global interaction

Computers and Structures 89 (2011) 422–434

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Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

Post-buckling behaviour and strength of cold-formed steel lipped channel columns experiencing distortional/global interaction Pedro B. Dinis, Dinar Camotim ⇑ Department of Civil Engineering and Architecture, ICIST/IST, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

a r t i c l e

i n f o

Article history: Received 18 November 2009 Accepted 23 November 2010 Available online 23 December 2010 Keywords: Cold-formed steel lipped channel columns Shell finite element analysis Distortional and global buckling Distortional/global mode interaction Elastic and elastic-plastic post-buckling Column strength

a b s t r a c t This paper reports the results of a numerical investigation concerning the elastic and elastic–plastic postbuckling behaviour of cold-formed steel lipped channel columns affected by distortional/global (flexural– torsional) buckling mode interaction. The results presented and discussed were obtained by means of analyses performed using the finite element code ABAQUS and adopting column discretisations into fine 4-node isoparametric shell element meshes. The columns analysed (i) are simply supported (locally/globally pinned end sections that may warp freely), (ii) have cross-section dimensions and lengths that ensure equal distortional and global (flexural–torsional) critical buckling loads, thus maximising the distortional/global mode interaction effects, and (iii) contain critical-mode initial geometrical imperfections exhibiting different configurations, all corresponding to linear combinations of the two ‘‘competing’’ critical buckling modes. After briefly addressing the lipped channel column ‘‘pure’’ distortional and global post-buckling behaviours, one presents and discusses in great detail a fair number of numerical results concerning the post-buckling behaviour and strength of similar columns experiencing strong distortional/global mode interaction effects. These results consist of (i) elastic (mostly) and elastic–plastic non-linear equilibrium paths, (ii) curves or figures providing the evolution of the deformed configurations of several columns (expressed as linear combination of their distortional and global components) and, for the elastic–plastic columns, (iii) figures enabling a clear visualisation of (iii1) the location and growth of the plastic strains and (iii2) the characteristics of the failure mechanisms more often detected in the course of this research work. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Most cold-formed steel members display very slender thinwalled open cross-sections, a feature making them highly susceptible to several instability phenomena, namely local, distortional and global (flexural or flexural–torsional) buckling – see Fig. 1(a)–(d). Moreover, depending on the member length and cross-section shape/dimensions, any of these buckling modes can be critical. However, since several commonly used cold-formed steel member geometries may lead to rather similar distortional and global buckling stresses, the corresponding post-buckling behaviour (elastic or elastic–plastic), ultimate strength and failure mechanism are likely to be strongly affected by the interaction between these two buckling modes. It has been well known for quite a long time that cold-formed steel members exhibit stable local and global elastic post-buckling behaviours with clearly different post-critical strength reserves: rather high in the first case and quite low in the second. On the ⇑ Corresponding author. Tel.: +351 21 8418403; fax: +351 21 8497650. E-mail address: [email protected] (D. Camotim). 0045-7949/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2010.11.015

other hand, fairly recent studies have shown that (i) the distortional post-buckling behaviour fits somewhere in the middle of the two previous ones (in kinematic and strength terms) and (ii) exhibits a non-negligible asymmetry with respect to the sense of the flange-stiffener motion (outward or inward) – e.g., see the works of Kwon and Hancock [1], Prola and Camotim [2], Camotim and Silvestre [3] or Silvestre and Camotim [4]. Concerning the mode interaction phenomena that may affect the column post-buckling behaviour and strength, those stemming from the nearly simultaneous occurrence of local and global buckling are, by far, the better understood – this is attested by the fact that their effects are already taken into account by virtually all current hot-rolled and cold-formed steel design codes, either through the well-known ‘‘plate effective width’’ concept (e.g., [5]) or by means of the much more recent (but increasingly popular) ‘‘Direct Strength Method’’ (e.g., [6,7]). On the other hand, the influence of local/distortional mode interaction effects on the post-buckling behaviour and strength of lipped channel columns has attracted the attention of several researchers in the recent past (e.g., [8– 13]) – it is worth noting that some of the investigations carried out have already led to the development and calibration of novel

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Fig. 1. Lipped-channel column (a) local, (b) distortional, (c) flexural–torsional and (d) flexural buckling mode (cross-section) shapes.

t bw bf

(a)

tred

t E = 210 GPa ν = 0.3

bw.eff

fy = 355 MPa bf.eff

bs

ceff

bw

bf

bs

t

bw.eff

bf.eff

ceff

t

tred

(mm) 150

(mm) 110

(mm) 17.5

(mm) 2.4

(mm) 92.6

(mm) 104.1

(mm) 16.3

(mm) 2.4

(mm) 1.29

(b)

Aeff (mm2 ) 648.4

Fig. 2. Lipped channel column (a) cross-section dimensions and elastic constants, and (b) effective cross-section geometry and dimensions (according to EC3 and for fy = 355 MPa).

applications (design curves) of the Direct Strength Method (e.g., [9,12–14]). However, there are very few studies addressing the influence of the distortional/global buckling mode interaction on the postbuckling behaviour and ultimate strength of cold-formed steel columns (or any other members, for that matter). Indeed, besides a preliminary version of the current paper [15], the authors are only aware of a very recent publication that reports an experimental study on fixed-ended cold-formed stainless steel lipped channel columns [16]. Nevertheless, one should mention that (i) Eurocode 3 (Part 1–3) includes a procedure to jointly account for global, local (through an ‘‘effective width’’ approach) and distortional (through a ‘‘reduced thickness’’ approach) buckling, mainly based on work done by Thomasson [17] and Höglund [18], and that (ii) the North-American and Australian/New Zealander Specifications adopt an ‘‘effective width’’ approach to handle distortional/global interactive buckling, based on experimental work carried out by Desmond et al. [19]. However, none of the above studies addresses explicitly the mechanics of the mode interaction phenomenon dealt with in this paper – this is probably the reason why those design approaches are extremely conservative. Therefore, the aim of this work is to present and discuss a set of numerical results concerning the (i) post-buckling behaviour (elastic and elastic–plastic), (ii) ultimate strength and (iii) failure mode nature of cold-formed steel lipped channel simply supported columns affected by distortional/global (flexural–torsional) mode interaction. In order to enable a thorough assessment of all possible mode interaction effects, one analyses columns with (i) the cross-section dimensions and material properties given in Fig. 2(a), leading to a distortional buckling load meaningfully (about 20%) lower than its local counterpart, thus ensuring that local/distortional interaction effects are not relevant, and (ii) a length selected to guarantee the coincidence between the distortional (D – multiple half-waves) and global (G – single half-wave) buckling loads, thus maximising the distortional/global interaction effects.1 In order to provide an idea of how the Eurocode 3 design approach handles a column with this cross-section, Fig. 2(b) shows the associ1 Moreover, the selection of the column cross-section dimensions, obtained through ‘‘trial-and-error’’ buckling analyses, also satisfies two additional conditions: (i) competing (‘‘pure’’) buckling modes with odd half-wave numbers, so that the maximum deformation occurs at mid-span (although this feature is by no means essential, it renders the presentation of the results much easier) and (ii) no higherorder distortional buckling mode ‘‘close’’ to the distortional and global modes under consideration – this last condition was not easy to enforce (for the cross-section dimensions chosen, there is a 12% buckling load gap).

