An Overall Interaction Concept for an alternative approach to steel members design

Journal of Constructional Steel Research 135 (2017) 199–212

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Journal of Constructional Steel Research

An Overall Interaction Concept for an alternative approach to steel members design Nicolas Boissonnade a,⁎, Marielle Hayeck b, Elsy Saloumi b, Joanna Nseir c a b c

Civil and Water Engineering Department, Laval University, Adrien Pouliot Bldg., Office 2928G, 1065, Avenue de la Médecine, Quebec G1V 0A6, Canada University of Applied Sciences of Western Switzerland – Fribourg, Switzerland Saint Joseph University, Beirut, Lebanon

a r t i c l e

i n f o

Article history: Received 4 February 2016 Received in revised form 17 February 2017 Accepted 27 February 2017 Available online xxxx Keywords: Steel section Steel member Design Overall Interaction Concept Resistance Stability

a b s t r a c t The present paper focuses on a new, alternative design philosophy: the Overall Interaction Concept (O.I.C.). This concept, based on the well-established resistance-instability interaction and the definition of a generalised relative slenderness, was thought and built to i) improve actual design practice, ii) increase accuracy, iii) advance simplicity and consistency, and iv) provide a sound framework for computer-assisted resistance predictions. This paper first details the bases and features of the O.I.C. approach, and provides mechanical interpretations of its application steps. Comprehensive sets of results at the cross-sectional level are then presented, for both H-shaped and hollow sections. Ayrton-Perry-based χ-λ design relationships for hollow structural shapes are proposed and shown to lead to more accurate, consistent and safe resistances when compared to Eurocode 3 rules, in addition to being significantly simpler in application. As for the behaviour and response of members, numerous F.E. results are reported, demonstrating the potential of a χ-λ approach to successfully apply to members under combined load cases. Also, the challenging case of coupled instabilities is investigated, and the O.I.C. approach is showed to be very efficient and appropriate. Further developments towards the derivation of full O.I.C. procedures to steel members are currently under way. © 2017 Published by Elsevier Ltd.

Abbreviations O.I.C. E.W.M. D.S.M. C.S.M. ρ λp ψ b h t kσ Mpl Mel λL λL+G

Overall Interaction Concept Effective Width Method Direct Strength Method Continuous Strength Method Plate reduction factor (according to the E.W.M.) Relative plate slenderness Ratio of longitudinal stresses at plate edges or end moment ratio Width of profile Height of profile Thickness of plate Plate buckling coefficient Plastic bending moment Elastic bending moment Generalised cross-section relative slenderness (includes influence of local buckling behaviour) Generalised member relative slenderness (includes influences of local and global buckling behaviour)

χL χL+G χFE χEC3 χproposal Rpl Rcr,L Rcr,G σcr,p ε εy αL β δ λ0

Generalised cross-section local buckling factor Generalised member local and global buckling factor Generalised buckling factor calculated numerically by finite elements Generalised buckling factor calculated according to Eurocode 3 equations Generalised buckling factor calculated according to proposed approach Load ratio to reach to “resistance” limit (plastic capacity) Load ratio to reach to cross-sectional (local) “stability” limit Load ratio to reach to member (global) “stability” limit Plate critical stress Strain Strain at first yield (elastic) Generalised imperfection factor (cross-section level) Factor accounting for strain hardening effects Factor accounting for post-buckling resistance reserves Non-dimensional length of plateau for resistance curve

1. Motivation – context ⁎ Corresponding author. E-mail address: [email protected] (N. Boissonnade).

http://dx.doi.org/10.1016/j.jcsr.2017.02.030 0143-974X/© 2017 Published by Elsevier Ltd.

The present paper focuses on a new, alternative design philosophy: the Overall Interaction Concept (O.I.C.). This concept, based on the

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well-established resistance-instability interaction and the definition of a generalised relative slenderness, was thought and built to i) improve actual design practice, ii) increase accuracy, iii) advance simplicity and consistency, and iv) provide a sound framework for computer-assisted resistance predictions. The idea of developing this new approach rose as a response to current specific problems – both conceptual and practical – and as an anticipation to the emergence of new materials (e.g. high or ultra-high strength steels) and design tools (i.e. software). As one of the current main issues in steel design, the cross-section classification concept bears many inconsistencies and practical difficulties. Cross-section classification consists in a preliminary step to section and member resistance checks, that intends at i) informing the designer on the possibility to resort to a plastic analysis (class 1 – plastic – sections possessing sufficient ductility and rotation capacity to allow developing a complete frame plastic collapse mechanism), and ii) orientating the designer to either plastic (classes 1 and 2), elastic (class 3) or effective (class 4) resistance checks. Classification of a section is achieved by means of b/t limit ratios which provide, knowing the actual stress distribution on each element and its “support conditions” (i.e. flange or web element), the class of the cross-section, determined as the class of its worst element. First of all, various papers have evidenced the inaccuracy of the proposed b/t limits, and both unsafe and over-conservative resistance predictions are reported, especially for hollow sections ([1,2]). Also, several discrepancies can be reported, e.g. as the class 3–4 border, where a section may happen to be classified as class 4 according to Eurocode 3 Part 1-1, thus requiring the calculation of effective properties following Eurocode 3 Part 1-5 design guidance, but eventually found to be class 3 and fully effective from the latter part of the code ([3]). This can be primarily attributed to the inaccurate definitions of the b/t limits as in Eurocode 3 (see Table 2). These limits, for the sake of simplicity, were basically derived for compression, mono-axial bending, and mono-axial bending with compression ([2]). Their background lies in i) the somewhat arbitrary adoption of plate slenderness limits λp = 0.5 and λp = 0.6 to define class 1–2 and class 2–3 limits ([4]), respectively, and in ii) the use of the so-called “Winter formulae” ([5]) to set class 3–4 limits. Additional background information may be found in [4]. Winter formulae for “internal compression plate elements” (webs) used for setting class 3–4 limits are as follows: ρ¼

λp −0:05ð3 þ ψÞ 2

λp

ρ¼

≤1 ðWinterÞ and

λp −0:055ð3 þ ψÞ 2

λp

ð1Þ

≤1 ðmodified WinterÞ

For “outstand compression plate elements” (flanges), the so-called “Winter formula” reads: ρ¼

