 571,6 571,6 1,16

0,75 856,8 664,0 647,6 686,0 0,880 0,898 596,1 596,1 1,151 928,2 664,0 638,0 692,3 0,846 0,922 612,4 612,4 1,13

1,5 1120,4 664,0 610,4 620,8 0,770 0,972 645,2 610,4 1,022 1358,1 664,0 578,4 588,5 0,699 1,000 664,0 578,4 1,02


0 855,3 664,0 664,0 653,7 0,881 0,922 612,0 612,0 1,070,5 1018,2 664,0 657,1 686,0 0,808 0,974 646,9 646,9 1,06

0,75 1116,2 664,0 647,6 673,5 0,771 0,998 662,5 647,6 1,041 1201,2 664,0 638,0 659,7 0,744 1,000 664,0 638,0 1,03

1,5 1420,8 664,0 610,4 616,5 0,684 1,000 664,0 610,4 1,012 1699,1 664,0 578,4 590,0 0,625 1,000 664,0 578,4 1,02


0 932,8 664,0 664,0 683,5 0,844 0,988 656,3 656,3 1,040,5 1040,4 664,0 657,1 657,1 0,799 1,000 664,0 657,1 1,00

0,75 1092,6 664,0 647,6 660,0 0,780 1,000 664,0 647,6 1,021 1172,8 664,0 638,0 657,1 0,752 1,000 664,0 638,0 1,03

1,5 1339,0 664,0 610,4 634,8 0,704 1,000 664,0 610,4 1,042 1553,1 664,0 578,4 572,6 0,654 1,000 664,0 578,4 0,99

ψ = -‐ 1

ψ = -‐ 0.5

ψ = 0

ψ = 0.5

HEB300 / L = 15 m / fy = 355 MPaψ = 1


A2.46


0 352,5 860,5 860,5 367,8 1,562 0,401 345,4 345,4 1,060,5 400,3 860,5 851,5 383,5 1,466 0,442 380,4 380,4 1,01

0,75 421,5 860,5 839,1 404,2 1,429 0,459 395,1 395,1 1,021 457,3 860,5 826,7 440,9 1,372 0,487 418,7 418,7 1,05

1,5 525,8 860,5 790,9 519,3 1,279 0,535 460,2 460,2 1,132 609,2 860,5 749,5 558,0 1,188 0,586 504,2 504,2 1,11


0 463,5 860,5 860,5 433,9 1,363 0,504 434,0 434,0 1,000,5 546,0 860,5 851,5 511,4 1,255 0,572 491,8 491,8 1,04

0,75 596,2 860,5 839,1 545,4 1,201 0,608 522,8 522,8 1,041 648,1 860,5 826,7 598,9 1,152 0,641 551,9 551,9 1,09

1,5 775,4 860,5 790,9 662,3 1,053 0,711 611,6 611,6 1,082 936,4 860,5 749,5 751,4 0,959 0,777 668,7 668,7 1,12


0 633,8 860,5 860,5 628,2 1,165 0,670 576,1 576,1 1,090,5 768,3 860,5 851,5 703,2 1,058 0,757 651,8 651,8 1,08

0,75 856,8 860,5 839,1 745,2 1,002 0,803 691,3 691,3 1,081 928,2 860,5 826,7 774,8 0,963 0,835 718,4 718,4 1,08

1,5 1120,4 860,5 790,9 817,6 0,876 0,901 774,9 774,9 1,062 1358,1 860,5 749,5 855,3 0,796 0,955 822,1 749,5 1,14


0 855,3 860,5 860,5 777,7 1,003 0,823 708,0 708,0 1,100,5 1018,2 860,5 851,5 839,3 0,919 0,892 767,6 767,6 1,09

0,75 1116,2 860,5 839,1 866,2 0,878 0,924 795,1 795,1 1,091 1201,2 860,5 826,7 842,6 0,846 0,947 815,1 815,1 1,03

1,5 1420,8 860,5 790,9 804,0 0,778 0,993 854,7 790,9 1,022 1699,1 860,5 749,5 756,9 0,712 1,000 860,5 749,5 1,01


0 932,8 860,5 860,5 851,6 0,960 0,892 767,5 767,5 1,110,5 1040,4 860,5 851,5 890,1 0,909 0,936 805,4 805,4 1,11

0,75 1092,6 860,5 839,1 888,3 0,887 0,954 821,0 821,0 1,081 1172,8 860,5 826,7 931,6 0,857 0,979 842,1 826,7 1,13

1,5 1339,0 860,5 790,9 892,1 0,802 1,000 860,5 790,9 1,132 1553,1 860,5 749,5 832,4 0,744 1,000 860,5 749,5 1,11

ψ = -‐ 1

ψ = -‐ 0.5

ψ = 0

ψ = 0.5

HEB300 / L = 15 m / fy = 460 MPaψ = 1


A2.47


0 542,6 664,0 664,0 443,2 1,106 0,635 421,5 421,5 1,050,5 681,9 664,0 657,1 507,5 0,987 0,708 469,9 469,9 1,08

0,75 725,9 664,0 647,6 556,7 0,956 0,726 482,2 482,2 1,151 866,2 664,0 638,0 525,1 0,876 0,774 514,2 514,2 1,02

1,5 1113,5 664,0 610,4 544,4 0,772 0,832 552,6 552,6 0,992 1457,3 664,0 578,4 598,2 0,675 0,882 585,7 578,4 1,03


0 713,3 664,0 664,0 565,3 0,965 0,773 513,2 513,2 1,100,5 955,1 664,0 657,1 644,0 0,834 0,859 570,4 570,4 1,13

0,75 1106,8 664,0 647,6 653,9 0,775 0,894 593,8 593,8 1,101 1288,0 664,0 638,0 679,5 0,718 0,925 614,4 614,4 1,11

1,5 1750,4 664,0 610,4 700,4 0,616 0,974 646,9 610,4 1,152 2425,1 664,0 578,4 670,1 0,523 1,000 664,0 578,4 1,16


0 974,0 664,0 664,0 665,1 0,826 0,936 621,5 621,5 1,070,5 1352,0 664,0 657,1 686,7 0,701 1,000 664,0 657,1 1,04

0,75 1693,6 664,0 647,6 690,9 0,626 1,000 664,0 647,6 1,071 1826,9 664,0 638,0 711,4 0,603 1,000 664,0 638,0 1,12

1,5 2463,6 664,0 610,4 701,8 0,519 1,000 664,0 610,4 1,152 3409,0 664,0 578,4 678,6 0,441 1,000 664,0 578,4 1,17


0 1311,1 664,0 664,0 689,1 0,712 1,000 664,0 664,0 1,040,5 1742,9 664,0 657,1 702,9 0,617 1,000 664,0 657,1 1,07

0,75 1983,4 664,0 647,6 727,5 0,579 1,000 664,0 647,6 1,121 2266,9 664,0 638,0 728,5 0,541 1,000 664,0 638,0 1,14

1,5 3002,2 664,0 610,4 686,8 0,470 1,000 664,0 610,4 1,132 4147,6 664,0 578,4 625,8 0,400 1,000 664,0 578,4 1,08


0 1432,0 664,0 664,0 636,2 0,681 1,000 664,0 664,0 0,960,5 1718,6 664,0 657,1 693,9 0,622 1,000 664,0 657,1 1,06

0,75 1931,2 664,0 647,6 713,8 0,586 1,000 664,0 647,6 1,101 2113,4 664,0 638,0 716,7 0,561 1,000 664,0 638,0 1,12

1,5 2685,7 664,0 610,4 695,2 0,497 1,000 664,0 610,4 1,142 3575,5 664,0 578,4 631,8 0,431 1,000 664,0 578,4 1,09

ψ = -‐ 1

ψ = -‐ 0.5

ψ = 0

HEB300 / L = 20 m / fy = 355 MPaψ = 1

ψ = 0.5


A2.48


0 542,6 860,5 860,5 509,8 1,259 0,546 469,6 469,6 1,090,5 681,9 860,5 851,5 589,9 1,123 0,624 537,3 537,3 1,10

0,75 725,9 860,5 839,1 615,5 1,089 0,645 555,3 555,3 1,111 866,2 860,5 826,7 632,3 0,997 0,702 603,8 603,8 1,05

1,5 1113,5 860,5 790,9 681,0 0,879 0,772 664,5 664,5 1,022 1457,3 860,5 749,5 745,9 0,768 0,834 717,9 717,9 1,04


0 713,3 860,5 860,5 612,2 1,098 0,679 584,3 584,3 1,050,5 955,1 860,5 851,5 714,0 0,949 0,784 674,3 674,3 1,06

0,75 1106,8 860,5 839,1 737,2 0,882 0,829 713,0 713,0 1,031 1288,0 860,5 826,7 765,3 0,817 0,869 747,8 747,8 1,02

1,5 1750,4 860,5 790,9 868,6 0,701 0,934 803,6 790,9 1,102 2425,1 860,5 749,5 807,8 0,596 0,983 845,7 749,5 1,08


0 974,0 860,5 860,5 759,3 0,940 0,853 733,9 733,9 1,030,5 1352,0 860,5 851,5 876,6 0,798 0,954 821,1 821,1 1,07

0,75 1693,6 860,5 839,1 913,1 0,713 1,000 860,5 839,1 1,091 1826,9 860,5 826,7 836,9 0,686 1,000 860,5 826,7 1,01

