Contribution to the design of steel I and H-sections members by means of the Overall Interaction Concept

(1)

Contribution to the design of steel I and H-sections

members by means of the Overall Interaction Concept

Thèse

Lucile Gérard

Doctorat en génie civil

Philosophiæ doctor (Ph. D.)

(2)

Contribution to the design of steel I and H-sections

members by means of the Overall Interaction Concept

Thèse

LUCILE GÉRARD

Sous la codirection de :

Nicolas Boissonnade, Directeur de recherche

Charles-Darwin Annan, Codirecteur de recherche

(3)

Résumé

La conception des profilés en acier est généralement influencée par l’apparition d’instabilités au niveau local et global de l’élément. Le voilement, le flambement, le déversement ont été source de nombreuses recherches dans le but d’optimiser les coûts des constructions en acier. Une estimation plus précise des charges de ruines réelles devrait permettre des gains, en pratique, de par l’allègement de structures typiquement conçues de façon excessivement sécuritaires. En effet, les règles de conception, proposées par les normes, mènent à une estimation des résistances locales et globales des profilés en I et en H, dont la précision peut être améliorée, et certains chercheurs ont ainsi orienté leur travail vers une optimisation de ces règles.

Dans ce contexte, l’O.I.C. (Overall Interaction Concept) qui a été développé dans un premier temps pour les sections tubulaires en acier, permet la conception de tous types de géométries de sections, et cas de chargement, d’une manière simple et efficace. Les formules de résistances ont été développées à partir de résultats expérimentaux et numériques, ceux-ci permettant de prendre en compte, de manière continue, l’interaction entre résistance et instabilité. Cette thèse s’inscrit dans la démarche de l’O.I.C., de par le développement de formules de conception pour les sections en I et en H.

Suite à une étude numérique réalisée à partir d’un modèle aux éléments finis, dont la précision a été préalablement vérifiée, des schémas de contraintes résiduelles et imperfections géométriques ont pu être recommandés, afin de garantir l’obtention de résistances fiables. Des études paramétriques furent menées sur des éléments courts, soumis à des cas de chargement simples, telles que la compression pure, la flexion d’axe fort, et la flexion d’axe faible, ainsi que des cas de chargements combinés. Des formulations prédisant la résistance, dans un format O.I.C., ont été ensuite proposées. Davantage d’analyses numériques ont été réalisées afin d’étudier la réponse globale des poutres en I et en H, sujettes au phénomène de déversement, et une équation prédisant leur résistance a été développée.

Les formulations proposées ont montré une meilleure fiabilité que celle obtenue selon les directives de l’Eurocode, tout particulièrement dans le cas des sections élancées de classe 4, dont les règles de conception préconisées présentent un manque de précision important. La formulation développée selon l’O.I.C., dans le cas des chargements combinés, bien que d’apparence complexe, permet d’obtenir une estimation de la capacité locale de façon très précise, là où la norme a montré d’importantes lacunes. La performance de la formulation a été préférée à la simplicité, pour assurer son apport vis-à-vis de la précision des normes actuelles. C’est pourquoi ce travail doit être considéré comme une première étape vers une formulation complète de conception des poutres en I et en H, et de futurs travaux pourraient s’intéresser à leur simplification.

(4)

Abstract

– iii –

Abstract

Design of steel profiles is generally ruled by the occurrence of local and global instabilities which have been of major interest for many years now to reduce costs within steel applications. Closer estimate of actual ultimate strengths shall allow for some savings in practice since some structures typically designed through conservative rules provided by standards such as the Eurocode shall be lightened. The design guidelines currently suggested in codes to predict local and global strengths of I and H-shapes are broadly known to exhibit some conservatism and researchers have been directing their efforts towards an optimisation of such design.

Accordingly, the O.I.C. (Overall Interaction Concept) which was first developed for tubular sections, provides a general approach for steel design with a common and straightforward formulation for all load cases and cross-section’s geometries. O.I.C. design equations are derived based on strengths mostly achieved by means of full non-linear analyses since F.E. simulations can provide accurate predictions of actual ultimate strengths and continuously account for the Resistance – Instability interaction. Through the present thesis, design formulae for I and H-shapes were developed as part of the development of the O.I.C.

Subsequently to an extensive numerical study through a F.E. model whose reliability was first established by means of comparisons with tests data, reasonable local geometrical imperfections as well as appropriate residual stresses patterns were chosen so that reliable ultimate strengths could be reached with the F.E. models. Then, parametric studies were carried out on short members subject to simple load cases such as simple axial force, major-axis bending or minor-axis bending and combined loadings. Observations on local strengths tendencies allowed the determination of key parameters so that O.I.C. based design proposals predicting the local strengths of hot-rolled and welded I and H-sections could be established. Numerical investigations then focused on the member strength of I and H-sections prone to suffer from Lateral Torsional Buckling so that an accurate design proposal including local/global coupling effects was eventually derived.

Accuracy showed great benefits from these O.I.C. based design proposals compared to Eurocode rules, especially for slender sections. The design for which coherence and accuracy was preferred to simplicity has shown drastic efficiency for simple and combined load cases. Owing to the complex strength tendencies observed for sections subject to combined loadings, no consequent simplification of the proposals could be achieved without losing the benefits in accuracy compared to the Eurocode one. This work shall be considered as a first step towards a more coherent and accurate design for steel I and H-sections since further work may be needed towards a simplification of such proposal.

(5)

As part of this thesis, the following conference and journal papers were achieved.

Conference Paper 1: L. Gérard, C. Arsenault, M. Kettler and N. Boissonnade, “The influence of geometrical and material imperfections on the stability and resistance of I and H sections”, Proceedings of the Annual Stability Conference of the Structural Stability Research Council, Baltimore, Maryland, April 10-13, 2018.

Conference Paper 2: L. Gérard, D. W. White, N. Boissonnade, “Design of steel beams affected by local/global coupled instabilities”, Proceedings of the Annual Stability Conference of the Structural Stability Research Council, Atlanta, Georgia, April 21-24, 2020.

Journal Paper 1: L. Gérard, L. Li, M. Kettler and N. Boissonnade, “Recommendations on the geometrical imperfections definition for the resistance of I-sections”, Journal of Constructional Steel Research, 2019. Journal Paper 2: L. Gérard, L. Li, M. Kettler, D. W. White and N. Boissonnade, “Recommendations on the material imperfections definition for the resistance of I-sections”, Submitted to Thin-Walled Structures, 2020.

Journal Paper 3: L. Gérard, L. Li, M. Kettler, N. Boissonnade, “Steel I-sections resistance under compression or bending by the Overall Interaction Concept”, Submitted to the Journal of Constructional Steel Research, 2019.