1 λp

−

0:188 λp

2

≤1

ð2Þ

Depending on the support conditions of the plate or element (i.e. web or flange) and its associated stress distribution, these assertions lead to Table 1 b/t limits.1 It shall be noted here that, for the sake of consistency with Eurocode 3 Part 1.5 rules for plate buckling ([6]), the “modified Winter formulae” have been used in Table 1. The recommendations of Eurocode 3, in turn, provide b/t limits as in Table 2. As can be seen, multiple differences can be evidenced, for which quite limited scientific justification could be made. In the particular

1 Note that ε = √(235/fy) intends at unifying b/t limits for various steel grades, acknowledging for the well-known more important relative influence of local buckling for higher steel grades.

case of b/t limits for webs, several authors ([3,7,8]) have reported significant unconservative resistance predictions triggered by the seemingly too optimistic b/t = 38 ε value at the class 2–3 border. This is clearly seen in the comparison between Tables 1 and 2, and further evidenced by Fig. 1a, where the comparison between different standards exhibits large and questionable differences – note in particular that Eurocode 3 proposes a 4 ε-wide class 3 range, from 38 ε to 42 ε, while other standards propose an end of class 3 at 38 or 34 ε, i.e. an end of the class 3 range well below the beginning of that of Eurocode 3. Also, it is now widely recognised that smooth and continuous resistance transitions along the b/t range shall be made available in modern design codes, in order to provide consistent design provisions from plastic to slender situations. In Eurocode 3 ([9]) in particular, the current rules shall be improved, as suggested in [3] in order to avoid the gap of resistance at the class 2–3 border (see Fig. 1b), which is mechanically meaningless and unacceptable. It may be noted here that several standards ([10,11,12]) have already included such continuous provisions. As another point suffering criticism, the assumption of “ideal support conditions” for the element plates comprised within the whole section brings further inadequate and inaccurate resistance predictions. The interaction between elements is indeed usually disregarded ([13]), each element being presumed to behave discretely; flanges are assumed to behave as under pinned-free support conditions, while webs are assumed as pinned-pinned. It is however clear that elements are interacting, and that this may lead to both over-conservative results (e.g. a weak web associated with strong flanges so that the web is close to clamped-clamped support conditions), or unsafe results (e.g. a slender, locally-buckled web may be attributed a “negative” stiffening effect to flange stability [14]). Several attempts to account for it directly ([15]) or indirectly through interdependent b/t limits for flanges and webs may be found in the literature. Early results ([8,16]) have shown that the assumption of “ideal support conditions” may lead to significant differences with respect to more rigorous modelling. In this respect, the possibility to consider the cross-section as a whole in the design procedure represents a great improvement, which the O.I.C. allows for (see Section 2.1). Eventually, one may report on major practical application difficulties of the cross-section classification concept, for sections under compression and biaxial bending or for the determination of the plastic neutral axis of hollow sections under My + Mz for example; the latter situations lead to disproportionate efforts regarding the information it provides, when related to the consecutive design checks, in which designers are primarily interested. Besides issues associated with the cross-section classification concept, the adoption of the Effective Width Method (E.W.M.) in major design standards is known to bring further practical difficulties. Indeed, while the assumption of neglecting parts of the elements that are most concerned with local buckling can make sense from a Structural Mechanics point of view, it triggers long and tedious calculations of the cross-section effective properties – sometimes even through an iterative process, cf. [17]. While the calculation of effective properties may be considered as affordable for large girders in sophisticated structures (e.g. bridges), this may be deemed as unacceptable in case of standard, simple building elements. This point is expected to become of greater importance in the near future through the increasing use of high strength steels, where the relative importance of instabilities is growing (especially local buckling) and situations where local-global coupled instabilities have to be accounted for are also met more often. In addition, the material response of high strength steels being known to sometimes be more non-linear than mild steels (i.e. no yield plateau), the concept of plastic resistance makes no sense anymore2; other similar cases include stainless steel members, cold-formed

2 Indeed, the “disappearance” of the plastic plateau prevents from using typical constant stress blocs diagrams, causing the usual determination of plastic resistance inappropriate.

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Table 1 b/t Limits as calculated from underlying concepts and modified Winter formulae. Class 1–2 limits

Class 2–3 limits

Class 3–4 limits

kσ

λp = 0.5

λp = 0.6

Plate in compression λp = 0.673

Plate in bending λp = 0.874

0.43 (flange) 4.0 (web) 23.9 (web)

9.32 ε ≈ 9 ε 28.42 ε ≈ 28 ε 69.47 ε ≈ 69 ε

11.18 ε ≈ 11 ε 34.10 ε ≈ 34 ε 83.36 ε ≈ 83 ε

12.54 ε ≈ 13 ε 38.25 ε ≈ 38 ε /

/ / 121.43 ε ≈ 121 ε

Table 2 b/t Limits proposed in Eurocode 3. Class 1–2 limits

Class 2–3 limits

Class 3–4 limits

kσ

λp = 0.5

λp = 0.6

Plate in compression λp = 0.673

Plate in bending λp = 0.874

0.43 (flange) 4.0 (web) 23.9 (web)

9ε 33 ε 72 ε

10 ε 38 ε 83 ε

14 ε 42 ε /

/ / 124 ε

tubes, carbon steel at high temperatures… Accounting for a more accurate stress-strain response as well as for potentially beneficial strain hardening effects is thus clearly desirable. This ascertainment has been underlying the development of the Continuous Strength Method (C.S.M., [18,19,20]), as an attempt to provide better predictions of the observed behaviour in such cases as well as material savings. As a last point, one may here refer to the treatment of beam-column design, that although having received wide attention through many years ([21]) and despite tremendous recent research efforts ([22]), remains relatively complex in practical applications; too, frequently-met cases of beam-column with intermediate restraints and/or non-ideal support conditions (i.e. non “fork” support conditions) are usually poorly addressed in design codes. Albeit the practical design of beamcolumns remains one of the most difficult and challenging problems for engineers and that no simple yet accurate design rules shall be foreseen, the current sets of rules proposed in major design codes cannot be accepted as fully satisfactory. Accounting for all this and despite recent efforts to overcome some of these points, the development of a new, alternative approach to deal with the practical design of steel members and sections as influenced by instabilities was initiated: the Overall Interaction Concept (O.I.C.). The approach basically relies on the resistance-instability interaction and its subsequent treatment through an extended Ayrton-Perry approach ([23]). Moreover, it makes use of generalised relative slenderness, both at the cross-section (local) and member (global) levels, by

means of so-called load ratios. The O.I.C. provides solutions to many above-listed issues, and in particular: i) Abandons the discrete and artificial cross-section classification concept; ii) Does not make use of the E.W.M. (i.e. no effective properties are needed); iii) Provides smooth and continuous resistance predictions from plastic to slender responses of sections and members; iv) Proposes an identical, mechanically-based concept for the design checks of sections and members; v) Allows all cross-section shapes (open or closed) to be treated similarly – the design procedures shall only differ in locally calibrated coefficients. The bases and features of the O.I.C. approach are detailed in Section 2; in particular, mechanical interpretations of its application steps are given in Section 2.2. Section 3 is dedicated to the application of the O.I.C. to cross-section resistance – both hollow and open sections, while Section 4 devotes to member buckling resistance. A specific section (Section 4.3) is devoted to local-global coupled instability effects. Conclusions and future development steps are proposed in Section 5. It shall perhaps be mentioned here that the derivation of full O.I.C. procedures is partly achieved (Section 3.1) and in progress; consequently, some of the results presented in this paper are only intended at

