Review: Interactive behaviour, failure and DSM design of cold-formed steel members prone to distortional buckling

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Review

Review: Interactive behaviour, failure and DSM design of cold-formed steel members prone to distortional buckling ⁎

Dinar Camotima, , Pedro B. Dinisa, André D. Martinsa, Ben Youngb a b

CERIS, ICIST, DECivil, Instituto Superior Técnico, Universidade de Lisboa, Portugal Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Coupling phenomena involving distortional buckling Cold-formed steel columns and beams Local-distortional interaction Local-distortional-global interaction Distortional-global interaction Experimental/numerical post-buckling and ultimate strength results Direct Strength Method (DSM) design

The aim of this work is to provide an overview of the current status of an extensive ongoing investigation on cold-formed steel columns and beams affected by mode coupling phenomena involving distortional buckling, namely local-distortional (L-D), local-distortional-global (L-D-G) and distortional-global (D-G) interaction. The investigation comprises experimental tests, numerical simulations and design proposals, intended to (i) acquire in-depth knowledge on the post-buckling behaviour (elastic and elastic-plastic), ultimate strength and failure mode nature of the members under consideration, and (ii) take advantage of the above knowledge to develop, calibrate and validate efficient Direct Strength Method (DSM)-based design approaches to predict their ultimate strength. Initially, column results are used to illustrate and help grasp some fundamental concepts, namely the characterisation of the (i) three above mode coupling phenomena, (ii) different sources of mode interaction that may cause ultimate strength erosion, (iii) global post-buckling behaviour and (iv) the most detrimental initial geometrical imperfections. Then, the paper addresses separately each mode coupling phenomenon, for columns, and only L-D interaction for beams. For columns undergoing L-D and L-D-G interaction, the research activity reported concerns experimental investigations, numerical (shell finite element) simulations and the development and/or assessment of DSM-based design approaches and/or guidelines. The picture is different for columns experiencing D-G interaction and beams affected by L-D interaction, due to the lack of available experimental results − nevertheless, even if only numerical results are reported, they enable unveiling rather interesting (and unexpected) behavioural features concerning the real nature of these mode coupling phenomena.

1. Introduction It is well known for a long time that cold-formed steel columns and beams exhibit (i) highly stable local and (ii) marginally stable global post-buckling behaviours. More recently, it was found that their distortional post-buckling behaviours (i) fit somewhere in between the two previous ones, both in kinematic and strength terms, and (ii) are characterised by asymmetry with respect to the compressed flangestiffener motion (outward or inward) (e.g., [1–3]). Concerning mode interaction phenomena that may affect the column and/or beam postbuckling behaviour and ultimate strength, those involving local and global buckling are, by far, the better understood − their effects are currently taken into account in the design of slender (hot-rolled or coldformed) steel members, through either the classical “plate effective width” concept or the more recent Direct Strength Method (DSM) (e.g.,

⁎

[4]). In columns and beams prone to distortional buckling, couplings with local and/or global buckling, namely local-distortional (L-D), local-distortional-global (L-D-G) or distortional-global (D-G) interaction, are potential additional sources of failure load/moment erosion not yet adequately covered by any cold-formed steel specification around the world. The objective of this work is to provide an overview of the current status of an extensive research effort, carried out in the last few years at the Universities of Lisbon and Hong Kong, on mode interaction in coldformed steel columns (mostly) and beams susceptible to distortional buckling. The ongoing investigation, comprising experimental tests, numerical simulations and design proposals, is intended to (i) acquire in-depth knowledge on the post-buckling behaviour (elastic and elasticplastic), ultimate strength and failure mode nature of cold-formed steel columns and beams experiencing L-D, L-D-G and D-G interaction, (ii)

Corresponding author. E-mail address: [email protected] (D. Camotim).

http://dx.doi.org/10.1016/j.tws.2017.07.011 Received 17 February 2017; Received in revised form 22 June 2017; Accepted 12 July 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.

Please cite this article as: Camotim, D., Thin-Walled Structures (2017), http://dx.doi.org/10.1016/j.tws.2017.07.011

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obtain and/or collect experimental and numerical ultimate strength data, and (iii) use this information to develop, calibrate and validate efficient (safe and accurate) DSM-based design approaches to predict the ultimate strengths of the members under consideration.

(II)

2. Outline of the paper The paper begins by providing some fundamental concepts, namely the characterisation of (i) the three mode coupling phenomena (involving distortional buckling) dealt with in this work, (ii) the different sources of mode interaction that may cause ultimate strength erosion, (iii) the mechanics of column global post-buckling behaviour, (iv) the most detrimental initial geometrical imperfections (those included in the numerical simulations) and (v) the current DSM. In order to help grasp some of the above concepts, illustrative post-buckling results concerning simply supported lipped channel columns are presented and discussed. Then, the main results available for each of the four member interaction problems addressed in this work, namely (i) L-D interaction in columns, (ii) L-D-G interaction in columns, (iii) D-G interaction in columns and (iv) L-D interaction in uniformly bent beams, are reported in a separate section, which includes also the most relevant references for the problem under scrutiny. For columns undergoing L-D and L-D-G interaction the research activity reported comprises experimental investigations, numerical (shell finite element) simulations and the development and/or assessment of DSM-based design approaches and/or guidelines − note that the most relevant (and complete) part of the research carried out until now concerns columns undergoing L-D interaction. For columns experiencing D-G interaction and beams affected by L-D coupling, the picture is quite different: due to the lack of available experimental results, only numerical ones are reported. However, such numerical results make it possible to unveil quite interesting (and unexpected) behavioural features that shed fresh light on the real nature of these mode coupling phenomena. All the numerical results presented were obtained by either Generalised Beam Theory (GBT − buckling analyses) or shell finite element analyses (SFEA − buckling and post-buckling analyses), using the codes GBTUL [5] and ABAQUS [6] − the ABAQUS SFE model adopted is briefly described next.

(III)

(IV)

(V)

2.1. Abaqus shell finite element model The member buckling and elastic/elastic-plastic post-buckling analyses were performed employing shell finite element models similar to those used earlier by the authors. Nevertheless, for the sake of completion, the main characteristics of these models are summarized next [7–9]:

integration rule) − previous studies showed that length-to-width ratios of about 1.0 provide accurate results. Support conditions. The members analysed are (i) either simply supported or fixed-ended (columns) and (ii) simply supported (beams). In fixed-ended columns, all the end cross-section global and local displacements and rotations are prevented (as happens in the vast majority of column experimental tests) − obviously, the loaded end section axial translation(s) must be free. The simply supported columns and beams exhibit end cross-sections either (i) locally and globally pinned, free to warp and prevented from twisting (SCA support condition) or (ii) locally fixed, globally pinned and prevented from warping and twisting (SCB support condition) − the latter corresponds to physically attaching rigid plates to the member end cross-sections, which is often done in experimental tests. Loading. In the columns, equal compressive loads, corresponding to an unit uniform stress distribution, are applied at both column end-sections − the ABAQUS value provides the average stress acting on the column. In the beams, equal bending moments are applied at the end-sections, either by means of sets of nodal concentrated forces statically equivalent to an unit moment (SCA support condition) or directly at the rigid end plate centroids (SCB support condition) − in either case, the ABAQUS value provides the applied end moments. Material model. The carbon steel material behaviour, deemed isotropic and homogeneous, is modelled as linear elastic (buckling analysis) or elastic-perfectly plastic (post-buckling analysis). In the latter case, the well-known Prandtl-Reuss model is adopted − it is based on J2-flow plasticity theory and combines von Mises's yield criterion with its associated flow rule. No strain-hardening is considered. Initial imperfections. The members analysed contain only initial geometrical imperfections, whose shape and amplitude are defined in Section 3.4. These initial geometrical imperfections are incorporated in the analysis automatically, as linear combinations of the competing buckling modes, obtained from preliminary linear buckling analyses based on the finite element mesh adopted in the post-buckling analysis − in some cases, capturing non-critical buckling mode shapes requires altering the member wall thickness. Both residual stresses and corner strength effects are disregarded − as mentioned by several authors (e.g., [10]), the combined influence of strain-hardening, residual stresses and corner strength effects has little impact on the member load-carrying capacity.

3. Fundamental concepts 3.1. Mode coupling phenomena involving distortional buckling

(I) Discretisation. The member mid-surface is discretised into fine S4 element (ABAQUS nomenclature meshes − 4-node isoparametric shell elements with the shear stiffness obtained by a full

The curves shown in Figs. 1(a), 2(a) and 3(a) concern simply supported (SCA support condition) lipped channel columns with the cross-

Fig. 1. (a) L-D column buckling curve and (b) LL-D = 27 cm column “combined” L-D buckling mode.

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Fig. 2. (a) L-D-G column buckling curve and (b) LL-D-G = 152 cm column “combined” L-D and G (flexural-torsional) buckling modes.

Fig. 3. (a) D-G column Pcr vs. L curve and (b) LD-G = 222 cm column D and G buckling modes.

3.2. Types of mode interaction Table 1 Lipped channel column cross-section dimensions, lengths and coupling phenomena. bw (mm) 100 92 150

bf (mm) 50 65 110

bl (mm) 5 10 17.5

t (mm) 1.0 1.2 2.4

L (cm) 27 152 222

Mode interaction involving distortional buckling may occur in members exhibiting geometries (cross-section shape and dimensions, and length) such that either (i) the competing critical buckling stresses are very close (the criterion usually adopted is fcr.max /fcr.min ≤ 1.10, where fcr.max and fcr.min are the largest and smallest critical buckling stresses involved), which characterises the so-called “true interaction”, or (ii) the local and/or distortional critical buckling stresses are visibly lower than the remaining one( s), provided that the yield stress exceeds the highest critical buckling stress by a large enough amount (i.e., fy > fcr.max), which characterises the so-called “secondary bifurcation interaction”. It is worth noting that, as reported in [12] in the context of L-D-G coupling, no secondary bifurcation interaction occurs when global buckling is critical, because of the associated minute post-critical strength reserve, which precludes reaching applied stress levels close to the critical local and/or distortional buckling stresses. Naturally, the most pronounced secondary bifurcation interaction effects arise when local buckling has the smallest critical stress, due to the ensuing large post-critical strength reserve − when fcr.min = fcrD the post-critical strength reserve is just moderate (but still visible). In order to illustrate and provide a better grasp of the difference between the two types of mode interaction mentioned in the previous paragraph, Fig. 4(a)–(b), taken from Young et al. [13], display the elastic equilibrium paths (P vs. d, where P is the applied load1 and d is a generic local displacement) and six cross-section buckled configurations concerning lipped channel columns exhibiting true L-D-G interaction and secondary distortional/global-bifurcation L-D-G interaction, re-

Coupling L-D L-D-G D-G

section dimensions and lengths given in Table 1, selected to illustrate the L-D, L-D-G and D-G mode interaction phenomena [11]. Each curve provides the variation of the critical buckling load Pcr with the column length L (logarithmic scale). As for Figs. 1(b), 2(b) and 3(b), they display the (i) local and/or distortional and (ii) global (flexural-torsional) buckling modes of the columns with the lengths given in Table 1, selected to maximise the mode interaction effects, which is ensured by virtually coincident competing critical buckling loads. Indeed, the columns have identical L-D (Pcr.L-D = 21.1 kN), L-D-G (Pcr.L-D-G = 48.0 kN) and D-G (Pcr.D-G = 203.9 kN) critical loads, associated with the simultaneous occurrence of the following buckling modes: (i) 3 half-wave local + single half-wave distortional (see Fig. 1(b)), (ii) 19 half-wave local + 3 half-wave distortional + single half-wave global (see Fig. 2(b)) and (iii) 3 half-wave distortional + single half-wave global (see Fig. 3(b)).

1

3

Note that P = f × A, where A is the column cross-section area.

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Fig. 4. Equilibrium paths and cross-section buckled shapes of lipped channel columns exhibiting (a) true and (b) secondary distortional/global-bifurcation L-D-G interaction.

spectively − the latter corresponds to a situation in which Pcr.min = PcrL and Pcr.max = PcrD ≈ PcrG. In the first case, coupling starts at the early loading stages and evolves as loading progresses − local, distortional and global (flexural-torsional) deformations develop along the whole equilibrium path, provided that the column contains initial geometrical imperfections with L, D and G components. In the second case, on the other hand, the deformation is essentially local up to the vicinity of the critical distortional/global buckling stress level (only this initial geometrical imperfection component increases), when visible distortional and global deformations begin to emerge and subsequently develop. Of course, this only occurs in elastic-plastic columns if the squash load (Py = A fy) is “high enough” to allow for the above emergence and/or development (otherwise, plasticity kicks in and precipitates a local failure before distortional and global deformations become significant).

Fig. 5. Column global elastic post-buckling behaviour − P/Pcr vs. |β| equilibrium paths.