(6)

Table of Contents – v –

Résumé ... ii

Abstract ... iii

Table of Contents ... v

List of Figures ... x

List of Tables ... xxiii

Notations ... xxviii

Acknowledgments ... xxxi

Introduction ... 1

1 State-of-the-art ... 12

1.1 General review on buckling ... 12

1.1.1 Basics ... 12

1.1.1.1 Brief historical review ... 12

1.1.1.2 Definition of stability ... 14

1.1.1.3 Characterization of buckling behaviours ... 16

1.1.2 Local buckling ... 18

1.1.2.1 Theory of plate buckling ... 18

1.1.2.2 Influence of a plate boundary conditions ... 20

1.1.2.3 Post-buckling reserves ... 23

1.1.2.4 Effective Width Method (E.W.M.) ... 24

1.1.2.5 Formulae available for plate effective width ... 26

1.1.3 Lateral Torsional Buckling ... 29

1.1.3.1 Theoretical expressions of elastic critical buckling bending moment Mcr ... 29

1.1.3.2 Uniform and non-uniform torsion ... 33

1.1.3.3 Inclusion of pre-buckling deflections into theoretical expressions ... 35

1.1.4 Historical review on geometrical and material imperfections ... 36

1.1.4.1 Local geometrical imperfections ... 36

1.1.4.2 Global geometrical imperfections ... 38

1.1.4.3 Residual stresses ... 41

1.1.4.4 Conclusions ... 57

1.2 Non-linear behaviour ... 57

(7)

Table of Contents

1.2.2 Plastic design and material non-linearity (M.N.A.) ... 58

1.2.2.1 From elastic to plastic design ... 58

1.2.2.2 Fundamentals of plastic design ... 59

1.2.2.3 Limitations in the use of plastic design ... 63

1.2.3 Geometrical second-order effects (G.N.A. / G.N.I.A.) ... 64

1.3 Buckling curves ... 66

1.3.1 Basics of buckling curves design ... 67

1.3.2 Formulation of buckling curves ... 68

1.3.2.1 Mathematical series formulae ... 68

1.3.2.2 Merchant-Rankine approach ... 69

1.3.2.3 Ayrton-Perry proposal ... 70

1.3.2.4 Modified Ayrton-Perry proposal for Lateral Torsional Buckling behaviour ... 71

1.4 Available design recommendations in standards ... 73

1.4.1 Concept of cross-section classification for local behaviour ... 73

1.4.2 Design according to Eurocode 3 ... 75

1.4.3 Design according to the Canadian standard ... 79

1.4.4 Shortcomings ... 83

1.5 Modern approaches ... 84

1.5.1 The Direct Strength Method (D.S.M.) ... 84

1.5.2 The Continuous Strength Method (C.S.M.) ... 85

1.5.3 The Overall Interaction Concept (O.I.C.) ... 87

1.5.4 Design proposals for the local resistance of open-sections ... 88

1.5.5 Design proposals for open-sections member strengths under major-axis bending ... 89

1.5.6 Design proposals accounting for coupled instabilities ... 92

1.6 Summary and conclusions ... 94

2 Finite Element model description and validation ... 97

2.1 Finite Element model characteristics ... 97

2.2 Mesh density studies ... 101

2.2.1 Local behaviour ... 101

2.2.2 Global behaviour ... 108

2.3 Finite Element model validation against experimental data ... 114

2.3.1 Local behaviour ... 115

2.3.2 Global behaviour including L.T.B. and local buckling ... 121

(8)

– vii –

3 Numerical studies on the influence of imperfections on the local resistance of

open-sections ... 135

3.1 Introduction ... 135

3.2 Input data ... 135

3.3 1st_{parametric study: influence of geometrical imperfections ... 148}

3.3.1 Comparison with experimental tests ... 149

3.3.2 Use of sinusoidal functions ... 157

3.3.2.1 Influence of the half-wave length ... 157

3.3.2.2 Influence of the half-wave amplitude ... 163

3.3.3 Use of the 1st_{buckling mode shape ... 168}

3.3.4 Recommendations for the introduction of geometrical imperfections in F.E. models ... 171

3.3.5 Conclusions ... 172

3.4 2nd_{parametric study: influence of material imperfections ... 172}

3.4.1 Introduction ... 172

3.4.2 Influence of residual stresses on hot-rolled profiles cross-section resistance ... 172

3.4.3 Influence of residual stresses on welded profiles cross-section resistance ... 180

3.4.3.1 Influence of welded residual stresses pattern shape ... 180

3.4.3.2 Brief comparison between hot-rolled and welded residual stresses ... 184

3.4.4 Recommendations for the introduction of material imperfections in F.E. models ... 187

4 Local resistance and design of open-sections under simple load cases ... 190

4.1 Introduction ... 190

4.2 F.E. model basic features and assumptions ... 191

4.3 Influence of yield strength ... 195

4.4 Design proposal for the local resistance of open-sections under pure compression ... 198

4.4.1 Bases of the proposed approach ... 198

4.4.2 Design proposal ... 199

4.4.2.1 Results for hot-rolled sections ... 200

4.4.2.2 Results for welded sections ... 203

4.5 Design proposal for the local resistance of open-sections under major-axis bending .... 209

4.5.1 Hot-rolled sections ... 209

4.5.2 Welded sections ... 211

4.5.2.1 First proposal ... 211

4.5.2.2 Second proposal ... 214

(9)

Table of Contents

4.6.1 Hot-rolled sections ... 218

4.6.2 Welded sections ... 220

4.7 Conclusions ... 222

5 Local resistance and design of I and H-sections under combined load cases ... 225

5.1 Input data ... 225

5.2 Influence of warping restraints for combined load cases ... 228

5.3 Influence of yield stress on hot-rolled sections strength ... 237

5.4 Design proposals for hot-rolled sections ... 238

5.4.1 Basics of the design proposal ... 238

5.4.2 Numerical results and design proposals ... 239

5.4.2.1 First design proposal ... 245

5.4.2.2 Worked examples using the first design proposal ... 251

5.4.2.3 Simplified version of the first design proposal ... 252

5.4.2.4 Second design proposal ... 253

5.4.3 Design proposals efficiencies ... 254

5.5 Design proposal for welded sections ... 265

5.5.1 Numerical results ... 265

5.5.2 Design proposal ... 269

5.5.3 Accuracy of the design proposal compared to EC3 predictions ... 272

6 Numerical studies on the resistance of open-sections against Lateral Torsional

Buckling ... 280

6.1 Introduction ... 280

6.2 Influence of imperfections ... 280

6.3 Determination of L.T.B. strengths including local buckling effects ... 298

6.3.1 Input data ... 298

6.3.2 Local/Global ultimate resistance ... 309

6.3.2.1 Brief preliminary study ... 309

6.3.2.2 Results and first observations ... 313

6.3.3 Development of an appropriate method for local/global coupling behaviour ... 319

6.3.3.1 Bases of the method ... 319

6.3.3.2 Design of hot-rolled sections ... 322

(10)

– ix –

7 Worked examples ... 333

7.1 Local strength of a hot-rolled W1000X296 under axial force ... 333

7.1.1 Design by means of the O.I.C. proposal ... 334

7.1.1.1 Determination of the local capacity ... 334

7.1.1.2 Cross-section verification ... 335

7.1.2 Design by means of the Eurocode ... 336

7.1.2.1 Cross-section classification ... 336

7.1.2.2 Class 4 section: determination of the effective area ... 337

7.1.2.3 Determination of the effective axial strength ... 338

7.1.3 Summary ... 339

7.2 Local strength of a hot-rolled W610X241 under combined loading ... 339

7.2.2.2 Determination of the axial strength ... 343

7.2.2.3 Determination of the minor-axis bending strength ... 344

7.2.3 Summary ... 345

7.3 Local strength of a welded HEAS01 under uniform major-axis bending ... 346

7.3.2.2 Class 4 section: determination of the effective properties ... 351

7.3.2.3 Determination of the bending strength ... 352

7.3.3 Summary ... 353

7.4 Conclusions ... 353

Conclusions ... 354

(11)

List of Figures

Figure 1: Pont de Québec (Canada), see [2]. ... 2

Figure 2: Viaduc de Garabit, Cantal. Pictures from [3] on the left and [4] on the right. ... 2

Figure 3: Timmerhuis building in Rotterdam, pictures from [5] on the left and [6] on the right. ... 3

Figure 4: Local buckling of a column. ... 3

Figure 5: Bending moment as a function of the rotation, cross-section classification concept. ... 5