Fig. 1. a) Bending capacity of a cold-formed rectangular hollow section as a function of flange (leading) b/t ratios according to various standards – b) Gap of resistance at the class 2–3 border in bending.

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Fig. 2. Basic principles and application steps of the O.I.C.

evidencing its potential and features. Complete and comprehensive design equations will be made available in a near future. As far as possible, the notations adopted herein are those of Eurocode 3. 2. The Overall Interaction Concept (O.I.C.) 2.1. Principles and application steps The O.I.C. is an alternative approach to the current design methods predicting steel member resistance capacity. Applicable to both sections and members, it is based on the well-known “resistance-instability” interaction concept (see Fig. 2). Various ways of dealing with this resistance-instability interaction have been derived in the past, and are available in the literature, as well as in design recommendations. In a non-exhaustive manner, one may cite the proposals from Ayrton-Perry [23], Merchant-Rankine [24], Winter [25], Schafer [26] or Gardner [18]. As a classical feature, many of these approaches make use of a so-called “relative slenderness” that is meant to characterise the relative influence of instability on pure resistance. The O.I.C., too, is a relative slenderness-based approach, but its definition has been generalised as follows:

λrel

sffiffiffiffiffiffi Rpl ¼ Rcr

ð3Þ

where Rpl represents the factor by which the actual loading has to be multiplied to reach the (plastic) resistance limit (i.e. the influence of

instability and imperfections is completely disregarded – step 1), while Rcr is the load ratio used to reach the elastic buckling load (instability limit, i.e. allowable stress is infinite – step 2). Obviously, the generalised expression (3) allows λrel to take the balance between resistance and instability (step 3), but also allows, thanks to the use of load ratios, to treat combined load cases as straightforwardly as simple ones. In addition, provided that Rcr is relative to either local or global instabilities, the O.I.C. can be applied identically to cross-section and/or to member design (cf. Fig. 3). The presence of imperfections is also accounted for through the use of the interaction curve (step 4), that provides χ = f °(λrel) factors standing as the penalty owing to instabilities and imperfections on the pure resistance response. Similarly to most modern design codes, use of non-dimensional axes is kept, and note that Rpl is relative to the plastic resistance (i.e. χ = 1.0 cases – no influence of instabilities – lead to plastic resistances of the sections/members). As can be seen in Fig. 3, the influence of local buckling on the crosssection resistance is evaluated first, and, in a second step, used in the definition of λL+G to eventually lead to χL+G that stands as the overall penalty factor to be applied to the cross-section plastic capacity so as to account for the detrimental influences of cross-sectional and member instabilities. Note that the final resistance check (step 5) i) includes partial safety factors ϕ or γM, and ii) requires Rb,L+G to be greater than unity, unlike what is usually met in design standards (lower than unity). The latter simply means that the ultimate load ratio – paying due allowance to all previously-cited influences – is such that the predicted carrying capacity shall be greater than the applied loading on the member. 2.2. Mechanical interpretation – specific features The O.I.C. format, as described in the previous paragraphs, defines so-called χ-factors, that stand as penalty factors owing to buckling effects on the plastic capacity (of the section/member). Through adequate definitions of χ = f °(λ) relationships, one may include the influence of imperfections, strain hardening effects, post-buckling reserves, … Obviously, as is done in many codes (e.g. flexural buckling), multiple curves χ = f °(λ) shall be contemplated, with respect to many key parameters, amongst which the level of imperfections is of significant relevance. In

Fig. 3. O.I.C. design flow chart – Section and member resistance.

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particular, equations characterising the cross-sectional response shall be differentiated from those applying to member resistance – see Sections 3 and 4. As one of the major features of the O.I.C., the direct determination of the design resistance is worth being emphasised. Indeed, once λ and χ have been evaluated, the final resistance check (step 5) is straightforward, whatever the complexity of the loading (no need for interaction formulae). Also, since a smooth and continuous prediction of resistance is ensured from plastic to slender sections, the cross-section classification preliminary step and its many issues are no more necessary and disappear from the design procedure. In addition, the particular case of slender cross-sections which, currently, implies long and tedious effective properties calculations in practice, is covered in a simple, effective and direct way. This shall also be seen as a major improvement brought to current and future daily practice, as many more design situations are expected to fall in the slender range in the near future with the emergence of high strength steels.3 Also, thanks to the adoption of load ratios in the definition of the generalised relative slenderness, unique sets of design provisions may be proposed, regardless of the cross-section slenderness (i.e. no different sets of rules as a function of the cross-section class), unlike in most design standards. Coupled local-global instabilities are also treated in a very simple way, as Fig. 3 shows – in principle, the adopted approach is similar to the one of Eurocode 3 ([9]). Further to resistance continuity aspects, consistency with existing recognised design rules for particular situations (e.g. flexural buckling, lateral torsional buckling) is guaranteed since an identical format (socalled Ayrton-Perry format) is already used for such cases. Another important improvement lies in the consideration of the cross-section as a whole, i.e. including all cross-sectional elements in the determination of R-factors at the cross-sectional level. As a consequence, possible plate interactions are considered within the cross-section, with a potential beneficial influence on the resistance. This leads to smaller, more favourable λ values, and associated more economic χ values. This also sometimes allow for an elastic-plastic resistance of slender sections, usually restricted to elastic effective capacities; in the context of an increasing use of high strength steels, this shall lead to an interesting economy potential. Eventually, the O.I.C. approach offers the possibility to account for beneficial strain hardening effects and for non-constant distributions of the yield stress across the section. The former typically concerns elements made of materials exhibiting a pronounced non-linear response at low strains (aluminium, stainless steel …), while the latter can for example be relative to cold-formed sections for which the corner parts of the section usually exhibit higher actual yield stress values compared to their flat faces counterparts. On the assumption of a classical yield plateau to provide the reference plastic capacity (cf. definition of Rpl), these beneficial influences may be accounted for through χL ≥ 1.0 values – see also Section 3.1. Besides, the response of members subjected to varying bending moment distributions may also be adequately associated with χL+G values higher than 1.0, as will be shown in Section 4.