3.3. Mechanics of column global post-buckling behaviour

deformation modes (4 global and 2 distortional), shows the variation of the buckling load Pb with the column length L (L > 40 cm). Fig. 6(b) displays the column modal participation diagram (for single half-wave buckling − nw = 1), showing the contributions of each deformation mode to the column buckling modes. Lastly, Fig. 6(c) show the L = 152, 200, 600 cm column buckled crosssections and also the in-plane shapes of the 3 contributing deformation modes. These results lead to the following conclusions: (ii1) The curve descending branch concerns two buckling modes: flexural-torsional-distortional (2 + 4 + 6 − 100 < L ≤ 500 cm) and flexural-torsional (2 + 4 − 500 < L < 2000 cm). The participation of the anti-symmetric distortional mode 6 gradually fades as the column length grows − it finally vanishes for L ≈ 500 cm. (ii2) Since the six column lengths are located inside the above two intervals, the corresponding buckling modes have different natures. While the L6 column buckles in a flexural-torsional mode (44–56% participations of modes 2–4), the remaining ones exhibit flexural-torsional-distortional buckling with variable participations of modes 2, 4, 6: 39%, 48%, 13% (L1), 43%, 51%, 6% (L2), 45%, 52%, 3% (L3), 47%, 52%, 1% (L4), and 49%, 50%, 1% (L5). (ii3) A visible (higher than 5%) participations of mode 6 in the column so-called “global” buckling mode lowers its postbuckling stiffness and causes an unstable behaviour − see the L1-L2 columns. Note that mode 6 is virtually undetectable in the ABAQUS results (see Fig. 5). (ii4) In the L = 152 cm column, three critical buckling modes occur simultaneously: a 19 half-wave pure local mode, a 3 half-wave pure symmetric distortional mode and a single half-wave “mixed” (flexural-torsional-distortional) mode, combining 30% of mode 2, 38% of mode 4 and 32% of mode 6 − the latter was

The “pure global” post-buckling behaviour of columns with the cross-section dimensions given in the second row of Table 1 is investigated in this section. Twelve columns with six lengths are analysed: (i) L1 = 200 cm (Pcr = 29.4 kN), (ii) L2 = 250 cm (Pcr = 19.6 kN), (iii) L3 = 300 cm (Pcr = 14.3 kN), (iv) L4 = 350 cm (Pcr = 11.0 kN), (v) L5 = 400 cm (Pcr = 8.8 kN) and (vi) L6 = 600 cm (Pcr = 4.9 kN) − note that the critical buckling loads of the L1-L6 columns are comprised between 61% and 10% of the Pcr value associated with L-D-G interaction (Pcr = 48.0 kN for LL-D-G = 152 cm). Each column pair contains positive and negative critical-mode (global) initial geometrical imperfections: clockwise or counter-clockwise mid-span web chord rotations equal to β0 = ± 0.018 rad − in the L1 columns, this value corresponds exactly to a mid-span flange-lip corner vertical displacement equal to L/1000 (i.e., 2mm). Fig. 5 shows the upper portions (P/Pcr > 0.6) of the six column postbuckling equilibrium paths P/Pcr vs. |β| − also shown are two column mid-span cross-section deformed configurations at β = 0.26 rad. These post-buckling results lead to the following conclusions [11]: (i) The L3-L6 columns exhibit the expected stable global post-buckling behaviour: small post-critical strength reserve and perfectly symmetric equilibrium paths. Conversely, the L1 and L2 columns exhibit clearly unstable symmetric post-buckling behaviours (limit points at P/Pcr < 0.91). Note also that the post-buckling behaviours of the L5 and L6 columns are virtually identical and clearly “more stable” of than those of columns L4 and (mostly) L3. (ii) Generalised Beam Theory (GBT − e.g., [14]) provides the explanation for the distinct post-buckling behaviours exhibited by the L1-L2 and L3-L6 columns. The curve shown in Fig. 6(a), obtained from GBT analyses carried out with code GBTUL [5] and including 6

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Fig. 6. GBT-based (a) Pb vs. L curves (L > 40 cm), (b) modal participation diagrams and (c) L = 152, 200, 600 cm column cross-section buckled shapes and deformations modes 2, 4, 6.

1 mm − local), flange-lip corner vertical displacement (vD = 1 mm − distortional3) and also flange-lip corner vertical displacement, caused by a mid-span web chord rotation (vG = 1 mm − global). (ii) To scale down the above pure modes, thus leading to the following magnitudes for the local, distortional and global imperfections: wL.0 = 0.1t, vD.0 = 0.1t and vG.0 = L/1000.4 (iii) A given initial geometrical imperfection shape is obtained as a linear combination of the scaled competing buckling modes shapes, with coefficients CL.0, CD.0 and/or CG.0 satisfying the conditions (CL.0)2 + (CD.0)2 = 1, (CL.0)2 + (CD.0)2 + (CG.0)2 = 1 or (CD.0)2 + (CG.0)2 = 1, respectively for L-D, L-D-G and D-G interaction. A better “feel” and visualisation of the initial imperfection shapes may be obtained by considering an unit radius circle or sphere drawn in the CL.0-CD.0 plane, CL.0-CD.0-CG.0 space or CD.0CG.0 plane, as shown in Figs. 7(a), 8(a) and 9(a) for columns: each “acceptable” imperfection shape lies on the circle or sphere and can be defined by either one angle (θ) or two angles (α and θ). Also shown are the buckled mid-span cross-sections corresponding to

previously termed “global”, a designation retained in this work for simplicity. Therefore, the D-G and L-D-G interaction phenomena addressed in this work really involve flexural-torsional-distortional critical buckling modes, even if they are termed “global”. Moreover, the presence of antisymmetric distortion (deformation mode 6) in the “global” buckling mode is responsible for a very significant decrease in post-buckling stiffness, with respect to that exhibited by columns buckling in flexuraltorsional critical modes, which often causes an unstable behaviour. Nevertheless, it should be noted that fixing/clamping the columns end supports attenuates the post-buckling stiffness drop to an extent that the unstable behaviour ceases to occur − this issue will be further addressed in Sections 6.1 and 6.2. 3.4. Most detrimental initial geometrical imperfections A very important issue in mode interaction studies is to assess how the initial geometrical imperfection shape influences the post-buckling behaviour and strength of the structural system under scrutiny − i.e., to perform an imperfection-sensitivity study. In particular, it is essential to identify the most detrimental initial imperfection shape, in the sense that it leads to the lowest member strengths. Due to the presence of two or three competing critical buckling modes, the commonly used approach of considering critical-mode initial imperfections ceases to be well defined − infinite shapes satisfy this condition. Therefore, it was necessary to obtain and compare equilibrium paths of members with initial geometrical imperfections (i) spanning the whole critical-mode shape range and (ii) sharing a common amplitude (otherwise, no meaningful comparison can be made). A systematic approach to define such a set of initial geometrical imperfections was devised, which involves the performance of the following procedures [11]:

3

Note that the pure distortional buckling mode involves also web transverse bending. While the global amplitude adopted has been used in the past and can be viewed as consensual among the technical and scientific community, the same is not true for its local and distortional counterparts. The reason for choosing these local and distortional amplitudes is that earlier investigations showed that they lead to numerical failure loads correlating very well with their DSM predictions provided by the currently codified local and distortional design curves. Although the authors are well aware that larger local and distortional amplitudes are quite common, these initial geometrical imperfection amplitudes have been used in practically all the numerical investigations concerning interaction phenomena involving distortional buckling in cold-formed steel members. This is due to a kind of “tradition” and also to the fact that altering the initial imperfection amplitude only changes the member interactive behaviour quantitatively, and not by a significant amount. Very recently, the authors [42] carried out an imperfection-sensitivity study in the context of columns undergoing L-D interaction that, to a large extent, confirmed the above assertions. Indeed, it was shown that the failure load of lipped channel columns exhibiting moderate-to-high slenderness (λcr > 1.25) is virtually insensitive to the initial geometrical imperfection shape (local or distortional) and amplitude. Although the picture naturally changes for non-slender columns (λcr ≤ 1.25), it was found that the most imperfection-sensitive columns (those containing local initial geometrical imperfections) exhibit a fairly small failure load erosion due to L-D interaction and, therefore, the 0.1t amplitude is quite adequate to obtain meaningful failure loads of columns undergoing L-D interaction. 4

(i) To determine the “pure”2 critical buckling mode shapes, normalised to exhibit unit mid-span mid-web flexural displacement (wL =

2 In general, the L, D and G buckling modes are not “pure”, in the sense that they exhibit small participations of cross-section deformations modes that are not the dominant ones − e.g., the L and G buckling modes always exhibit small symmetric (L) and anti-symmetric (G) distortional deformations (barely perceptible in the ABAQUS results). However, in the context of this shell finite element investigation they are treated as “pure” buckling modes.

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Fig. 7. (a) Initial imperfection representation in the CL.0-CD.0 plane and (b) θ = 0°, 90°, 180° and 270° column initial imperfection shapes.

this is a very perceptive illustration of column L-D interaction. (iii) In spite of the three half-wave pure local initial imperfections, the θ = 90° and θ = 270° columns evolve towards a common predominantly distortional single half-wave deformed configuration involving inward flange-lip motions (that exhibited by the θ = 180° column), which implies that they must undergo web bending reversal at mid-span along their equilibrium paths. The reversal occurs at the central (θ = 90°) or outer (θ = 270°) web regions, as shown in Fig. 11: the outward wave(s) are gradually “swallowed” by the inward one(s), until a sudden switch to an “almost singlewave” configuration occurs at a limit point. The final configuration is the θ = 180° column one, displayed in Fig. 12. The θ = 270° column limit point is more “abrupt” because the bending reversal involves the two outer half-waves, much more “confined” (by the supports) than the central one. (iv) The post-buckling equilibrium paths of columns with intermediate θ values (not shown in Figs. 10 and 11) merge into one of the two curves concerning the pure distortional initial imperfections: the 300° ≥ θ ≥ 60° paths merge into the θ = 180° curve and the remaining ones into the θ = 0° one − note that the pure distortional initial imperfections are the most detrimental, in the sense that the corresponding equilibrium paths lie below all the others (i.e., exhibit lower strengths).

the individual buckling modes. Finally, Figs. 7(b), 8(b) and 9(b) provide the pure L (inward or outward mid-span web bending), D (inward or outward mid-span flange-lip motions) and G (clockwise or counter-clockwise mid-span cross-section rotations) initial geometrical imperfection shapes. After having defined “the full set of possible initial geometrical imperfections”, it becomes possible to assess and compare the elastic post-buckling behaviours of columns containing them, in order to (i) obtain numerical evidence of the occurrence of interaction and (ii) identify the most detrimental shape. This was done for a very large spectrum of initial imperfections shapes (24 per plane − 15° apart). This procedure is briefly illustrated here for the columns undergoing L-D interaction − the interested reader may find additional results in the works of Dinis et al. [7] and Dinis and Camotim [15–17]. The equilibrium paths displayed in Fig. 10(a) (P/Pcr vs. |v|/t) and 10(b) (P/Pcr vs. |w|/t) correspond to columns containing pure distortional (θ = 0°, 180°) and pure local (θ = 90°, 270°) initial geometrical imperfections. Fig. 11, on the other hand, concerns only the θ = 90°, 270° columns and shows the web deformed configuration evolution at five points located along the associated postbuckling paths. The observation of these figures leads to the following conclusions: (i) The θ = 90°, 270° paths (i1) are quite “irregular”, (i2) exhibit limit points (the θ = 270° one is more “abrupt” and involves a minor “snap-back”) and, for P/Pcr > 1.1, (i3) converge to the θ = 180° equilibrium path. (ii) The θ = 0°, θ = 180° paths are much “smoother” than the θ = 90°, 270° ones and evolve always in a monotonic fashion. Nevertheless, their web deformed configuration evolutions, displayed in Fig. 12, clearly show that they consist of combinations of a single distortional half-wave with three local half-waves (the two curves labelled “DM” exhibit identical sinusoidal shapes and amplitudes) −

3.5. Direct Strength Method (DSM) The Direct Strength Method (DSM) is already included in the Australian/New Zealand [18], North American [4] and Brazilian [19] cold-formed specifications for the design of cold-formed steel columns and beams. The currently codified [4] DSM design/strength curves for columns concern failures in local, distortional, global and local-global (interactive) modes. They read, respectively (e.g., [20,21]),

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Fig. 8. (a) Initial imperfection representation in the CL.0-CD.0-CG.0 space and (b) pure local (θ = 0° + α = 90° or 270°), distortional (θ = 0° + α = 0° or 180°) and “global” (α = 0° + θ = 90° or 270°) initial imperfection shapes.

Fig. 9. (a) Initial imperfection representation in the CD.0-CG.0 plane and (b) θ = 0°, 90°, 180° and 270° column initial imperfection shapes.

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Fig. 10. Post-buckling equilibrium paths of columns with pure distortional (θ = 0°, 180°) and local (θ = 90°, 270°) initial imperfections: (a) P/Pcr vs. |v|/t and (b) P/Pcr vs. |w|/t paths.

Fig. 11. Equilibrium paths and web deformed configuration evolutions of the θ = 90° and θ = 270° columns.

Fig. 12. Web deformed configuration evolutions of the θ = 0° and θ = 180° columns.

⎨ PnL = Py ⎩

( ) PcrL Py

0.4

⎡1 − 0.15 ⎢ ⎣

( ) PcrL Py

0.4

⎤ if λL > 0.776 ⎥ ⎦

(1)

if λD ≤ 0.561

⎧ PnD = Py ⎨ PnD = Py ⎩

As for the originally codified DSM design/strength curves for beams, they mimic their column counterparts: also concern failures in local, distortional, global and local-global (interactive) modes, and read, respectively,

if λL ≤ 0.776

⎧ PnL = Py

( ) PcrD Py

0.6

⎡1 − 0.25 ⎢ ⎣

( ) PcrD Py

0.6

⎤ if λD > 0.561 ⎥ ⎦

⎨ MnL = My ⎩

(2)

2

⎨ PnG = ⎩

if λ G > 1.5

⎨ PnL = PnG ⎩

( )

⎣

− 0.15

PcrL 0.4 ⎤ PnG

( )

McrL My

⎡1 − 0.15 ⎢ ⎣

( ) McrL My

0.4

⎤ if λL > 0.776 ⎥ ⎦

⎦

if λL > 0.776

( )

0.5

⎨ MnD = My ⎩

(4)

⎧ MnG = McrG ⎪ 10 MnG = 9 My 1 − ⎨ ⎪ MnG = My ⎩

(

where (i) λL = (Py/PcrL)0.5, λD = (Py/PcrD)0.5 and λG = (Py/PcrG)0.5 are the local, distortional, global slenderness, (ii) Py is the squash load and (iii) PcrL, PcrD and PcrG are the critical local, distortional and global buckling loads. Note that Eq. (4) is obtained by replacing Py and λL = (Py/PcrL)0.5 by PnG and λL = (PnG/PcrL)0.5 in Eq. (1).