ated ‘‘effective area’’, which has been calculated for a yield stress fy = 355 MPa and must be considered in the safety checking against global (flexural–torsional) buckling – it corresponds to a 33% cross-section area reduction (Aeff/A = 0.67). A fairly large number of columns are analysed and they only differ in the initial geometrical imperfection configuration. The various configurations consist of linear combinations of the competing distortional and global buckling mode shapes with amplitudes (mid-span flange-lip corner vertical displacements) of (i) 10% of the wall thickness t (distortional mode) and (ii) L/1000 (global mode), values that are often adopted for cold-formed steel members and fall below the typical allowable geometrical tolerances, namely (i) b/500, prescribed by ECCS [20] for plates, and (ii) L/750, global tolerance recently stipulated in Europe. Although investigating the influence of the initial imperfection amplitudes on the column post-buckling behaviour and strength is outside of the aim and scope of this work, a very limited imperfection-sensitivity study is included in the paper – for further insight on this topic, the interested reader is referred to investigations carried out (i) by Schafer and Peköz [21] and Dubina and Ungureanu [22], in the context of cold-formed steel members, and (ii) by Maiorana et al. [23], concerning steel girder webs. All numerical results presented were yielded by finite element analyses carried out in the code ABAQUS [24] that (i) adopt member discretisations into fine 4-node isoparametric shell element meshes (preliminary convergence/ accuracy studies showed that it suffices to discretise the cross-section mid-line into 24 finite elements – 10 in the web, 6 in each flange and 1 in each stiffener – Fig. 4(b) illustrates the meshes adopted, which correspond to an element width approximately equal to 15 mm and a length/width ratio roughly equal to 1), (ii) model the simply supported conditions by imposing null transverse displacements at all end section nodes, as illustrated in Fig. 3(a) (note also that, to preclude a spurious longitudinal rigidbody motion, the axial displacement was also prevented at the mid-span mid-web node) and (iii) simulate the axial compression loading through compressive forces p, uniformly distributed along both column end-section mid-lines, as depicted in Fig. 3(b) (the load parameter is P = pA/t, where A and t are the column cross-section area and wall thickness). Detailed accounts of all the relevant modelling issues can be found in previous works by Dinis et al. [11] and Dinis and Camotim [25]. Column buckling analyses are performed at the outset, in order (i) to select the column length that maximised the D/G interaction, and also (ii) to obtain the associated buckling mode shapes, required to define the initial geometrical imperfections. Next, one addresses

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p=σt u1=u2=0 p=σt G

x3

x2

x1

(a)

(b) u1=u2=0

Fig. 3. Lipped channel column (a) end-section support conditions and (b) loading procedure adopted in the shell finite element simulations.

Pb (kN)

400

ABAQUS

300

CUFSM

DM

200

100 LD=76cm

L(cm)

LD/G=222cm

0 6 10

100

1000

(a)

GM

(b)

Fig. 4. (a) Column buckling curves and (b) distortional and global (flexural–torsional) buckling mode shapes of the column with LD/G = 222 cm.

the pure distortional and global elastic post-buckling behaviours, which are deemed not affected by D/G interaction. Finally, the numerical results concerning the column post-buckling behaviour and ultimate strength under D/G interaction are presented and discussed. They comprise (i) several elastic (mostly) and elastic–plastic non-linear (post-buckling) equilibrium paths, (ii) curves and figures providing the evolution, along the elastic paths, of the deformed configurations of several columns (expressed as combinations of their distortional and global components) and, for the elastic–plastic columns, (iii) figures enabling a clear visualisation of (iii1) the location and growth of the plastic strains and (iii2) the characteristics of the failure mechanisms detected in the course of this research work – by analysing members with different yield stresses, it is also possible to assess how the D/G mode interaction effects vary with the yieldto-critical stress ratio. 2. Buckling behaviour – length selection The curves depicted in Fig. 4(a) provide the variation, with the column length L (logarithmic scale), of (i) the ABAQUS critical load Pcr and (ii) the single-wave buckling load Pb.1, provided by CUFSM [26] finite strip analyses. As for Fig. 4(b), it shows the ABAQUS distortional and flexural–torsional buckling mode shapes of the column with L = 222 cm. These buckling results prompt the following remarks: (i) The ABAQUS buckling curve exhibits three distinct zones, corresponding to (i1) 1–4 half-wave local buckling, (i2) 1–3 halfwave distortional buckling and (i3) single half-wave global buckling. (ii) The black and white dots identify the practically coincident minimum single half-wave distortional critical loads yielded by the CUFSM and ABAQUS analyses (Pcr.D = 203.7 kN), for LD = 76 cm.

(iii) As clearly shown in Fig. 4(a), the LD/G = 222 cm has virtually identical distortional and global critical loads (Pcr.D = 203.6 kN and Pcr.G = 203.9 kN), associated with three and single half-wave buckling modes, respectively (see Fig. 4(b)). Obviously, the post-buckling behaviour and ultimate strength of such column will be highly affected by distortional/global mode interaction.