3. Application to cross-section resistance 3.1. Hollow sections The present paragraph first proposes a series of results on the application of the O.I.C. to the cross-sectional resistance of hollow structural shapes. Accordingly, only the cross-section application steps in Fig. 3 are of concern. Also, attention is brought to the fact that the critical load ratio Rcr,L shall be considered, characterising the influence of local buckling (of the cross-section as a whole). 3 High strengths steels indeed cause the relative importance of buckling to become more important, compared to lower steel grades.

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Rcr,L may be calculated by means of classical linear buckling theory, assuming the plates of the cross-section to be simply-supported and keeping the least favourable plate to provide the value of Rcr,L:  Rcr;L ¼ min

σ cr;p;i

σ Ed; max;i

 ð4Þ plate element i

where σcr,p,i represents the critical plate buckling stress of element i within the cross-section and σEd,max,i refers to the actual stress distribution on element i. Alternatively, Rcr,L may be efficiently and more favourably calculated by numerical tools considering the complete cross-section. As for the Rpl factor, use of analytical Eurocode 3 equations or more accurate proposals ([27]) may be contemplated – again, more economic and efficient tools are available ([28]). In the particular case of cold-formed tubes, for which no yield plateau is usually observed, the concept of plastic resistance makes no sense anymore, and so does a value of Rpl in such cases. However, resorting to typical conventional 0.2% proof stresses, combined with the adoption of appropriate interaction curves, allows keeping the same approach and application steps, while accounting for the rounded material response and the associated beneficial strain hardening effects. Once Rcr,L and Rpl factors have been evaluated, by either analytical or computer-assisted means, the remaining step for a complete application of the O.I.C. to hollow sections lies in the adoption of suitable interaction curves. This last step involves adequate inclusion of several key parameters in the definition of the interaction curve, such as the fabrication process, the cross-section shape (rectangular or square hollow section), the type of loading (simple or combined), the steel grade…. Similarly to what is currently used in Eurocode 3 for columns flexural buckling, choice has been made here to use parametric χL = f °(λL) curves as based on an “extended” Ayrton-Perry format ([29]). Extension of the well-known traditional format was necessary to characterise the specific response of sections. As schematically represented in Fig. 4, modifications to the standard Ayrton-Perry format have been brought as follows: i) Use of a λ0 parameter to define a variable χL ≥ 1.0 plateau length; ii) Adoption of factor β to allow for χL ≥ 1.0 values, as a consequence of positive strain hardening effects; iii) Calibration of δ expressions so as to suitably account for postbuckling resistance reserves, at the cross-sectional level; iv) Appropriate definition of “generalised imperfection factor α”, to apply to hollow sections. It should be emphasised here that a smooth continuity from plastic to slender capacities is naturally associated with this extended AyrtonPerry format. To derive and assess the χL = f °(λL, λ0, β, δ, α) expressions, many relevant test data have been gathered, both from own tests and from literature ([8]). Also, an extensive data set from shell F.E. non-linear analyses was used ([8]), containing N42,000 results with respect to various cross-section shapes and dimensions, steel grades, manufacturing processes, cross-section slenderness and loading. Careful and detailed analysis of these results ([8]) allowed to derive expressions for the λ0, α, β and δ parameters, as a function of i) the h/b ratio, ii) the relative axial compression n = NEd/Npl,Rd. and iii) the cross-sectional slenderness λL ([8]). Typical expressions for simple load cases are reproduced in Table 3. Full sets of design expressions have also been developed for combined loading situations ([8]). Note that no dependency on the steel grade was kept in the different expressions, as negligible differences (lower than 3%) were observed in all cases. In all situations, the h/b ratio has appeared of prime importance. Fig. 5 plots the comparison between results obtained from validated shell F.E. models [8] and the proposed interaction curves, for various h/b ratios, in the case of hot-rolled tubular sections. As can be observed,

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Fig. 4. Extension of Ayrton-Perry format to cross-sectional resistance.

rectangular hollow sections (h/b ≥ 1.0) reach higher relative section resistance compared to square hollow sections possessing the same relative slenderness, particularly in the slender range. The level of restraint offered by the narrow faces of the rectangular section to the wider ones is indeed shown to provide an increased cross-section resistance through stress redistributions once local buckling develops in the more buckling-prone plates. Consequently, the cross-section resistance is increased with the h/b ratio, and square sections consequently exhibit

the lowest resistance to compression owing to simultaneous buckling of the constitutive plates. Besides, the resistance limit was arbitrarily limited to χL = 1.0 (i.e. β = 1.0) since, for hot-rolled tubes, strain hardening reserves were small and only observed for unrealistic deformation levels. Also, it can be seen that in the χL ∈ [λ0; 1.0] range where the influence of imperfections is strong on the cross-section response, the square sections are seen to be less penalised by local instabilities. Oppositely, once

Table 3 Proposed O.I.C. design expressions for simple load cases (hollow sections). General format qffiffiffiffiffiffiffi Rpl λL ¼ Rcr;L δ

ϕL ¼ 0:5  ð1 þ α L  ðλL −λ0 Þ þ λL  βÞ β qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ ϕL þ ϕL 2 −λL β

χL ¼

Hot-rolled hollow sections λ0 ¼ 0:35 and χL ≤ 1.0 i.e. β = 1.0 Parameters

αL

δ

Compression Major-axis bending Minor-axis bending

0.15 0.1δ + 3/200 0.08

−0.4h/b + 1.45 −0.4h/b + 0.25 0.65

Cold-formed hollow sections – 1st approach λ0 ¼ 0 Parameters

δ

αΛ

β

Compression

−0.4h/b + 1.45

0.1δ + 3/40

−0:15  λL þ 1:15

Major-axis bending

−0.4h/b + 0.25

0.1δ + 7/200

−0:20  λL þ 1:20

Minor-axis bending

0.65

0.1

−0:20  λL þ 1:20

Cold-formed hollow sections – 2nd approach λ0 ¼ 0:40 and εεy ¼



Parameters

0:40

1:5

λL λL ≤λ0

λL N λ0 (β = 1.0) δ

αL

Compression

χ L ¼ 1:15−

0:15 ðε=ε y Þ0:6

−0.4h/b + 1.45

0.1δ + 7/40

Major-axis bending

χ L ¼ 1:20−

0:20 ðε=ε y Þ0:6

0.4h/b + 0.25

0.25δ − 1/80

Minor-axis bending

χ L ¼ 1:20−

0:20 ðε=ε y Þ0:6

0.65

0.15

N. Boissonnade et al. / Journal of Constructional Steel Research 135 (2017) 199–212

Fig. 5. Hot-rolled hollow sections shapes in compression (F.E. results for S355 steel).