1,5 2463,6 860,5 790,9 873,0 0,591 1,000 860,5 790,9 1,102 3409,0 860,5 749,5 834,8 0,502 1,000 860,5 749,5 1,11


0 1311,1 860,5 860,5 868,0 0,810 0,973 836,8 836,8 1,040,5 1742,9 860,5 851,5 900,7 0,703 1,000 860,5 851,5 1,06

0,75 1983,4 860,5 839,1 911,7 0,659 1,000 860,5 839,1 1,091 2266,9 860,5 826,7 836,0 0,616 1,000 860,5 826,7 1,01

1,5 3002,2 860,5 790,9 872,7 0,535 1,000 860,5 790,9 1,102 4147,6 860,5 749,5 804,8 0,455 1,000 860,5 749,5 1,07


0 1432,0 860,5 860,5 868,0 0,775 1,000 860,5 860,5 1,010,5 1718,6 860,5 851,5 900,7 0,708 1,000 860,5 851,5 1,06

0,75 1931,2 860,5 839,1 911,7 0,667 1,000 860,5 839,1 1,091 2113,4 860,5 826,7 836,0 0,638 1,000 860,5 826,7 1,01

1,5 2685,7 860,5 790,9 872,7 0,566 1,000 860,5 790,9 1,102 3575,5 860,5 749,5 804,8 0,491 1,000 860,5 749,5 1,07

ψ = -‐ 1

ψ = -‐ 0.5

HEB300 / L = 20 m / fy = 460 MPaψ = 1

ψ = 0.5

ψ = 0


A2.49


0 204,9 664,0 664,0 227,3 1,800 0,320 212,2 212,2 1,070,5 247,9 664,0 657,1 272,4 1,637 0,373 247,8 247,8 1,10

0,75 273,0 664,0 647,6 301,6 1,560 0,403 267,3 267,3 1,131 300,7 664,0 638,0 327,8 1,486 0,433 287,7 287,7 1,14

1,5 364,9 664,0 610,4 354,3 1,349 0,498 330,8 330,8 1,072 443,0 664,0 578,4 381,6 1,224 0,565 375,4 375,4 1,02


0 269,1 664,0 664,0 283,5 1,571 0,398 264,4 264,4 1,070,5 342,6 664,0 657,1 355,8 1,392 0,487 323,3 323,3 1,10

0,75 392,9 664,0 647,6 408,3 1,300 0,543 360,4 360,4 1,131 434,0 664,0 638,0 442,3 1,237 0,584 387,6 387,6 1,14

1,5 544,7 664,0 610,4 482,6 1,104 0,675 448,3 448,3 1,082 678,5 664,0 578,4 510,4 0,989 0,756 502,0 502,0 1,02


0 365,5 664,0 664,0 365,6 1,348 0,529 351,1 351,1 1,040,5 476,9 664,0 657,1 447,7 1,180 0,657 436,5 436,5 1,03

0,75 547,1 664,0 647,6 549,3 1,102 0,722 479,2 479,2 1,151 600,8 664,0 638,0 584,6 1,051 0,763 506,8 506,8 1,15

1,5 741,3 664,0 610,4 617,5 0,946 0,848 563,0 563,0 1,102 908,6 664,0 578,4 634,3 0,855 0,916 608,2 578,4 1,10


0 487,1 664,0 664,0 484,3 1,168 0,680 451,8 451,8 1,070,5 610,4 664,0 657,1 520,6 1,043 0,788 523,5 523,5 0,99

0,75 678,9 664,0 647,6 580,9 0,989 0,835 554,3 554,3 1,051 740,2 664,0 638,0 600,6 0,947 0,870 577,4 577,4 1,04

1,5 891,9 664,0 610,4 630,7 0,863 0,935 621,1 610,4 1,032 1079,1 664,0 578,4 657,6 0,784 0,989 657,0 578,4 1,14


0 533,8 664,0 664,0 537,2 1,115 0,748 497,0 497,0 1,080,5 612,8 664,0 657,1 574,2 1,041 0,818 543,4 543,4 1,06

0,75 652,7 664,0 647,6 609,5 1,009 0,848 563,3 563,3 1,081 709,4 664,0 638,0 626,9 0,967 0,886 588,1 588,1 1,07

1,5 829,5 664,0 610,4 639,1 0,895 0,948 629,7 610,4 1,052 981,6 664,0 578,4 663,9 0,822 1,000 664,0 578,4 1,15

ψ = -‐ 1

ψ = -‐ 0.5

ψ = 1HEB300 / L = 25 m / fy = 355 MPa

ψ = 0.5

ψ = 0


A2.50


0 204,9 860,5 860,5 229,5 2,049 0,256 220,5 220,5 1,040,5 247,9 860,5 851,5 264,2 1,863 0,302 259,5 259,5 1,02

0,75 273,0 860,5 839,1 321,9 1,775 0,327 281,3 281,3 1,141 300,7 860,5 826,7 351,6 1,692 0,354 304,5 304,5 1,15

1,5 364,9 860,5 790,9 391,1 1,536 0,412 354,8 354,8 1,102 443,0 860,5 749,5 458,9 1,394 0,476 409,5 409,5 1,12


0 269,1 860,5 860,5 290,1 1,788 0,323 278,0 278,0 1,040,5 342,6 860,5 851,5 356,9 1,585 0,393 337,8 337,8 1,06

0,75 392,9 860,5 839,1 424,3 1,480 0,438 377,2 377,2 1,121 434,0 860,5 826,7 419,7 1,408 0,478 411,1 411,1 1,02

1,5 544,7 860,5 790,9 538,8 1,257 0,571 491,0 491,0 1,102 678,5 860,5 749,5 628,2 1,126 0,660 567,6 567,6 1,11


0 365,5 860,5 860,5 363,9 1,534 0,413 355,2 355,2 1,020,5 476,9 860,5 851,5 470,7 1,343 0,532 457,8 457,8 1,03

0,75 547,1 860,5 839,1 552,8 1,254 0,599 515,2 515,2 1,071 600,8 860,5 826,7 629,9 1,197 0,644 554,1 554,1 1,14

1,5 741,3 860,5 790,9 685,2 1,077 0,742 638,2 638,2 1,072 908,6 860,5 749,5 741,5 0,973 0,827 711,4 711,4 1,04


0 487,1 860,5 860,5 493,4 1,329 0,548 471,8 471,8 1,050,5 610,4 860,5 851,5 600,3 1,187 0,663 570,9 570,9 1,05

0,75 678,9 860,5 839,1 659,0 1,126 0,716 616,4 616,4 1,071 740,2 860,5 826,7 692,3 1,078 0,758 652,1 652,1 1,06

1,5 891,9 860,5 790,9 747,6 0,982 0,840 723,1 723,1 1,032 1079,1 860,5 749,5 787,6 0,893 0,913 785,3 749,5 1,05


0 533,8 860,5 860,5 579,2 1,270 0,608 522,8 522,8 1,110,5 612,8 860,5 851,5 624,3 1,185 0,684 588,1 588,1 1,06

0,75 652,7 860,5 839,1 678,2 1,148 0,718 617,6 617,6 1,101 709,4 860,5 826,7 716,6 1,101 0,762 655,4 655,4 1,09

1,5 829,5 860,5 790,9 749,9 1,018 0,839 722,2 722,2 1,042 981,6 860,5 749,5 802,9 0,936 0,913 785,7 749,5 1,07

0,000

ψ = -‐ 1

ψ = -‐ 0.5

ψ = 0

ψ = 0.5

ψ = 1HEB300 / L = 25 m / fy = 460 MPa


A2.51


0 15920,7 1709,8 1709,8 1868,6 0,328 1,000 1709,8 1709,8 1,090,5 17927,1 1709,8 1675,7 1805,8 0,309 1,000 1709,8 1675,7 1,08

0,75 19322,3 1709,8 1631,6 1728,3 0,297 1,000 1709,8 1631,6 1,061 20490,2 1709,8 1587,5 1633,9 0,289 1,000 1709,8 1587,5 1,03

1,5 23867,4 1709,8 1472,8 1604,3 0,268 1,000 1709,8 1472,8 1,092 28490,7 1709,8 1360,2 1413,6 0,245 1,000 1709,8 1360,2 1,04


0 21009,8 1709,8 1709,8 1885,8 0,285 1,000 1709,8 1709,8 1,100,5 24596,1 1709,8 1675,7 1763,8 0,264 1,000 1709,8 1675,7 1,05

0,75 27216,9 1709,8 1631,6 1804,4 0,251 1,000 1709,8 1631,6 1,111 29566,5 1709,8 1587,5 1778,4 0,240 1,000 1709,8 1587,5 1,12

1,5 36826,4 1709,8 1472,8 1695,0 0,215 1,000 1709,8 1472,8 1,152 48132,6 1709,8 1360,2 1469,5 0,188 1,000 1709,8 1360,2 1,08


0 29281,3 1709,8 1709,8 1885,8 0,242 1,000 1709,8 1709,8 1,100,5 36033,2 1709,8 1675,7 1763,8 0,218 1,000 1709,8 1675,7 1,05

0,75 41686,6 1709,8 1631,6 1804,4 0,203 1,000 1709,8 1631,6 1,111 46052,5 1709,8 1587,5 1778,4 0,193 1,000 1709,8 1587,5 1,12