3. Distortional and global post-buckling behaviours Numerical results concerning the pure distortional and global column post-buckling behaviours (i.e., without D/G mode interaction) are first presented. Fourteen columns with seven lengths are analysed: (i) LD = 76 cm (Pcr = 203.6 kN), (ii) LG.1 = 300 cm (Pcr = 119.9 kN), (iii) LG.2 = 400 cm (Pcr = 72.0 kN), (iv) LG.3 = 450 cm (Pcr = 58.8 kN), (v) LG.4 = 500 cm (Pcr = 49.3 kN), (vi) LG.5 = 600 cm (Pcr = 26.6 kN), and (vii) LG.6 = 900 cm (Pcr = 20.9 kN) – note that columns LG.1 and LG.6 have critical loads equal to, respectively, 59% and 10% of that associated with D/G interaction: LD/G = 222 cm (Pcr.D = 203.6 kN). For each length, the analyses involve columns containing either positive or negative critical-mode geometrical imperfections, i.e., (i) LD columns having distortional imperfections of magnitude ±10% of the wall thickness t (positive and negative mean mid-span inward and outward flange-lip motions, respectively), and (ii) LG columns with global imperfections having midspan web chord rotations equal to ±0.016 rad2 (positive and negative mean clockwise and counter-clockwise rotations, respectively). Fig. 5(a) and Fig. 5(b) show the upper portions (P/Pcr > 0.6) of the post-buckling equilibrium paths (i) P/Pcr vs. v/t (v is the midspan top flange-lip corner vertical displacement), for the two LD 2 Note that, in the LG.1 columns, this magnitude corresponds to a global initial imperfection associated with mid-span flange-lip corner vertical displacements equal to L/1000 (i.e., 3 mm).

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LD Columns P / Pcr 1.2 v=−4.0 t

v0=±0.1t

0.8

0.6 0

0.6 10

20

0.6

LG.2

β >0

β<0

v>0

10

LG.2 LG.1

LG.1

v<0

LG.1

1

LG.2

0.8

20

β=+0.03rad

1.2

1

v=+3.8 t ⏐v⏐/ t

β0=±0.016rad P / Pcr LG Columns

0.3

(a)

0.3

0

⏐β⏐(rad)

0.6

(b)

Fig. 5. Curves providing the column (a) distortional (P/Pcr vs. v/t) and (b) global (P/Pcr vs. b) elastic post-buckling behaviours.

Pb (kN ) 300 ABAQUS

L(cm)

222

300

900

2

4

6

GBTUL − 6 modes

200

(nw =1)

GBT Buckling Mode (nw=1)

+

100

LD/G= 222cm

0 1000

200

(a) 1.0

GBT Deformation Mode

pi 6

0.5

4 2

0

(b)

L (cm)

200

1000

L (cm)

(c)

Fig. 6. GBT-based column buckling: (a) Pb vs. L curves (L > 100 cm), (b) modal participation diagrams (nw = 1) and (c) in-plane shapes of 3 buckling (L = 222, 300, 900 cm) and deformation modes.

columns, and (ii) P/Pcr vs. b (b is the mid-span web chord rotation), for the twelve LG columns – also shown are a few deformed configurations of column mid-span cross-sections at advanced postbuckling stages. These post-buckling results lead to the following comments: (i) The LD and LG.4–LG.6 columns exhibit the expected stable distortional and global post-buckling behaviours (e.g., [27]): while the LD columns exhibit a fair amount of post-critical strength reserve and a clearly visible asymmetry (the inward column is a bit stiffer), the LG.4–LG.6 ones display little postcritical strength (the LG.6 columns are a little stiffer) and perfectly symmetric equilibrium paths. (ii) However, the LG.1–LG.3 columns exhibit an unexpected unstable symmetric post-buckling behaviour – the equilibrium path limit points occur for 0.87 6 P/Pcr 6 0.94 and the peak load takes place for progressively higher b values as the column length increases (LG.1 ? LG.3). (iii) In order to try to understand why the LG columns exhibit two different post-buckling behaviours, it was decided to perform buckling analyses of those columns using Generalised Beam Theory (GBT – e.g., [28,29]), hoping that its modal features would shed new light on this issue – indeed, this was the case. The curves shown in Fig. 6(a) provide the variation of the buckling load Pb with the column length L (L P 222 cm): (iii1) one yielded by ABAQUS analyses and already shown in Fig. 4(a), and (iii2) the other obtained from GBT analyses carried out in GBTUL [30,31] and including 6 single half-wave deformation modes (4 global and 2 distortional) – dashed curve. As for Fig. 6(b), it displays the GBT-

based column modal participation diagram (for single halfwave buckling), providing the contributions of each GBT deformation mode to the column buckling modes.3 Finally, Fig. 6(c) shows the buckling mode shapes yielded by the GBT analyses for columns with lengths L = 222, 300, 900 cm, as well as the in-plane shapes of the 3 deformation modes that participate in them. The following conclusions can be drawn from these GBT buckling results: (iii.1) The GBT curve descending branch involves, in fact, two distinct buckling modes: (iii1) distortional–flexural–torsional (2 + 4 + 6), for 222 < L < 700 cm, and (iii2) flexural–torsional (2 + 4), for 700 6 L < 2000cm – note that the participation of the anti-symmetric distortional mode 6 progressively fades as the column length grows, until it vanishes for L  700 cm. (iii.2) Since the six LG column lengths are located inside the two intervals identified in the previous item, their buckling modes have different natures: while (iii1) the LG.6 column buckling mode is flexural–torsional (participation of 70% and 30% from modes 2 and 4, respectively), (iii2) the LG.1–LG.5 columns buckle in distortional–flexural–torsional modes with varying participations from modes 2, 4 and 6, namely 44%, 31% and 25% (LG.1), 55%, 36% and 9% (LG.2), 58%, 36% and 6% (LG.3), 60%, 37% and 3% (LG.4), and 63%, 36% and 1% (LG.5). 3 The participation of a given deformation mode in a column buckling mode is obtained on the basis of the corresponding mid-span cross-section deformed configurations (e.g., [28]).