⎡1 − 0.22 ⎢ ⎣

( ) McrD My

0.5

⎤ if λD > 0.673 ⎥ ⎦

⎨ MnL = MnG ⎩

(6)

if McrG < 0.56 My 10My 36McrG

)

if 2.78 My ≥ McrG ≥ 0.56 My if McrG > 2.78 My

(7)

if λL ≤ 0.776

⎧ MnL = MG

8

(5)

if λD ≤ 0.673 McrD My

(3)

if λL ≤ 0.776

⎧ PnL = PnG PcrL 0.4 ⎡1 PnG

( )

0.4

⎧ MnD = My

⎧ PnG = Py 0.658 λG if λ G ≤ 1.5 0.877 Py λ 2 G

if λL ≤ 0.776

⎧ MnL = My

McrL 0.4 ⎡1 MnG

( )

⎣

− 0.15

McrL 0.4 ⎤ MnG

( )

⎦

if λL > 0.776

(8)

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knowledge, the only results available are the tests reported by Yang and Hancock [34] and the numerical study carried out by Martins et al. [35]. This section provides an overview of the recent research activity concerning cold-formed steel plain and stiffened columns affected by LD interaction. Initially, experimental tests performed at The University of Hong Kong are addressed and the results obtained commented. Next, an ABAQUS SFEA investigation aimed at characterising and comparing the behaviour and failure of plain and stiffened lipped channel columns is presented and discussed, making it possible to (i) assess how the L-D interaction relevance varies with the ratios RDL = PcrD/PcrL and Ry = Py/PcrMax (PcrMax = max{PcrD; PcrL}) ratios and (ii) gather substantial failure load data. Finally, the available experimental and numerical failure loads are used to (i) improve/develop existing/new DSM design approaches specifically developed to handle L-D interaction and (ii) assess the merits of the corresponding failure load estimates.

Fig. 13. Possible beam DSM local and distortional design/strength curves prescribed by AISI [4].

where (i) λL = (My/McrL)0.5, λD = (My/McrD)0.5 and λG = (My/McrG)0.5 are the local, distortional, global slenderness, (ii) My is the yield moment and (iii) McrL, McrD and McrG are the critical local, distortional and global buckling moments. As before, Eq. (8), it is obtained by replacing My and λL = (My/McrL)0.5 by MnG and λL = (MnG/McrL)0.5 in Eq. (5). However, the currently codified beam local, distortional and global design/strength curves [4] differ slightly from those given in Eqs. (5)–(7): the “yield plateaus” are replaced by linear approximations whose expressions depend on (i) the failure mode nature (local, distortional, global) and (ii) whether the cross-section has first yield in compression or tension (the first case covers also the symmetric crosssections), thus reflecting the existence of a (previously disregarded) cross-section inelastic bending strength reserve [21,22]. For illustrative purposes, Fig. 13 shows possible local and distortional beam DSM design curves prescribed by AISI [4], which already take into account the cross-section inelastic bending strength reserve in the low slenderness range (stocky beams).

4.1. Tests performed at The University of Hong Kong Although there exist several experimental studies specifically aimed at investigating L-D interaction in fixed-ended cold-formed steel plain or stiffened columns, for a long time specimens providing clear experimental evidence of this coupling phenomenon were relatively scarce. Recently, after acquiring in-depth knowledge on the behavioural aspects characterising column L-D interaction, it became possible to plan and perform carefully, at The University of Hong Kong (UHK), test campaigns concerning fixed-ended plain lipped channel [26] and rack-section [27] columns. The test campaigns involved (i) 16 lipped channel (C) and (ii) 10 rack-section (R) columns brake-pressed from high strength and mild zinc-coated structural steel sheets − the zinc coating was removed, by acid etching, prior to the tests. Table 2 provides the specimen dimensions (bw, bf, bl, bs − web/flange/lip/stiffener widths, wall thickness t and length L), critical local/distortional/global buckling and squash loads (PcrL, PcrD, PcrG, Py), load ratios PcrD/PcrL, PcrG/PcrMax, Py/PcrMax, and column failure loads PExp. The specimen end cross-sections were milled flat electronically and welded to 25mm thick steel plates, ensuring full contact with the test machine end bearings. The material properties were obtained from tensile coupon tests and two initial displacements were measured at mid-height prior to testing, providing information about the deformations caused by major-axis bending and cross-section distortion. A servo-controlled hydraulic testing machine applied the compressive axial force, through steel plates welded to the column ends − the top end plate was bolted to the associated bearing plate, which was fully restrained against warping, twisting and major and/or minor axis bending. This setting corresponds to “fully fixed-ended” support conditions. The readings of the load cell and displacement transducers were recorded at regular intervals during the tests. Fig. 14(a)–(b) concern a lipped channel specimen and provide front and side views of the test rig and a typical set-up for testing a fixed-ended column. All C and R specimens tested failed in L-D interactive modes − Fig. 15(a)–(b) show the deformed configurations, near collapse, of two C and two R column specimens, providing very clear experimental evidence of the occurrence of both local and distortional deformations. However, note that these L-D interactions stem from secondary distortional-bifurcation L-D interaction, and not from the near coincidence of PcrL and PcrD: PcrD/PcrL varies from 1.73 to 2.71 (C) and 1.31 to 1.46 (R). Interaction occurs because Py/PcrD is large enough to allow for the development of L-D coupling before plasticity becomes relevant: Py/PcrD varies from 1.09 to 6.10 (C) and 1.57 to 2.44 (R).6

4. Local-distortional interaction in columns Considerable research work has been devoted to investigate L-D interaction in cold-formed steel columns,5 particularly in the last decade, including experimental investigations, numerical simulations and design proposals − the last ones consist mainly of DSM developments/improvements. Most of the above research concerns columns with “plain” cross-sections (no intermediate stiffeners) − mainly (but not exclusively) lipped channels. For instance, the works of Kwon and Hancock [24], Loughlan et al. [25], Young et al. [26] and Dinis et al. [27] succeeded in providing experimental evidence of L-D interaction in columns. On the other hand, Silvestre et al. [28], Dinis and Camotim [29], Kumar [30] and Martins et al. [8] employed SFEA to investigate this coupling phenomenon and obtained ultimate strength data that led to the establishment of DSM-based design approaches able to handle the failure load erosion caused by L-D interaction. In “plain” crosssection columns, local buckling is almost always triggered by the web, where most of the L-D interaction takes place. This ceases to be true in the presence of web intermediate stiffeners (e.g., “v-shaped” stiffeners in lipped channels), as local buckling is bound to be triggered by the flanges, thus altering the L-D interaction features. Kwon and Hancock [24], Yap and Hancock [31] and He et al. [32] reported experimental evidence of flange-triggered L-D interaction, while Martins et al. [33] conducted a numerical study to assess the influence of L-D interaction on the behaviour and design of web-stiffened lipped channel (WSLC) columns exhibiting several combinations of the ratios between the local buckling, distortional buckling and yield stresses. Concerning webflange-stiffened lipped channel (WFSLC) columns, in which local buckling is again mostly triggered by the web, there is considerably less information on L-D interaction − indeed, to the authors’ best

6 These test results were surprising at the time, due the separated critical local and distortional buckling loads. In fact, it may be said that interpreting/understanding this experimental evidence provided motivation for the work reported by Martins et al. [8].

5

L-D interaction also plays an important role in the behaviour and strength of perforated columns − e.g., [23].

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Table 2 Specimen dimensions, squash and critical loads, yield-to-critical load ratios, failure loads (mm, kN). Specimen

bw

bf

bl

bs

L

t

Py

PcrL

PcrD

PcrG

PcrD PcrL

PcrG PcrMax

Py PcrMax

PExp

Obs.

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8 C-9 C-10 C-11 C-12 C-13 C-14 C-15 C-16

104.9 101.8 94.9 152.5 152.4 154.5 110.5 118.5 118.6 203.7 204.7 182.1 181.1 182.5 238.9 238.2

81.6 82.1 80.9 131.2 131.0 141.0 82.2 82.0 81.9 192.7 191.9 113.1 112.5 122.5 162.7 162.3

15.2 15.5 16.9 14.7 15.8 15.0 16.1 15.7 15.9 19.0 20.6 19.2 20.2 19.4 21.1 23.1

– – – – – – – – – – – – – – – –

2498 2499 2499 1425 1426 1426 1148 1149 1148 1852 1849 1030 1026 1027 1946 1947

0.962 0.984 0.978 1.227 1.227 1.206 1.459 1.453 1.469 1.478 1.459 1.934 1.915 1.974 1.966 1.947

154 157 152 320 322 331 221 225 228 458 454 290 287 309 400 398

24 26 28 33 33 31 78 70 72 42 41 99 97 108 79 78

43 46 51 57 64 54 144 138 143 76 87 243 263 252 158 174

100 98 88 1147 1161 1223 802 900 916 2057 2087 4416 4414 4880 2952 2964

1.80 1.73 1.80 1.73 1.92 1.76 1.84 1.97 1.98 1.83 2.14 2.46 2.71 2.34 2.00 2.24

2.34 2.14 1.71 20.00 18.19 22.57 5.56 6.51 6.39 26.94 24.04 18.21 16.79 19.40 18.67 17.00

3.60 3.44 2.97 5.58 5.03 6.10 1.53 1.63 1.59 5.99 5.22 1.20 1.09 1.23 2.53 2.28

39.9 42.1 42.0 68.1 71.0 67.7 109.0 102.8 103.6 92.9 94.7 145.2 146.1 142.5 129.7 131.6

L-D L-D L-D L-D L-D L-D L-D L-D L-D L-D L-D L-D L-D L-D L-D L-D

R-1 R-2 R-3 R-4 R-5 R-6 R-7 R-8 R-9 R-10

73.0 78.4 83.6 88.6 88.6 83.5 88.5 93.7 98.7 98.8

37.1 47.4 52.3 57.2 57.2 52.5 57.9 67.3 72.2 72.2

17.3 17.2 17.1 17.2 17.5 17.3 17.7 17.4 17.0 17.2

21.3 21.0 20.4 21.3 21.4 21.6 21.0 20.8 21.0 20.8

1300 1800 2100 2499 2500 1397 1599 1902 2501 2501

1.002 0.998 0.982 0.999 0.981 1.193 1.186 1.227 1.201 1.175

112 125 129 130 138 175 184 206 211 206

52 49 43 43 41 78 72 75 66 62

72 66 63 60 59 102 99 98 87 85

111 98 92 87 84 156 152 146 121 118

1.37 1.35 1.46 1.40 1.45 1.31 1.38 1.31 1.32 1.36

1.55 1.49 1.46 1.45 1.43 1.53 1.53 1.49 1.39 1.39

1.57 1.89 2.05 2.16 2.34 1.72 1.86 2.10 2.42 2.44

58.2 54.4 54.3 51.4 51.0 81.6 79.8 79.8 75.5 74.3

L-D L-D L-D L-D L-D L-D L-D L-D L-D L-D

Fig. 14. Test rig and typical test set-up: (a) front and (b) side views (lipped channel specimen).

[24], Kwon et al. [36], Yap and Hancock [31] and He et al. [32], for web-stiffened lipped channel columns, (iii) Yang and Hancock [34], for web-flange-stiffened lipped channel columns, and (iv) Yap and Hancock [38], for columns with complex-stiffened cross-sections.7

4.2. Other test results available in the literature Without claiming to be complete, experimental results concerning fixed-ended columns affected by L-D interaction have also been reported by (i) Kwon and Hancock [24], Loughlan et al. [25] and Kwon et al. [36,37], for plain lipped channel columns (there are also a few hat-section column results in Kwon et al. [36]), (ii) Kwon and Hancock

7 Due to the unusual cross-section shapes and small number of specimens tested, these test results were excluded from this study.

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Fig. 15. Experimental evidence of the occurrence of L-D interaction in (a) C and (b) R column tests.

et al. [25] − as shown by Martins et al. [8], local failures were misinterpreted as L-D interactive ones. It is worth mentioning that the authors are planning an experimental test program, to be carried out at The University of Hong Kong, involving WSLC and WFSLC columns with geometries selected to ensure SLB, TI and SDB L-D interaction − these test results will help “fill the gaps” in Table 3.

Table 3 Summary of the test results available in the literature on cold-formed steel columns affected by L-D interaction. SLB

N

PCS

WSLC Yap and Hancock [31] He et al. [32]

2 4

WFSLC Total

6

TI

N

SDB

N

Kwon and Hancock [24] Kwon et al. [36]

5

Loughlan et al. [25] Young et al. [26] Dinis et al. [27] Kwon et al. [37]

20 16 10 3

Kwon and Hancock [24] Yap and Hancock [31] He et al. [32] Kwon et al. [37]

3 9

He et al. [32]

5

8 7

Kwon et al. [37]

3

Yang and Hancock [34]

5

Yang and Hancock [34]

7

5

42

4.3. Numerical simulations This section goes briefly through the most relevant features concerning the post-buckling behaviour of columns exhibiting TI L-D interaction and addresses parametric studies performed to gather extensive failure load data. The results presented concern fixed-ended PCS [8], WSLC [33] and WFSLC [35] columns undergoing more or less severe L-D interaction − they exhibit several combinations of PcrD/PcrL, Py/PcrMax and slenderness values. The ABAQUS numerical simulations were performed using the SFE models addressed in Section 2.1 and also following some recommendations made by Schafer et al. [39], namely concerning (i) the number of elements adopted to discretize the lips (> 4) and (ii) the number of integration points through the thickness of the element in elastic-plastic analyses (> 5). Initially, six fixed-ended steel (E = 210 GPa; v = 0.30) column geometries associated with identical PcrL and PcrD values were identified: C + H + Z (same PcrL and PcrD), R, WSLC and WFSLC. Fig. 16(a)–(d) show the output of this procedure: each figure displays a Pcr vs. L (logarithmic scale) curve and the “mixed” L-D buckling mode corresponding to the selected column length LD-L. Due to the coincidence between PcrL and PcrD, the post-buckling behaviours of the above columns are strongly influenced by L-D interaction. However, because it was unveiled experimentally that L-D interaction may also be significant if PcrL and PcrD are apart (secondary bifurcation interaction), a systematic investigation was carried out to assess the relevance of L-D interaction. It involved columns with a wide variety of “L-D interaction levels” and its outcome is briefly addressed next. After determining, for each cross-section type (C, H, Z, R, WSLC,

64

Table 3 summarises the above fixed-ended column test results. After using the reported geometrical and material properties to evaluate the critical local/distortional/global buckling and squash loads of the tested specimens, the test results are divided into six sets, according to the (i) cross-section geometry (plain, web-stiffened lipped channel or web-flange-stiffened lipped channel − PCS, WSLC or WFSLC) and (ii) LD interaction nature (secondary local-bifurcation, true interaction or secondary distortional-bifurcation − SLB, TI or SDB) − N stands for the number of test results. It is readily concluded that there are only sizeable test result numbers for (i) PCS columns failing under SDB L-D interaction (23) and (ii) WSLC columns collapsing due to TI L-D interaction (27) − otherwise, the test results are either scarce or null. Nevertheless, 112 test results were collected, including the 26 addressed in Section 4.1 and the quite “dubious” 20 reported by Loughlan

Fig. 16. Pcr vs. L curves and “mixed” L-D buckling modes for (a) C + H + Z, (b) R, (c) WSLC and (d) WSFLC columns.