1,5 61579,2 1709,8 1472,8 1695,0 0,167 1,000 1709,8 1472,8 1,152 87105,9 1709,8 1360,2 1469,5 0,140 1,000 1709,8 1360,2 1,08


0 40384,4 1709,8 1709,8 1853,2 0,206 1,000 1709,8 1709,8 1,080,5 50559,0 1709,8 1675,7 1733,1 0,184 1,000 1709,8 1675,7 1,03

0,75 61183,1 1709,8 1631,6 1837,1 0,167 1,000 1709,8 1631,6 1,131 65559,0 1709,8 1587,5 1705,3 0,161 1,000 1709,8 1587,5 1,07

1,5 86642,3 1709,8 1472,8 1590,6 0,140 1,000 1709,8 1472,8 1,082 118774,0 1709,8 1360,2 1411,9 0,120 1,000 1709,8 1360,2 1,04


0 41383,3 1709,8 1709,8 1875,7 0,203 1,000 1709,8 1709,8 1,100,5 49552,3 1709,8 1675,7 1754,1 0,186 1,000 1709,8 1675,7 1,05

0,75 5464,8 1709,8 1631,6 1859,3 0,559 1,000 1709,8 1631,6 1,141 60694,5 1709,8 1587,5 1726,0 0,168 1,000 1709,8 1587,5 1,09

1,5 76406,2 1709,8 1472,8 1609,9 0,150 1,000 1709,8 1472,8 1,092 99226,9 1709,8 1360,2 1429,0 0,131 1,000 1709,8 1360,2 1,05

ψ = 1

ψ = 0.5

HEB500 / L = 2 m / fy = 355 MPa

ψ = 0

ψ = -‐ 0.5

ψ = -‐ 1


A2.52


0 15920,7 2215,6 2215,6 2429,2 0,373 1,000 2215,6 2215,6 1,100,5 17927,1 2215,6 2171,4 2347,6 0,352 1,000 2215,6 2171,4 1,08

0,75 19322,3 2215,6 2114,2 2419,7 0,339 1,000 2215,6 2114,2 1,141 20490,2 2215,6 2057,0 2124,1 0,329 1,000 2215,6 2057,0 1,03

1,5 23867,4 2215,6 1908,4 2085,6 0,305 1,000 2215,6 1908,4 1,092 28490,7 2215,6 1762,5 1696,3 0,279 1,000 2215,6 1762,5 0,96


0 21009,8 2215,6 2215,6 2451,5 0,325 1,000 2215,6 2215,6 1,110,5 24596,1 2215,6 2171,4 2293,0 0,300 1,000 2215,6 2171,4 1,06

0,75 27216,9 2215,6 2114,2 2526,2 0,285 1,000 2215,6 2114,2 1,191 29566,5 2215,6 2057,0 2311,9 0,274 1,000 2215,6 2057,0 1,12

1,5 36826,4 2215,6 1908,4 2203,6 0,245 1,000 2215,6 1908,4 1,152 48132,6 2215,6 1762,5 1763,4 0,215 1,000 2215,6 1762,5 1,00


0 29281,3 2215,6 2215,6 2451,5 0,275 1,000 2215,6 2215,6 1,110,5 36033,2 2215,6 2171,4 2293,0 0,248 1,000 2215,6 2171,4 1,06

0,75 41686,6 2215,6 2114,2 2363,8 0,231 1,000 2215,6 2114,2 1,121 46052,5 2215,6 2057,0 2311,9 0,219 1,000 2215,6 2057,0 1,12

1,5 61579,2 2215,6 1908,4 2203,6 0,190 1,000 2215,6 1908,4 1,152 87105,9 2215,6 1762,5 1763,4 0,159 1,000 2215,6 1762,5 1,00


0 40384,4 2215,6 2215,6 2401,3 0,234 1,000 2215,6 2215,6 1,080,5 50559,0 2215,6 2171,4 2245,7 0,209 1,000 2215,6 2171,4 1,03

0,75 61183,1 2215,6 2114,2 2380,4 0,190 1,000 2215,6 2114,2 1,131 65559,0 2215,6 2057,0 2209,7 0,184 1,000 2215,6 2057,0 1,07

1,5 86642,3 2215,6 1908,4 2061,1 0,160 1,000 2215,6 1908,4 1,082 118774,0 2215,6 1762,5 1829,5 0,137 1,000 2215,6 1762,5 1,04


0 41383,3 2215,6 2215,6 2438,4 0,231 1,000 2215,6 2215,6 1,100,5 49552,3 2215,6 2171,4 2280,3 0,211 1,000 2215,6 2171,4 1,05

0,75 5464,8 2215,6 2114,2 2138,2 0,637 1,000 2215,6 2114,2 1,011 60694,5 2215,6 2057,0 2243,8 0,191 1,000 2215,6 2057,0 1,09

1,5 76406,2 2215,6 1908,4 2092,9 0,170 1,000 2215,6 1908,4 1,102 99226,9 2215,6 1762,5 1886,3 0,149 1,000 2215,6 1762,5 1,07

ψ = 1

ψ = 0.5

ψ = 0

ψ = -‐ 0.5

HEB500 / L = 2 m / fy = 460 MPa

ψ = -‐ 1


A2.53


0 5811,3 1709,8 1709,8 1582,5 0,542 1,000 1709,8 1709,8 0,930,5 6556,5 1709,8 1675,7 1645,0 0,511 1,000 1709,8 1675,7 0,98

0,75 6933,4 1709,8 1631,6 1660,7 0,497 1,000 1709,8 1631,6 1,021 7513,3 1709,8 1587,5 1692,5 0,477 1,000 1709,8 1587,5 1,07

1,5 8784,9 1709,8 1472,8 1614,0 0,441 1,000 1709,8 1472,8 1,102 10552,8 1709,8 1360,2 1568,1 0,403 1,000 1709,8 1360,2 1,15


0 7672,0 1709,8 1709,8 1734,9 0,472 1,000 1709,8 1709,8 1,010,5 9005,5 1709,8 1675,7 1706,0 0,436 1,000 1709,8 1675,7 1,02

0,75 9815,0 1709,8 1631,6 1696,4 0,417 1,000 1709,8 1631,6 1,041 10867,8 1709,8 1587,5 1643,4 0,397 1,000 1709,8 1587,5 1,04

1,5 13630,9 1709,8 1472,8 1629,1 0,354 1,000 1709,8 1472,8 1,112 18082,5 1709,8 1360,2 1577,0 0,308 1,000 1709,8 1360,2 1,16


0 10721,8 1709,8 1709,8 1732,6 0,399 1,000 1709,8 1709,8 1,010,5 13240,3 1709,8 1675,7 1766,9 0,359 1,000 1709,8 1675,7 1,05

0,75 14797,2 1709,8 1631,6 1792,3 0,340 1,000 1709,8 1631,6 1,101 17016,3 1709,8 1587,5 1724,4 0,317 1,000 1709,8 1587,5 1,09

1,5 23022,3 1709,8 1472,8 1657,0 0,273 1,000 1709,8 1472,8 1,132 33154,0 1709,8 1360,2 1543,3 0,227 1,000 1709,8 1360,2 1,13


0 14960,1 1709,8 1709,8 1845,7 0,338 1,000 1709,8 1709,8 1,080,5 18773,3 1709,8 1675,7 1864,4 0,302 1,000 1709,8 1675,7 1,11

0,75 22120,4 1709,8 1631,6 1797,4 0,278 1,000 1709,8 1631,6 1,101 24313,8 1709,8 1587,5 1715,0 0,265 1,000 1709,8 1587,5 1,08

1,5 32817,0 1709,8 1472,8 1643,5 0,228 1,000 1709,8 1472,8 1,122 46715,3 1709,8 1360,2 1509,1 0,191 1,000 1709,8 1360,2 1,11


0 15620,3 1709,8 1709,8 1853,9 0,331 1,000 1709,8 1709,8 1,080,5 18653,1 1709,8 1675,7 1892,8 0,303 1,000 1709,8 1675,7 1,13

0,75 20137,0 1709,8 1631,6 1761,4 0,291 1,000 1709,8 1631,6 1,081 22878,8 1709,8 1587,5 1680,7 0,273 1,000 1709,8 1587,5 1,06

1,5 29082,5 1709,8 1472,8 1610,6 0,242 1,000 1709,8 1472,8 1,092 38796,2 1709,8 1360,2 1433,6 0,210 1,000 1709,8 1360,2 1,05

HEB500 / L = 3.5 m / fy = 355 MPa

ψ = -‐ 1

ψ = -‐ 0.5

ψ = 0

ψ = 0.5

ψ = 1


A2.54


0 5811,3 2215,6 1709,8 1733,3 0,617 0,909 2014,3 1709,8 1,010,5 6556,5 2215,6 1675,7 1787,1 0,581 0,925 2050,4 1675,7 1,07

0,75 6933,4 2215,6 1631,6 1720,0 0,565 1,000 2215,6 1631,6 1,051 7513,3 2215,6 1587,5 1687,0 0,543 1,000 2215,6 1587,5 1,06

1,5 8784,9 2215,6 1472,8 1603,1 0,502 1,000 2215,6 1472,8 1,092 10552,8 2215,6 1360,2 1509,8 0,458 1,000 2215,6 1360,2 1,11


