• Author / Uploaded
• mohan

Citation preview

         

χLT

1,2 1,0 0,8 0,6 0,4 0,2 0,0

0,00

0,50

1,00

1,50

2,00

2,50

3,00 λLT

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

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

Mahatma Gandhi

ii

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.

iii

iv

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 fullscale 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

v

vi

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

vii

viii

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  

ix

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  

x

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  

xi

xii

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' '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'

xiii

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'3.7'–'Schematic'representation'of'Step'2'...................................................................................................................'27! Figure'3.8'–'Schematic'representation'of'Step'3'...................................................................................................................'27! Figure'3.9'–'Schematic'representation'of'Step'4'...................................................................................................................'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!

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'3.7'–'Schematic'representation'of'Step'2'...................................................................................................................'27! Figure'3.8'–'Schematic'representation'of'Step'3'...................................................................................................................'27! Figure'3.9'–'Schematic'representation'of'Step'4'...................................................................................................................'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'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'

xvi

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'

xvii

xviii

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§ion&.....................................&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&

xix

xx

Chapter 1 Introduction

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 lateraltorsional 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 Isection 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 2

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 3

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 overconservative 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. 4

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.

5

6

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 Isection 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).

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 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

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 prebuckling deformations are neglected) − they read (note the change in sign of the Nt terms) EIz vIV + Nt vʹ′ʹ′+My φ ʹ′ʹ′= 0

(2.1)

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

(2.2)

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 v(x) = A1 sin (π/L x)

(2.3)

φ(x) = A2 sin (π/L x)

(2.4)

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

(2.5)

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

(2.6)

(2.7)

9

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. 30.0

Mcr (Nt) /Mcr (0)

25.0 20.0 15.0 10.0 5.0 0.0 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

β = Νt/Μy 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 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

19

4.4 IPE 300 IPE 500 HEB 300 HEB 500

4.0

Mcr (Nt) /Mcr (0)

3.6 3.2 2.8 2.4 2.0 1.6 1.2

1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

β = Νt/Μ 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 nonuniform 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.

20

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 minoraxis 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.

21

22

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.

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 24

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

(3.1)

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

25

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 crosssection 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:

26

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

Figure 3.6 – Schematic representation of Step 1

subsequent measurements are made on the four specimen segments. 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

Figure 3.7 – Schematic representation of Step 2

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.

Figure 3.8 – Schematic representation of Step 3

Step 4: After knowing the beam chord position, all measured displacements are related to the horizontal axis depicted in Figure 3.9. Figure 3.9 – Schematic representation of Step 4

Step 5: It is assumed that the beam ends (reference points) share the same position with respect to the z-axis. Figure 3.10 – Schematic representation of Step 5

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.

27

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

28

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

29

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 minoraxis 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 crosssection 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. 30

(b) Mid-span lateral displacements (WA 200) (a) Mid-span vertical displacements (TK 100)

(d) Loads applied by the hydraulic jacks (load cells C6A) (c) End support inclinations (one inclinometer KB–10EB and two TT50 LVDT’s) Figure 3.15 – Measuring device systems

31

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

b [mm] h [mm] tf [mm] tw [mm] r [mm]

IPE 200 (measured) 101.4 203.3 8.1 6.1 −

IPE 200 (nominal) 100 200 8.5 5.6 12

Δ (%) + 1.4 + 1.7 - 4.7 + 8.9

HEA 160 (measured) 162.3 153.8 8.8 6.7 −

HEA 160 (nominal) 160 152 9 6 15

Δ (%) + 1.4 + 1.2 - 2.2 + 11.7

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)

32

(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

E [GPa] fy [MPa] fu [MPa] εu

IPE200 208 320 446 30.57

HEA160 212 307 449 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).

33

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.

34

6 Top Flange Bottom Flange

10 9

Vertical displacement [mm]

Vertical displacement [mm]

11

8 7 6 5 4 3 2 1 0

Top Flange Bottom Flange

5 4 3 2 1 0

0

1000

2000

3000

4000

0

1000

Axial position of measurement [mm]

2000

3000

4000

Axial position of measurement [mm]

(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 18

Lateral displacement [mm]

Lateral displacement [mm]

3

2

1

0

16 14 12 10 8 6 4 2 0

0

1000

2000

3000

4000

Axial position of measurement [mm]

0

1000

2000

3000

4000

Axial position of measurement [mm]

(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

35

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

300

Applied Tension [kN]

270 240 210 180 150 120 90

Jacks (Side 1) Jacks (Side 2)

60 30 0 0

100

200

300

400

500

Time [s] 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

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.

37

250

250

Applied Tension [kN]

Applied Tension [kN]

300

200 150 100 LVDT_Support2 LVDT_Support1 Incl._Support2 Incl._Support1

50

200 150 100

0 -10

-5

0

5

50 0

10

0

5

End section flexural rotation θy [°]

10

15

25

30

35

40

Mid-span vertical displacement v [mm]

(a)

(b1)

250

Applied Tension [kN]

250

Applied Tension [kN]

20

200 150 100 50

200 150 100 50 0

0 0

5

10

15

20

0

2

4

6

8

10

12

14

16

18

20

Torsional rotation θx [°]

Mid-span lateral displacement u [mm] (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,

38

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

Applied Tension [kN]

160

120

80 Jacks (Side 1) Jacks (Side 2)

40

0 0

600

1200

1800

2400

3000

3600

Time [s] 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.