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(iii.3) The level of participation of mode 6 in the column critical buckling mode provides the explanation for the different post-buckling behaviours exhibited by the LG.1–LG.6 columns. Indeed, its presence (iii1) reduces the column post-critical strength (see the column sequence LG.6–LG.1) and (iii2) is responsible for the surprising unstable post-buckling behaviour of the LG.1–LG.3 columns (note that the participation of mode 6 is higher than 5% for these columns) – column LG.4 may be viewed as the ‘‘transition’’ between the stable and unstable post-buckling behaviours (see Fig. 5). (iii.4) At this stage, it worth noting that the contribution of mode 6 to the column critical buckling mode would remain virtually undetected in the absence of the GBT analysis – this statement can be clearly attested by looking at the member and cross-section deformed configurations shown in Figs. 4(b) and 6(c), obtained by means of ABAQUS shell finite element and CUFSM finite strip analyses.4 (iii.5) Since the content of item (iii.1) may have considerable implications in the design of cold-formed steel columns affected by D/G coupling, it was decided to investigate this matter further, by analysing lipped channel columns with other cross-section geometries and also susceptible to this interaction phenomenon. The results obtained confirm both (iii1) the participation of mode 6 in the so-called ‘‘global’’ column buckling modes and (iii2) the unstable post-buckling behaviour exhibited by the columns whose critical buckling mode has a visible participation of this deformation mode. This statement is illustrated by the buckling analysis of columns having bw = 110 mm, bf = 90 mm, bs = 10 mm and t = 1 mm (C1 column), and bw = 100 mm, bf = 70 mm, bs = 10 mm and t = 1.5 mm (C2 column). These columns (iii1) buckle in distortional–flexural–torsional modes for 272 < L < 700 cm (C1 column) and 154 < L < 400 cm (C2 column), (iii2) have a mode 6 contribution to the LD/G column critical buckling mode of 29% (C1 column) and 38% (C2 column), and (iii3) exhibit an unstable post-buckling behaviour for 272 < L < 500 cm (C1 column) and 154 < L < 300 cm (C2 column), i.e., roughly for columns with LD/G < L < 2LD/G – note that, for this length range, the contribution of mode 6 to the critical buckling mode is always higher then 5% (as is the case for the columns with the cross-section geometry given in Fig. 2(a)). (iii.6) In the GBT nomenclature, the LG.1–LG.5 column critical buckling mode is designated as ‘‘mixed’’, since it combines (‘‘mixes’’) deformation modes of different natures (2 + 4 and 6 – global and distortional). Note that the existence of a ‘‘mixed’’ buckling mode does not imply the occurrence of a buckling mode interaction phenomenon – the latter corresponds to the simultaneous (or nearly so) occurrence of two or more critical buckling modes (regardless of whether they are ‘‘mixed’’ or not). (iii.7) In the L = 222 cm column, two critical buckling modes occur simultaneously: (iii1) a three half-wave pure symmetric distortional mode and (iii2) a single half-wave ‘‘mixed’’ distortional–flexural–torsional 4 The sole ‘‘sign’’ of the presence of the anti-symmetric distortional mode 6 is a practically imperceptible web double-curvature bending (see Fig. 6(c)) that can only be detected if one knows about it beforehand.

mode, combining modes 2 (6%), 4 (19%) and 6 (55%) (i.e., mode 6 is predominant) – the latter was termed ‘‘global’’ earlier, a designation that will be retained for simplicity. Therefore, the D/G mode interaction phenomenon investigated in this work really involves the two competing critical buckling modes just described. 4. Post-buckling behaviour under D/G mode interaction One now investigates the elastic and elastic–plastic post-buckling behaviours of simply supported columns with L = 222 cm (Pcr = 203.6 kN), which are strongly affected by the interaction between nearly coincident distortional (three half-waves) and ‘‘global’’ (single half-wave) buckling modes.5 4.1. Initial geometrical imperfections A very important issue in mode interaction investigations is to assess how the initial geometrical imperfection shape influences the post-buckling behaviour and strength of the structural system under scrutiny. Indeed, the commonly used approach of including critical-mode imperfections ceases to be well defined, due to the presence of two competing buckling modes that may be combined arbitrarily. Thus, in order to obtain column equilibrium paths that (i) cover the whole D/G critical-mode imperfection shape range and (ii) can be meaningfully compared, one adopts the following approach, which accounts for the fact that the two competing (‘‘pure’’) buckling modes exhibit odd half-wave numbers: (i) To determine the ‘‘pure’’ critical buckling mode shapes, normalised to exhibit unit mid-span flange-lip corner vertical displacements: (i1) a distortional mode with vD = 1 mm, associated with a mid-web flexural displacement (measured with respect to the web chord) of wD = 0.265 mm, and (i2) a ‘‘global’’ one also with vG = 1 mm, implying a mid-span web chord rotation equal to bG = 0.005 rad. (ii) To scale down the pure modes, thus leading to the following magnitudes for the distortional and ‘‘global’’ imperfections: vD.0 = 0.1t and vG.0 = L/1000 (in this case, 0.1t = 0.24 mm, L/ 1000 = 2.22 mm). At this point, it is worth mentioning that, in order to assess the influence of the initial imperfection amplitudes on the column elastic post-buckling behaviour under D/G interaction, a few results were obtained with other imperfection magnitudes. (iii) A given initial geometrical imperfection shape is obtained as a linear combination of these scaled buckling modes shapes, with coefficients CD.0 and CG.0 satisfying the condition (CD.0)2 + (CG.0)2 = 1. A better visualisation and ‘‘feel’’ of the initial imperfection shape is obtained by considering the unit radius circle drawn in the CD.0–CG.0 plane that is shown in Fig. 7(a): each ‘‘acceptable’’ imperfection shape lies on this circle and corresponds to an angle h, measured counterclockwise from the horizontal (CD.0) axis and defining a CG.0/CD.0 ratio (CD.0 = cos h and DG.0 = sin h). Fig. 7(b) provides the pure D and G initial imperfection shapes (h = 0°, 90°, 180° and 270°) – note that (iii1) h = 0° and h = 180° correspond to inward and outward flange-lip motions, and that (iii2) h = 90° and h = 270° are associated with clockwise and counterclockwise cross-section rotations. In this work, one considers (mostly) initial imperfections corresponding to 15° intervals. 5 As just seen, this interaction really involves a three half-wave symmetric distortional buckling mode and a ‘‘mixed’’ buckling mode combining anti-symmetric distortion, major axis flexure and torsion (distortional–flexural–torsional mode).

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vG.0=L/1000

β G.0 90°

vD.0 =0.1t

CG.0 1

180°

−1

270°

180°

0°

0°

θ 1

vD.0 =− 0.1t

− wD.0

r =1

0

−1

wD.0

90°

CD.0

−β G.0

θ = 0º

θ = 180º

θ = 90º

θ = 270º

vG.0=− L/1000 270°

(a)

(b)

Fig. 7. (a) Initial imperfection representation in the CD.0–CG.0 plane and (b) initial imperfection shapes associated with h = 0°, 90°, 180° and 270°.

θ=180

0

θ=179

θ=179 θ=165

θ 165 ; 90

0≤θ≤180

P / Pcr 1

θ=180

θ 27 45 90

27 … 179 w<0

0.6

0.6

0

1 … 26

w>0 ⏐w⏐/ t

1

2

θ=0;180

1 0.8

0.4

(a)

0≤θ≤180

P / Pcr θ=0

0.8 θ θ

1

θ 26 15 1

1 15 26

θ= 27 ; 45

2

θ=0;180

θ=0

1 … 26

θ 27 … 179

(b) 0.4 0

θ=27−179

θ

(c1) θ=1−26

β >0

0.2

β (rad) 0.4

(c2)

Fig. 8. (a) P/Pcr vs. w/t and (b) P/Pcr vs. b equilibrium paths (0° 6 h 6 180° columns), and (c) limit deformed shapes of the (c1) h = 27–179° and (c2) h = 1–26° columns (curve descending branches).