0 7672,0 2215,6 1709,8 1797,7 0,537 1,000 2215,6 1709,8 1,050,5 9005,5 2215,6 1675,7 1755,1 0,496 1,000 2215,6 1675,7 1,05

0,75 9815,0 2215,6 1631,6 1643,4 0,475 1,000 2215,6 1631,6 1,011 10867,8 2215,6 1587,5 1623,1 0,452 1,000 2215,6 1587,5 1,02

1,5 13630,9 2215,6 1472,8 1489,4 0,403 1,000 2215,6 1472,8 1,012 18082,5 2215,6 1360,2 1454,2 0,350 1,000 2215,6 1360,2 1,07


0 10721,8 2215,6 1709,8 1750,5 0,455 1,000 2215,6 1709,8 1,020,5 13240,3 2215,6 1675,7 1728,9 0,409 1,000 2215,6 1675,7 1,03

0,75 14797,2 2215,6 1631,6 1683,3 0,387 1,000 2215,6 1631,6 1,031 17016,3 2215,6 1587,5 1662,5 0,361 1,000 2215,6 1587,5 1,05

1,5 23022,3 2215,6 1472,8 1525,6 0,310 1,000 2215,6 1472,8 1,042 33154,0 2215,6 1360,2 1388,3 0,259 1,000 2215,6 1360,2 1,02


0 14960,1 2215,6 1709,8 1784,7 0,385 1,000 2215,6 1709,8 1,040,5 18773,3 2215,6 1675,7 1762,6 0,344 1,000 2215,6 1675,7 1,05

0,75 22120,4 2215,6 1631,6 1843,2 0,316 1,000 2215,6 1631,6 1,131 24313,8 2215,6 1587,5 1694,9 0,302 1,000 2215,6 1587,5 1,07

1,5 32817,0 2215,6 1472,8 1555,3 0,260 1,000 2215,6 1472,8 1,062 46715,3 2215,6 1360,2 1415,3 0,218 1,000 2215,6 1360,2 1,04


0 15620,3 2215,6 1709,8 1823,0 0,377 1,000 2215,6 1709,8 1,070,5 18653,1 2215,6 1675,7 1797,0 0,345 1,000 2215,6 1675,7 1,07

0,75 20137,0 2215,6 1631,6 1704,2 0,332 1,000 2215,6 1631,6 1,041 22878,8 2215,6 1587,5 1683,2 0,311 1,000 2215,6 1587,5 1,06

1,5 29082,5 2215,6 1472,8 1544,5 0,276 1,000 2215,6 1472,8 1,052 38796,2 2215,6 1360,2 1405,5 0,239 1,000 2215,6 1360,2 1,03

HEB500 / L = 3.5 m / fy = 460 MPaψ = 1

ψ = 0.5

ψ = 0

ψ = -‐ 0.5

ψ = -‐ 1


A2.55


0 3219,6 1709,8 1709,8 1449,9 0,729 0,855 1462,0 1462,0 0,990,5 3645,8 1709,8 1675,7 1572,6 0,685 0,877 1499,8 1499,8 1,05

0,75 3877,3 1709,8 1631,6 1604,5 0,664 0,887 1517,1 1517,1 1,061 4193,6 1709,8 1587,5 1622,9 0,639 0,899 1537,8 1537,8 1,06

1,5 4923,5 1709,8 1472,8 1522,4 0,589 0,922 1576,3 1472,8 1,032 5942,3 1709,8 1360,2 1451,8 0,536 1,000 1709,8 1360,2 1,07


0 4249,4 1709,8 1709,8 1743,9 0,634 0,966 1651,7 1651,7 1,060,5 5010,4 1709,8 1675,7 1774,1 0,584 0,988 1688,7 1675,7 1,06

0,75 5498,1 1709,8 1631,6 1788,0 0,558 1,000 1709,8 1631,6 1,101 6073,8 1709,8 1587,5 1726,3 0,531 1,000 1709,8 1587,5 1,09

1,5 7654,1 1709,8 1472,8 1663,5 0,473 1,000 1709,8 1472,8 1,132 10211,5 1709,8 1360,2 1566,9 0,409 1,000 1709,8 1360,2 1,15


0 5930,5 1709,8 1709,8 1803,5 0,537 1,000 1709,8 1709,8 1,050,5 7348,9 1709,8 1675,7 1829,7 0,482 1,000 1709,8 1675,7 1,09

0,75 8121,8 1709,8 1631,6 1745,7 0,459 1,000 1709,8 1631,6 1,071 9457,0 1709,8 1587,5 1644,9 0,425 1,000 1709,8 1587,5 1,04

1,5 12782,5 1709,8 1472,8 1569,6 0,366 1,000 1709,8 1472,8 1,072 18401,0 1709,8 1360,2 1402,1 0,305 1,000 1709,8 1360,2 1,03


0 8270,2 1709,8 1709,8 1815,3 0,455 1,000 1709,8 1709,8 1,060,5 10358,1 1709,8 1675,7 1752,6 0,406 1,000 1709,8 1675,7 1,05

0,75 11775,3 1709,8 1631,6 1641,3 0,381 1,000 1709,8 1631,6 1,011 13356,1 1709,8 1587,5 1584,6 0,358 1,000 1709,8 1587,5 1,00

1,5 17964,3 1709,8 1472,8 1529,8 0,309 1,000 1709,8 1472,8 1,042 25670,9 1709,8 1360,2 1371,6 0,258 1,000 1709,8 1360,2 1,01


0 8712,5 1709,8 1709,8 1824,4 0,443 1,000 1709,8 1709,8 1,070,5 10331,0 1709,8 1675,7 1761,3 0,407 1,000 1709,8 1675,7 1,05

0,75 11307,8 1709,8 1631,6 1655,3 0,389 1,000 1709,8 1631,6 1,011 12596,4 1709,8 1587,5 1609,5 0,368 1,000 1709,8 1587,5 1,01

1,5 15956,9 1709,8 1472,8 1537,5 0,327 1,000 1709,8 1472,8 1,042 21339,4 1709,8 1360,2 1372,9 0,283 1,000 1709,8 1360,2 1,01

ψ = 1

ψ = 0

ψ = -‐ 0.5

ψ = -‐ 1

ψ = 0.5

HEB500 / L = 5 m / fy = 355 MPa


A2.56


0 3219,6 2215,6 1709,8 1791,9 0,830 0,801 1774,0 1709,8 1,050,5 3645,8 2215,6 1675,7 1707,6 0,780 0,828 1835,1 1675,7 1,02

0,75 3877,3 2215,6 1631,6 1685,9 0,756 0,841 1863,1 1631,6 1,031 4193,6 2215,6 1587,5 1657,5 0,727 0,856 1896,6 1587,5 1,04

1,5 4923,5 2215,6 1472,8 1544,1 0,671 0,884 1958,5 1472,8 1,052 5942,3 2215,6 1360,2 1438,7 0,611 0,912 2021,3 1360,2 1,06


0 4249,4 2215,6 1709,8 1800,2 0,722 0,923 2045,1 1709,8 1,050,5 5010,4 2215,6 1675,7 1713,0 0,665 0,952 2108,7 1675,7 1,02

0,75 5498,1 2215,6 1631,6 1690,5 0,635 0,966 2139,8 1631,6 1,041 6073,8 2215,6 1587,5 1661,1 0,604 0,979 2169,8 1587,5 1,05

1,5 7654,1 2215,6 1472,8 1543,9 0,538 1,000 2215,6 1472,8 1,052 10211,5 2215,6 1360,2 1533,0 0,466 1,000 2215,6 1360,2 1,13


0 5930,5 2215,6 1709,8 1862,4 0,611 1,000 2215,6 1709,8 1,090,5 7348,9 2215,6 1675,7 1771,4 0,549 1,000 2215,6 1675,7 1,06

0,75 8121,8 2215,6 1631,6 1747,9 0,522 1,000 2215,6 1631,6 1,071 9457,0 2215,6 1587,5 1717,3 0,484 1,000 2215,6 1587,5 1,08

1,5 12782,5 2215,6 1472,8 1594,9 0,416 1,000 2215,6 1472,8 1,082 18401,0 2215,6 1360,2 1583,6 0,347 1,000 2215,6 1360,2 1,16


0 8270,2 2215,6 1709,8 1943,0 0,518 1,000 2215,6 1709,8 1,140,5 10358,1 2215,6 1675,7 1848,0 0,462 1,000 2215,6 1675,7 1,10

0,75 11775,3 2215,6 1631,6 1823,6 0,434 1,000 2215,6 1631,6 1,121 13356,1 2215,6 1587,5 1791,6 0,407 1,000 2215,6 1587,5 1,13

1,5 17964,3 2215,6 1472,8 1664,0 0,351 1,000 2215,6 1472,8 1,132 25670,9 2215,6 1360,2 1583,8 0,294 1,000 2215,6 1360,2 1,16


0 8712,5 2215,6 1709,8 1921,1 0,504 1,000 2215,6 1709,8 1,120,5 10331,0 2215,6 1675,7 1783,8 0,463 1,000 2215,6 1675,7 1,06