Figure 6: Axial strength prediction by means of “buckling” curves. ... 7

Figure 7: Principles and application steps of the Overall Interaction Concept. ... 8

Figure 8: Equilibrium characteristics [36]. ... 14

Figure 9: Instability by bifurcation. ... 15

Figure 10: Instability by divergence of equilibrium. ... 15

Figure 11: Local buckling of an I-section under pure bending moment (left) and axial force (right). ... 16

Figure 12: The 3 first buckling modes of a column. ... 17

Figure 13: Lateral torsional buckling of a W1000X350 under uniform major-axis bending. ... 17

Figure 14: Elastic buckling of a plate. ... 19

Figure 15: Variation of k for several buckling modes as a function of the plate aspect ratio a / b for a plate under pure compression supported on four sides. ... 19

Figure 16: Load path behaviour of a plate under edge compression. ... 20

Figure 17: Buckling coefficient for several boundary conditions and stress distributions [37]. ... 21

Figure 18: Buckling of an axially compressed plate supported on four edges. ... 23

Figure 19: Typical load path of an axially compressed column (on the left) and plate (on the right). ... 24

Figure 20: Concept of the Effective Width Method [37]. ... 25

Figure 21: Concept of the plastic effective width. ... 26

Figure 22: Von Karman design using the Effective Width Method. ... 27

Figure 23: C1 parameter suggested for several bending moment distributions according to [43]. ... 31

Figure 24: Influence of the load application point on the behaviour against Lateral Torsional Buckling.32 Figure 25: Values of C1 and C2 parameters suggested by [43] for different bending moment distributions. ... 33

(12)

List of Figures

– xi –

Figure 27: Distribution of uniform and non-uniform torsion on a clamped-free profile under an external

torsional moment [37]. ... 35

Figure 28: Type 1 imperfection pattern suggested by Boissonnade and Somja [52]. ... 39

Figure 29: Residual stresses measurements on a hot-rolled specimen [66] and welded specimen [67]. ... 42

Figure 30: Residual stresses patterns for hot-rolled sections from a) ECCS [70], b) Galambos and Ketter [72]. ... 42

Figure 31: Residual stresses pattern for hot-rolled sections from Young [73]. ... 43

Figure 32: Parabolic residual stresses pattern for hot-rolled sections from a) Lindner et al. [59] [74] and b) Boissonnade and Somja [52]. ... 44

Figure 33: Residual stresses patterns for welded sections from: a) ECCS [70] and b) Wang et al. ... 45

Figure 34: Residual stresses measures of hot-rolled sections from [72]. ... 46

Figure 35: Residual stresses patterns with respect to column shapes (h / b £ 1.2). ... 50

Figure 36: Residual stresses patterns with respect to beam shapes (h / b > 1.2). ... 52

Figure 37: Residual stresses measures on welded sections, Hasham and Rasmussen [93]. ... 54

Figure 38: Residual stresses measures on a welded section from Prawel [58]. ... 54

Figure 39: Residual stresses shapes for welded sections adapted from ECCS recommendations [70] (left) and from a “Best-fit” of Prawel measures [57], [58] (right). ... 55

Figure 40: Stress distribution of an open-section under major-axis bending with elastic-perfectly plastic constitutive law. ... 59

Figure 41: Formation of a plastic hinge at a critical section of a steel profile. ... 60

Figure 42: Moment-rotation relation of an I-section. ... 60

Figure 43: Development of the bending moment applied as a function of the rotation q and formation of plastic hinges. ... 61

Figure 44: Influence of the steepness of the bending moment. ... 62

Figure 45: Main parameters influencing the rotational capacity of a beam. ... 63

Figure 46: Geometrical second-order effects. ... 64

Figure 47: Second-order effects due to local buckling in a short member under axial force. ... 66

Figure 48: Second-order effects due to local buckling in a short member under uniform major-axis bending. ... 66

(13)

List of Figures

Figure 50: Buckling curves for local behaviour. ... 68

Figure 51: Eurocode cross-section classification concept for bending. ... 74

Figure 52: Eccentricity of the axial load due to the effective properties of a Class 4 section under N + My. ... 78

Figure 53: Buckling curves provided by the European, Canadian and American standards. ... 82

Figure 54: Local and distortional tests results to ultimate strengths as a function of their respective relative slenderness compared to D.S.M. proposals. ... 85

Figure 55: Deformation capacity as a function of the plate relative slenderness through C.S.M. proposal [8]. ... 86

Figure 56: Proposal from Kettler for a continuous transition between Class 2 and Class 4 strength predictions. ... 88

Figure 57: Condition for the peak load validity. ... 98

Figure 58: Web-to-flange junction modelling of a hot-rolled section. ... 99

Figure 59: Constraints imposed to the F.E. model. ... 100

Figure 60: Loading situations applied to the F.E. model. ... 100

Figure 61: Mesh density influence on IPE sections under simple axial force and major-axis bending from left to right (L.B.A.). ... 105

Figure 62: Mesh density influence on HEA sections under simple axial force and major-axis bending from left to right (L.B.A.). ... 105

Figure 63: Mesh density influence on hot-rolled IPE sections under simple axial force and major-axis bending from left to right (G.M.N.I.A.). ... 106

Figure 64: Mesh density influence on hot-rolled HEA sections under simple axial force and major-axis bending from left to right (G.M.N.I.A.). ... 106

Figure 65: Mesh density influence on welded IPE sections under simple axial force and major-axis bending from left to right (G.M.N.I.A.). ... 107

Figure 66: Mesh density influence on welded HEA sections under simple axial force and major-axis bending from left to right (G.M.N.I.A.). ... 107

Figure 67: Load-displacements curves of a welded IPES under axial force (left) and hot-rolled HEA160 under major-axis bending (right). ... 108

(14)

List of Figures

– xiii –

Figure 70: Mesh density influence on a welded IPES under uniform major-axis bending. ... 113

Figure 71: Load-displacements curves of a welded IPES under uniform major-axis bending for L1 (left) and L2 (right). ... 113

Figure 72: Load-displacements curves of a hot-rolled HEA300 under uniform major-axis bending for L3 (left) and L4 (right). ... 114

Figure 73: Material properties considered for the Finite Element model validation study. ... 115

Figure 74: Residual stresses pattern for Finite Element model validation study. ... 116

Figure 75: Load application used through the experimental test series [9]. ... 117

Figure 76: Finite Element model for F.E. model validation against test data. ... 118

Figure 77: Ratio between the local reduction factor from experimental tests and numerical simulations, for each specimen. ... 121

Figure 78: Constitutive law adopted in the F.E. models for the comparison with Dux and Kitipornchai experimental tests. ... 122

Figure 79: Load cases investigated by Dux and Kitipornchai [32]. ... 123

Figure 80: Residual stresses pattern used in the F.E. model. ... 125

Figure 81: Ultimate load achieved by F.E. simulation over the one reached experimentally (see [32]) with assumed 30mm stiffeners. ... 125

Figure 82: Ratios of ultimate loads achieved numerically over the experimental ones (see [32]) for several stiffener thicknesses. ... 126

Figure 83: Characteristics of the specimens tested by Fahnestock and Sause (1998) and Salem and Sause (2004). ... 128

Figure 84: Material law considered for the F.E. model validation against Fahnestock, Salem and Sause tests. ... 129

Figure 85: Residual stresses considered in the F.E. model validation based on Fahnestock, Salem and Sause experiments. ... 130

Figure 86: Comparison between experimental and numerical ultimate maximum bending moments. ... 131

Figure 87: Experimental and numerical load-displacements curves for Specimen No.1 and Specimen No.2. ... 132

Figure 88: Experimental and numerical load-displacements curves for Specimen No.3 and Specimen No.4. ... 132

(15)

List of Figures

Figure 89: Experimental and numerical load-displacements curves for Specimen No.5 and Specimen No.6.