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Fig. 17. L-D interactive failure modes and plastic strains in (a) C, (b) H, (c) Z, (d) R, (e) WSLC and (f) WFSLC columns.

Moreover, Schafer [40] proposed two distinct strategies to estimate the failure loads of columns experiencing L-D interaction: replacing Py by either (i) PnD in the PnL equations (NLD approach − PnLD) or (ii) PnL in the PnD equations (NDL approach − PnDL). These approaches have already been employed by several authors − e.g., both were considered by He et al. [32], while Yang and Hancock [34] only took into account the former. Later, Silvestre et al. [28] proposed a novel DSM-based design approach, developed in the context of fixed-ended C columns undergoing TI L-D interaction (MNDL approach − PMnDL), which was subsequently shown to cover also H, Z and R columns by Dinis and Camotim [29].9 Recently, the above findings were extended and generalised, by assessing the performance of the MNDL approach for PCS [8], WSLC [33] and WFSLC [35] columns affected also by SDB or SLB LD interaction. This approach (i) coincides with fnD in the low-to-moderate distortional slenderness range (λD < 1.5) and, for more slender columns (λD ≥ 1.5), (ii) involves the definition of a modified local strength f*nL, dependent on the critical half-wave length ratio LcrD/LcrL (obtained from simply supported column signature curves) and estimates the column ultimate strength by replacing fnL with f*nL in the NDL equations − this modified local strength leads to PnD and PnDL estimates for LcrD/LcrL ≤ a and LcrD/LcrL ≥ b, where “a” and “b” are integers whose values depend on the cross-section shape (see details in [35]). Next, a brief overview of the numerical failure loads reported by Martins et al. [8,33,35], which are subsequently used to assess the quality of the corresponding DSM predictions, is presented and discussed. They concern a representative sample of the columns analysed: plain C, WSLC and WFSLC columns affected by various levels of L-D interaction, exhibiting RDL = 0.40–0.60–0.80–1.00–1.30–1.60–2.00 and having 9 distinct yield stresses, to cover a wide critical (local or distortional) slenderness range (1.00–1.25–1.50–1.75–2.00–2.50–3.00–3.25–3.50) − the figures include (i) failure load data concerning columns exhibiting pure distortional or pure local initial imperfections, labelled “Imp D” and “Imp L”, and also (ii) the failure modes of the RDL = 0.58C, 0.55 WFSLC and 2.40 C columns. Figs. 18(a1)–(b4) and 19(a1)–(b3) plot fU/fy against λL and λD for the identified column sets. For each RDL value, the numerical fU/fy values are compared with their DSM predictions10: (i) fnL or fnD, and

WFSLC), the most detrimental initial geometrical imperfection shape (see Section 3.4), parametric studies were performed to gather numerical failure load data of columns exhibiting more or less severe L-D interaction8: PcrD/PcrL varying between 0.40 and 2.40 − each value is associated with different critical (local or distortional) slenderness values, namely 1.00–1.25–1.50–1.75–2.00–2.50–3.00–3.25–3.50. The failure load data obtained were subsequently used to (i) identify the combinations of PcrD/PcrL and slenderness for which L-D interaction leads to a visible ultimate strength erosion, (ii) assess the merits of the currently codified DSM design curves in predicting L-D interactive failure loads and (iii) provide validation and/or guidance concerning the search for DSM-based design approaches able to handle adequately the failure load erosion due to L-D interaction – the last two aspects are addressed in Section 4.4. The results obtained provide clear evidence that the column failure load may be substantially eroded by TI and SDB L-D interaction − the latter provide numerical corroboration for the experimental observations reported by Young et al. [26] and Dinis et al. [27]. As for the SLB L-D interaction, it was found to cause negligible failure load erosion. Fig. 17(a)–(f) display the collapse mode shapes and plastic strain distributions of C, H, Z, R, WSLC and WFSLC columns affected by TI L-D interaction − in the last figure, the presence of a large number of local half-waves with very small amplitudes is barely perceptible.

4.4. Direct Strength Method (DSM) design It is nowadays widely accepted that the currently codified DSM design curves/expressions (see Section 3.5) provide an efficient methodology to design cold-formed steel columns and beams failing in distortional (PnD), global (PnG) and local or local-global interactive (PnL) modes. The method has now been shown to constitute a general approach to obtain efficient (safe, accurate and reliable) estimates of the ultimate strength of cold-formed steel columns and beams on the sole basis of the steel yield stress and appropriate elastic critical buckling stresses. For columns, the DSM local and distortional design curves, which were calibrated against mostly fixed-ended column results, are provided by “Winter-type” expressions − see Eqs. (1) and (2).

9 This approach was developed, calibrated and validated on the basis of SFEA failure loads concerning fixed-ended columns with RDL comprised between 0.90 and 1.10, all included in the TI L-D interaction region [8]. 10 Since the NDL curve is column-dependent (it varies with PcrL and PcrD), the curves in

8 For details about these numerical studies see Martins et al. [8,33,35] − PCS, WSLC and WFSLC columns, respectively.

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Fig. 18. Variation of fU/fy and corresponding DSM local and distortional strength predictions with (a) λL and (b) λD for (1)-(4) RDL = 0.40–0.60–0.80–1.00 (C, WSLC and WFSLC columns).

(ii) fnDL. Since the fMnDL values depend on the LcrD/LcrL ratio, which varies with the cross-section, the corresponding strength curves are not shown, to improve the readability of the figures. The observation of these figure pairs makes it possible to extract the following conclusions:

values (the overestimation grows with RDL), even if distortional collapses occur − Fig. 20(a)–(c) show the failure modes and plastic strains of RDL = 0.58 C, RDL = 0.50 WSLC, RDL = 0.55 WFSLC columns (λcr = 3.50). (iii) On the other hand, the fnL values estimate fairly accurately the ultimate strengths of the stocky columns with RDL > 1.00, which exhibit local collapses. For instance, Fig. 21(a1)–(c1) show the local collapse modes of three λcr = 1.25 columns (RDL = 2.40 C, RDL = 2.00 WSLC and RDL = 1.60 WSLC − the last two are amplified 10 times)11 − the longitudinal normal stress redistribution (shift from the mid-width regions towards the corners) is perceptible in these figures by looking at the yielded zones near the web-flange and flange-lip corners. Practically all their failure loads are well overestimated by the current DSM local and distortional design curves, thus providing clear evidence of the occurrence of significant SDB L-D interaction. Fig. 21(a2)–(c2) depict the L-D interactive collapses of columns differing from those in Fig. 21(a1)–(c1) only for a

(i) Although the numerical fU/fy values are well aligned along “Winter-type” curves, those concerning the WFSLC columns decrease faster with λD than those concerning the C and WSLC columns. This is due to the lower post-critical stiffness exhibited by the WFSLC columns. (ii) For the RDL = 0.40 columns, the fnD values provide fairly accurate failure load estimates in the whole slenderness range (no perceptible failure load erosion due to L-D interaction). The same occurs for the RDL = 0.60 and RDL = 0.80 C and WSLC columns. However, this is no longer true for the slender (λD ≥ 3.00) WFSLC columns: the DSM distortional design curve overestimates the fU/fy

(footnote continued) Figs. 18(a1)–(b4) and 19(a1)–(b3) are all different − conversely, the L and D curves are always the same. Moreover, note that f = P/A (A is the column cross-section area).

11

13

Note that λcr = min{λcrD; λcrL}.

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Fig. 19. Variation of fU/fy and corresponding DSM local and distortional strength predictions with (a) λL and (b) λD for (1)–(3) RDL = 1.30–1.60–2.00 (C, WSLC and WFSLC columns).

Fig. 20. Failure modes and plastic strains of the (a) RDL = 0.58 C, (b) RDL = 0.50 WSLC and (c) RDL = 0.55 WFSLC columns.

Fig. 21. Failure modes and plastic strains of the (a) RDL = 2.40C, (b) RDL = 2.00 WSLC and (c) RDL = 1.60 WFSLC columns with (1) λcr = 1.25 and (2) λcr = 3.50.

much higher yield stress (λcr = 3.5). The λcr value for which L-D interaction ceases to be relevant naturally increases with RDL. Moreover, the RDL > 1.00 column failure loads tend to be less overestimated by fnL as RDL increases (switch from TI to SDB L-D interaction) and λL decreases, since L-D interaction become less relevant − the number of accurate estimates (local failures) grows slowly with RDL − see Fig. 19(a1)–(a3).

(iv) In view of the content of the previous item, most RDL = 0.80, RDL = 1.00, RDL = 1.30, RDL = 1.60 column failure loads are not accurately predicted by the current DSM local or distortional design curves − it is necessary to resort to the MNDL approach to obtain accurate predictions. (v) In the RDL = 1.00 columns L-D interaction occurs in the whole slenderness range, as shown in Fig. 18(a4)–(b4) − these columns 14

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Fig. 22. Plots of (a) fU/fnD vs. λL, (b) fU/fnL vs. λD, (c) fU/fnDL vs. λD and (d) fU/fMnDL vs. λD, for all the numerical failure loads concerning (1) plain (C, H, Z, R), (2) WSLC and (3) WFSLC columns.

are affected by strong TI L-D interaction. The high failure load erosion is readily detected by comparing the numerical failure loads with their fnL and fnD estimates. (vi) The results of the C and WSLC columns are qualitatively very similar: for identical RDL and slenderness values, the fU/fy values almost coincide, even if the former are generally a bit higher.12 On the other hand, again for identical RDL and slenderness values, the WFSLC fU/fy values are generally higher (stocky columns) and

lower (slender columns) than their C and WSLC counterparts − the failure load erosion observed in the slender columns stems from the lower post-critical stiffness of the WFSLC columns. Therefore, it seems logical to expect that it will be possible to handle jointly the design of C and WSLC columns failing in L-D interactive modes. However, the same is not true for WFSLC columns − relevant differences were found in the high critical slenderness range (λD > 2.50).13

12 The failure loads of the two column sets were obtained considering the most detrimental distortional initial imperfections, which are distinct in the two columns: inward flange-lip motions in the WSLC columns and outward flange-lip motions in the C columns.

13 In the previous North American cold-formed steel specification [41], the DSM “prequalified columns” covered only C (with simple or complex lips), WSLC, H, Z and R crosssections – WFSLC columns were not dealt with.

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Fig. 23. Plots (a) fExp/fnDL vs. λD and (b) fExp/fMnDL vs. λD concerning the available experimental results of (1) plain cross-section and (2) web and web-flange-stiffened lipped channel columns.

deviation and minimum/maximum values equal to 1.00, 0.10, 0.79 and 1.22, respectively).