buckling becomes predominant, the square sections find themselves with the worst penalty compared to rectangular sections, owing to higher restraints brought by adjacent plates – the flanges become less critical components. As for cold-formed sections, an identical procedure was followed, and the corresponding expressions from λ0, α, β and δ are summarised in Table 3 as well. In the particular case of cold-formed sections, however, strain hardening effects have a positive influence on the cross-section capacity for λL b 0.4, so that an additional alternative approach for the development of interaction curves has been derived. It consists in relying on a strainbased approach at low cross-sectional slenderness (λL ≤ 0.4), while keeping an Ayrton-Perry format at intermediate and large slenderness values. It simply sets the strain governing the cross-section's response for λL ≤ 0.4, and a relationship between the relative strain level and λL was established as follows: ε ¼ εy

  0:40 1:5 for λL ≤0:40 λL

ð5Þ

where εy corresponds to the strain level at first yield; then, a relationship between ε/εy and χL was calibrated: 0:15  for λL ≤0:40 χ L ¼ 1:15−  ε=ε y

ð6Þ

205

Coefficients in Eqs. (5) and (6) were fitted according to the numerical results plotted in Fig. 7a and b. As can be seen, excellent and safe accordance of the proposed approach with reference F.E. results is observed, over the whole range of cross-sectional slenderness (Fig. 6b). Identical observations and conclusions could be drawn for the 1st approach, where similar accuracy and performance could be reported – both the 1st and 2nd approach take due account of strain hardening effects, see Fig. 6a. In order to further assess the performance of the proposed approach, systematic comparisons with Eurocode 3 design rules have been made. Examples of results are given in Figs. 8 to 10. Fig. 8a and b first relate to the cross-sectional resistance of hot-rolled rectangular and square hollow sections under major-axis bending moment My,Ed. Fig. 8a plots the cross-section slenderness λL vs. the χFE/χEC3 or χFE/χproposal ratios; the latter allow assessing the performance of both Eurocode 3 and the proposed design procedures, in terms of safety and accuracy. Fig. 8b proposes an alternative plotting of the same results, however in histogram format, so as to better assess the merits of both analytical predictions, namely owing to the large number of results and the dense scatters reported. In particular, it can be observed that: i) The proposed approach provides much more accurate, safe and consistent resistance predictions than Eurocode 3; ii) At intermediate slenderness ranges λL ∈ [0.3;0.6], Eurocode 3 provisions typically exhibit quite inconsistent results, as a consequence of the adoption of discrete behavioural classes and their associated resistance drop from Mpl to Mel; in contrast, the proposed O.I.C. approach leads to smooth, continuous and consistent predictions; iii) For very stocky sections (λL b 0.3), non-negligible strain hardening reserves leads to χL values significantly above unity, up to 40% benefits. Careful and detailed analysis of the corresponding results allowed to evidence that their beneficial influence shall not be safely considered in design recommendations, as being associated with high ε/εy strain ratios that were deemed unrealistic for practical applications. Consequently, results with χFE/χEC3 or χFE/χproposal ratios above 1.3 should be disregarded (red circles in Fig. 8a and b); iv) Eurocode 3 predictions are usually safe-sided, except for large λL values, for which systematic unsafe predictions down to χFE/χEC3 = 0.85 are reported (indicating about 15% unsafety). These unsafe results may not be compensated by safety factors γM = 1.0 nor γM = 1.1, as currently prescribed in Eurocode 3 and in most national annexes.

Fig. 6. Cold-formed hollow sections shapes in compression (F.E. results for S355 steel) – a) 1st approach (modified Ayrton-Perry) – b) 2nd approach (strain-based at low slenderness).

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Fig. 7. Cross-sectional reduction factor χL as a function of relative strain demand.

Figs. 9a and b are similar to Figs. 8a and b; they however relate to cold-formed hollow sections in compression, and to the 2nd design approach proposed here. Relatively similar trends are observed: i) The proposed approach yields excellent results, both more accurate and consistent than Eurocode 3. In particular, remarkably lower scattered resistance predictions are obtained, as indicated by the narrow distributions of results in the histogram of Fig. 9b; ii) Eurocode 3 is seen to lead to many unconservative results in the whole λL N 0.5 range, with an average χFE/χEC3 = 0.85 (see circles in Fig. 9a and b). This is of particular significance, both in terms of importance of the unsafe values (Fig. 9a) and in amount of results concerned (Fig. 9b), and this clearly suggests that Eurocode 3 rules shall be improved; iii) As a particular point, for λL b 0.4 cases, the proposed O.I.C. approach leads to more economical results, thanks to the possibility to account for strain hardening effects and thus to reach higher resistances than χL = 1.0 (i.e. higher that Npl = A.fy0.2% as based on the conventional 0.2% proof stress). As a last example, Fig. 10a and b report on results with NEd + My,Ed + Mz,Ed combined loading situations, where axial compression mobilizes about 20% of the cross-section capacity – loading was applied with all loads increased proportionally up to failure, with an initially-estimated NEd,ini value so as to take out nearly 20% of the section's resistance at

failure. These figures display the same tendencies as Fig. 9a and b, however exacerbated. In particular, highly unsafe values (up to 25%) as well as significantly over-conservative ones for Eurocode 3 rules are observed (N60% over-conservative predictions). In addition, one may notice that these bad predictions are related to λL values typically characterising the range of sections often used in practical applications – i.e. 0.4 ≤ λL ≤ 1.0. Oppositely, the proposed approach yields excellent, accurate and consistent resistance predictions.