0,75 11307,8 2215,6 1631,6 1717,6 0,443 1,000 2215,6 1631,6 1,051 12596,4 2215,6 1587,5 1653,8 0,419 1,000 2215,6 1587,5 1,04

1,5 15956,9 2215,6 1472,8 1641,9 0,373 1,000 2215,6 1472,8 1,112 21339,4 2215,6 1360,2 1524,6 0,322 1,000 2215,6 1360,2 1,12

ψ = 0.5

ψ = 0

ψ = -‐ 0.5

ψ = -‐ 1

ψ = 1HEB500 / L = 5 m / fy = 460 MPa


A2.57


0 1617,7 1709,8 1709,8 1290,3 1,028 0,682 1166,9 1166,9 1,110,5 1855,8 1709,8 1675,7 1320,6 0,960 0,724 1238,1 1238,1 1,07

0,75 1995,2 1709,8 1631,6 1378,3 0,926 0,745 1273,3 1273,3 1,081 2162,6 1709,8 1587,5 1391,4 0,889 0,766 1310,4 1310,4 1,06

1,5 2572,0 1709,8 1472,8 1410,2 0,815 0,809 1382,7 1382,7 1,022 3144,9 1709,8 1360,2 1487,7 0,737 0,851 1454,4 1360,2 1,09


0 2133,1 1709,8 1709,8 1501,6 0,895 0,820 1401,7 1401,7 1,070,5 2555,8 1709,8 1675,7 1696,2 0,818 0,869 1485,3 1485,3 1,14

0,75 2744,8 1709,8 1631,6 1717,1 0,789 0,886 1514,5 1514,5 1,131 3144,9 1709,8 1587,5 1733,2 0,737 0,915 1564,3 1564,3 1,11

1,5 4014,9 1709,8 1472,8 1646,5 0,653 0,958 1637,4 1472,8 1,122 5407,7 1709,8 1360,2 1562,0 0,562 1,000 1709,8 1360,2 1,15


0 2961,4 1709,8 1709,8 1743,1 0,760 0,978 1671,5 1671,5 1,040,5 3719,8 1709,8 1675,7 1788,1 0,678 1,000 1709,8 1675,7 1,07

0,75 4175,0 1709,8 1631,6 1710,1 0,640 1,000 1709,8 1631,6 1,051 4811,6 1709,8 1587,5 1660,4 0,596 1,000 1709,8 1587,5 1,05

1,5 6469,0 1709,8 1472,8 1533,1 0,514 1,000 1709,8 1472,8 1,042 9180,6 1709,8 1360,2 1482,7 0,432 1,000 1709,8 1360,2 1,09


0 4097,2 1709,8 1709,8 1759,4 0,646 1,000 1709,8 1709,8 1,030,5 5137,4 1709,8 1675,7 1804,8 0,577 1,000 1709,8 1675,7 1,08

0,75 5542,3 1709,8 1631,6 1726,0 0,555 1,000 1709,8 1631,6 1,061 6571,7 1709,8 1587,5 1675,8 0,510 1,000 1709,8 1587,5 1,06

1,5 8732,3 1709,8 1472,8 1547,4 0,443 1,000 1709,8 1472,8 1,052 12336,9 1709,8 1360,2 1496,5 0,372 1,000 1709,8 1360,2 1,10


0 4373,3 1709,8 1709,8 1780,3 0,625 1,000 1709,8 1709,8 1,040,5 5134,1 1709,8 1675,7 1826,2 0,577 1,000 1709,8 1675,7 1,09

0,75 5691,0 1709,8 1631,6 1746,6 0,548 1,000 1709,8 1631,6 1,071 6198,7 1709,8 1587,5 1695,8 0,525 1,000 1709,8 1587,5 1,07

1,5 7781,2 1709,8 1472,8 1565,8 0,469 1,000 1709,8 1472,8 1,062 10335,5 1709,8 1360,2 1514,3 0,407 1,000 1709,8 1360,2 1,11

ψ = 1HEB500 / L = 8 m / fy = 355 MPa

ψ = 0.5

ψ = 0

ψ = -‐ 0.5

ψ = -‐ 1


A2.58


0 1617,7 2215,6 1709,8 1393,3 1,170 0,597 1321,7 1321,7 1,050,5 1855,8 2215,6 1675,7 1572,8 1,093 0,643 1424,7 1424,7 1,10

0,75 1995,2 2215,6 1631,6 1620,8 1,054 0,667 1477,2 1477,2 1,101 2162,6 2215,6 1587,5 1713,6 1,012 0,692 1533,7 1533,7 1,12

1,5 2572,0 2215,6 1472,8 1646,7 0,928 0,743 1646,7 1472,8 1,122 3144,9 2215,6 1360,2 1535,7 0,839 0,795 1761,7 1360,2 1,13


0 2133,1 2215,6 1709,8 1701,7 1,019 0,735 1628,4 1628,4 1,040,5 2555,8 2215,6 1675,7 1593,7 0,931 0,796 1763,4 1675,7 0,95

0,75 2744,8 2215,6 1631,6 1641,2 0,898 0,818 1811,7 1631,6 1,011 3144,9 2215,6 1587,5 1732,4 0,839 0,856 1895,5 1587,5 1,09

1,5 4014,9 2215,6 1472,8 1664,9 0,743 0,912 2020,4 1472,8 1,132 5407,7 2215,6 1360,2 1535,7 0,640 0,963 2134,5 1360,2 1,13


0 2961,4 2215,6 1709,8 1956,9 0,865 0,909 2013,5 1709,8 1,140,5 3719,8 2215,6 1675,7 1751,3 0,772 0,970 2150,1 1675,7 1,05

0,75 4175,0 2215,6 1631,6 1660,6 0,728 0,996 2205,6 1631,6 1,021 4811,6 2215,6 1587,5 1753,0 0,679 1,000 2215,6 1587,5 1,10

1,5 6469,0 2215,6 1472,8 1684,6 0,585 1,000 2215,6 1472,8 1,142 9180,6 2215,6 1360,2 1554,2 0,491 1,000 2215,6 1360,2 1,14


0 4097,2 2215,6 1709,8 1931,9 0,735 1,000 2215,6 1709,8 1,130,5 5137,4 2215,6 1675,7 1815,7 0,657 1,000 2215,6 1675,7 1,08

0,75 5542,3 2215,6 1631,6 1647,1 0,632 1,000 2215,6 1631,6 1,011 6571,7 2215,6 1587,5 1631,8 0,581 1,000 2215,6 1587,5 1,03

1,5 8732,3 2215,6 1472,8 1616,7 0,504 1,000 2215,6 1472,8 1,102 12336,9 2215,6 1360,2 1586,8 0,424 1,000 2215,6 1360,2 1,17


0 4373,3 2215,6 1709,8 1883,2 0,712 1,000 2215,6 1709,8 1,100,5 5134,1 2215,6 1675,7 1769,9 0,657 1,000 2215,6 1675,7 1,06

0,75 5691,0 2215,6 1631,6 1605,6 0,624 1,000 2215,6 1631,6 0,981 6198,7 2215,6 1587,5 1590,7 0,598 1,000 2215,6 1587,5 1,00

1,5 7781,2 2215,6 1472,8 1575,9 0,534 1,000 2215,6 1472,8 1,072 10335,5 2215,6 1360,2 1546,8 0,463 1,000 2215,6 1360,2 1,14

ψ = 1HEB500 / L = 8 m / fy = 460 MPa

ψ = 0.5

ψ = 0

ψ = -‐ 0.5

ψ = -‐ 1


A2.59


0 1207,1 1709,8 1709,8 1006,7 1,190 0,585 1000,1 1000,1 1,010,5 1401,1 1709,8 1675,7 1089,0 1,105 0,636 1087,0 1087,0 1,00

0,75 1490,6 1709,8 1631,6 1205,4 1,071 0,656 1122,0 1122,0 1,071 1651,7 1709,8 1587,5 1316,1 1,017 0,689 1178,1 1178,1 1,12

1,5 1986,8 1709,8 1472,8 1432,1 0,928 0,744 1271,3 1271,3 1,132 2456,2 1709,8 1360,2 1440,8 0,834 0,798 1364,4 1360,2 1,06


0 1590,6 1709,8 1709,8 1300,9 1,037 0,723 1235,5 1235,5 1,050,5 1933,6 1709,8 1675,7 1473,1 0,940 0,790 1350,1 1350,1 1,09

0,75 2247,5 1709,8 1631,6 1564,2 0,872 0,835 1427,3 1427,3 1,101 2410,8 1709,8 1587,5 1590,3 0,842 0,854 1459,9 1459,9 1,09

1,5 3111,2 1709,8 1472,8 1556,3 0,741 0,913 1560,7 1472,8 1,062 4219,5 1709,8 1360,2 1469,8 0,637 0,965 1650,0 1360,2 1,08


0 2199,5 1709,8 1709,8 1622,8 0,882 0,897 1533,3 1533,3 1,060,5 2798,2 1709,8 1675,7 1755,4 0,782 0,964 1648,9 1648,9 1,06

0,75 3294,8 1709,8 1631,6 1766,5 0,720 1,000 1709,7 1631,6 1,081 3639,5 1709,8 1587,5 1703,5 0,685 1,000 1709,8 1587,5 1,07

1,5 4882,6 1709,8 1472,8 1629,0 0,592 1,000 1709,8 1472,8 1,112 6891,0 1709,8 1360,2 1504,3 0,498 1,000 1709,8 1360,2 1,11