4.4.1. Assessment of the numerical and experimental ultimate strength estimates Based on the numerical ultimate strengths obtained, it is possible to draw conclusions on the quality of their DSM predictions. Fig. 22(a)–(d) plot fU/fnD vs. λL, fU/fnL vs. λD, fU/fnDL vs. λD and fU/fMnDL vs. λD14 for the plain cross-section (C, H, Z, R), WSLC, and WFSLC columns − Fig. 22(a)–(b) include only columns with RDL ≤ 1.0 and RDL > 1.0, respectively, while Fig. 22(c)–(d) include all the columns analysed. The observation of these figures prompts the following remarks:

Attention is now turned to the quality assessment of the estimates of the experimental failure loads available in the literature concerning plain and stiffened column tests (see Sections 4.1 and 4.2).15 For clarity, the results of plain cross-section columns are addressed first (see Fig. 23(a1)–(b1)). Then, the results concerning the WSLC and WFSLC columns, presented in Fig. 23(a2)–(b2), are dealt with. Fig. 23(a1)–(b1) plot fExp /fnDL and fExp/fMnDL against λD − Fig. 23(a1) includes all experimental results and Fig. 23(b1) includes only results concerning specimens with 0.70 < RDL < 1.60, i.e., inside the domain of application of the MNDL approach (see items (iii) and (iv) commenting Figs. 18 and 19) − this comprises all the tests reported by Kwon and Hancock [24], Dinis et al. [27] and Kwon et al. [36], as well as specimen A-8-41000 from Kwon et al. [37]. The first plot shows that the NDL approach generally predicts safely the experimental ultimate strengths (moreover, no fExp/fnDL value is below 0.86), but several predictions are excessively safe − the fExp/fnDL mean and standard deviation are equal to 1.13 and 0.20. If the specimens reported by Loughlan et al. [25], exhibiting “doubtful” L-D interactive collapses, are removed, the fExp/fnDL indicators improve to 1.05 and 0.18, respectively.16 The second plot shows that the fMnDL estimates are on the unsafe side, as reflected by the overall fExp/fMnDL mean and standard deviation equal to 0.93 and 0.10 − removing the single test result reported by Kwon et al. [37] and all hat column test results reported by Kwon et al. [36] would improve these indicators to 0.98 and 0.07.17

(i) The current DSM local and distortional design curves are unable to predict adequately the failure load erosion due to L-D interaction. The local curve estimates exhibit very poor indicators, with the sole exception of the stocky columns with high RDL, which collapse in pure local modes − the worst indicators concern the WFSLC columns: mean and standard deviation equal to 0.77 and 0.22, and minimum of 0.39. The same applies to the fnD predictions, although to a lesser extent: all unsafe estimates in Fig. 22(a1)–(a3) concern either columns experiencing TI L-D interaction or slender WFSLC columns (see Fig. 19(b1)–(b3)) − note also how the WFSLC column “estimate cloud” differs from its plain and WSLC counterparts. (ii) Although the fnDL values provide safe ultimate strength estimates for practically all columns analysed, a large fraction are excessively underestimated, particularly in the moderate-to-high slenderness range. The fnDL estimates exhibit better indicators for the WFSLC columns than for their plain and WSLC counterparts. (iii) The fMnDL values are the best ultimate strength failure estimates (see Fig. 22(d1)–(d3) and the corresponding fU/fMnDL indicators). The worst amongst them are associated with the WFSLC columns, even if they still constitute very good predictions (mean, standard

15 Most available tests results on column L-D interaction were obtained from studies that did not have this coupling phenomenon as their key objective − the works of Yang and Hancock [34], Young et al. [26] and Dinis et al. [27] are exceptions to this rule. 16 In the authors’ opinion, these specimens failed in pure local modes. 17 In the tests reported by Kwon et al. [36,37] the specimen fixed-ended support conditions are a bit “suspicious”, since they were achieved through polyester resin capping system. If this arrangement is not capable of ensuring fully fixed-ended columns at high load levels (as the authors believe), it seems logical to expect lower experimental failure loads.

14 The apparently “illogical” fU/fND vs. λL and fU/fNL vs. λD plots are included (instead of the “logical” fU/fND vs. λD and fU/fNL vs. λL ones) for clarity, i.e., to improve “readability” – although both plot pairs contain the same information, in the latter the points corresponding to the various ratio values are located on the same vertical line and “on top of each other”.

16

17

+ + + + + + + + + + + + + + + + + D D D D D D D D D D D D D D D D D + + + + + + + + + + + + + + + + + L L L L L L L L L L L L L L L L L 46.2 44.8 44.2 39.8 39.7 61.7 59.2 49.5 47.5 51.1 50.6 42.6 46.2 66.8 60.7 55.5 52.1 1/4394 − 1/32503 − 1/66891 − 1/4043 − 1/12075 1/29860 − 1/3586 − 1/1888 − 1/2927 − 1/4596 1/15769 − 1/66891 − 1/6501 − 1/2974 − 1/2756 − 1/4202 − 1/10826 0.945 0.575 0.575 0.385 0.805 − 0.175 0.108 − 0.865 − 1.780 0.260 0.105 0.475 − 0.050 − 0.208 − 1.238 0.733 − 0.735 1.94 2.38 2.33 2.85 3.41 2.46 2.84 3.22 3.61 1.62 1.88 2.37 2.29 2.06 2.37 2.79 3.17 1.23 1.14 1.16 1.11 1.13 1.17 1.18 1.18 1.25 1.37 1.39 1.30 1.35 1.18 1.24 1.21 1.16 1.19 1.14 1.16 1.10 1.13 1.06 1.11 1.10 1.11 1.34 1.31 1.17 1.30 1.03 1.06 1.01 1.02 58.3 51.6 52.9 46.0 40.5 69.7 64.0 59.0 54.8 65.4 61.2 53.7 53.9 78.9 71.3 64.1 61.5 56.6 51.7 52.9 46.4 40.4 59.4 54.2 50.2 43.7 63.8 57.4 48.2 51.7 69.1 60.9 53.0 53.0 47.6 45.2 45.7 42.0 35.7 63.2 60.3 55.3 48.7 47.6 43.9 41.2 39.8 67.0 57.5 53.7 54.0 104.0 112.5 113.1 121.1 126.5 143.0 151.2 158.5 164.6 97.3 105.5 116.9 113.2 135.2 141.0 149.3 162.4 1395 1651 1649 1951 2300 1896 2004 2302 2603 1401 1602 1699 1899 1851 2100 2402 2750 0.985 0.997 1.001 1.001 0.976 1.193 1.203 1.194 1.171 0.983 0.989 0.987 0.986 1.204 1.174 1.176 1.204 12.5 12.4 12.5 12.5 12.5 12.2 11.9 12.0 11.8 12.8 12.7 12.4 12.8 12.4 12.5 11.9 12.3 53.5 57.5 57.6 62.6 68.7 70.8 70.9 75.7 82.1 58.0 63.3 62.7 68.5 73.2 78.4 83.3 88.5 LC1 LC2-1 LC2-2 LC3 LC4 LC5 LC6 LC7 LC8 LC9 LC10 LC11 LC12 LC13 LC14 LC15 LC16

56.7 61.4 61.4 66.2 71.0 72.3 78.3 82.9 87.7 48.2 52.6 63.9 57.3 63.3 68.4 73.4 78.3

Δ0 (mm) Py Pcr . max Pcr . max Pcr . min Pcr . int Pcr . min

PcrG (kN) PcrD (kN) PcrL (kN) Py (kN) L (mm) t (mm) bs (mm) bf (mm)

The amount of research activity devoted to investigate L-D-G interaction in cold-formed steel columns is relatively scarce and concerns exclusively lipped channel columns (mostly fixed-ended). Indeed, experimental tests and numerical simulations on this topic have been reported by Young and Rasmussen [43], Young and Yan [44], Kwon et al. [37], Dinis and Camotim [17], Dinis et al. [12,45,46], Santos et al. [47–49], Young et al. [13,50], Cava [51] and Cava et al. [52]. However, the above works contain little more than incipient design considerations and it was not until very recently that a properly validated DSM design approach to handle columns failing in local-distortional-global interactive modes was proposed by Dinis et al. [53,54] − however, this design approach was developed exclusively in the context of lipped channel columns and further research work is currently under way to assess whether it can be readily extended to columns exhibiting other cross-section shapes (quite promising numerical results were very recently reported in [55], concerning hat, zed and rack-section columns). This section provides an overview of the recent research activity concerning cold-formed steel lipped channel columns affected by L-D-G interaction. Initially, experimental tests performed at The University of Hong Kong (UHK) are addressed and the results obtained commented. This experimental study provides (i) clear evidence of the occurrence of L-D-G interaction and (ii) the means to validate a previously developed ABAQUS SFE model. This model is then employed to carry out a parametric study aimed at gathering additional numerical failure load data to help developing a DSM-based design approach for lipped channel columns failing in L-D-G interactive modes. The columns analysed exhibit either (i) geometries of specimens tested at UHK or (ii) crosssection dimensions and lengths recently identified by Cava [51]. Finally, the experimental and numerical failure loads reported in this paper, as well as others collected from the literature, are used to draw fairly definite conclusions concerning the establishment of a DSM-based design approach that can handle efficiently (accurately, safely and reliably) triple interactive failures in cold-formed steel lipped channel columns − a total of 52 experimental and 893 numerical failure loads are considered for this purpose.

bw (mm)

5. Local-distortional-global interaction in columns

Column specimens

Table 4 Specimen geometries, squash and critical (L, D, G) buckling loads, load ratios, initial geometrical imperfection amplitudes, experimental failure loads and observed failure mode natures.

δ0/L

PExp (kN)

Failure mode

Fig. 23(a2)-(b2) show similar plots for the WSLC and WFSLC columns. Again, only the experimental results to which the MNDL approach is applicable are included in Fig. 23(b2) − specimens (i) SWC2000_1 and SWC2000_2, reported in [27], (ii) WSC20-1000-1, WSC20-1000-2 and WSC20-1000-3, reported in [32], (iii) LC800a, LC800b, LC800sa and LC800sb, reported in [34], and (iv) B-6-2-400, B6-1-400 and B-8-1-400, reported in [36] are excluded. The first plot shows that the NDL approach provides satisfactory predictions (fExp/ fnDL mean and standard deviation equal to 0.97 and 0.15), even if most of them (32 out of 53) are unsafe and the scatter is fairly high. Moreover, note that some fExp/fnDL values are clearly (i) above 1.0 (the 4 tests of Yang and Hancock [34] and 2 tests of Kwon and Hancock [24] or (ii) below 1.0 (specimen B-6-1-1200 of Kwon et al. [37] is the “extreme case”). Fig. 23(b2) shows the fMnDL values are generally worst estimates than the fnDL ones: much larger numbers and amounts of overestimations, as reflected by the fExp/fMnDL mean (0.83) and standard deviation (0.13). Even so, the failure loads reported in [24,34] are better predicted by the fMnDL values than by the fnDL ones. It is concluded that the MNDL approach predicts the WSLC and WFSLC column experimental failure loads less successfully than their plain cross-section column counterparts − recall that this design approach predicted equally well the numerical failure loads concerning the two sets of columns. The authors are currently finalising a more indepth investigation on the DSM-based design of columns affected by LD interaction [42] − hopefully, the outcome of this study will pave the way for the codification of an efficient DSM design approach for such columns.

G G G G G G G G G G G G G G G G G

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Fig. 24. (a) Fixed-ended column test set-up, (b) coupon stress-strain curves, (c) displacement transducer locations (mid-height cross-section), and (d) mid-height initial displacement measurements.

Fig. 25. Experimental evidence of L-D-G interaction in specimens (a) LC7 and (b) LC13 (front and back views), and (c) load vs. axial shortening curve concerning specimen LC5.

specimens with t = 1.0 mm, labelled LC1-4, LC9-12, and eight specimens with t = 1.2 mm, labelled LC5-8, LC13-16 − to check the test repeatability, two nearly identical LC2 specimens were tested. The inside corner radius was 2.0 mm in all cases and the specimen end sections were welded to 25 mm thick steel end plates, ensuring full contact with the test machine end bearings (fixed end supports) − see Fig. 24(a). The specimen material properties were obtained from tensile coupon tests. Fig. 24(b) shows the stress-strain curves of two specimen tensile coupon tests − note that there is practically no strain-hardening. To assess the specimen initial and deformed configurations, seven displacement transducers were used to measure the column midheight cross-section deformation: three in the web, one in each

5.1. Tests performed at The University of Hong Kong The experimental investigation carried out at The University of Hong Kong, which was fully reported by Young et al. [13], involved lipped channel specimens brake-pressed from high strength zinccoated structural steel sheets of grades G500 and G550 with nominal thickness and yield stress (i) t = 1.2 mm and fy = 500 MPa (G500) and (ii) t = 1.0 mm and fy = 550 MPa (G550). All measured specimen cross-section dimensions and lengths are given in Table 4, together with the associated column critical buckling loads (E = 210 GPa and ν = 0.3) and squash loads (Py = A fy, where the areas A were computed on the basis of the average measured cross-section dimensions − the rounded corners were disregarded): eight 18

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interaction with L or D modes [12]. Finally, note that the failure loads of the repeated tests (specimens LC-2-1 and LC-2-2) differ by 1.3%, thus showing quite good test repeatability.

5.2. Other test results available in the literature Besides the 17 experimental failure loads obtained in the UHK, concerning specimens with 1.39 ≥ Pcr.max/Pcr.min ≥ 1.11 and 3.61 ≥ Py/Pcr.max ≥ 1.62, there are other test results available in the literature, namely (i) 2 tests reported by Young and Rasmussen [43], for which Pcr.max/Pcr.min and Py/Pcr.max are equal to (2.33; 0.94) and (1.47; 1.46), (ii) 5 tests reported by Kwon et al. [37], such that 1.56 ≥ Pcr.max/Pcr.min ≥ 1.17 and 2.27 ≥ Py/Pcr.max ≥1.27, and (iii) two test campaigns carried out at the Federal University of Rio de Janeiro (COPPE-UFRJ) and reported by Santos et al. [47–49], involving 12 tests with Pcr.max /Pcr.min ≤ 1.10 and 1.19 ≤ Py/Pcr.ma x ≤ 2.26, and 16 tests such that 1.08 ≤ Pcr.max/Pcr.min ≤ 1.44 and 1.01 ≤ Py/Pcr.max ≤ 1.82.18 It is worth noting that a close inspection of the first COPPE-UFRJ test campaign showed that fully fixed end support conditions had not

Fig. 26. COPPE-UFRJ test deformed configurations at onset of collapse: (a)–(c) first test campaign (specimens C5, C8, C10 [47]) and (d)-(f) second test campaign (specimens CP2, CP18, CP28 [48,49]).