3.2. Open sections This paragraph is relative to the cross-sectional resistance of open sections, as characterised by means of an O.I.C. approach. Again, the following results rely on the calculation of Rpl and Rcr,L factors to evaluate λL, and both Rpl and Rcr,L were evaluated numerically, except for case of simple load cases where direct, exact and easy analytical determination of Rpl was preferred. Rcr,L was computed by means of CUFSM software ([30]), so as to get accurate values; it shall be mentioned here that a specific treatment of the web-to-flange junction area was made for a more realistic modelling ([31]). Again, alternative analytical determination of Rcr,L is possible by means of classical plate buckling stress σcr,p expressions ([5]), however with less accuracy on each plate's true support conditions – note that this approach was used to get the “Eurocode 3” results in the following plots.

Fig. 8. Accuracy of proposal vs. Eurocode 3, hot-rolled sections under major axis bending moment.

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Fig. 9. Accuracy of proposal vs. Eurocode 3, cold-formed sections under compression.

Figs. 11 to 12 propose numerical F.E. results relative to usual hotrolled open section shapes in simple compression or under major-axis bending. Fig. 11a reports on the F.E.-predicted response of all European traditional shapes (IPEs, HEAs, HEBs and HEMs) under axial compression, with varying steel grades from S235 to S690. It is observed that: i) According to the F.E. resistance predictions, relatively few profiles exhibit a loss in capacity owing to local buckling – usually, sections with λL N 0.7 with a high yield stress. In addition, the overall χL penalty factor never reaches values below 0.8, indicating a relatively limited influence of local buckling on the overall cross-sectional response, although many of these sections belong to the “class 4” (slender) range by far, namely for S690 steel grade; ii) Some strain hardening effects are visible at low slenderness (λL b 0.3), mostly for S235 steel grade and the most compact sections (HEBs and HEMs). Similarly to hot-rolled tubes, these beneficial results can be shown to occur at very large strains, whose levels were deemed unrealistic for practical applications and thus not considered – i.e. χL ≤ 1.0 is recommended; iii) Except for leading to increasing λL values, the influence of the yield stress is seen to have quite little influence on the results, and this parameter can reasonably be omitted in the formulation of χL = f °(λL) equations;

iv) Comparison with the design proposal of Li ([32]) shows that the latter analytical proposal proposes slightly unsafe resistance predictions, although adopting the same definition of λrel but not an Ayrton-Perry format. Fig. 11b, also relative to open sections under axial compression, however plots in “O.I.C. axes” results obtained for the North American structural shapes database (203 different section shapes), again as a function of the steel grade. Except for the fact that these dimensions lead to λL values as high as 1.80 for 100 ksi steel, all previous tendencies and features can be observed as well, namely regarding the slightly unsafe predictions of the proposal of Li ([32]) and the limited influence of the material's yield stress. Fig. 12 plots numerical results obtained for the European's sections database submitted to major-axis bending My,Ed, differentiated following the section's type (IPE, HEA, HEB or HEM – all steel grades S235, S460 and S690 are included and merged). It shows that: i) Although many sections belong to class 3 or 4 (especially for S690 steel), nearly all sections are shown to reach their plastic capacity, or close to it – recall that χL = My,Ed/Mpl,y,Rd. = 1.0 indicates the attainment of the plastic resistance, Rpl being relative to a plastic distribution of stresses within the cross-section. This observation is remarkable, since for such class 3 or 4 sections,

Fig. 10. Accuracy of proposal vs. Eurocode 3, cold-formed sections under combined NEd + My,Ed + Mz,Ed load cases, n = 0.2.

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Fig. 11. Behaviour of hot-rolled open section shapes in compression (European and North American databases).

Eurocode 3 as well as many design codes limits the resistance to the elastic capacity, i.e. a drop of some 12% to 15% corresponding to χL values as low as 0.85, which are not observed here on the “true” cross-sectional response; similarly to [3], this suggests that improvements in the design specifications shall be made; ii) As expected, the most compact HEM sections are seen to be the ones more (positively) affected by strain hardening effects – χL ≥ 1.0 can be observed; iii) Resistance predictions from Eurocode 3 design rules are seen to be both unconservative at λrel values comprised between 0.3 and 0.5 and over-conservative beyond λrel = 0.5, as a result of the class 2–3 resistance gap mentioned earlier. Detailed analysis of these results indeed show that the sudden drop from the plastic capacity Mpl,Rd to the elastic capacity Mel,Rd is responsible for such discrepancies. Also, one may recall here that Eurocode 3 does not allow to account for strain hardening reserves, i.e. χL max = 1.0 so that the ultimate cross-section bending resistance is strictly limited to Mpl,Rd. = Wpl,y fy.

4. Application to member resistance 4.1. Hollow sections The present section deals with member behaviour and resistance, i.e. both cross-sectional and member instabilities are likely to occur (and

Fig. 12. Behaviour of European hot-rolled open section shapes under major-axis bending.

interact), and are dealt with here. Consequently, the complete O.I.C. procedure as described in Fig. 3 applies, and both cross-section and member design checks are considered. However, Section 4.3 being dedicated to local-global coupled instabilities aspects, the present paragraph is preponderantly relative to situations as influenced by global instability, only i.e. by major and/or minor-axis flexural buckling in the case of hollow sections. Accordingly, focus is here on: i) The extent of yielding within sections and along the member, as influenced by the development of flexural buckling; ii) The influence of various bending moment distributions on the member's carrying capacity; iii) In case of combined load cases, the influence of load distribution, especially regarding the level of axial compression. In a first step, the influence of bending moment distribution is investigated, as depicted in Fig. 13a. It summarises a very large number of shell non-linear F.E. results ([33]), where NEd + My,Ed + Mz,Ed combined loading situations are considered with different linear bending moment distributions along the member, characterised by end moment ratio ψ.4 Cases with ψ = 1 are then relative to constant bending moment situations, while ψ = 0 cases relate to triangular bending moment distributions. Obviously, bending moment distributions ψ = 0 provide higher member capacities than ψ = 1 situations, in every case. Such results indicate that ψ is a key parameter in such beam-columns' response and that χG = f°(λG) relationships have to be made dependant on this parameter – such as most current design approaches recommend, cf. [22]. Fig. 13a also depicts a reasonably reduced scatter, accounting for the fact that results for very different situations are reported (various crosssection shapes and dimensions, steel grades, load combinations…); this further evidences the ability of the O.I.C. to capture the behaviour of beam-columns in a straightforward manner, i.e. without resorting to so-called member interaction formulae. Provided the governing parameters are identified, the possibility to derive accurate and suitable, mechanically-based interaction curves seems obvious, and is actually under completion. Fig. 13b is relative to in-plane situations (either NEd + My,Ed or NEd + Mz,Ed), and characterises the influence of the cross-section shape. It clearly evidences the influence of the height-to-width h/b ratio, as was observed for cross-section resistance. The same mechanical interpretation applies here too, i.e. weaker plates in the cross-section are restrained by the stronger ones, depending on the cross-section geometry and on the various stress distributions. Again, very little scatters are

4

ψ = Mend min./Mend max. and −1 ≤ ψ ≤ 1.