... 133

Figure 90: Experimental and numerical load-displacements curves for Specimen No.7. ... 133

Figure 91: Definition of section dimensions. ... 136

Figure 92: Material properties considered in the F.E. model. ... 136

Figure 93: Triangular (ECCS [70]) and parabolic residual stresses patterns for hot-rolled sections. ... 137

Figure 94: Rectangular residual stresses pattern for welded sections adapted from ECCS [70] (left) and trapezoidal from [74] (right). ... 138

Figure 95: Definition of local imperfections on hot-rolled and welded sections. ... 139

Figure 96: Local imperfections on a HEAA hot-rolled (left) and welded (right). ... 139

Figure 97: Web and flanges buckling lengths for hot-rolled (on the left) and welded (on the right) sections. ... 141

Figure 98: Various numbers of half-waves in local imperfections. ... 142

Figure 99: Sine-shape imperfections of cases a), c), d), e), f), g), h) and i) specified in Table 35 on the upper left; case b) on the upper right; case j) on the lower image. ... 143

Figure 100: Magnified initial imperfections of an IPE140 section; case a) (left); case b) (middle); case j) (right). ... 145

Figure 101: Ratio between ultimate load provided by numerical simulations with various geometrical imperfection types and respective ultimate load from experimental test [9]. ... 153

Figure 105: Influence of the end-plates on the agreement reached between numerical and experimental capacities. ... 156

Figure 106: Results from numerical simulations on I-sections under pure compression, using a sine-shape imperfection with different periods. ... 158 Figure 107: Results from numerical simulations on I-sections under major-axis bending, using a sine-shape

(16)

List of Figures

– xv –

Figure 108: Results from numerical simulations on H-sections under pure compression, using a sine-shape imperfection with different periods. ... 160 Figure 109: Results from numerical simulations on H-sections under major-axis bending, using a

sine-shape imperfection with different periods. ... 160 Figure 110: Influence of the period of the sinusoidal imperfection on hot-rolled I-sections (upper) and

H-sections (lower) S460 steel, axial compression and major-axis bending. ... 161 Figure 111: Magnified imperfect shape of case –hw / P_pp / A_pp_200. ... 162 Figure 112: Variation of buckling modes respecting to the plate buckling coefficient as a function of the

plate aspect ratio a / b for a plate supported on both longitudinal edges under pure compression. 163 Figure 113: Results from numerical simulations on I-sections under pure compression, using a sine-shape

imperfection with different amplitudes. ... 164 Figure 114: Results from numerical simulations on I-sections under major-axis bending, using a sine-shape

imperfection with different amplitudes. ... 165 Figure 115: Results from numerical simulations on H-sections under pure compression, using a sine-shape

imperfection with different amplitudes. ... 165 Figure 116: Results from numerical simulations on H-sections under major-axis bending, using a

sine-shape imperfection with different amplitudes. ... 166 Figure 117: Histograms which highlight the half-wave amplitude influence on hot-rolled I-sections (upper

histogram) and H-sections (lower histogram), steel S460, under axial compression and major-axis bending. ... 167 Figure 118: Results from numerical simulations on I-sections under pure compression and pure major-axis

bending, using the 1st_{eigenmode as initial imperfection with different amplitudes. ... 169}

Figure 119: Results from numerical simulations on H-sections under pure compression and pure major-axis bending, using the 1st_{eigenmode as initial imperfection with different amplitudes. .... 169}

Figure 120: Histograms which highlight the influence of the 1st_{buckling mode shape amplitude on}

hot-rolled I-sections (upper histogram) and H-sections (lower histogram), steel S460, under axial compression and major-axis bending. ... 170 Figure 121: Results from numerical simulations on I-sections under pure compression, for various types

of hot-rolled residual stress pattern. ... 174 Figure 122: Results from numerical simulations on I-sections under major-axis bending, for various types

(17)

List of Figures

Figure 123: Results from numerical simulations on H-sections under pure compression, for various types

of hot-rolled residual stress pattern. ... 175

Figure 124: Results from numerical simulations on H-sections under major-axis bending, for various types of hot-rolled residual stress pattern. ... 176

Figure 125: Influence of hot-rolled residual stresses pattern on I-sections (upper) and H-sections (lower) – S460 steel, pure compression and major-axis bending. ... 177

Figure 126: Load-displacement curves at the central node at mid-length, for hot-rolled patterns, HEAS, steel Fy460. ... 179

Figure 127: Combined stresses at first stage and yielded zones at peak load of a HEAS, Fy460, under axial force. ... 181

Figure 128: Combined stresses at first stage and yielded zones at peak load of a HEAS, Fy460, under major-axis bending. ... 182

Figure 129: Results from numerical simulations on I-sections under pure compression and pure major-axis bending, using various types of welded residual stresses pattern. ... 183

Figure 130: Results from numerical simulations on H-sections under pure compression and pure major-axis bending, using various types of welded residual stresses pattern. ... 183

Figure 131: Influence of the welded residual stresses pattern on I and H-sections – S460 steel, axial compression (left) and major-axis bending (right). ... 184

Figure 132: Hot-rolled and welded residual stresses patterns considered in the comparison. ... 185

Figure 133: Hot-rolled and welded residual stresses influence. ... 185

Figure 134: Ratio between welded and hot-rolled sections local reductions factors, for H-shapes. ... 186

Figure 135: Recommendations for residual stresses patterns. ... 188

Figure 136: Material properties considered in the Finite Element model. ... 191

Figure 137: Loading situations considered in the F.E. model. ... 192

Figure 138: Plate slenderness ratios of the sections studied and Eurocode limits for sections classification. ... 193

Figure 139: Longitudinal geometrical imperfections shape introduced. ... 194

Figure 140: Residual stresses patterns for a) hot-rolled sections; b) welded sections. ... 194

Figure 141: Yield stress influence on the local strength of hot-rolled and welded sections under axial force. ... 196

(18)

List of Figures

– xvii –

Figure 143: Yield stress influence on the local strength of hot-rolled and welded sections subject to

minor-axis bending. ... 197

Figure 144: Section’s dimensions. ... 200

Figure 145: Design proposal for hot-rolled open-sections under pure compression. ... 201

Figure 146: Accuracy of O.I.C. and EC3 design proposals for hot-rolled sections under pure compression. ... 203

Figure 147: Design proposal for welded open-sections under pure compression. ... 207

Figure 148: Accuracy of O.I.C. and EC3 design proposals for welded sections under pure compression. ... 208

Figure 149: Design proposal for hot-rolled open-sections under major-axis bending. ... 209

Figure 150: Accuracy of O.I.C. and EC3 design proposals for hot-rolled sections under major-axis bending. ... 211

Figure 151: Design proposal for welded open-sections under major-axis bending. ... 212

Figure 152: Relationship between x and µW,My obtained at certain values of local relative slenderness. 213 Figure 153: Accuracy of O.I.C. and EC3 design proposals for welded sections under major-axis bending. ... 214

Figure 154: Second design proposal for welded open-sections under major-axis bending. ... 216

Figure 155: Accuracy of O.I.C. second design proposal for welded sections under major-axis bending. ... 217

Figure 156: Accuracy of O.I.C. second design proposal for welded sections under major-axis bending as a function of k. ... 218