Fig. 27. Numerical failure modes of the (a) L and (b) G columns (the former is amplified 8 times).

flange and one in each lip − see Fig. 24(c). A data acquisition system recorded the applied load and displacement transducer readings at regular intervals during the tests. Concerning the specimen initial configurations, two displacements were measured at mid-height prior to testing: Δ0 and δ0 (see Fig. 24(d)) − all values are given in Table 4. While the Δ0 values concern the initial distortional deformation, the δ0 ones may stem from various combinations of initial minor-axis flexure, torsional rotation and, to a lesser extent, local and/or distortional deformations − positive δ0 values mean minoraxis bending curvatures towards the lips. No initial displacement profiles were measured. The experimental failure loads obtained are given in Table 4, together with the observed failure mode natures − Fig. 25(a)–(b) show the deformed configurations near collapse of specimens LC7 and LC13. As for Fig. 25(c), it depicts the applied load vs. axial shortening equilibrium path of specimen LC5 − several other equilibrium paths, corresponding to other specimens and transducer measurements can be found in [13]. All specimens failed in L-D-G interactive modes − Fig. 25(a)-(b), clearly show the simultaneous occurrence of local, distortional and global deformations. However, these deformations do not stem from the near coincidence of PcrL, PcrD and PcrG − Table 4 shows that the ratio between the highest and lowest buckling loads varies between 1.11 and 1.39. Instead, the interaction occurs because (i) the squash load is “sufficiently larger” than the highest column critical buckling load (the corresponding ratio varies between 1.61 and 3.37) and (ii) global buckling is never critical − numerical studies show that columns whose lowest critical buckling load is global fail in pure global modes, with no visible

been achieved and prompted the performance of the second test campaign, in which the flaw detected (deficient welding) was corrected. Although no visible collapse mode differences were observed in the two sets of specimens tested at COPPE-UFRJ, as can be attested by looking at Fig. 26(a)–(f), the ultimate strengths (fExp) obtained in the second test campaign are generally noticeably larger, due to the absence of fully fixed end supports in the first test campaign. Indeed, a rotational restraint/spring (even if very stiff) entails non-negligible drops in the column global (mostly) and distortional critical buckling loads − due to the large number of half-waves involved, the impact on the local critical buckling is very small. Since the column collapse is mainly governed by global buckling, a drop in the associated critical load causes an ultimate strength decrease [53,54]. In spite of what was mentioned in the previous sentences, the first COPPE-UFRJ test campaign failure loads were also included in the data considered in this work, thus bringing the total number of experimental failure loads to 52, even if it is expected that their failure loads will lie below the remaining ones (experimental and numerical), which concern really fully fixed-ended columns. Finally, one last word to mention a difference between the specimen failure modes observed at the UHK and COPPE-UFRJ tests: while local deformations are clearly visible in the former (see Fig. 25(a)-(b)), they

18 The first two sets of experimental failure loads were taken from publications reporting research work that either (i) was not specifically intended to investigate L-D-G interaction or (ii) did not involve only lipped channel specimens. The failure loads collected concern lipped channel specimens either (i) with close L, D and G critical buckling loads or (ii) for which L, D and G deformations were visually observed.

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Fig. 28. Illustrative (a) equilibrium paths P/Pcr vs. v/t (fy/fcr = 1.0; 2.0; 3.0; ∞) and (b) deformed shape and plastic strains at collapse (fy/fcr = 3.0) − bw = 75 mm, bf = 65 mm, bs = 11 mm, t = 1.1 mm, L = 2350 mm.

Fig. 29. DSM design curves against interactive failures involving global deformations and plots of fU/fy against λG for the column failure loads obtained (a) experimentally and (b) numerically.

Fig. 30. Plots of the ultimate-to-predicted strength ratios (a) fU/fnLG, (b) fU/fnDG, (c) fU/fnLDG against λG.

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Fig. 31. (a) Pcr vs. L curves, (b) GBT modal participation diagrams and (c) 6 GBT deformations mode in-plane shapes.

interactive failures involving distortional buckling. Following the procedure adopted to handle local-global interactive failures, it is possible to develop DSM-based design approaches to estimate the ultimate strength of columns failing in local-distortional or distortional-global interactive modes, by replacing (i) fy with fnD in the fnL equations (NLD approach) or (ii) fy with fnG in the fnD equations (NDG approach), as first suggested by Schafer [40]. A modified version of the first procedure was proposed by Silvestre et al. [28], in the context of lipped channel columns undergoing L-D local interaction, and later extended to H, Z and R columns under the same circumstances [29] − see Section 4.4. As for the second procedure, it was employed by Yap and Hancock [31], in the context of web-stiffened lipped channel columns experiencing local-distortional interaction. Moreover, these authors carried the same reasoning one step further and argued that it may be possible to predict the failure loads of cold-formed steel columns affected by L-D-G interaction using fnLDG values, obtained from the fnL expressions through the replacement of fy with fnDG. Thus, the available column nominal strengths against interactive failures involving global buckling are fnLG, fnDG and fnLDG. Fig. 29(a) compares the three above DSM column nominal strengths (fnLG, fnDG, fnLDG), plotted against λG21, with the failure stress ratios fExp/ fy concerning the 52 available experimental failure loads. On the other hand, Fig. 29(b) displays similar results for the 893 numerical failure loads obtained previously. Finally, Fig. 30(a)–(c) plot, for the whole set of columns, the ultimate-to-predicted strength ratios fU/fnLG, fU/fnDG, fU/fnLDG against λG − note that experimental ultimate strengths fExp are also termed fU. The observation of these results leads to the following conclusions:

are barely perceptible in the latter (see Fig. 26(a)–(f)). This is due to a combination of geometrical and material characteristics, namely the relative values of the L, D and G critical buckling loads, initial geometrical imperfections and yield stresses. In order to illustrate this assertion, Fig. 27(a)-(b) show the numerical collapse modes of columns with (i) fcrL = fcr.min, fcrD = 1.12 fcr.min, fcrG = fcr.max = 1.27 fcr.min and fy = 1.20 fcr.max (L column − identical to specimen C28 of [48]), and (ii) fcrG = fcr.min, fcrD = 1.06 fcr.min, fcrL = fcr.max = 1.26 fcr.min and fy = 1.24 fcr.max (G column − similar to the UHK LC16 specimen, but with a larger thickness to ensure global critical buckling): while the former combines local, distortional and global deformations, the latter exhibits only global (mostly) and distortional deformations. 5.3. Numerical simulations The numerical failure loads addressed in this work correspond to 893 columns: (i) 134 reported by Dinis et al. [12], for columns with 1.00 ≤ Pcr.max/Pcr.min ≤ 1.10 and 0.53 ≤ Py/Pcr.max ≤ 6.24, (ii) 391 obtained recently by Cava et al. [52], for columns with 1.27 ≥ Pcr.max/ Pcr.min ≥ 1.02 and λG = (fy/fcrG)0.5 varying from 0.5 to 2.5 in 0.5 intervals, and (iii) 368 reported by Dinis et al. [53,54], for columns with geometries of the specimens tested at the UHK (1.39 ≥ Pcr.max/Pcr.min ≥ 1.11) and selected from those identified by Cava [51] (1.27 ≥ Pcr.max/ Pcr.min ≥ 1.04) − the yield stresses fy were chosen to enable covering a wide and fairly evenly populated (with fU/fy values) slenderness λG range,19 which implies the consideration of several unrealistically high values (for the sake of completion). The failure loads are determined for columns containing global (critical-mode) initial geometrical imperfections with amplitude L/1000, value in line with the measurements made in the specimens tested at the UHK.20 Fig. 28(a)–(b), concerning columns with bw = 75 mm, bf = 65 mm, bs = 11 mm, t = 1.1 mm, L = 2350 mm and various yield stresses, illustrate the equilibrium paths that it was necessary to determine in order to obtain the numerical failure load data.

(i) As anticipated, the experimental ultimate strengths obtained from the first COPPE-UFRJ test campaign fall below all the remaining ones. Moreover, the experimental ultimate strengths reported by Kwon et al. [37] are also generally lower than those reported by the other authors, again probably due to the inability to ensure fully fixed end support conditions (see footnote 13 in Section 4.4.1). (ii) With the exception of the lower experimental ultimate strength values addressed in the previous item, the numerical and experimental fU/fy values correlate very well − moreover, all these

5.4. Direct Strength Method (DSM) Design The currently codified DSM column design curve against localglobal interactive failures (fnLG) is obtained through the replacement of fy by fnG in the expressions providing the local strength fnL (see Section 3.5). However, no similar strength curves are yet available for

21 Obviously, the joint representation of the three design curves is made under the assumption that λL ≈ λD ≈ λG, which is a crude approximation is some cases. A more accurate account would require a different plot for each design curve, thus making their comparison much less clear (or maybe even impossible). Nevertheless, it should be noted that the closeness between the NDG and NLDG curves in Fig. 29 may be quite misleading − the real difference between the corresponding column failure load predictions can be observed in Fig. 30.

19 Note that, for instance, Cava et al. [52] considered the same five λG values for all the columns analysed, which led to fU /fy values located along five “vertical lines”. 20 This initial geometrical imperfection shape was found to be the most detrimental, in the sense that it leads to the lowest column strength and failure load [12,16,45,52].

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Fig. 32. F columns: (a) P/Pcr vs. |w|/t and (b) P/Pcr vs. β paths, and (c) limit deformed shapes for θ = 0°, 15° ≤ θ ≤ 165°, θ = 180°.

Fig. 33. P columns: (a) P/Pcr vs. |w|/t and (b) P/Pcr vs. β paths, and (c) limit deformed shapes for θ = 0°, 15° ≤ θ ≤ 165°, θ = 180°.

values are nicely aligned along a “Winter-type” curve with small vertical dispersion. (iii) Since the sequence and closeness of the L, D and G critical buckling loads are so diverse, it may be argued that they have been selected (nearly) randomly. Nevertheless, the corresponding cloud of numerical fU/fy values is quite “homogeneous”, thus implying that the ultimate strength erosion stemming from the triple interaction does not vary substantially with those parameters. (iv) The fnLG values provide mostly safe predictions of the experimental and numerical failure loads and, with the exception of the experimental values obtained from the COPPE-UFRJ first test campaign, the relatively few overestimations are never more than mildly pronounced. As for the fnDG and fnLDG values, they underestimate all but the experimental fU values of the COPPE-UFRJ first tests series − obviously, the fnLDG values are the smallest ones. A large number of underestimations are clearly excessive, particularly those concerning the numerical fU values of the most slender columns. (v) Concerning the fU/fnLG ratio, only (v1) values associated with the COPPE-UFRJ first test series are below 0.89 and (v2) the means of the test results reported by Santos et al. [47] and Kwon et al. [37] fall below 1.0 (0.86 and 0.92, respectively). Moreover, the mean

and standard deviation values of the whole sets of experimental and numerical fU/fnLG values are 1.04/0.15 and 1.15/0.10. If the COPPE-UFRJ first test series is excluded, the first set of indicators improves to 1.14/0.09. The LRFD (Load and Resistance Factor Design) resistance factors ϕc (prescribed by the North American cold-formed steel specification [4]) evaluated for the use of fnLG to predict the ultimate strength of lipped channel columns undergoing L-D-G interaction provide evidence of the excellent capability of the currently codified NLG approach. Indeed: (i) When all the failure load data are considered, the values obtained are ϕc = 0.85 (experimental), ϕc = 1.00 (numerical) and ϕc = 0.99 (experimental and numerical) − given the huge disparity between the numbers of experimental and numerical failure loads available, the last two values are practically identical. Recall that the value ϕc = 0.85 is recommended, for compression members, by the current North American specification. (ii) When the COPPE-UFRJ first test series results are excluded, one has ϕc = 0.92 (experimental) and ϕc = 1.00 (experimental and numerical).

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Fig. 34. Elastic “global” post-buckling equilibrium paths P/Pcr vs. β of the LG.1, LG.2 and LG.3 (a) F and (b) P columns.

Fig. 35. θ = 90 °F column (a) P/Pcr vs. β elastic-plastic paths, (b) plastic strain and deformed configuration evolution (for fy/fcr ≈ 1.5) and (c) variation of PU/Pcr with θ and fy/fcr.

6. Distortional-global interaction in columns

knowledge acquired from this comparative study is essential in order to be able to correlate meaningfully experimental results obtained from specimens tested under the two end support conditions.

The amount of research available on cold-formed steel columns experiencing D-G interaction is rather scarce. Indeed, to the authors’ best knowledge, the works addressing the influence of this coupling phenomenon on the post-buckling behaviour and ultimate strength of cold-formed steel columns consist of (i) experimental studies on racksection uprights with and without holes [56,57] and, more recently, web-stiffened lipped channel columns [58] 22, and (ii) the shell finite element investigations on simply supported and fixed-ended lipped channel columns [11,16], and simply supported and warping prevented rack-section uprights with and without holes [61]. It is fair to argue that a lot of research work still lacks before a well-founded design approach against D-G interactive failures can be proposed. In particular, it is indispensable to identify which combinations of the ratios between the critical buckling stresses involved (fcrD and fcrG) and the yield stress cause significant interaction effects. However, it should be noted that the authors have currently under way a numerical investigation concerning the behaviour, strength and DSM-based design of cold-formed steel columns affected by D-G interaction [62]. This section addresses a specific problem, namely the comparison between the buckling and post-buckling behaviours of lipped channel columns experiencing D-G interaction and exhibiting the support conditions usually adopted in experimental investigations, namely (i) fixedended columns (mostly adopted in North America, Australia and Hong Kong) and (ii) columns with the end cross-sections locally fixed, globally pinned and prevented from warping and twisting, herein termed “simply supported” (mostly adopted in Europe) − see Fig. 31(a). The designation of the two columns will be abbreviated to “F” (fixed-ended) and “P” (simply supported SCB condition − pinned) columns. The

6.1. Buckling behaviour The Pcr vs. L (logarithmic scale) curves depicted in Fig. 31(a) concern F and P columns with the cross-section dimensions given in the figure and were obtained by means of ABAQUS SFEA. This figure also shows critical buckling loads obtained with GBTUL [5], including the following 11 deformation modes: 4 global (1–4), 2 distortional (5–6) and 5 local (7–11). Fig. 31(b) displays the corresponding GBT modal participation diagrams, providing the contributions of each deformation mode to the column buckling modes. Lastly, Fig. 31(c) shows the in-plane shapes of the 6 most relevant deformation modes. The observation of these buckling results shows that the short-to-intermediate P and F columns only differ in the length associated with the transition from distortional to “global” buckling: LD-G = 130 cm (P columns) and LD-G = 145 cm (F columns). This means that P and F columns with lengths below L = 130 cm have identical critical buckling loads and mode shapes. Moreover, the LD-G columns exhibit practically identical distortional and global critical buckling loads: PcrD = 40.5 kN (fcrD = 227.8 MPa) and PcrG = 40.8 kN, for the F column, and PcrG = 40.9 kN (fcrG = 229.9 MPa) and PcrD = 41.1 kN, for the P column − naturally, the post-buckling behaviours and ultimate strengths of such columns are expected to be strongly affected by D-G interaction. However, recall that the so-called column “global” buckling mode is, indeed, a flexuraltorsional-distortional one (see Section 3.3): participations from the antisymmetric distortional mode 6, major-axis flexural mode 2 and torsion mode 4 with values 43% + 26% + 31% (F column) and 26% + 59% + 15% (P column) − the amount of anti-symmetric distortion gradually decreases with the column length, until it vanishes. Moreover, note also that the difference between the “non-critical” minor-axis flexural buckling load (Pb. F) and Pcr is considerably smaller in the LD-G P column

22 It is still worth noting the tests reported by the Rossi et al. [59,60] on cold-formed stainless steel lipped channel columns.