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Fig. 13. Behaviour of hot-rolled tubes under combined loading situations.

Fig. 14. Influence of relative axial compression level (hot-rolled tubes).

observed, as an excellent indicator of the potential to accurately yet simply and efficiently take this into account. Eventually, Fig. 14a and b isolate the influence of the level of relative compression n = NEd/Nb,Rd at failure,5 for NEd + My,Ed or NEd + Mz,Ed situations. They point out: i) The – expected – much higher influence of instability effects for n = 0.7 cases, as indicated by much higher λL+G values for an identical subset of data; ii) Again, the relative little scatters as well as clear tendencies (especially for n = 0.3 situations), which, combined with sorting through ψ and h/b ratios allow to reach very clear and consistent trends; these results further confirm the capacity of the O.I.C. to correctly predict the response of hollow structural shapes members, whatever the loading distribution. Further developments are under way to derive relevant interaction curves ([33]). 4.2. Open sections This paragraph proposes F.E. results on open section members (i.e. with I of H-shaped cross-sections). As in Section 4.1, little attention is 5 F.E. results reported here can only be relative to n = 0.3 or n = 0.7 cases for which 0.3 and 0.7 ratios are only indications on the true level of axial force at failure, the NEd + MEd combined loading being applied proportionally from an initial NEd + MEd set.

paid to the influence of local buckling, and the results presented in the following are relative to cases where cross-sectional instability has a negligible influence. Unlike hollow structural shapes members, open section members may be significantly influenced by torsional deformations, and in particular experience Lateral Torsional Buckling (L.T.B.) in addition to or coupled to flexural buckling. Accordingly, Fig. 15a and b provide F.E. results obtained for open section members (IPE 200, IPE 500 and HEB 300) under the most complex NEd + My,Ed + Mz,Ed loading situations ([34]), where torsional deformations where prevented for Fig. 15a results, and left free for Fig. 15b results. As is clearly observed, the influence of torsional deformations, essentially through the development of lateral torsional buckling here, has a very significant influence on the distribution of results. Consequently, the derivation of an O.I.C.-based approach for wide flange section members shall pay due account for the influence of L.T.B., since members affected by L.T.B. exhibit a much more complex response – this particular feature is well-known and has been quite extensively studied and further translated into design rules, see [9] or [22] for example. Fig. 15a presents a large set of results in O.I.C. axes, i.e. λL+G in horizontal axis and χL+G along the vertical axis. Many different loading situations and combinations are reported in this single graph, as being constituted of NEd + My,Ed + Mz,Ed situations with varying levels of each internal force; differentiation is made with respect to bending moment distributions (ψ = 1.0, 0 or −1.0). Quite clearly, the little scatters

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Fig. 15. Influence of torsional deformations (Lateral Torsional Buckling).

observed indicate that i) the use of a generalised member relative slenderness and of a global χL+G penalty factor for combined loading situations is appropriate, and that ii) bending moment distributions, through ratio ψ, have a strong influence on the overall member resistance, as was shown for tubes. For a given value of ψ, detailed analysis of the results show that NEd/My,Ed or NEd/Mz,Ed ratios are the other leading parameters responsible for the remaining scatters. Last, it is also worth noting that, as expected, ψ = 1.0 results are much more influenced by instability than their ψ = − 1.0 counterparts, as indicated by quite different χL+G values, much lower than unity in the first case, and χL+G ≈ 1.0 for ψ = −1.0 up to λL+G ≈ 0.8 – the latter response is more characteristic of a plastic response of the member, i.e. governed by cross-section resistance. In contrast, the same data set, however with free torsional deformations, as reported in Fig. 15b, exhibits much less clear tendencies, even sorted by means of ψ ratios, as depicted. The influence of L.T.B. can be shown to be responsible for these – supposedly – inconsistent results, and further investigations are currently being led in this respect. 4.3. Coupled instabilities This last paragraph focuses on so-called “coupled instabilities”, where both local and global buckling influence the member's resistance and interact with each other. When considered under combined load

cases, these situations represent the most difficult practical design situations, as many complex phenomena are involved, and multiple interactions develop. Such cases are certainly the ones where the use of the O.I.C. is the most rewarding, thanks to its direct, simple and straightforward approach. In this respect, additional shell F.E. simulations ([35]) where performed with Fig. 16a slender sections. All sections' dimensions have been chosen so as the sections to be quite significantly affected by local buckling, whatever the loading considered. In addition, these sections have been built in order to get either i) a single plate to be the most affected by local buckling and lead the slender response of the crosssection (left sections on Fig. 16a), or ii) nearly equally weak webs and flanges so that a slender plate shall not expect significant restraints from adjacent ones (right sections in Fig. 16a). Fig. 16b reports on the obtained results at the cross-sectional level (i.e. application of the O.I.C. to get χL penalty factors, see Fig. 3). As indicated by quite severe χL values, the detrimental influence of local buckling on the sections' capacities is seen to be rather strong, in all cases and especially for open sections. The relatively wide range in λL values is explained by various NEd + My,Ed + Mz,Ed loading proportions ([35]). The results obtained for open section members are reported in Fig. 17a, which reports O.I.C.-calculated λL + G values as a function of shell F.E. ultimate load calculations of χL+G. For the sake of clarity, results are again sorted as a function of ψ. Owing to the quite complex

Fig. 16. Cross-sectional response of slender sections (various shapes).

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Fig. 17. O.I.C.-plots with proposed approach for coupled instabilities under NEd + My,Ed + Mz,Ed – a) Open sections – b) Hollow sections.