Figure 157: Design proposal for hot-rolled sections under minor-axis bending and effectiveness of the design proposal. ... 219

Figure 158: Design proposal for welded open-sections under minor-axis bending. ... 221

Figure 159: Cumulative frequencies with O.I.C. and EC3 design proposals for welded sections under minor-axis bending. ... 222

Figure 160: Constitutive laws for the following yield limits: 355, 690 and 770 MPa. ... 225

Figure 161: F.E. models for N + My load case (left) and N + My + Mz (right). ... 226

Figure 162: Loading situations applied to the F.E. model. ... 226

Figure 163: Interaction between biaxial bending and axial compression. ... 227

(19)

List of Figures

Figure 165: Load-displacement curve of an IPE300 under pure compression for a G.M.N.A. analysis. 231 Figure 166: End-plate thickness influence on plastic capacities for n » 0.80 and abiax = 70 and n » 0.75

and abiax = 30. ... 232

Figure 167: End-plate thickness influence on plastic capacities for n » 0.70 and abiax = 15 and n » 0.65 and abiax = 0. ... 233

Figure 168: End-plate influence on local strengths of welded I-shapes. ... 234

Figure 169: End-plate influence on local strengths of welded slender H-shapes and hot-rolled W-shapes. ... 235

Figure 170: Yielding of flanges and local buckling of the web of a slender I-section. ... 236

Figure 171: Yield stress influence on open-sections’ strengths under combined load cases for low levels of axial force. ... 237

Figure 172: Yield stress influence on open-sections’ strengths under combined load cases for high levels of axial force. ... 238

Figure 173: Approximation of Dirac function by means of continuous functions. ... 239

Figure 174: Finite Element results for hot-rolled open-sections under combined loading. ... 240

Figure 175: Finite Element results for hot-rolled open-sections under combined loading. ... 241

Figure 176: Influence of biaxiality on hot-rolled sections under My + Mz (0% axial compression). ... 242

Figure 177: Influence of biaxiality on hot-rolled sections under N + My + Mz (about 27% axial compression). ... 242

Figure 180: Design proposal for respectively abiax = 0 and abiax = 30 with 60% axial force applied, from left to right. ... 247

Figure 181: Design proposal for respectively abiax = 50 and abiax = 70 with 60% axial force applied, from left to right. ... 248

Figure 182: Design proposal for abiax = 90 with 60% axial force applied. ... 248 Figure 183: Design proposal for respectively abiax = 0 and abiax = 30 with 80% axial force applied, from

(20)

List of Figures

– xix –

Figure 184: Design proposal for respectively abiax = 50 and abiax = 70 with 80% axial force applied, from left to right. ... 249 Figure 185: Design proposal for abiax = 90 with 80% axial force applied. ... 250 Figure 186: Design proposal for respectively abiax = 30 and abiax = 50 with 0% axial compression, from

left to right. ... 250 Figure 187: Design proposal for abiax = 70 with 0% axial compression. ... 251 Figure 188: Accuracy of O.I.C. and EC3 design proposals for hot-rolled sections for n = 0 to n = 0.15. ... 255 Figure 189: Accuracy of O.I.C. and EC3 design proposals for hot-rolled sections for n = 0.15 to n = 0.5. ... 256 Figure 190: Accuracy of O.I.C. and EC3 design proposals for hot-rolled sections for n = 0.5 to n = 0.85. ... 256 Figure 191: Accuracy of O.I.C. and EC3 design proposals for hot-rolled sections for n > 0.85. ... 257 Figure 192: Accuracy of O.I.C. and EC3 strengths predictions as a function of lL for n = 0 to n = 0.15

(hot-rolled). ... 260 Figure 193: Accuracy of O.I.C. and EC3 strengths predictions as a function of lL for n = 0.15 to n = 0.5

(hot-rolled). ... 261 Figure 194: Accuracy of O.I.C. and EC3 strengths predictions as a function of lL for n = 0.5 to n = 0.85

(hot-rolled). ... 261 Figure 195: Accuracy of O.I.C. and EC3 strengths predictions as a function of lL for n > 0.85 (hot-rolled). ... 262 Figure 196: Accuracy of O.I.C. and EC3 strengths predictions as a function of the axial force proportion

n (hot-rolled). ... 263 Figure 197: Overall welded sections tested under combined loadings. ... 265 Figure 198: Influence of biaxiality on welded sections under My + Mz (0% axial compression). ... 266 Figure 199: Influence of biaxiality on welded sections under N + My + Mz (about 23% axial compression). ... 267 Figure 200: Influence of biaxiality on welded sections under N + My + Mz (about 56% axial compression). ... 268 Figure 201: Influence of biaxiality on welded sections under N + My + Mz (about 67% axial compression). ... 268

(21)

List of Figures

Figure 202: Accuracy of O.I.C. and EC3 design proposals for welded sections for n = 0 to n = 0.25. .. 272 Figure 203: Accuracy of O.I.C. and EC3 design proposals for welded sections for n = 0.25 to n = 0.6. 273 Figure 204: Accuracy of O.I.C. and EC3 design proposals for welded sections for n = 0.6 to n > 0.8. . 274 Figure 205: Accuracy of O.I.C. and EC3 strengths predictions as a function of lL for n = 0 to n = 0.25

(welded). ... 275 Figure 206: Accuracy of O.I.C. and EC3 strengths predictions as a function of lL for n = 0.25 to n = 0.6

(welded). ... 275 Figure 207: Accuracy of O.I.C. and EC3 strengths predictions as a function of lL for n = 0.6 to n > 0.8

(welded). ... 276 Figure 208: Accuracy of O.I.C. and EC3 strengths predictions as a function of the axial force proportion

n (welded). ... 277 Figure 209: Parameters used in the definition of global imperfections. ... 281 Figure 210: Local/Global imperfection shapes for a hot-rolled section considered for Lateral Torsional

Buckling. ... 282 Figure 211: Imperfections amplitude influence on specimens’ strength using a unique half-wave, with

parabolic residual stresses (left) and triangular shape (right). ... 287 Figure 212: Imperfections amplitude influence on specimens’ strength using multiple half-waves, with

parabolic residual stresses (left) and triangular shape (right). ... 287 Figure 213: Agreement between numerical and experimental resistances for the hot-rolled specimens

tested in [32], using measured imperfections. ... 288 Figure 214: Agreement between numerical and experimental strengths for welded specimens tested in

[33]. ... 290 Figure 215: Agreement between experimental and numerical strengths for the welded specimens tested in

[33] as a function of the global relative slenderness. ... 291 Figure 216: Global reduction factor over the global relative slenderness for each specimen tested in [33]

except specimen No.4. ... 292 Figure 217: Imperfections amplitude influence on the resistance of a W1000X883 made of Fy355. ... 293 Figure 218: Imperfections amplitude influence on the resistance of a HEA300 made of Fy355. ... 293 Figure 219: Imperfections amplitude influence on the resistance of an invented section, IPES made of

(22)

List of Figures

– xxi –

Figure 221: F.E. model used to obtain global ultimate capacities – free of any local buckling effects. . 307 Figure 222: Deformed shape at peak load of a welded IPES04S. ... 308 Figure 223: Global strengths of members subjected to uniform major-axis bending. ... 310 Figure 224: Load-displacements curves of a HEA280 for three different member lengths. ... 311 Figure 225: Major-axis bending applied as a function of displacements and torsional twists for a HEAS01,