4.2. Elastic mode interaction Initially, one presents post-buckling results concerning columns with 13 initial imperfection shapes corresponding to 0° 6 h 6 180° and separated by 15° intervals6 – moreover, in order to clarify some behavioural aspects, the h = 1°, 26°, 27°, 179°, 181°, 359° columns were also analysed. Fig. 8(a) and Fig. 8(b) shows the upper parts of the most representative column post-buckling equilibrium paths (i) P/Pcr vs. w/t, where w is the mid-web flexural displacement at mid-span (measured with respect to the web chord), and (ii) P/Pcr vs. b, where b is the mid-span web chord rigid-body rotation. As for Fig. 5(c), it shows the limit deformed configurations of the h = 27–179° and h = 1–26° columns, which correspond to the curve descending branches (advanced post-buckling stages). In order to shed some light on issues raised by the close scrutiny of the curves shown in Fig. 8(a) and Fig. 8(b), additional post-buckling results are presented in Figs. 9 to 14 – they consist of (i) equilibrium paths previously displayed in Fig. 8(a), complemented by illuminating column mid-span cross-section deformed configurations, (ii) ultimate load values and (iii) figures providing the mode coupling evolution along those equilibrium paths. The close observation of the postbuckling results presented in Figs. 8(a)–(c) leads to the following remarks (and also further results):

6 Except for the 181° and 349° columns, addressed next, no post-buckling results concerning columns with initial imperfection shapes defined by 180° < h < 360° are presented - since the column post-buckling behaviour is symmetric with respect to the deformed configuration global component sign, regardless of whether there is mode interaction or not (see Fig. 5(b)), the 0° 6 h 6 180° column results provide all the necessary information.

(i) All column equilibrium paths (P/Pcr vs. w/t and P/Pcr vs. b) exhibit well-defined limit points, which occur almost always for quite small w/t and b values. The exceptions are the h = 0°, 180° column equilibrium paths (pure distortional initial imperfections), which have limit points associated with (i1) slightly larger w/t values and (i2) null b values – they correspond to a singular post-buckling behaviour that will be addressed further ahead. (ii) The comparison between the equilibrium paths P/Pcr vs. b of the (ii1) h = 90° column (pure ‘‘global’’ initial imperfections with b0 = 0.005 rad), shown in Fig. 8(a), and (ii2) LG.1 column (b0 = 0.016 rad), depicted in Fig. 5(b), shows clearly the adverse effect (strength erosion) due to the D/G mode interaction. Indeed, in spite of the considerably smaller initial imperfection magnitude (about one third), the limit point of the h = 90° column equilibrium path corresponds to a lower applied load level (P/Pcr = 0.81 vs. P/Pcr = 0.87). (iii) The equilibrium paths displayed in Fig. 8(a) can be grouped in three categories, each one corresponding to a different post-buckling behaviour – they are identified next and addressed separately in the sequel: (iii.1) h = 0°, 180° equilibrium paths, corresponding to pure distortional imperfections. As mentioned before, these two columns exhibit a singular post-buckling behaviour, characterised by the fact that their cross-sections exhibit no rigid-body rotations (i.e., b = 0). (iii.2) 1° 6 h 6 26° equilibrium paths, associated with predominantly distortional imperfections with outward outer half-waves. (iii.3) 27° 6 h 6 179° equilibrium paths, corresponding to all the remaining imperfection shapes.

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CG

27 ≤θ ≤ 179

40 0.45

θ =90

20

1

-60

-40

-20

θ =30

θ =135

0

θ =27

θ =165

CD -80

θ =60 θ =45

D

θ =179

20

Fig. 9. Mode coupling ratio CG/CD evolution along the equilibrium paths (27° 6 h 6 179° columns).

90

Pu /Pcr

1

120

0.8 27

θ (º)

30

120 150 180

0.6 0

60

90

75

105

135

θ (º)

Pu / Pcr

θ (º)

Pu / Pcr

27 30 45 60 75 90

0.928 0.909 0.864 0.836 0.817 0.807

105 120 135 150 165 179

0.803 0.807 0.819 0.839 0.870 0.920

Fig. 10. Variation of Pu/Pcr with the initial imperfection shape (27° 6 h 6 179° columns).

P / Pcr θ=1,359 1 II III 2 3 I 1 0.8

θ=179,181

IV

4

(a) 1

0

1

2

3

4

θ =359

1

2

3

4

θ =179

I

II

III

IV

θ =181

I

II

III

IV

⏐w⏐/ t

0.6 2

θ =1

1

2

P / Pcr 1 0.8

θ =359

θ=1

0.6 θ=179

θ =181

(b)

0.4

0.5 0.25

0

0.25

β (rad) 0.5

(c)

Fig. 11. (a) P/Pcr vs. w/t and (b) P/Pcr vs. b equilibrium paths and (c) mid-span cross-section deformed configuration evolution (h = 1°, 359° and h = 179°, 181° columns).

1 ≤θ ≤ 26

CG 40

θ (º) 20

θ =26 θ =15 θ =1

CD

(a)

1 15 26

Pu / Pcr 0.944 0.937 0.928

span web bending – since the 27°6h6179° category includes the vast majority of equilibrium paths, thus providing a more meaningful characterisation of the D/G interaction effects, it will be addressed first. Then, one tackles the 1°6h626° category, which constitutes another particular case.

(b) 0

20

40

60

Fig. 12. (a) Evolution of the mode coupling ratio CG/CD and (b) variation of Pu/Pcr with the initial imperfection shape (1° 6 h 6 26° columns).

At this stage, it is worth noting that the first two equilibrium paths concern a rather unexpected D/G interactive behaviour, which will be the last to be dealt with. Moreover, the equilibrium paths belonging to the last two categories merge into common post-buckling curves associated with (iii1) clockwise mid-span web chord rotations and (iii2) either inward (27°6h6179°) or outward (1°6h626°) mid-

4.2.1. Columns with 27°6h6179° initial geometrical imperfections These equilibrium paths correspond to initial geometrical imperfections whose global components involve always clockwise cross-section rigid-body rotations (in view of the global post-buckling symmetry, it is not necessary to consider counter-clockwise rotations)7 – such component is either (i) the only one (h = 90°) or (ii) combined with a distortional one exhibiting inward 7 Recall that the GBT buckling analysis showed that these ‘‘global’’ components combine (i) a clockwise torsion rotation, (ii) a downward major axis bending displacement and (iii) downward flange-lip motions associated with anti-symmetric distortion. Since they are all ‘‘linked’’ through the buckling mode shape, it suffices to mention only the torsion rotation b (the most ‘‘visible’’ one).