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Fig. 36. θ = 90°P column (a) P/Pcr vs. β elastic-plastic paths, (b) plastic strain and deformed configuration evolution (for fy/fcr≈1.5) and (c) variation of PU/Pcr with θ and fy/fcr.

(iii) The pure “global” imperfections (θ = 90°) are the most detrimental for both the F and P columns − they lead to the lowest strengths. (iv) There is a major difference between the F and P column elastic equilibrium paths: while all the latter have well-defined limit points, occurring generally for loads levels below Pcr and small w/t and β values, the former always exhibit a considerable post-critical strength (no limit point was detected). Moreover, all P column deformed mid-span cross-sections show clear evidence of minoraxis flexure (web in tension). (v) The behavioural features mentioned in the previous item stem for the joint influence of three factors: (v1) participation of mode 6 in the column “global” buckling mode, (v2) absence of end support minor-axis bending rotation restraints and (v3) relative closeness between the minor-axis flexural buckling load Pb. F and Pcr (see Section 6.1) − the interaction with minor-axis flexural buckling is clearly visible in Fig. 33(c). (vi) In order to provide mechanical insight on the marked behavioural differences between the P and F columns, the pure “global” elastic post-buckling behaviours of longer columns with the same crosssection (deemed not affected by D-G interaction) are depicted in Fig. 34(a)–(b). The additional F and P columns have lengths LG.1 = 150 cm, LG.2 = 300 cm and LG.3 = 600 cm, and contain pure “global” initial geometrical imperfections associated with midspan flange-lip corner vertical displacements equal to LDG /1000. It is observed that: (vi1) The LG.1-LG.3 F columns exhibit the expected stable postbuckling behaviours and their equilibrium paths virtually coincide, all with a fair amount of post-critical strength reserve. (vi2) While the LG.3 P column exhibits a marginally stable postbuckling behaviour (minute post-critical strength), its LG.1 and LG.2 counterparts exhibit a limit point, more pronounced in the shorter column. As mentioned above, these limit points occur due to the joint influence of (i) the closeness between Pb.F and Pcr.FT, (ii) the absence of minor-axis end reaction moments and (iii) the presence of mode 6. This explains why such limit points do not occur in the three F columns (Pb.F /Pcr.FT always high and minor-axis end reaction moments) and LG.3 P column (no mode 6, even if Pb.F /Pcr.FT is low and there are no end reaction moments). (vi3) The presence of mode 6 acts as a “triggering device”: it causes “effective centroid shifts” (towards the web) that lead to minor-axis flexure (eccentric compression), thus providing the “sparkle” that initiates the interaction with minor-axis flexural buckling, made possible by the relative closeness between Pb.F and Pcr − Pb.F/Pcr.FT = 2.2 and 1.9 in the LG.1 and LG.2 P columns, respectively. In the LG.3 P column, which has even closer Pb.F and Pcr values (Pb.F/Pcr.FT = 1.6), the mode 6 “sparkle” is no longer there.

than in its F counterpart − Pb. F/Pcr = 2.24 (P column) and 7.23 (F column) − see Camotim and Dinis [63]. 6.2. Post-buckling behaviour This section addresses briefly the most relevant features of the elastic and elastic-plastic post-buckling behaviour of F and P lipped channel columns with the LD-G lengths determined in Section 6.1, i.e., strongly affected by D-G interaction. Initially, elastic imperfectionsensitivity analyses are performed to unveil the most detrimental critical-mode initial geometrical imperfection shapes for the F and P columns − in both cases, such initial geometrical imperfection are linear combinations of the 3 half-wave distortional and single half-wave “global” buckling mode shapes, normalised to exhibit amplitudes vD.0 = 0.1t = 0.1 mm and vG.0 = L/1000 = 1.45 or 1.3 mm, respectively. In order to ensure a common amplitude of all initial geometrical imperfections considered, the linear combination coefficients CD.0 and CG.0 satisfy (CD.0)2 + (CG.0)2 = 1, which means that each initial geometrical imperfection shape may be associated with an angle θ, such that CD.0 = cosθ and CG.0 = sinθ (see Section 3.4 − Fig. 9) − since the column global post-buckling behaviour is symmetric, only 13 initial imperfection shapes need to be considered (those for which 0° ≤ θ ≤ 180° in 15° intervals). Figs. 32(a)–(b) and 33(a)–(b) show the upper parts (P/ Pcr > 0.4) of the F and P column elastic equilibrium paths P/Pcr vs. |w|/t and P/Pcr vs. β −w and β are the mid-span mid-web flexural displacement (measured with respect to the web chord) and web chord rigidbody rotation. Figs. 32(c) and 33(c) display the deformed configurations of the θ = 0°, 15° ≤ θ ≤ 165° and 180° columns at advanced loading (post-buckling) stages − for clarity purposes, their deformed mid-span cross-sections are also shown. The observation of these elastic post-buckling results prompts the following remarks: (i) Almost all F and P column P/Pcr vs. |w|/t and P/Pcr vs. β equilibrium paths merge into common curves, associated with mid-span (i1) clockwise web chord rotations and (i2) outward flange-lip motions (inward mid-web transverse flexure). The θ = 0° and θ = 180° columns (pure distortional imperfections) are the exceptions − they correspond to a singular post-buckling behaviour23 that was first unveiled in [16], exhibits no D-G interaction (only distortional deformations − see Figs. 32(c) and 33(c)) and is currently under investigation by means of GBT geometrically nonlinear analyses. (ii) The limit deformed configuration of the merging 15° ≤ θ ≤ 165° column curves couples the competing critical buckling modes, as it combines (ii1) a 3 half-wave distortional component (outward midspan flange-lip motions) with (ii2) a single half-wave “global” component − see Figs. 32(c) and 33(c).

23 Indeed, columns with θ values close to 0° or 180° exhibit post-buckling behaviours similar to those shown for the θ = 15° and θ = 165° columns [16,63].

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Fig. 37. MU/My vs. λD plots concerning (a) SCA and (b) SCB C, H, Z beams, together with the current and proposed DSM distortional design curves.

Fig. 38. RDL = 1.0 Z beam affected by L-D interaction: (a) elastic and elastic-plastic (λcr = 1.0, 2.5, 3.5) M/Mcr vs. (v + v0)/t paths and (b) deformed configurations at the onset of collapse for the (1) SCA and (2) SCB beams with λcr = 1.0 and λcr = 3.5.

illustrated in Fig. 35(a). In the θ = 90° column with fy/fcr ≈ 1.5, (iv1) yielding begins at the bottom lip free end zone in the column mid-span region (diagram I in Fig. 35(b)), and (iv2) collapse occurs after the full yielding of the web-flange corners at the column central region − see the diagram III detail in Fig. 35(b). (v) None of the P columns exhibits a non-negligible elastic-plastic strength reserve: failure occurs either in the elastic range or almost immediately after the onset of yielding (at the bottom lip mid-span zone in all columns going beyond the elastic range prior to failure − see diagram I in Fig. 36(b)). Moreover, the columns with the highest yield stress (fy /fcr ≈ 3.3) remain elastic until failure − the collapse is fully governed by geometrically non-linear effects stemming from D-G interaction. (vi) The failure load erosion due to D-G interaction is relevant for both the F and P columns − the amount of erosion is more substantial in the P columns. Indeed, the F column failure loads are comprised between 0.67 Pcr and 1.19 Pcr − in the θ = 90° column, Pcr is only marginally exceeded (PU/Pcr = 1.019) for the higher yield stress (fy/fcr ≈ 3.3). On the other hand, the P column failure loads are comprised between 0.72 Pcr and 0.95 Pcr − in the θ = 90° column, the drops with respect to Pcr are 28% (fy/fcr ≈ 1.1), 20% (fy/fcr ≈ 1.5) and 15% (fy/fcr ≥ 2.4). Note that there is no benefit in increasing the yield stress beyond fy/fcr ≈ 2.4, since the collapse takes place in the elastic regime.

Finally, Figs. 35(a) and 36(a) show the upper parts (P/Pcr > 0.4) of the elastic-plastic equilibrium paths P/Pcr vs. β of four F and four P columns containing pure “global” initial imperfections and exhibiting yield-to-critical stress ratios fy/fcr ≈ 1.1, 1.5, 2.4, 3.3 (and also the elastic paths shown in Figs. 32(a) and 33(a) − fy/fcr = ∞). As for Figs. 35(b) and 36(b), they concern the fy/fcr ≈ 1.5 F and P columns and display plastic strain diagrams corresponding to equilibrium states located along the post-buckling paths, including the collapse mechanisms − to enable a better visualisation of the plastic strains at collapse mechanisms, back views of the web mid-span region are also provided. Figs. 35(c) and 36(c) plot the failure load ratios PU/Pcr concerning the various θ and fy/fcr combinations considered (additional information can be found in [63]). The observation of these post-buckling results leads to the following comments: (i) Naturally (after the elastic results), the F and P columns exhibit distinct elastic-plastic post-buckling behaviours. (ii) The characteristics of the F column elastic-plastic post-buckling behaviour and collapse mechanism depend on the fy/fcr value − e.g., the onset of yielding varies between 0.67 Pcr and 0.88 Pcr in the yield stress range considered. (iii) In the F columns with fy/fcr close to 1.0, first yielding occurs for a fairly uniform normal stress distribution and, thus, precipitates an “abrupt” collapse − yielding occurs simultaneously in a large portion of the “most deformed cross-section”, whose location depends on the initial imperfection shape (mid-span for θ = 90°). (iv) In the F columns with higher fy/fcr values, first yielding occurs when the column normal stress distribution is already highly “nonuniform” and, therefore, does not lead to an immediate collapse. Instead, collapse occurs after a mild snap-through phenomenon, followed by a subsequent strength increase up to a limit point, as

7. Local-distortional interaction in beams The amount of research available on beams undergoing L-D interaction is also quite scarce, as most of the available results dealing with this coupling phenomenon concern cold-formed steel fixed-ended columns. Indeed, the authors are only aware of very few numerical and 25

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Fig. 39. MU/My vs. (a) λL or (b) λD plots for SCA and SCB Z-beams exhibiting (1)–(4) RDL = 0.50–0.70–0.85–1.00.

In principle, it should be possible to base the development of a DSMbased design approach for beams experiencing L-D interaction on the available beam strength curves concerning the local (Eq. (5)) and distortional (Eq. (6)) failures: (i) the NLD approach (MnLD), proposed by Silvestre et al. [64], and (ii) the NDL approach (MnDL), following an idea put forward by Schafer [40] for columns (see Section 4.4). However, a recent numerical investigation by Landesmann and Camotim [72] provided solid evidence that the currently codified DSM beam distortional design curve overestimates the vast majority of failure moments of nonstocky lipped channel beams with end-sections locally/globally pinned and free to warp. In fact, the overwhelming majority of the results employed to develop the above distortional design curve (and, by the way, also the local one) concern fairly stocky beams [73]. However, the fact that all these specimens exhibited small-to-moderate distortional slenderness values (0.68–1.53), plus the findings reported in [72], led the authors to precede the investigation on L-D interaction by a numerical study on the distortional strength of simply supported lipped channel beams under uniform bending (see Section 7.1). The beams analysed exhibit (i) lipped channel (C), hat (H) and zed (Z) cross-sections and (ii)

experimental studies addressing the influence of L-D interaction on the post-buckling behaviour and ultimate strength of cold-formed steel beams. The former concern simply supported lipped channel beams exhibiting flange or web-triggered L-D interaction under uniform major-axis bending with 0.85 ≤ McrL/McrD ≤ 1.15 [64] and McrL/McrD ≈ 1.0 [15,65], and also (ii) Z-section beams with and without intermediate stiffeners subjected to 4-point bending [66–68] − recall that McrL and McrD are the beam critical local and distortional buckling moments. As for the experimental investigations, they are restricted to the tests conducted by Bernard et al. [69,70] on simply supported thinwalled profiled steel decks with and without intermediate stiffeners (“vshaped and ”flat-hat”) under minor-axis bending. However, a systematic investigation, aimed at identifying which combinations of the ratios involving McrL or McrD and the yield moment lead to sizeable/ relevant L-D interaction effects is still lacking and is indispensable to achieve an efficient (safe and reliable) DSM-based design approach − the ultimate goal of this ongoing research effort. The most recent fruits bore by this research effort, which are not included in this work, can be found in [71].

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Fig. 40. MU/My vs. (a) λL or (b) λD plots for SCA and SCB Z-beams exhibiting (1)–(4) RDL = 1.30–1.60–1.80–2.00.

post-buckling behaviour and ultimate strength. This effort led to 30 sets of geometries for each combination of cross-section type and end support condition [9]. (ii) Generate randomly25 17 distortional slenderness values per beam geometry selected and determine the ultimate strength of all beams by using the SFEA model briefly described in Section 2.1. (iii) Develop and propose an alternative “Winter-type” curve, by solving an appropriate optimization problem, which is cast in the form

two simply supported end conditions, differing in the end cross-section local displacement/rotation and warping restraint: either free (SCA) or prevented (SCB) – see Section 2.1. While the C and H-beams are subject to major-axis bending, the Z-beams are under skew bending causing flange uniform compression (the most unfavourable case). 7.1. Distortional failure of beams under uniform bending This section summarises the procedure adopted to develop a proposal for new DSM-based distortional design curves to estimates failure moments of simply supported beams under uniform bending – more details can be found in the recent work of Martins et al. [9]. Such design curves were obtained by means of the following procedure:

24 As these beams have small-to-moderate post-critical strength reserve, it suffices to ensure that McrD is well below McrL and McrG. 25 For each beam geometry selected, 17 distinct distortional slenderness values were generated randomly, following a continuously uniform distribution in the interval [0.25, 4.0]. In this way, the optimization procedure discussed next involves beam slenderness values with the same “weight”.