L = 1.9m. ... 312 Figure 226: Yielded zones at peak load of welded section HEAS01, L = 1.9m, under uniform My. ... 313 Figure 227: cL+G – lG graph for hot-rolled sections as a function of the height-to-width ratio. ... 315 Figure 228: cL+G – lG graph for hot-rolled sections as a function of sections strong axis to weak axis yield

modules. ... 316 Figure 229: cL+G – lG graph for welded sections as a function of the height-to-width ratio. ... 316 Figure 230: cL+G – lG graph for welded sections as a function of sections strong axis to weak axis yield

modules. ... 317 Figure 231: cL+G – lG graph for welded sections as a function of µW,My. ... 318 Figure 232: Torsional twists reached at peak load on the upper flange of hot-rolled and welded sections. ... 319 Figure 233: Eccentricities between the loading and the deformed shape at mid-span cross-section. ... 320 Figure 234: Resulting minor-axis bending in both flanges. ... 321 Figure 235: Influence of local effects and local/global interactions on the member strength of hot-rolled

sections. ... 323 Figure 236: Design of the local/global coupling factor fL/G for hot-rolled sections. ... 324 Figure 237: Design proposal and F.E. results for the global strength of hot-rolled sections. ... 325 Figure 238: Influence of local effects and local/global interactions on the member strength of welded

sections. ... 326 Figure 239: Design of the local/global coupling factor fL/G for welded sections. ... 327 Figure 240: Degree of biaxiality as a function of the global relative slenderness achieved with welded

sections. ... 328 Figure 241: Section geometrical properties and loading applied within the first worked example. ... 334 Figure 242: Effective area of a W100X296 made of steel S770 and subject to axial force. ... 338 Figure 243: Section geometrical properties and loading applied within the second worked example. ... 340

(23)

List of Figures

Figure 244: Effective area of a W610X241 made of steel S770 and subject to N + Mz. ... 344 Figure 245: Main geometrical properties of a HEAS01 and loading applied on the section. ... 346 Figure 246: Effective area of a HEAS01 subject to simple major-axis bending. ... 351

(24)

List of Tables

– xxiii –

List of Tables

Table 1: Web local buckling coefficient accounting for web/flanges interaction from Seif et al. [25]. ... 22 Table 2: Parabolic residual stresses for hot-rolled sections (see Figure 32b). ... 47 Table 3: Summary of the residual stresses measurements for hot-rolled sections with h / b £ 1.2. ... 48 Table 4: Summary of the residual stresses measurements for hot-rolled sections with h / b > 1.2. ... 49 Table 5: Ratio between the measured residual stresses and the ones suggested on Figure 35 for h / b £ 1.2. ... 50 Table 6: Ratio between the measured residual stresses and the ones suggested on Figure 36 for h / b > 1.2. ... 52 Table 7: Summary of the residual stresses measurements for welded sections. ... 55 Table 8: Ratio between the measured residual stresses and the ones suggested through the patterns on

Figure 39. ... 56 Table 9: Buckling curves imperfection factors of European standard. ... 75 Table 10: Distribution of buckling curves according to the section height-to-width limit. ... 76 Table 11: C.S.A. values for columns buckling curve coefficient n. ... 80 Table 12: Canadian recommendations for bending moment resistance for Class 1, 2 and 3. ... 80 Table 13: First design suggested by Villette for hot-rolled sections under major-axis bending. ... 90 Table 14: Second design proposal suggested by Villette for hot-rolled sections under major-axis bending. ... 90 Table 15: Imperfection factors suggested for hot-rolled sections. ... 91 Table 16: Equation of the imperfection factor aLT suggested by Couto et al. [60]. ... 94 Table 17: Comparison of Euler, Von Karman, and Winter formulae for principal criterion. ... 94 Table 18: Comparison of the D.S.M., C.S.M. and the O.I.C. design proposals. ... 95 Table 19: Mesh types studied for columns local resistances. ... 101 Table 20: Mesh types studied for beams local resistances. ... 102 Table 21: Mesh types at the initial configuration prior to loading and their amplified deformation at peak

load, for HEA160 under major-axis bending. ... 103 Table 22: Mesh types at the initial step and their amplified deformation at peak load, for IPES under axial

force. ... 104 Table 23: Mesh types studied for columns member strengths. ... 109

(25)

List of Tables

Table 24: Mesh types studied for beams member strengths. ... 109 Table 25: Mesh types of an IPES for the lengths L1 and L2. ... 110

Table 26: Mesh types of an IPES for the lengths L3 and L4. ... 111

Table 27: Geometrical properties and imperfections measured on the specimens. ... 119 Table 28: Test specimens characteristics and ultimate load provided experimentally and numerically. 120 Table 29: Measured geometrical properties and loading parameter b of each test. ... 124 Table 30: Initial bows and torsional twists measured by Dux and Kitipornchai. ... 124 Table 31: Agreement between experimental tests and numerical simulations, for each specimen tested by

Dux and Kitipornchai. ... 127 Table 32: Characteristics of the welded specimens tested. ... 129 Table 33: Geometrical imperfections measured by Fahnestock and Sause and Salem and Sause. ... 130 Table 34: Concordance between experimental tests and numerical simulations, for each specimen tested

by Fahnestock, Salem and Sause. ... 131 Table 35: Definition of the various types of imperfections considered and their associated parameters. ... 140 Table 36: Values of input parameters for each I-section geometry considered. ... 144 Table 37: Values of input parameters for each H-section geometry considered. ... 144 Table 38: Values of amplitudes applied to scale the 1st_{buckling mode shape. ... 145}

Table 39: 1st_{eigenmode shapes of the studied sections under pure compression. ... 147}

Table 40: 1st_{eigenmode shapes of the studied sections under major-axis bending. ... 148}

Table 41: Main characteristics of the specimens tested by Kettler [9]. ... 150 Table 42: Imperfections amplitudes measured on the specimens tested and corresponding sets of

amplitudes obtained from measured geometries. ... 152 Table 43: Statistical results from the comparison of the hot-rolled patterns for sections under major-axis

bending. ... 178 Table 44: Statistical results from the comparison of the hot-rolled patterns for sections under pure

compression. ... 178 Table 45: Statistical results from the comparison of the constant and trapezoidal patterns. ... 184 Table 46: Statistical results from the comparison between hot-rolled and welded patterns, for H-shapes.

(26)

List of Tables

– xxv –

Table 48: Definition of parameters considered in the modified Ayrton-Perry formulation for design proposal. ... 202 Table 49: Welded sections geometry and their corresponding µW from µW = 1.51 to µW = 4.00. ... 205 Table 50: Welded sections geometry and their corresponding µW from µW = 7.00 to µW = 16.20. ... 206 Table 51: Definition of parameters considered in the modified Ayrton-Perry formulation for design

proposal. ... 208 Table 52: Leading parameters tested for hot-rolled sections under major-axis bending. ... 210 Table 53: Definition of parameters considered in the modified Ayrton-Perry formulation for design

proposal. ... 210 Table 54: Relation between x and µW,My. ... 213 Table 55: Definition of parameters considered in the modified Ayrton-Perry formulation for design

proposal. ... 213 Table 56: Leading parameters tested for welded sections under major-axis bending. ... 215 Table 57: Design curve parameters for the 2nd_{design proposal for welded sections under major-axis}

bending. ... 216 Table 58: Definition of parameters considered in the modified Ayrton-Perry formulation for design

proposal. ... 220 Table 59: Definition of parameters considered in the modified Ayrton-Perry formulation for design

proposal. ... 221 Table 60: Level of axial force applied and degrees of biaxial bending tested. ... 228 Table 61: Adopted end-plate thickness allowing to reach the full plastic capacity for each cross-section. ... 233 Table 62: O.I.C. based design proposal for local strengths of hot-rolled sections for lL > l0. ... 246