P.B. Dinis, D. Camotim / Computers and Structures 89 (2011) 422–434

429

Fig. 13. (a) P/Pcr vs. w/t equilibrium paths and (b) mid-span cross-section deformed configuration evolution (h = 26° and h = 27° columns).

(27°6h<90°) or outward (90°
which combines participations of (i1) a three halfwave distortional component with outward mid-span flange-lip motions and (i2) a single half-wave global component that are roughly equal to one and two thirds of the total deformed configuration (31% and 69%, to be precise) – this coupled buckling mode shape can be visualised in Fig. 8(c1).10 (ii) All equilibrium paths exhibit a limit point prior to merging into the ‘‘common curve’’ and one observes that, generally speaking, the column limit load decreases as the initial imperfection global component becomes more dominant – this can be confirmed by looking at the table included in Fig. 10, which provides the variation of the column ultimate load ratio Pu/Pcr with its initial imperfection shape (i.e., with h). However, one observes that the minimum ultimate load (Pu/Pcr = 0.803) occurs for the h = 105° column, i.e., the one combining 96.5% and 25.9% of the pure global and distortional imperfections11 – nevertheless, one must mention that the Pu/Pcr value remains practically constant for 90° 6 h 6 120° (0.5% between the minimum and maximum values – see curve detail in Fig. 10), i.e., analysing a column with a pure global imperfection (h = 90°) will certainly provide a rather accurate estimate of the minimum Pu/Pcr value – in other words, the pure global imperfection may be viewed, for practical purposes, as the most detrimental one, in the sense that it maximises the strength erosion due to the D/G interaction.12 (iii) Although it is evident that the initial imperfection global component plays a crucial role in the column post-buckling behaviour (note that all equilibrium paths exhibit limit points), it is also obvious that there must exist a plausible explanation for the qualitative and quantitative differences exhibited by the 90° < h 6 179° and 1° 6 h < 90° column equilibrium paths, clearly visible in Fig. 8(a) and Fig. 8(b) – (iii1) while the former tend to the common curve in a ‘‘regular’’ fashion (the amplitudes of both the global and distortional components grow monotonically), (iii2) the latter either tend to that same common curve ‘‘irregularly’’ (27° 6 h < 90° – occurrence of distortional component

10 The mid-span cross-section deformed configuration associated with this coupled buckling mode shape can also be viewed in Figs. 11(c) (h = 179° – IV) and 13(b) (h = 27° – IV). 11 It is interesting to notice that 105° is very close to the average between 27° and 179°. 12 Even if this assessment concerns only the 27° 6 h 6 179° interval, it is valid for any h – as shown in Fig. 8(a) and Fig. 8(b) and discussed ahead, the h = 0°, 180° and 1° 6 h 6 26° column ultimate loads are higher.

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reversals) or do not tend to it at all (1° 6 h < 27° – they tend to another common curve, which will be addressed later). (iv) All the distinct post-buckling behaviours described in the previous item stem from the influence exerted by the outer half-wave flange-lip motions (distortional feature) on the major axis flexure (global feature). Indeed, (iv1) while inward flange-lip motions reduce the cross-section major moment of inertia, thus facilitating the occurrence of the corresponding flexure, (iv2) the outward flange-lip motions have precisely the opposite effect. This assessment justifies (or is confirmed by) the following facts: (iv.1) The ‘‘regularity’’ of the 90° < h6179° column equilibrium paths, due to the converging effects of the initial imperfection global and distortional components. (iv.2) The distortional component amplitude reversals occurring in the 27° < h 6 90° column equilibrium paths, due to the opposing effects of the initial imperfection global and distortional components, with the former prevailing over the latter. (iv.3) The ‘‘peculiarity’’ of the 1° < h 6 26° column equilibrium paths, due to the opposing effects of the initial imperfection global and distortional components, with the latter prevailing over the former. (iv.4) The ‘‘equality/symmetry’’ between the equilibrium paths and mid-span cross-section deformed configurations concerning the column pairs h = 179°, 181° and h = 1°, 359°, clearly shown in Figs. 11(a)–(c) – for clarity, the first three cross-section deformed configurations of each column are amplified 20, 5 and 3 times, respectively. It is worth noting that the initial imperfections in each pair have the same distortional component and opposite-sign global components. 4.2.2. Columns with 1° 6 h 6 26° initial geometrical imperfections These equilibrium paths correspond to initial imperfections whose global components involve always clockwise cross-section rigid-body rotations, combined with a distortional one exhibiting inward (1° 6 h 6 26°) mid-span flange-lip motions. The joint observation of all these equilibrium paths leads to the following remarks: (i) Like those dealt with previously (27°6h6179°), they all merge into a common curve – this merging occurs only in their descending branches (after the limit points) and the common curve corresponds to the deformed configuration shown in Fig. 8(c2).13 However, the characteristics of this deformed configuration change continuously as post-buckling progresses, as attested by the equilibrium paths shown in Fig. 8(a) and Fig. 8(b) (particularly the latter) and also by the evolution of the mode coupling ratio CG/CD displayed in Fig. 12(a) – indeed, one notices that the amplitude of the deformed configuration distortional component starts decreasing at a growing rate along the P/Pcr vs. w/t equilibrium path descending branches (see Fig. 8(a)). This also implies a global component growth rate increase, clearly visible in the corresponding P/Pcr vs. b equilibrium paths (their common curve becomes almost horizontal – see Fig. 8(b)).14

13 The mid-span cross-section deformed configuration associated with this coupled buckling mode shape can also be viewed in Figs. 11(c) (h = 1° – 4) and 13(b) (h = 26° – 4). 14 Since the outer half-wave outward flange-lip motions decrease, the occurrence of major axis flexure becomes ‘‘easier’’, which explains the global component (i.e., b) growth rate increase.