39

140

120

120

Applied Tension [kN]

Applied Tension [kN]

140

100 80 60 40

LVDT_Support2 LVDT_Support1 Incl._Support2 Incl._Support1

20

100

-5

0

5

60 40 20

0 -10

80

0

10

0

End section flexural rotation θy [°]

5

10

15

25

30

35

40

45

50

Mid-span vertical displacement v [mm]

(a)

(b1)

140

140

120

120

Applied Tension [kN]

Applied Tension [kN]

20

100 80 60 40

100 80 60 40 20

20 0 -0.5

0

0.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5

2.0

Torsional rotation θx [°]

Mid-span lateral displacement u [mm] (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:

40

(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

41

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 crosssection contour. The second aspect concerns the allowance for longitudinal displacements ensuring that the end 42

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)

43

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)

44

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

45

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

46

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 47

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. 160

300 Applied Tension [kN]

Applied Tension [kN]

140 250 200 150 LVDT_Support2 LVDT_Support1 Incl._Support2 Incl._Support1 Numerical_Support2 Numerical_Suppor1

100 50 0

120 100 80 60 40

Numerical

20

Experimental

0 -10

-5

0

5

10

0

End section flexural rotation θy [°]

5

10

15

20

25

30

35

40

45

50

Mid-span vertical displacement v [mm]

(a)

(b1)

160

300

Applied Tension [kN]

Applied Tension [kN]

140 120 100 80 60 40 20 0 -0.5

250 200 150 100 50

Experimental Numerical

0

0.0

0.5

1.0

0

1.5

5

10

15

Mid-span lateral displacement u [mm]

Torsional rotation θx [°]

(b2)

(b3)

20

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)

48

300

160 Applied Tension [kN]

Applied Tension [kN]

140 120 100 80 60

LVDT_Support2 LVDT_Support1 Incl._Support2 Incl._Support1 Numerical_Sup2 Numerical_Sup1

40 20 0

250 200 150 100 50

Experimental Numerical

0 -10

-5

0

5

10

0

End section flexural rotation θy [°]

10

30

40

Mid-span vertical displacement v [mm]

(a)

(b1)

300

160 140

250

Applied Tension [kN]

Applied Tension [kN]

20

200 150 100 50

Experimental Numerical

5

10

15

100 80 60 40 Experimental Numerical

20 0 -0.5

0 0

120

20

0.0

0.5

1.0

1.5

2.0

Torsional rotation θx [°]

Mid-span lateral displacement u [mm] (b2)

(b3)

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.

49

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.

50

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 nonlinear 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.

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 opensection 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

52

!

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 midspan vertical displacement (δ), and the beam deformed configuration at the brink of the LTB collapse.

F

δ

Figure 4.4 – Beam numerical F-δ equilibrium path and deformed configuration at the brink of the LTB collapse

53

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 54

(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).

55

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 cross56

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)

57

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)

58

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 59

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, crosssection 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

60

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 crosssection 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 lateral61

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).

62

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 (EC3ENV-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.

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

64

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 65

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).

1.2 1.1

β=0 β = 0.5 β = 0.75 β=1 β = 1.5 β=2

1.0

Mu / Mpl [-]

0.9 0.8 0.7 0.6

EC3-1-1 curve b ( ψ = 0)

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

λLT 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:

66

1.2 1.1

β=0 β = 0.5 β = 0.75 β=1 β = 1.5 β=2

1.0

Mu / Mpl [-]

0.9 0.8 0.7 0.6

EC3-1-1 curve b ( ψ = 1)

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

λLT Figure 5.2 − Comparison between the Mu /Mpl,Rk (numerical) and Mb,Rd /Mpl,Rk (proposed design approach) values for ψ=1

1.2 1.1

β=0 β = 0.5 β = 0.75 β=1 β = 1.5 β=2

1.0

Mu / Mpl [-]

0.9 0.8 0.7 0.6

EC3-1-1 curve b ( ψ = 0.5)

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

λLT Figure 5.3 − Comparison between the Mu /Mpl,Rk (numerical) and Mb,Rd /Mpl,Rk (proposed design approach) values for ψ=0.5

67

1.2 1.1

β=0 β = 0.5 β = 0.75 β=1 β = 1.5 β=2

1.0

Mu / Mpl [-]

0.9 0.8 0.7 0.6

EC3-1-1 curve b ( ψ = 0)

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

λLT Figure 5.4 − Comparison between the Mu /Mpl,Rk (numerical) and Mb,Rd /Mpl,Rk (proposed design approach) values for ψ=0

1.2 1.1

β=0 β = 0.5 β = 0.75 β=1 β = 1.5 β=2

1.0

Mu / Mpl [-]

0.9 0.8 0.7 0.6

EC3-1-1 curve b ( ψ = - 0.5)

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

λLT Figure 5.5 − Comparison between the Mu /Mpl,Rk (numerical) and Mb,Rd /Mpl,Rk (proposed design approach) values for ψ = − 0.5

68

1.2 1.1

β=0 β = 0.5 β = 0.75 β=1 β = 1.5 β=2

1.0

Mu / Mpl [-]

0.9 0.8 0.7 0.6

EC3-1-1 curve b ( ψ = -1)

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

λLT 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. 69

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%

70

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%

71

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

Meff,Ed = Wcomp σcomp,Ed

(5.1)

with Wcomp denoting the cross-section elastic modulus concerning the most compressed fibre and σcomp,Ed is calculated by means of the expression

σcomp,Ed = MEd /Wcomp,Ed − ψvec NEd /A

(5.2)

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

72

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 lateraltorsional 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 73

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)

74

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

ψ=1

Average

St. Dev.

Max

Min

Beam Number esults.

23%

45%

195%

− 42%

270

(a)

ψ = −1

Average

St. Dev.

Max

Min

Beam Number

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.

75

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.

76

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.  

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

78

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