0 3022,7 1709,8 1709,8 1772,2 0,752 1,000 1709,8 1709,8 1,040,5 3809,2 1709,8 1675,7 1813,4 0,670 1,000 1709,8 1675,7 1,08

0,75 4361,2 1709,8 1631,6 1658,5 0,626 1,000 1709,8 1631,6 1,021 4864,5 1709,8 1587,5 1588,3 0,593 1,000 1709,8 1587,5 1,00

1,5 6430,6 1709,8 1472,8 1524,0 0,516 1,000 1709,8 1472,8 1,032 9042,3 1709,8 1360,2 1502,0 0,435 1,000 1709,8 1360,2 1,10


0 4138,3 1709,8 1709,8 1789,9 0,643 1,000 1709,8 1709,8 1,050,5 4955,2 1709,8 1675,7 1831,6 0,587 1,000 1709,8 1675,7 1,09

0,75 5454,2 1709,8 1631,6 1708,3 0,560 1,000 1709,8 1631,6 1,051 6069,5 1709,8 1587,5 1604,2 0,531 1,000 1709,8 1587,5 1,01

1,5 7646,2 1709,8 1472,8 1585,0 0,473 1,000 1709,8 1472,8 1,082 9926,9 1709,8 1360,2 1517,0 0,415 1,000 1709,8 1360,2 1,12

ψ = 0

HEB500 / L = 10 m / fy = 355 MPaψ = 1

ψ = 0.5

ψ = -‐ 0.5

ψ = -‐ 1


A2.60


0 1207,1 2215,6 1709,8 1099,1 1,355 0,495 1097,0 1097,0 1,000,5 1401,1 2215,6 1675,7 1307,3 1,258 0,547 1211,2 1211,2 1,08

0,75 1490,6 2215,6 1631,6 1443,0 1,219 0,568 1259,0 1259,0 1,151 1651,7 2215,6 1587,5 1509,9 1,158 0,604 1337,5 1337,5 1,13

1,5 1986,8 2215,6 1472,8 1589,0 1,056 0,665 1474,2 1472,8 1,082 2456,2 2215,6 1360,2 1568,2 0,950 0,730 1617,9 1360,2 1,15


0 1590,6 2215,6 1709,8 1433,5 1,180 0,622 1378,2 1378,2 1,040,5 1933,6 2215,6 1675,7 1785,7 1,070 0,699 1548,2 1548,2 1,15

0,75 2247,5 2215,6 1631,6 1688,8 0,993 0,753 1669,2 1631,6 1,041 2410,8 2215,6 1587,5 1565,8 0,959 0,777 1721,7 1587,5 0,99

1,5 3111,2 2215,6 1472,8 1492,5 0,844 0,853 1889,3 1472,8 1,012 4219,5 2215,6 1360,2 1427,5 0,725 0,922 2042,1 1360,2 1,05


0 2199,5 2215,6 1709,8 1677,2 1,004 0,802 1777,3 1709,8 0,980,5 2798,2 2215,6 1675,7 1821,4 0,890 0,891 1973,6 1675,7 1,09

0,75 3294,8 2215,6 1631,6 1722,6 0,820 0,940 2082,1 1631,6 1,061 3639,5 2215,6 1587,5 1597,1 0,780 0,965 2138,6 1587,5 1,01

1,5 4882,6 2215,6 1472,8 1522,3 0,674 1,000 2215,6 1472,8 1,032 6891,0 2215,6 1360,2 1456,1 0,567 1,000 2215,6 1360,2 1,07


0 3022,7 2215,6 1709,8 1730,7 0,856 0,940 2083,2 1709,8 1,010,5 3809,2 2215,6 1675,7 1688,3 0,763 1,000 2215,6 1675,7 1,01

0,75 4361,2 2215,6 1631,6 1647,0 0,713 1,000 2215,6 1631,6 1,011 4864,5 2215,6 1587,5 1606,6 0,675 1,000 2215,6 1587,5 1,01

1,5 6430,6 2215,6 1472,8 1567,3 0,587 1,000 2215,6 1472,8 1,062 9042,3 2215,6 1360,2 1528,9 0,495 1,000 2215,6 1360,2 1,12


0 4138,3 2215,6 1709,8 1888,1 0,732 1,000 2215,6 1709,8 1,100,5 4955,2 2215,6 1675,7 1841,9 0,669 1,000 2215,6 1675,7 1,10

0,75 5454,2 2215,6 1631,6 1678,4 0,637 1,000 2215,6 1631,6 1,031 6069,5 2215,6 1587,5 1621,2 0,604 1,000 2215,6 1587,5 1,02

1,5 7646,2 2215,6 1472,8 1596,9 0,538 1,000 2215,6 1472,8 1,082 9926,9 2215,6 1360,2 1522,2 0,472 1,000 2215,6 1360,2 1,12

ψ = -‐ 1

ψ = 0

HEB500 / L = 10 m / fy = 460 MPaψ = 1

ψ = 0.5

ψ = -‐ 0.5


A2.61


0 745,8 1709,8 1709,8 702,3 1,514 0,421 720,2 720,2 0,980,5 898,6 1709,8 1675,7 880,8 1,379 0,483 825,6 825,6 1,07

0,75 963,6 1709,8 1631,6 1004,3 1,332 0,507 866,5 866,5 1,161 1098,0 1709,8 1587,5 1094,0 1,248 0,552 943,9 943,9 1,16

1,5 1365,3 1709,8 1472,8 1191,9 1,119 0,627 1072,1 1072,1 1,112 1738,6 1709,8 1360,2 1329,1 0,992 0,705 1205,0 1205,0 1,10


0 981,3 1709,8 1709,8 916,8 1,320 0,530 906,7 906,7 1,010,5 1248,8 1709,8 1675,7 1119,2 1,170 0,629 1075,5 1075,5 1,04

0,75 1477,3 1709,8 1631,6 1323,1 1,076 0,695 1188,3 1188,3 1,111 1619,2 1709,8 1587,5 1382,8 1,028 0,729 1246,5 1246,5 1,11

1,5 2149,3 1709,8 1472,8 1451,9 0,892 0,822 1405,4 1405,4 1,032 2953,8 1709,8 1360,2 1554,2 0,761 0,902 1542,2 1360,2 1,14


0 1346,9 1709,8 1709,8 1184,4 1,127 0,701 1198,6 1198,6 0,990,5 1786,7 1709,8 1675,7 1435,4 0,978 0,823 1406,6 1406,6 1,02

0,75 1996,1 1709,8 1631,6 1526,5 0,926 0,864 1477,2 1477,2 1,031 2368,3 1709,8 1587,5 1550,3 0,850 0,920 1572,3 1572,3 0,99

1,5 3178,6 1709,8 1472,8 1538,9 0,733 0,993 1697,5 1472,8 1,042 4416,6 1709,8 1360,2 1505,7 0,622 1,000 1709,8 1360,2 1,11


0 1828,7 1709,8 1709,8 1481,6 0,967 0,853 1458,8 1458,8 1,020,5 2360,1 1709,8 1675,7 1671,8 0,851 0,944 1613,8 1613,8 1,04

0,75 2694,1 1709,8 1631,6 1679,7 0,797 0,981 1678,1 1631,6 1,031 3031,5 1709,8 1587,5 1611,8 0,751 1,000 1709,8 1587,5 1,02

1,5 3995,3 1709,8 1472,8 1549,7 0,654 1,000 1709,8 1472,8 1,052 5538,4 1709,8 1360,2 1521,4 0,556 1,000 1709,8 1360,2 1,12


0 1987,2 1709,8 1709,8 1595,4 0,928 0,921 1574,1 1574,1 1,010,5 2345,4 1709,8 1675,7 1687,5 0,854 0,981 1676,9 1675,7 1,01

0,75 2501,7 1709,8 1631,6 1695,5 0,827 1,000 1709,8 1631,6 1,041 2842,1 1709,8 1587,5 1627,9 0,776 1,000 1709,8 1587,5 1,03

1,5 3570,7 1709,8 1472,8 1568,2 0,692 1,000 1709,8 1472,8 1,062 4725,0 1709,8 1360,2 1540,6 0,602 1,000 1709,8 1360,2 1,13

ψ = -‐ 1

HEB500 / L = 15 m / fy = 355 MPaψ = 1

ψ = 0.5

ψ = -‐ 0.5

ψ = 0


A2.62


0 745,8 2215,6 1709,8 739,5 1,724 0,343 760,5 760,5 0,970,5 898,6 2215,6 1675,7 937,6 1,570 0,398 882,5 882,5 1,06

0,75 963,6 2215,6 1631,6 1020,5 1,516 0,420 931,2 931,2 1,101 1098,0 2215,6 1587,5 1115,9 1,420 0,463 1026,0 1026,0 1,09

1,5 1365,3 2215,6 1472,8 1278,1 1,274 0,538 1191,3 1191,3 1,072 1738,6 2215,6 1360,2 1453,8 1,129 0,621 1376,2 1360,2 1,07


0 981,3 2215,6 1709,8 918,0 1,503 0,427 945,0 945,0 0,970,5 1248,8 2215,6 1675,7 1238,2 1,332 0,523 1158,5 1158,5 1,07