Table 63: Participation rate of the proportion of axial compression by means of exponential functions. ... 247 Table 64: O.I.C. based design proposal for N + Mz, n = 0.80, for lL > l0. ... 252

Table 65: O.I.C. based design proposal for N + My + Mz, n = 0.60, abiax = 50 for lL > l0. ... 252

Table 66: Ayrton-Perry parameters of the simplified design proposal. ... 253 Table 67: Second design proposal based on the O.I.C. approach for local strengths of hot-rolled sections

(27)

List of Tables

Table 68: Accuracy of cL,O.I.C. / cL,F.E. ratio for hot-rolled sections under various load cases, using the 1st

O.I.C. proposal and its simplified version. ... 259 Table 69: Accuracy of cL,Ref. / cL,F.E. ratio for hot-rolled sections under various load cases, using the 2nd

O.I.C. proposal and the Eurocode rules. ... 259 Table 70: Design proposal based on the O.I.C. approach for the local strength of welded sections for

lL > l0. ... 271

Table 71: Maximum global imperfection amplitudes measured by Dux and Kitipornchai as opposed to the ones suggested by Boissonnade and Somja. ... 284 Table 72: Maximum global imperfection amplitudes measured by Fahnestock, Salem and Sause as

opposed to the ones suggested by Boissonnade and Somja. ... 284 Table 73: Characteristics associated to cross-sections considered within the investigation on L.T.B.

strength. ... 300 Table 74: Characteristics associated to cross-sections considered within the investigation on L.T.B.

strength. ... 306 Table 80: Dimensions of the sections considered in the brief preliminary study. ... 309 Table 81: Definition of O.I.C. parameters used in both graphs of Figure 223. ... 309 Table 82: Formulation of the local/global coupling factor fL/G for hot-rolled sections. ... 324 Table 83: Definition of parameters considered in the formulation for global design of hot-rolled sections. ... 325 Table 84: Formulation of the local/global coupling factor fL/G for welded sections. ... 327

(28)

– xxvii –

Table 86: Formulation of the degree of biaxial bending abiax for welded sections. ... 328 Table 87: Determination of the local relative slenderness of a hot-rolled I-shape under axial force. ... 334 Table 88: Definitions of parameters used in the O.I.C. proposal for a hot-rolled section local strength under

N, for lL > l0. ... 335

Table 89: Comparison between the O.I.C. and Eurocode proposals reliability, for the first worked example. ... 339 Table 90: Determination of the local relative slenderness of a hot-rolled I-shape subject to N + Mz. .... 340 Table 91: O.I.C. based design proposal for local strengths for a hot-rolled sections subject to N + Mz for

lL > l0. ... 341

Table 92: Comparison between the O.I.C. and Eurocode proposals reliability, for the second worked example. ... 345 Table 93: Values of the triggering factors involved in the O.I.C. proposal for local strengths of hot-rolled

sections. ... 347 Table 94: Determination of the local relative slenderness of a welded slender H-shape subject to uniform

My. ... 347 Table 95: Comparison of the reliability reached with the O.I.C. and Eurocode for the local strength of a

(29)

Notations

O.I.C. Overall Interaction Concept E.W.M. Effective Width Method

D.S.M. Direct Strength Method C.S.M. Continuous Strength Method

A.I.S.C. American Institute of Steel Construction C.S.A. Canadian Standards Association

E.N. European Standard F.E.M. Finite Element Method

G.M.N.I.A.. Geometrically and Materially Non linear Analysis including Imperfections G.M.N.A.. Geometrically and Materially Non linear Analysis

L.B.A. Linear Buckling Analysis M.N.A. Materially Non-linear Analysis

r Plate reduction factor (according to the E.W.M.) lp Relative plate slenderness

y Ratio of longitudinal stresses at plate edges or end moment ratio b Width of profile

h Height of profile t Thickness of plate tf Thickness of flanges tw Thickness of the web ks Plate buckling coefficient

Npl Plastic axial force Nu Ultimate axial strength

(30)

Notations

– xxix –

My,el Elastic bending moment about the major-axis bending Mz,el Elastic bending moment about the minor-axis bending My,u Ultimate bending moment about the major-axis bending Mz,u Ultimate bending moment about the minor-axis bending Wy,pl Strong axis section plastic modulus

Wz,pl Weak axis section plastic modulus Wy,el Strong axis section elastic modulus Wz,el Weak axis section elastic modulus

lL Generalised cross-section relative slenderness (includes influence of local buckling behaviour)

lG

Generalised member relative slenderness (includes influence of global buckling behaviour)

lL+G Generalised member relative slenderness (includes influences of local and global buckling behaviour)

cL Generalised cross-section local buckling factor cG Generalised member global buckling factor

cL+G Generalised member local and global buckling factor

cF.E. Generalised buckling factor determined numerically by means of Finite Elements cExp. Generalised buckling factor obtained from experimental test

cEC3 Generalised buckling factor calculated according to Eurocode 3 equations cO.I.C. Generalised buckling factor calculated according to the proposed approach

Rpl Load ratio to reach the “resistance” limit (plastic capacity) Rcr,L Load ratio to reach the cross-sectional (local) “stability” limit Rcr,G Load ratio to reach the member (global) “stability” limit

Rb,L Load ratio to reach the cross-sectional (local) ultimate capacity Rb,G Load ratio to reach the member (global) ultimate capacity Rb,L+G Load ratio to reach the member (local/global) ultimate capacity

(31)

Notations

scr,p Plate critical stress

e Strain

ey Strain at first yield (elastic)

aL Generalised imperfection factor (cross-section level) aG Generalised imperfection factor (member level)

b Factor accounting for strain-hardening effects

d Factor accounting for post-buckling resistance reserves l0 Non-dimensional length of plateau for resistance curve

(32)

Acknowledgments

– xxxi –

Acknowledgments

First of all, I would like to thank my supervisor Nicolas Boissonnade, for his continuous availability and help throughout this project. I would not have been able to understand all this amount of knowledge without all our meetings which allowed me to benefit from your impressive expertise on the subject. I wish you my best thoughts and deeply hope the O.I.C. will belong to future standards.

Markus Kettler and Donald W. White allowed me to take a step back from the thesis and gain in perspective. It has been really enlightening to benefit from their respective point of views and profound knowledge. Thank you, Professor White, for these two months at Georgia Tech which have truly been an enjoyable experience.

I would like to thank my O.I.C. partners: Caroline, Amir, Liya, Valentin, Anne-Sophie and Jeanne. It has been a real pleasure to get to know you and work together. Thank you, Caroline, for the discussions and for your support in more difficult times. Many thanks to Jeanne who has accompanied me and given me her crucial help within the final steps of my thesis.

Some warm thanks to Catherine, Valérie and Julia for your esteemed friendship. Sharing this office with you Julia was a true chance. I am also grateful to my friends from France who came to visit me in Canada at different times during my three-year-stay. Their precious company gave me the comfort I needed to keep up with my thesis.

Deep thanks to my parents and brothers, who managed to give me precious strength from their side of the ocean and with their many visits in Canada. Thank you for your patience and continuous support which have helped me so much.

I am truly grateful to Jeanne, Liya and Quentin for taking the time to read some parts of my thesis. Thank you mum, I hope you developed a keen interest in steel structures while tracking down all possible mistakes within this thesis. As for Clémence who read the whole report and suggested many improvements, thank you a lot for your contribution.