Thus, no coupled buckling mode shape can be inferred from (or linked to) the common curve concerning these equilibrium paths. (ii) Whenever the initial outer half-wave outward flange-lip motions are large enough (i.e., for h < 27°), their post-buckling growth (amplification) ‘‘retards’’ the dominant appearance of the (destabilising) deformed configuration global component. This leads to slightly higher limit loads (see table in Fig. 12(b) and compare its values with those appearing in Fig. 10), that tend to occur for visibly larger w values (e.g., compare the limit point locations of the h = 1° and h = 179° column P/Pcr vs. w/t equilibrium paths). (iii) The amplitude reversals of the column deformed configuration distortional components cease abruptly, as shown in Fig. 8(a). Indeed, no further equilibrium states could be detected, most likely because the column outer half-wave flanges-lips were about to ‘‘snap’’ from an outward position to an inward one – obviously, such dynamic behaviour could not be captured by the ABAQUS (static) geometrically non-linear analyses carried out. (iv) To enable a better grasp of the qualitative and quantitative differences between the two sets of column equilibrium paths dealt with (27° 6 h 6 179° and 1° 6 h 6 26°), Fig. 13 shows the h = 26°, 27° column (iv1) P/Pcr vs. w/t equilibrium paths and (iv2) mid-span cross-section deformed configuration evolution. Despite the tiny initial imperfection difference, the two column post-buckling behaviours are quite distinct – e.g., (iv1) the equilibrium path shapes and (iv2) the distortional components of the mid-span cross-section deformed configurations along the descending branches (the h = 26° and h = 27° column flange-lip assemblies move inward and outward, respectively). 4.2.3. Columns with h = 0°, 180° initial geometrical imperfections Finally, one addresses the equilibrium paths of the columns containing pure distortional initial imperfections, as they exhibit rather peculiar post-buckling behaviours, both characterised by (i) the highest limit loads (Pu/Pcr = 0.945, 0.961, for h = 0°, 180°, respectively), (ii) much less pronounced limit points (see Fig. 7(a)) and (iii) the total absence of torsional rotations (see Fig. 8(b)). This is rather surprising, since (i) there are no D/G mode interaction effects of the type found for all other initial imperfection shapes (no torsional rotations) but, at the same time, (ii) the two column equilibrium paths are not similar to the purely distortional ones shown in Fig. 5(a) (occurrence of limit points). Then, in order to understand the nature of the singular column post-buckling behaviour described above, it is convenient to look at Fig. 14(a) and Fig. 14(b), showing (i) the two column equilibrium paths, (ii) mid-span cross-section deformed configurations (amplified twice) at three equilibrium states and also (iii) the column overall deformed configurations at the descending branch equilibrium states. Indeed, it is possible to observe that: (i) The h = 0° and h = 180° column P/Pcr vs. w/t equilibrium paths are slightly different – in particular, the former column exhibits a lower limit load. (ii) While the first mid-span cross-section deformed configurations (1–I) are ‘‘purely distortional’’, the second and third ones (2–3 and II–III) indicate the presence of minor axis flexure (web in tension) – this is clearly confirmed by the column overall deformed configurations shown in Fig. 14(b), which correspond to the equilibrium states 3–III. (iii) Therefore, the h = 0° and h = 180° columns are affected by a different type of distortional/global mode interaction phenomenon, which (iii1) involves only minor axis flexure and (iii2) is not caused by the closeness of two buckling loads.

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P / Pcr θ=0 1

θ=180 III II

I

1

2

3

θ =0

1

2

3

θ =180

I

II

III

0.8

⏐w⏐/ t

0.6 1

0

θ =180

θ =0

(b)

(a)

1

Fig. 14. (a) P/Pcr vs. w/t paths and mid-span cross-section deformed configuration evolution, and (b) post-peak deformed configurations (h = 0°, 180° columns).

IMP(D0–D5;G1) IMP(D0–D5;G2)

Pu /Pcr

P /Pcr 1

0.8

0.8

w<0

3

IMP(D1–D5;G0)

0.6

IMP(D0–D5;G3)

2

1

w>0 ⏐w⏐/ t

0.4 0

G3

θ=105

1

2

3

D5 D4 D3 D2 D1

0.2t 0.15t 0.125t 0.1t 0.075t

D0 (θ=90)

0

G2

G1

L/750 L/1000 L/1250 0.768 0.797 0.817 0.770 0.799 0.820 0.771 0.801 0.821 0.773 0.803 0.823 0.774 0.804 0.825 0.775

0.806

G0 (θ=0) 0 0.916 0.929 0.936 0.944 0.953

0.827

Fig. 15. Variation of (i) the P/Pcr vs. w/t paths and (ii) Pu/Pcr with the initial imperfection amplitudes, for h = 105°, 90° and 0° columns.

Indeed, this mode interaction stems from the occurrence of horizontal shifts (towards the web) of the column cross-section effective centroids, responsible for the development of minor axis flexure (the axial compression becomes gradually more eccentric) – these effective centroid shifts stem from the progressive ‘‘weakening’’ (axial stiffness drop) of the flange-lip assemblies, due to the presence of rather high longitudinal compressive normal stresses (e.g., [32]).15 (iv) The lower limit load exhibited by the h = 0° column is due to the fact that it is associated with two half-waves involving outward flange-lip motions, which develop higher compressive stresses than their inward counterparts (e.g., [2,4]).16 (v) This distortional/global mode interaction did not occur in the column single half-wave distortional post-buckling presented in Fig. 5(a) (the equilibrium paths exhibit no limit points) – this is due to the much shorter column length (about one third of that considered now), which corresponds to a much higher global (minor axis flexure) buckling load and, therefore, precludes the occurrence of interaction (the cross-section ‘‘weakening’’ is insufficient to trigger it).

4.2.4. Influence of the initial imperfection amplitudes In order to make it possible to assess the influence of the initial imperfection amplitudes on the column elastic post-buckling behaviour addressed above, Fig. 15 shows (i) the P/Pcr vs. w/t equilibrium paths and (ii) the variation of Pu/Pcr for columns with imperfection shapes corresponding to h = 105° (the most detrimental ones) and 15 different amplitudes. The initial imperfection magnitudes stem from the combination of (i) five distortional imperfection amplitudes (vD.0 = 0.075t, 0.1t, 0.125t, 0.15t, 0.2t – amplitudes D1 to D5) and (ii) three ‘‘global’’ ones (vG.0 = L/1250, 15 This type of distortional/global mode interaction is qualitatively similar to the well-know local-plate/global one, which is currently incorporated in virtually all steel design codes through the ‘‘plate effective width’’ concept. 16 This limit load difference only exists because the number of distortional halfwaves is odd, which brings about the different behaviours exhibited by the outward and inward ones. Moreover, this difference will obviously decrease as the (odd) halfwave number increases.