NEd + My,Ed + Mz,Ed coupled instabilities considered here, the observed results are remarkably consistent, whatever the bending moment distribution, which is again noteworthy. As for hollow sections (Fig. 17b), similar trends are observed, and identical comments can be made. These last sets of results further evidence the many advantages of the O.I.C. approach, more direct, simple (no Effective Width Method) yet mechanically-based. It may be recalled here that such local-global coupled instabilities situations are expected to be much more frequently met in design practice applications, namely through an increased use of high strength steels. 5. Conclusions and further implications within Structural Engineering – Ongoing and future steps 5.1. Further implications within Structural Engineering As was shown in Section 2, the major features associated with the O.I.C. approach can be summarised as follows: i) Use of an identical concept for the resistance verifications of sections and members; ii) Avoidance of complex and tedious calculations (effective properties), no need for cross-section classification steps and their many issues; iii) Treatment of combined loading situations as straightforwardly as simple ones (no interaction formulae); iv) Concept substantiates many design formulae (design provisions simpler and shorter); v) Distribution of resistance is smooth and continuous with slenderness, both at the cross-section and member level; vi) Economy and performance: several typical aspects lie at the basis of accurate and advanced resistance predictions, such as considering the cross-section as a whole (i.e. with beneficial plate interactions) or accounting for possible strain hardening effects. Also, of importance regarding economy are the direct prediction of resistance, the strong mechanical background and the readability of the approach, which guarantees an easy, consistent and efficient use for practitioners. All these features and advantages qualify the O.I.C. as a strong and effective alternative approach for the design of steel members, in the perspective of an introduction in design standards. Further, the use of the O.I.C. in design shall result in various major changes in design habits. Firstly, the O.I.C. design philosophy is explicitly oriented towards the use of numerical tools (semi-analytical approach): key R-factors in the

definition of the overall slenderness can indeed be calculated analytically through classical Structural Mechanics theories or textbooks,6 or be calculated by means of dedicated software.7 This immediately brings significant simplifications and help to the designer, since intelligence is put on mechanically-based powerful tools rather than on approximate formulae. Knowledge then arises from tools, and this represents a major innovation for designers. It is also somewhat surprising to realise that Structural Analysis software are existing and intensively used for N 40 years now, while approximate, hand formulae are still in use to characterise the resistance of elements (i.e. without resorting to any software-based assistance). Direct design approaches such as the O.I.C. shall also allow for a more efficient and rational use of high strength steel members. Indeed, further than exacerbated instabilities, serviceability aspects should play a key role in the rational exploitation of high strength steel members, especially members mostly acted by bending. Since the Young's modulus is barely improved for high strength steels, unlike the yield stress, the structural relevance of using steel grades higher than S355 for hotrolled long products is poor in design practice: as soon as a simplysupported beam is considered, many practical situations are governed by serviceability conditions (deflections), which renders the use of high strength steel for standard hot-rolled cross-section shapes pointless. Nevertheless, considering built-up sections in high-strength steel made with thinner flanges and webs can be shown to be capable of rebalancing the resistance vs. deflection equation so that ultimate limit states regain prevalence, taking the full benefits of a higher steel grade ([36]). This, however, can only be reached at the costs of i) slightly more expensive manufacturing costs (welding) and ii) exacerbated inference of local buckling (slender sections). Currently, this route is avoided by designers since the application of the E.W.M. and the long and tedious calculations of effective properties are necessary. In contrast, application of the O.I.C. yields direct resistance predictions, cancelling this obstacle and thus potentially leading to a much more efficient and rational use of high strength steel through optimised slender sections, thanks to the easy treatment of local-global instability coupling. Associated with this last assertion is a new thinking of section sizes: new, innovative, tailored and performance-based cross-section shapes in high strength steel will lead to more economical solutions ([36]), to 6 For example, Rpl can be evaluated by means of plastic resistance interaction equations, and Rcr,L can be calculated through the determination of the least favourable critical stress of each individual plate within the cross-section. 7 Several such free software are currently available ([31]) and provide immediate Rcr results.

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complement usual hot-rolled shapes. Involving low-thickness welded plates, such cross-section shapes shall significantly benefit from the O.I.C. approach and its associated tools. Several research works relative to all above-listed points are currently in progress. 5.2. Conclusions – Ongoing and future steps The present paper introduced the essentials of an alternative design approach for steel sections and members: the Overall Interaction Concept (O.I.C.). The paper first presented the bases, principles and application steps of the proposed approach, its strong mechanical background and main advantages and features. In particular, the benefits brought by its simple, direct and straightforward application steps, along with the avoidance of several usual concepts (cross-section classification, Effective Width Method…), have been detailed. The concept is shown to be applicable to the design of both sections and members, whatever the cross-section shape (open or tubular section) and distribution of loading (simple or combined loading). The O.I.C. is also perfectly suited to so-called “local-global” coupled instabilities, and proposes direct and continuous resistance predictions from plastic to slender capacities, possibly accounting for strain hardening effects. Section 3 presented a comprehensive set of results at the crosssectional level, for both hollow and wide-flange sections. AyrtonPerry-based χ-λ design relationships for hollow structural shapes were also presented and shown to lead to more accurate, consistent and safe resistances when compared to Eurocode 3 rules, in addition to being significantly simpler in application. Many results for open sections were given as well, for various loading situations; they further evidence the ability of the O.I.C. to simply yet efficiently capture the parameters governing open sections' resistance. Section 4 was devoted to the behaviour and response of members, and mostly to beam-columns members. Numerous F.E. results were reported, demonstrating the potential of a χ-λ approach to successfully apply to members under combined load cases. Also, the challenging case of coupled instabilities was investigated, and the O.I.C. approach was again shown to be very efficient and appropriate. Further developments towards the derivation of full O.I.C. procedures to steel members are currently under way, namely regarding open sections and member behaviour. Acknowledgements CIDECT, by means of financial and technical support within research project “HOLLOPOC”, is gratefully acknowledged. References [1] Boissonnade N., Jaspart J.-P., Oerder R., Weynand K.; “A new design model for the resistance of steel semi-compact cross-sections”, Proceedings of the 5th European Conference on Steel Structures, Eurosteel 2008, Graz, 6 p., September 3–5, 2008. [2] Y. Chen, X. Cheng, D. Nethercot, An overview study on cross-section classification of steel H-sections, J. Constr. Steel Res. 80 (January 2013) 386–393. [3] Greiner R., Kettler M., Lechner A., Freytag B., Linder J., Jaspart J.-P., Boissonnade N., Bortolotti E., Weynand K., Ziller C., Oerder R.; “SEMI-COMP: plastic member capacity of semi-compact steel sections – a more economic design”, Research Fund for Coal and Steel, European Commission, (ISBN 978-92-79-11113-6, 139 pages), 2009. [4] G. Sedlacek, M. Feldmann, The b/t ratios controlling the applicability of analysis models in Eurocode 3, Part 1.1, Background Document 5.09 for Chapter 5 of Eurocode 3, Part 1.1, Aachen, 1995.

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