(i) Identify/select beam geometries (cross-section dimensions and lengths) (i1) ensuring pure distortional buckling and failure modes under uniform bending24 and (i2) associated with distinct halfwave numbers and web-to-flange and flange-to-lip width ratios, since these features play important roles in the beam distortional 27

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Fig. 41. Plots (a) MU/MnL vs. λL, (b) MU/MnD vs. λD, (c) MU/MnLD vs. λD and (d) MU/MnDL vs. λD of all numerical failure moments of SCA and SCB (1) C, (2) H and (3) Z beams with 0.50 ≤ RDL ≤ 2.00.

MnD =

−2 ⎧ My + (1 − Cyd )(Mp − My ) if λD ≤ 0.673 − − b ⎨ (1 − aλD ) λD c My if λD > 0.673 ⎩

figures makes it possible to conclude that: (9)

(i) The “vertical dispersion” is much higher for the SCB beams, due to the different inelastic strength reserves exhibited by the various beams − such strength reserve is practically null in all the SCA beams. (ii) The proposed DSM design/strength curves provide better predictions of the simply supported beam distortional failure moments than the currently codified curve (Eq. (6)). However, the authors acknowledge that there is a dependence on the beam geometry and critical buckling mode half-wave number that cannot be explicitly taken into account by the DSM without compromising its roots and elegance. Indeed, such dependence cannot be captured solely by the

where Mp is the plastic moment, a, b and c are the solution of the optimization problem, and Cyd = (0.673/λD)0.5 ≤ 3. Fig. 37(a)–(b) plot, against λD, the MU/My values concerning the C, H and Z beams with SCA and SCB end support conditions − also depicted are the proposed design curves, together with the corresponding a, b and c values. Since the SCA and SCB beams exhibit substantially different distortional post-buckling behaviours, they should be designed by means of distinct strength curves. The observation of the above 28

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this accuracy is heavily dependent on the flange-lip width ratio bf/ bl and critical distortional half-wave length, as discussed in [9,74]. The vast majority of such beams fail in pure distortional modes − only the most slender ones exhibit minute local deformations, stemming from SLB L-D interaction, which means that there is virtually no failure moment erosion due to this type of L-D interaction. For beams with higher RDL values, namely RDL = 0.85 (see Fig. 39(b3)), the proposed MnD strength curve gradually ceases to provide safe failure moment ultimate estimates for the slender beams, which exhibit visible local deformations prior to collapse. This is because such beams correspond to the transition between beams not affected by L-D interaction and beams experiencing (iv1) SLB L-D interaction (RDL = 0.70) or (iv2) TI L-D interaction (RDL = 0.85). However, the failure moments of all the beams affected by L-D interaction are accurately predicted by the MnLD (and MnDL) values. (v) Naturally, neither the currently codified DSM local design curve nor the proposed DSM distortional strength curve are able to capture the failure moment erosion exhibited by the RDL = 1.00 beams (see Fig. 39(a4)–(b4)) − this is because TI L-D interaction occurs in the whole slenderness range. Once again, the DSM NLD (and NDL) approach predicts accurately the failure moment erosion intrinsic to this coupling phenomenon. (vi) The currently codified MnL values only provide accurate failure moment estimates for stocky beams with RDL > 1.00 − the number of accurate predictions grows with RDL and λL, i.e., when the beam collapse occurs in progressively more predominantly local modes (with no trace of L-D interaction) – see Fig. 40(a1)–(a4). On the other hand, the MnL (and MnD) values overestimate the failure moments of the RDL = 1.30–1.601.80–2.00 slender beams, thus providing clear evidence of the occurrence of L-D interaction, now due to SDB L-D interaction. Unlike the SLB L-D interaction (see item (iv)), this type of L-D interaction cannot be ignored in design. For such beams the proposed DSM NLD (and NDL) approach also provides high quality predictions.

distortional slenderness, which automatically implies excessive beam failure moment underestimations (especially in SCB beams). 7.2. Numerical simulations The numerical failure moments addressed here comprise (i) 43 beam geometries for each combination of cross-section shape and end support conditions, selected by means of GBT buckling analyses to ensure 0.50 ≤ RDL ≤ 2.00, where RDL = McrD/McrL, and (ii) 11 yield stresses per beam type, to cover a wide slenderness range (0.50–3.50). The failure moments were determined for beams containing criticalmode initial geometrical imperfections with amplitude equal to 0.1t. For instance, Fig. 38(a1)–(a2) show the Mcr vs. (v + v0)/t equilibrium paths (v is the mid-span compressed/top flange-lip corner vertical displacement and v0 its initial value) of SCA and SCB RDL = 1.0 Z beams with 4 critical slenderness values: 1.0; 2.5; 3.5; ∞ (the last stands for elastic behaviour). Fig. 38(b1)–(b2) display beam deformed configurations and plastic strain diagrams at the onset of collapse for the λcr = 1.0 and λcr = 3.5 beams. The observation of these results prompts the following remarks: (i) The SCA and SCB beam post-buckling behaviours are clearly distinct: while the former exhibits a rapid stiffness erosion until an elastic limit point is reached (no elastic-plastic strength reserve), the latter involves much slower stiffness degradation and is associated with a relevant elastic-plastic strength reserve. (ii) Despite the different behaviours associated with the two support conditions, Fig. 38(b) shows that, regardless of the beam yield stress (Fig. 38(b1)–(b2) − Z beams with λcr = 1.0; 3.5), local and distortional deformations are visible in the compressed flange at the onset of collapse, thus evidencing the occurrence of L-D interaction. 7.3. Direct Strength Method (DSM) design The main results of a first contribution towards the development of an efficient DSM design approach for cold-formed steel simply supported uniformly bent beams affected by L-D interaction [74] are addressed next. Although such results concern SCA and SCB C, H and Z beams, only a representative sample of those dealing with the Z beams analysed are presented and discussed in this work (details on the C and H beams analysed can be found in [74]) − they are displayed in Fig. 39(a1)-(b4) and 40(a1)-(b4), and concern beams with RDL = 0.50–0.70–0.85–1.00–1.30–1.60–1.80–2.00. These figures plot MU/My against the critical slenderness (λL or λD) for the above 8 beam sets − obviously, the SCA and SCB beams in each figure do not share the same geometry (they have different web-to-flange and flange-to-lip width ratios, as will be discussed next). For each RDL value, the numerical MU/ My values are compared with their DSM predictions, namely (i) MnL (currently codified design curve − Eq. (5)) and MnD (currently codified and proposed design curves − Eqs. (6) and (9)) and (ii) MnDL and MnLD, which account for L-D interaction – the MnDL values/curve are not shown, as they virtually coincide with the MnLD ones − however, the merits of both will be assessed later (in Fig. 41(a1)-(d3), concerning C, H and Z beams). The observation of these results makes it possible to extract the following conclusions (similar ones apply to C and H beams):

On the basis of the simply supported beam numerical failure moments obtained, the quality of their DSM-based predictions is assessed next, with the help of the plots (i) MU/MnL vs. λL (Fig. 41(a1)–(a3)), (ii) MU/MnD (curve proposed by Martins et al. [9]) vs. λD (Fig. 41(b1)–(b3)), (iii) MU/MnLD vs. λD (Fig. 41(c1)–(c3)) and (iv) MU/MnDL vs. λD (Fig. 41(d1)–(d3)) – Fig. 41(1), (2) and (3) concern C, H and Z beams, respectively. Note that the SCA C beam results reported by Silvestre et al. [64]26 are also shown (grey squares). The observation of these plots prompts the following comments: (i) The results reported by Silvestre et al. [64], concerning SCA C beams with small-to-moderate slenderness values, follow the same trends as those reported here − see Fig. 41(a1)–(d1). (ii) The numerical-to-predicted failure moment ratio “clouds” (MU/ MnL, MU/MnD, MU/MnLD and MU/MnDL) are very similar for the vast majority of the six pairs of plots displayed in Fig. 41(a1)–(d3). The exceptions are the MU/MnD ratios of the SCA and SCB C beams, which exhibit much better quality than those concerning the other beams considered − see the statistical indicators presented in the above figures. (iii) The pairs of plots in Fig. 41(a1)–(b3) clearly show that the currently codified local and proposed distortional DSM strength curves are unable to predict adequately the failure moment erosion due to L-D interaction − the only fairly accurate MnL estimates

(i) All the numerical MU/My values are well aligned along “Wintertype” curves. (ii) There is a strong qualitative resemblance between the responses of beams and columns affected by L-D interaction (see Section 4.4). (iii) The currently codified DSM distortional curve provides substantial failure moment overestimations for both SCA and SCB beams. (iv) Generally speaking, the proposed MnD strength curve (Eq. (9)) provides accurate failure moment estimates for beams with RDL«1.0 (e.g., RDL = 0.50; 0.70 − see Fig. 39(b1)–(b2)). However,

26 These results concern simply supported lipped channel beams with RDL values comprised between 0.85 and 1.20. However, 30 beams (out of 90) were excluded because local bucking is triggered by the web (situation outside of the scope of this work).

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doctoral scholarship SFRH/BD/87746/2012.

concern stocky beams with RDL > 1.00, which fail in pure local modes. Both sets of predictions exhibit poor indicators: means well below 1.0 and very high standard deviations. Even if the indicators of the proposed DSM distortional design curve are visibly better than those obtained with the currently codified one (not shown here), this curve still cannot cope with L-D interaction − naturally, the most accurate estimates concern stocky beams with RDL < 1.00, which fail in pure distortional modes. (iv) The MnDL and MnLD approaches provide the best failure moment estimates, an assertion that can be confirmed by merely looking at Fig. 41(c1)–(d3) − both exhibit similarly good accuracy (see the corresponding statistical indicators). Nevertheless, better predictions are found for the SCA beams, due to the less pronounced dependence on the flange-lip width ratio bf/bl, as reported by Martins et al. [74] − recall that none of the DSM approaches depends on this ratio. It is still worth noting that all overestimations concern beams with large flange-lip width ratios. Although both approaches provide similarly accurate results for 0.50 ≤ RDL ≤ 2.0, the NDL approach should be adopted. This is because the effects of the “secondary distortional bifurcation L-D interaction”, which affect the most slender beams, gradually cease to occur as RDL increases beyond 2.0. In other words, the beam behaviour tends to a “purely” local one (no interaction occurs), which is adequately captured by the NDL approach – since λDL is very low (below 0.673) for these beams, the NDL and NL approaches coincide (see [71]).

References [1] L.C. Prola, D. Camotim, On the distortional post-buckling behavior of cold-formed lipped channel steel columns, in: Proceedings of SSRC Annual Stability Conference (Seattle, 24-27/4), 2002, pp. 571–590. [2] L.C. Prola, D. Camotim, On the distortional post-buckling behaviour of cold-formed lipped channel steel beams, in: S.L. Chan, J.G. Teng, K.F. Chung (Eds.), Advances in Steel Structures (ICASS’02 − 9-11/12), Elsevier, Amsterdam, 2002, pp. 331–339. [3] N. Silvestre, D. Camotim, Local-plate and distortional post-buckling behavior of cold-formed steel lipped channel columns with intermediate stiffeners, J. Struct. Eng. (ASCE) 132 (4) (2006) 529–540. [4] AISI (American Iron and Steel Institute), North American Specification (NAS) for the Design of Cold-Formed Steel Structural Members − AISI-S100-S116, Washington DC, 2016. [5] R. Bebiano, N. Silvestre, D. Camotim. GBTul − a code for the buckling analysis of cold-formed steel members, in: R. LaBoube, W.-W. Yu (eds.) 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8. Concluding remarks This work provided an overview of the current status of an ongoing investigation on cold-formed steel columns and beams affected by mode coupling phenomena involving distortional buckling, taking place at the Universities of Lisbon and Hong Kong, which comprises experimental tests, numerical simulations and design proposals. This investigation is intended (i) to acquire in-depth knowledge on the postbuckling behaviour, ultimate strength and failure mode nature of elastic-plastic columns and beams and (ii) take advantage of this knowledge to develop, calibrate and validate efficient DSM-based design approaches to predict adequately (safely and accurately) their ultimate strengths. After using column results to illustrate and help grasp some fundamental concepts in mode interaction problems, the paper addressed separately several coupling phenomena, namely (i) L-D interaction in columns, (ii) L-D-G interaction in columns, (iii) D-G interaction in columns and (iv) L-D interaction in beams. In the first two cases, experimental investigations, numerical simulations and DSM-based design approaches were reported. In the last two cases, only numerical results were presented (there are no available experimental results), even if a fair amount of design considerations were made for the beams undergoing flange-triggered L-D interaction. Finally, one last word to mention that, due to space limitations, this paper did not include any results concerning a line of research newly open by the authors and aimed at acquiring fresh knowledge on the mechanics underlying the elastic post-buckling behaviour of columns and beams affected by coupling phenomena involving distortional buckling. It employs a recently developed (enhanced) geometrically non-linear GBT formulation [75] and, up to now, results dealing with the post-buckling behaviour of lipped channel columns and beams undergoing L-D and D-G interaction were already reported [76–79] − in fact, the results concerning beams affected by L-D interaction are published in this journal special issue. Acknowledgments The third author gratefully acknowledges the financial support of FCT (Fundação para a Ciência e a Tecnologia − Portugal), through the 30

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