The last but not the least, my deepest gratitude to Quentin, who was willing to join me and move to Canada to enjoy this experience with me. Thank you for your love, your support and your reasoned advice which have helped me in a tremendous way. We were lucky Canada is such a diverse country you were able to have you own exciting Nordic adventure in Nunavut.

(33)

(34)

Introduction

– 1 –

Introduction

The use of steel as a construction material has significantly expanded over the years: steel presenting great balance between weight and strength, engineers and architects have gradually resorted to this material for the design of bridges and buildings.

The main characteristics of steel structures may be listed as follows:

• By guaranteeing sufficient strength even for long spans, steel allows to have wide open-spaces in buildings, so that it raises its usefulness on an esthetical level;

• The rapid erection of steel structures may drastically decrease a construction cost which may justify its use over concrete;

• Steel presents the advantage of high ductility which is of great use in seismic areas since it favours energy dissipation. Its behaviour against fatigue and the redistribution of forces to the more restrained part of a steel structure once an element has deformed may also be mentioned as great advantage of steel used in seismic areas;

• Although the great inherent strength of concrete in fire situations may represent an obvious benefit, the advantage of steel ductility to allow for a progressive collapse of the structure instead of the brittle one developed by concrete structures represents an attractive aspect. One could resort to steel hollow shapes filled with concrete to benefit from both their interesting properties. These characteristics justify why steel may be chosen instead of concrete. However, some efforts still need to be made to increase steel strength against fire or to make steel reuse less marginal. Although steel structures present less environmental impacts than concrete ones, steel reuse and steel recycling are still subject to improvements. After observing the peculiar practice of steel reuse, Dunant et al. [1] presented a survey on the possible options to make steel reuse profitable. The authors mentioned that for now, in the United Kingdom, the reutilisation of steel components may be too expensive if these elements are not provided from a site close to the construction one.

Steel has been widely used in construction and several of these works may be cited herein. As a major steel construction, the Canadian prestigious bridge built in Québec to cross the St Laurent (see Figure 1) may be cited here. Le Pont de Québec whose construction was achieved in 1917 was designed by Theodore Cooper, an American civil engineer. His erection was source of many problems since it failed and collapsed twice respectively in 1907 and 1916, causing many casualties. The bridge has now reached its end of life and experts should search for solutions to repair or replace it.

(35)

Introduction

Figure 1: Pont de Québec (Canada), see [2].

Open-sections such as column and beam shapes are frequently used in bridge and building constructions. Their use may typically be noticed in the construction of the Viaduc de Garabit in France which was designed by Leon Boyer and realised by Gustave Eiffel (see Figure 2).

Figure 2: Viaduc de Garabit, Cantal. Pictures from [3] on the left and [4] on the right.

Steel I and H-sections have been used in constructions of buildings as well and they are nowadays included as a main apparent architectural object. Timmerhuis building, which was designed by the prestigious architects firm OMA and raised in Rotterdam, may be used as a key reference for the use of such sections (see Figure 3).

(36)

Introduction

– 3 –

Figure 3: Timmerhuis building in Rotterdam, pictures from [5] on the left and [6] on the right.

The design of such structures relies on the interaction between an element strength and behaviour against instability if one shall focus on this aspect of the structure – other elements such as connections shall be cautiously designed as well. Under compression forces, elements made of steel sections such as I and H-shapes are prone to suffer from local buckling (see Figure 4) as well as from member buckling (e.g. flexural buckling or Lateral Torsional Buckling).

Figure 4: Local buckling of a column.

Finding efficient design provisions for steel structures represents one of today’s main challenges in Civil Engineering. The role of the designer is to determine the ultimate load-carrying capacity of a steel element

(37)

Introduction

by considering the interaction between pure resistance and stability. The occurrence and types of instabilities rely on the shape of the section, its slenderness as well as on the member’s one, the yield strength, the boundary conditions etc.. In addition, due to various interactions between buckling modes, addressing the design of slender elements made of slender sections may become complex and it may be difficult to find a great compromise between strength-gain (through a close estimate of the actual capacity) and simplicity of the design formulation. For many years now, the inaccuracies in codes design have been pointed out – some of them being already addressed by some authors. The issues in the standards led to improvements suggested by several researchers such as Schafer et al. [7] for the design of cold-formed slender sections, Gardner et al. [8] for the design of compact sections, Kettler [9] for the local strength of open-sections or Taras [10] as for the resistance of members subject to Lateral Torsional Buckling. However, a more drastic review of the codes may be needed to improve the parts of the codes which present lacks of accuracy and whose mechanical background may be updated. The Overall Interaction Concept (O.I.C.) represents a first solution for a complete reassessment of the design of steel structures. As pointed out for many years now by lots of authors, existent standards rules require some improvements both as of their potential to predict reliable strengths but in their efficiency as well. One may cite in a non-exhaustive way the following major issues: (i) the use of cross-section classification leads to complex calculation for slender sections, (ii) high strength steel which leads to the use of highly slender sections prone to suffer from local instabilities is currently not thoroughly addressed in codes, (iii) codes design for combined load cases relies on interaction formulae which may break the continuity in the design and (iv) the interaction between buckling modes is included into interaction factors whose determination is not straightforward.

Thus far, standards as EN1993-1-1 [11] in Europe, A.I.S.C. [12] in the US and C.S.A.-S16 [13] in Canada all make use of the concept of cross-section classification which may be considered as one of the main aspects researchers have tried to improve in the past few years. According to the Eurocode, the concept of sections classification relies on the distribution of sections within four classes depending on their slenderness and rotation capacity. The different bending moment resistances – section rotation curves associated to the four classes suggested in the Eurocode are described on Figure 5. The rotation capacity characterises in which extent a section may undergo large rotations while supporting its plastic capacity. The characteristics about each class may be defined as follows:

• Class 1 (compact sections) characterizes sections which are able to reach their full plastic bending moment Mpl and then develop large rotations while maintaining Mpl until a plastic mechanism may form;

(38)

Introduction

– 5 –

• Class 2 represents compact sections which can reach their full plastic bending moment but with restricted deformation capacity: these sections may not support sufficient rotations to allow for the full development of a plastic mechanism;

• Class 3 refers to semi-compact sections which are only able to reach their elastic bending moment although some codes do take their plastic strength reserve into account;

• Class 4 represents the most slender cross-sections which are highly sensitive to local buckling. They are thus able to attain only an “effective” elastic bending moment.

The classification concept is used within several codes guidelines and, even though some differences exist, its utilisation remains quite similar between the different standards.

Figure 5: Bending moment as a function of the rotation, cross-section classification concept.

Section classification has shown many deficiencies, especially the resistance gap between compact and semi-compact cross-sections which was addressed by Kettler during his thesis [9]. In addition, this concept does not provide efficient rules for the design of slender sections (Class 4) which may be quite complex especially for non-symmetric sections. The inaccuracies shown by the standards will be deeply developed further in the report throughout the State-of-the-Art.

In the purpose of providing improved design guidelines, some investigations focused on the local strength of doubly-symmetric open-sections. Among these studies, two major works can be distinguished: Kettler’s work [9] which led to a design formulation providing a continuous transition between compact and slender elements for which many experimental and numerical studies were carried out, and the work of Li [14] which was dedicated to the extension of the Direct Strength Method (D.S.M.) to hot-rolled and welded H-profiles. Further to these two main works, Seif [15] also contributed to the design of steel open-sections

M

_el,y

M

_pl,y

M

y Rotation

θ

Class 1 Class 2 Class 3 Class 4 M_y M_y

Rotational capacity for Class 1

Rotational capacity for Class 2

PRODUCED BY AN AUTODESK STUDENT VERSION