0,75 1477,3 2215,6 1631,6 1471,7 1,225 0,592 1311,4 1311,4 1,121 1619,2 2215,6 1587,5 1543,3 1,170 0,629 1394,1 1394,1 1,11

1,5 2149,3 2215,6 1472,8 1585,5 1,015 0,738 1634,4 1472,8 1,082 2953,8 2215,6 1360,2 1545,6 0,866 0,839 1858,2 1360,2 1,14


0 1346,9 2215,6 1709,8 1602,7 1,283 0,577 1278,1 1278,1 1,250,5 1786,7 2215,6 1675,7 1732,2 1,114 0,712 1577,0 1577,0 1,10

0,75 1996,1 2215,6 1631,6 1731,2 1,054 0,761 1686,9 1631,6 1,061 2368,3 2215,6 1587,5 1644,1 0,967 0,831 1842,1 1587,5 1,04

1,5 3178,6 2215,6 1472,8 1578,0 0,835 0,930 2060,0 1472,8 1,072 4416,6 2215,6 1360,2 1463,2 0,708 1,000 2215,6 1360,2 1,08


0 1828,7 2215,6 1709,8 1682,2 1,101 0,738 1635,7 1635,7 1,030,5 2360,1 2215,6 1675,7 1818,1 0,969 0,852 1886,7 1675,7 1,08

0,75 2694,1 2215,6 1631,6 1801,6 0,907 0,902 1998,1 1631,6 1,101 3031,5 2215,6 1587,5 1725,6 0,855 0,941 2085,2 1587,5 1,09

1,5 3995,3 2215,6 1472,8 1621,5 0,745 1,000 2215,6 1472,8 1,102 5538,4 2215,6 1360,2 1535,8 0,632 1,000 2215,6 1360,2 1,13


0 1987,2 2215,6 1709,8 1716,5 1,056 0,804 1782,2 1709,8 1,000,5 2345,4 2215,6 1675,7 1855,2 0,972 0,882 1953,5 1675,7 1,11

0,75 2501,7 2215,6 1631,6 1838,4 0,941 0,909 2013,9 1631,6 1,131 2842,1 2215,6 1587,5 1760,8 0,883 0,958 2122,1 1587,5 1,11

1,5 3570,7 2215,6 1472,8 1654,6 0,788 1,000 2215,6 1472,8 1,122 4725,0 2215,6 1360,2 1567,1 0,685 1,000 2215,6 1360,2 1,15

ψ = -‐ 1

ψ = 0

HEB500 / L = 15 m / fy = 460 MPaψ = 1

ψ = 0.5

ψ = -‐ 0.5


A2.63


0 542,6 1709,8 1709,8 537,0 1,775 0,327 559,1 559,1 0,960,5 681,9 1709,8 1675,7 719,8 1,584 0,393 672,2 672,2 1,07

0,75 744,7 1709,8 1631,6 776,2 1,515 0,421 719,4 719,4 1,081 866,2 1709,8 1587,5 882,1 1,405 0,470 804,3 804,3 1,10

1,5 1113,5 1709,8 1472,8 1078,5 1,239 0,557 952,3 952,3 1,132 1457,3 1709,8 1360,2 1255,1 1,083 0,649 1109,3 1109,3 1,13


0 713,3 1709,8 1709,8 727,1 1,548 0,407 696,1 696,1 1,040,5 955,1 1709,8 1675,7 922,9 1,338 0,519 887,8 887,8 1,04

0,75 1074,3 1709,8 1631,6 1055,6 1,262 0,568 970,4 970,4 1,091 1288,0 1709,8 1587,5 1189,7 1,152 0,641 1096,7 1096,7 1,08

1,5 1750,4 1709,8 1472,8 1311,4 0,988 0,757 1293,6 1293,6 1,012 2425,1 1709,8 1360,2 1511,8 0,840 0,855 1462,5 1360,2 1,11


0 974,0 1709,8 1709,8 937,5 1,325 0,545 932,4 932,4 1,010,5 1352,0 1709,8 1675,7 1029,3 1,125 0,703 1201,5 1201,5 0,86

0,75 1627,4 1709,8 1631,6 1365,8 1,025 0,785 1341,9 1341,9 1,021 1826,9 1709,8 1587,5 1515,6 0,967 0,831 1421,3 1421,3 1,07

1,5 2463,6 1709,8 1472,8 1636,2 0,833 0,931 1591,8 1472,8 1,112 3409,0 1709,8 1360,2 1553,1 0,708 1,000 1709,8 1360,2 1,14


0 1311,1 1709,8 1709,8 1216,5 1,142 0,702 1201,0 1201,0 1,010,5 1742,9 1709,8 1675,7 1396,3 0,990 0,833 1425,1 1425,1 0,98

0,75 1929,6 1709,8 1631,6 1597,5 0,941 0,874 1494,9 1494,9 1,071 2266,9 1709,8 1587,5 1680,3 0,868 0,931 1592,1 1587,5 1,06

1,5 3002,2 1709,8 1472,8 1607,4 0,755 1,000 1709,8 1472,8 1,092 4147,6 1709,8 1360,2 1516,0 0,642 1,000 1709,8 1360,2 1,11


0 1432,0 1709,8 1709,8 1370,8 1,093 0,770 1316,2 1316,2 1,040,5 1718,6 1709,8 1675,7 1486,7 0,997 0,859 1468,1 1468,1 1,01

0,75 1947,6 1709,8 1631,6 1612,3 0,937 0,913 1560,3 1560,3 1,031 2113,4 1709,8 1587,5 1696,6 0,899 0,944 1614,6 1587,5 1,07

1,5 2685,7 1709,8 1472,8 1624,3 0,798 1,000 1709,8 1472,8 1,102 3575,5 1709,8 1360,2 1568,0 0,692 1,000 1709,8 1360,2 1,15

ψ = -‐ 0.5

ψ = -‐ 1

ψ = 0

HEB500 / L = 20 m / fy = 355 MPaψ = 1

ψ = 0.5


A2.64


0 542,6 2215,6 1709,8 597,3 2,021 0,262 581,6 581,6 1,030,5 681,9 2215,6 1675,7 729,4 1,803 0,319 706,3 706,3 1,03

0,75 744,7 2215,6 1631,6 812,6 1,725 0,343 759,6 759,6 1,071 866,2 2215,6 1587,5 922,3 1,599 0,387 857,5 857,5 1,08

1,5 1113,5 2215,6 1472,8 1107,6 1,411 0,468 1036,4 1036,4 1,072 1457,3 2215,6 1360,2 1296,7 1,233 0,560 1241,5 1241,5 1,04


0 713,3 2215,6 1709,8 756,7 1,762 0,331 733,1 733,1 1,030,5 955,1 2215,6 1675,7 959,4 1,523 0,417 925,0 925,0 1,04

0,75 1074,3 2215,6 1631,6 1151,3 1,436 0,462 1023,6 1023,6 1,121 1288,0 2215,6 1587,5 1349,5 1,312 0,536 1186,5 1186,5 1,14

1,5 1750,4 2215,6 1472,8 1588,8 1,125 0,660 1463,0 1463,0 1,092 2425,1 2215,6 1360,2 1635,1 0,956 0,779 1726,1 1360,2 1,20


0 974,0 2215,6 1709,8 979,1 1,508 0,424 938,9 938,9 1,040,5 1352,0 2215,6 1675,7 1291,3 1,280 0,579 1282,2 1282,2 1,01

0,75 1627,4 2215,6 1631,6 1585,5 1,167 0,668 1480,3 1480,3 1,071 1826,9 2215,6 1587,5 1559,4 1,101 0,722 1599,6 1587,5 0,98

1,5 2463,6 2215,6 1472,8 1506,6 0,948 0,846 1875,1 1472,8 1,022 3409,0 2215,6 1360,2 1402,1 0,806 0,949 2102,3 1360,2 1,03


0 1311,1 2215,6 1709,8 1291,0 1,300 0,571 1264,8 1264,8 1,020,5 1742,9 2215,6 1675,7 1465,4 1,127 0,715 1584,1 1584,1 0,93

0,75 1929,6 2215,6 1631,6 1804,9 1,072 0,764 1692,0 1631,6 1,111 2266,9 2215,6 1587,5 2082,1 0,989 0,835 1850,1 1587,5 1,31

1,5 3002,2 2215,6 1472,8 2171,6 0,859 0,938 2078,4 1472,8 1,472 4147,6 2215,6 1360,2 2327,8 0,731 1,000 2215,6 1360,2 1,71


0 1432,0 2215,6 1709,8 1434,7 1,244 0,630 1396,3 1396,3 1,030,5 1718,6 2215,6 1675,7 1674,4 1,135 0,730 1616,6 1616,6 1,04

0,75 1947,6 2215,6 1631,6 1640,0 1,067 0,794 1760,0 1631,6 1,011 2113,4 2215,6 1587,5 1571,6 1,024 0,834 1848,4 1587,5 0,99

1,5 2685,7 2215,6 1472,8 1506,9 0,908 0,937 2076,0 1472,8 1,022 3575,5 2215,6 1360,2 1468,0 0,787 1,000 2215,6 1360,2 1,08

HEB500 / L = 20 m / fy = 460 MPaψ = 1

ψ = 0.5

ψ = -‐ 0.5

ψ = -‐ 1

ψ = 0


A2.65


0 417,5 1709,8 1709,8 471,0 2,024 0,262 447,7 447,7 1,050,5 545,5 1709,8 1675,7 597,9 1,770 0,328 561,6 561,6 1,06

0,75 625,2 1709,8 1631,6 653,2 1,654 0,367 627,6 627,6 1,041 717,1 1709,8 1587,5 782,0 1,544 0,409 699,0 699,0 1,12

1,5 948,3 1709,8 1472,8 893,6 1,343 0,501 857,0 857,0 1,042 1269,3 1709,8 1360,2 1093,8 1,161 0,602 1029,7 1029,7 1,06


0 427,7 1709,8 1709,8 494,2 1,999 0,267 457,1 457,1 1,080,5 561,8 1709,8 1675,7 624,9 1,745 0,337 575,4 575,4 1,09

0,75 669,2 1709,8 1631,6 684,6 1,599 0,387 662,3 662,3 1,031 741,8 1709,8 1587,5 783,6 1,518 0,420 717,3 717,3 1,09

1,5 983,2 1709,8 1472,8 924,0 1,319 0,531 908,1 908,1 1,022 1316,1 1709,8 1360,2 1225,6 1,140 0,650 1111,4 1111,4 1,10


0 561,9 1709,8 1709,8 564,8 1,744 0,337 575,5 575,5 0,980,5 793,1 1709,8 1675,7 802,0 1,468 0,448 766,0 766,0 1,05

0,75 968,3 1709,8 1631,6 876,2 1,329 0,542 927,6 927,6 0,941 1108,5 1709,8 1587,5 1049,0 1,242 0,608 1039,8 1039,8 1,01

1,5 1533,3 1709,8 1472,8 1298,7 1,056 0,759 1298,3 1298,3 1,002 2134,0 1709,8 1360,2 1385,5 0,895 0,887 1516,4 1360,2 1,02


0 764,9 1709,8 1709,8 759,7 1,495 0,432 738,2 738,2 1,030,5 1111,3 1709,8 1675,7 1088,0 1,240 0,619 1058,3 1058,3 1,03

0,75 1364,2 1709,8 1631,6 1225,1 1,120 0,722 1234,3 1234,3 0,991 1529,0 1709,8 1587,5 1284,8 1,057 0,776 1326,6 1326,6 0,97

1,5 2075,4 1709,8 1472,8 1507,3 0,908 0,901 1540,9 1472,8 1,022 2871,1 1709,8 1360,2 1516,4 0,772 0,997 1705,3 1360,2 1,11


0 1120,7 1709,8 1709,8 1151,7 1,235 0,638 1090,7 1090,7 1,060,5 1373,7 1709,8 1675,7 1359,7 1,116 0,748 1279,3 1279,3 1,06

0,75 1490,2 1709,8 1631,6 1439,5 1,071 0,790 1350,9 1350,9 1,071 1720,6 1709,8 1587,5 1572,0 0,997 0,859 1469,0 1469,0 1,07

1,5 2217,3 1709,8 1472,8 1586,6 0,878 0,962 1644,3 1472,8 1,082 2975,0 1709,8 1360,2 1474,4 0,758 1,000 1709,8 1360,2 1,08

ψ = 1

ψ = 0.5

ψ = 0

ψ = -‐ 0.5

ψ = -‐ 1

HEB500 / L = 25 m / fy = 355 MPa


A2.66


0 417,5 2215,6 1709,8 472,5 2,304 0,208 461,8 461,8 1,020,5 545,5 2215,6 1675,7 604,2 2,015 0,264 584,2 584,2 1,03

0,75 625,2 2215,6 1631,6 733,9 1,882 0,296 656,6 656,6 1,121 717,1 2215,6 1587,5 850,3 1,758 0,332 736,4 736,4 1,15

1,5 948,3 2215,6 1472,8 1061,3 1,529 0,415 919,9 919,9 1,152 1269,3 2215,6 1360,2 1301,5 1,321 0,512 1135,3 1135,3 1,15


0 427,7 2215,6 1709,8 496,3 2,276 0,213 471,9 471,9 1,050,5 561,8 2215,6 1675,7 652,6 1,986 0,271 599,3 599,3 1,09

0,75 669,2 2215,6 1631,6 792,6 1,820 0,314 695,3 695,3 1,141 741,8 2215,6 1587,5 838,3 1,728 0,342 757,2 757,2 1,11

1,5 983,2 2215,6 1472,8 1038,6 1,501 0,427 946,7 946,7 1,102 1316,1 2215,6 1360,2 1305,6 1,297 0,544 1206,2 1206,2 1,08


0 561,9 2215,6 1709,8 650,4 1,986 0,271 599,4 599,4 1,080,5 793,1 2215,6 1675,7 841,7 1,671 0,361 799,5 799,5 1,05

0,75 968,3 2215,6 1631,6 976,0 1,513 0,422 934,7 934,7 1,041 1108,5 2215,6 1587,5 1171,6 1,414 0,483 1070,6 1070,6 1,09

1,5 1533,3 2215,6 1472,8 1529,8 1,202 0,640 1417,2 1417,2 1,082 2134,0 2215,6 1360,2 1514,4 1,019 0,790 1749,8 1360,2 1,11


0 764,9 2215,6 1709,8 811,3 1,702 0,350 776,3 776,3 1,040,5 1111,3 2215,6 1675,7 1147,5 1,412 0,487 1079,7 1079,7 1,060,75 1364,2 2215,6 1631,6 1328,6 1,274 0,591 1309,9 1309,9 1,01

1 1529,0 2215,6 1587,5 1546,0 1,204 0,650 1439,1 1439,1 1,071,5 2075,4 2215,6 1472,8 1604,5 1,033 0,797 1765,6 1472,8 1,092 2871,1 2215,6 1360,2 1568,8 0,878 0,924 2046,5 1360,2 1,15


0 1120,7 2215,6 1709,8 1137,5 1,406 0,496 1099,1 1099,1 1,030,5 1373,7 2215,6 1675,7 1416,4 1,270 0,607 1345,3 1345,3 1,05

0,75 1490,2 2215,6 1631,6 1480,0 1,219 0,652 1445,0 1445,0 1,021 1720,6 2215,6 1587,5 1719,0 1,135 0,730 1617,9 1587,5 1,08

1,5 2217,3 2215,6 1472,8 1589,8 1,000 0,857 1897,9 1472,8 1,082 2975,0 2215,6 1360,2 1502,8 0,863 0,974 2157,2 1360,2 1,10

ψ = 0

ψ = -‐ 0.5

ψ = -‐ 1

HEB500 / L = 25 m / fy = 460 MPaψ = 1

ψ = 0.5

A3 - 1

Annex 3: Measured Initial Geometrical Imperfections

This annex presents the initial geometrical imperfections recorded in the tested IPE 200 and HEA 160

beams. The initial displacements were measured along longitudinal lines passing through the

cross-section points indicated in Figures A3.1(a)-(b). The corresponding initial displacement profiles

are displayed in Figures (i) A3.2 to A3.4 (IPE 200 beam) and (ii) A3.5 to A3.7 (HEA 160 beam) −

these displacement profiles comprise the whole beam length (including the two outstand segments).

(a) (b)

Figure A3.1 – Cross-section points for which initial displacement profiles were measured: (a) IPE 200 and (b) HEA 160 beam


0 1000 2000 3000 4000

Vert

ical

dis

plac

emen

t [m

m]

0

1

2

3

4

5

6

7

8Point APoint BPoint C

Figure A3.2 – Initial vertical displacement longitudinal profiles measured in the IPE 200 beam top flange

Measurement Position along the Beam Length [mm]

A3 - 2


0 1000 2000 3000 4000

Late

ral d

ispl

acem

ent

[mm

]

0

2

4

6

8

10

12

14

16Point EPoint FPoint D

Figure A3.3 – Initial lateral displacement longitudinal profiles measured in the IPE 200 beam web


0 1000 2000 3000 4000

Vert

ical

dis

plac

emen

t [m

m]

0123456789

101112

Point GPoint HPoint I

Figure A3.4 – Initial vertical displacement longitudinal profiles measured in the IPE 200 beam bottom flange



A3 - 3


0 1000 2000 3000 4000

Vert

ical

dis

plac

emen

t [m

m]

0

1

2

3

4

5

6

7Point APoint BPoint C

Figure A3.5 – – Initial vertical displacement longitudinal profiles measured in the HEA 160 beam top flange


0 1000 2000 3000 4000

Late

ral d

ispl

acem

ent

[mm

]

0

1

2

3

4

5Point EPoint FPoint D

Figure A3.6 – Initial lateral displacement longitudinal profiles measured in the HEA 160 beam web



A3 - 4


0 1000 2000 3000 4000

Vert

ical

dis

plac

emen

t [m

m]

0

1

2

3

4

5Point GPoint HPoint I

Figure A3.7 – Initial vertical displacement longitudinal profiles measured in the HEA 160 beam bottom flange Measurement Position along the Beam Length [mm]

0,00 0,50 1,00 1,50 2,00 2,50 3,00 λLT I-Steel beams … beams under tension: Lateral torsional...

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Transcript of 0,00 0,50 1,00 1,50 2,00 2,50 3,00 λLT I-Steel beams … beams under tension: Lateral torsional...