L/1000, L/750 – amplitudes G1 to G3). The equilibrium paths associated with each imperfection combination are identified by designations of the type IMP(Di; Gj) – note that the h = 105° column previously analysed corresponds to IMP(D2; G2). In order to provide a better feel of the column imperfection-sensitivity, Fig. 15 also includes equilibrium paths associated with pure distortional (h = 0°) and pure global (h = 90°) initial imperfections – they are identified by the designations G0 and D0, respectively. The observation of all these equilibrium paths and ultimate strength values leads to the following remarks: (i) Varying the initial imperfection distortional and global amplitudes does not change the column post-buckling behaviour qualitatively. Indeed, all equilibrium paths still (i1) exhibit elastic limit points, (i2) merge into one of two common descending branches (depending on whether the initial imperfection is pure distortional or has a global component). Moreover, the lower ultimate loads continue to belong to the h = 105° columns, even if the differences with respect to their h = 90° column counterparts remain minute (<0.3%). (ii) Quantitatively speaking, the comparison between the results concerning the h = 90° (pure global imperfections) and h = 0° (pure distortional imperfections) columns shows that the global imperfection amplitude has a larger impact on the (elastic) ultimate strength. Indeed, the ultimate load reductions associated with the imperfect amplitude increases from G1 to G3 and from D1 to D5 are equal to 7% and 4%, respectively. (iii) Concerning the h = 105° columns, whose initial imperfections are predominantly global (distortional and global components with 25.9% and 96.5% of their maximum amplitudes), their behaviours are naturally quite close to those exhibited by their h = 90° counterparts. This is confirmed by the fact that increasing the initial imperfection global amplitude from G1 to G3, while keeping vD.0 – 0 fixed, also causes an ultimate load drop of about 7%. On the other hand, varying the initial imperfection distortional amplitude from D1 to D5, for a fixed vG.0 – 0, has almost no effect on the ultimate load (it always drops by less than 1%). The

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Fig. 16. (a) P/Pcr vs. w/t elastic–plastic equilibrium paths of the h = 90°, 26° and 0° columns, and (b) plastic strain and deformed configuration evolution for fy/rcr  1.1.

previous assertions are illustrated in the zoomed portion of Fig. 15, showing three equilibrium path sets, each of them corresponding to a different vG.0 value and comprising 5 paths (one per vD.0 value) – the differences are much smaller within each set than between separate sets. 4.3. Elastic–plastic mode interaction In this section, a few results dealing with the elastic–plastic post-buckling behaviour of simply supported lipped channel columns experiencing D/G mode interaction are presented. These results concern columns (i) containing mostly the 13 initial imperfection shapes dealt with before (0° 6 h 6 180 – 15° intervals and critical mode amplitudes of vD.0 = 0.1t and vG.0 = L/1000) and (ii) exhibiting three yield-to-critical stress ratios, namely fy/ rcr  1.1, 1.7, 2.5, which correspond to yield stresses equal to fy = 235, 355, 520 MPa, respectively – recall that rcr = 209.5 MPa. For comparative purposes, some elastic results obtained earlier are presented again – they may be viewed as corresponding to an infinite yield stress (i.e., fy = fy/rcr = 1). Fig. 16(a1)–(a3) show the upper portions (P/Pcr > 0.6) of twelve equilibrium paths P/Pcr vs. w/t, describe the post-buckling behaviours of columns (i) containing the most significant initial imperfections, i.e., (i1) the pure global (h = 90° – the choice of this particular initial imperfection shape is due to the fact that the elastic column was found to exhibit one of the lowest limit loads17), (i2) 17 Recall that the h = 105° column has the lowest limit load, differing 0.5% from the h = 90° column one.

the h = 26° and (i3) the pure distortional (h = 0°) ones, and (ii) exhibiting different yield-to-critical stress ratios. As for Fig. 16(b1)–(b3), they concern the columns with fy/rcr  1.1 and display nine plastic strain diagrams, corresponding to equilibrium states located along their post-buckling paths (as indicated in Fig. 16(a1)–(a3)) and including the column collapse mechanisms. Finally, Table 1 provides the column ultimate load ratios (Pu/Pcr) associated with the h and fy combinations considered in this study. The observation of these results prompts the following remarks: (i) Out of the twelve column set analysed (i1) only those with fy/ rcr  1.1 exhibit (a minute) elastic–plastic strength reserve and (i2) those with fy/rcr  2.5 remain elastic up until the ultimate (limit) load is reached – moreover, in the columns with fy/rcr  1.7, the onset of yielding triggers the column failure. (ii) In all h = 27–179° columns, yielding starts at the bottom lip mid-span zone, as illustrated in Fig. 16(b1) (diagram I – h = 90° column). Collapse occurs almost immediately after and is caused by the full yielding of the mid-span bottom web-flange corner, leading to the formation of a ‘‘distortional plastic hinge’’ that precipitates the collapse of the mid-span bottom flange-lip assembly (see diagram II in Fig. 16(b1), which also shows a yield line crossing the whole mid-span cross-section bottom flange). Along the equilibrium path descending branch, yielding spreads progressively along the lower web and bottom flange-lip central regions, while all other column areas remain elastic (see diagram III in Fig. 16(b1)).

P.B. Dinis, D. Camotim / Computers and Structures 89 (2011) 422–434 Table 1 Variation of the ultimate load ratio Pu/Pcr with h and fy/rcr. h (°) 0 15 30 45 60 75 90

fy/rcr 1.1

1.7

2.5

1

0.850 0.826 0.807 0.797 0.764 0.740 0.721

0.931 0.921 0.912 0.863 0.831 0.811 0.797

0.945 0.937 0.909 0.864 0.836 0.817 0.807

0.945 0.937 0.909 0.864 0.836 0.817 0.807

h (°) 90 105 120 135 150 165 180

fy/rcr 1.1

1.7

2.5

1

0.721 0.711 0.716 0.726 0.754 0.792 0.854

0.797 0.792 0.797 0.807 0.831 0.864 0.940

0.807 0.803 0.807 0.819 0.839 0.870 0.960

0.807 0.803 0.807 0.819 0.839 0.870 0.961

(iii) In all h = 1–26° columns the plastic strain evolution is qualitatively quite similar to the one described in the previous item – see Fig. 16(b2) for the h = 26° column. There is one important difference, though: yielding starts and spreads around the column regions located near the outer half-wave crests, where the largest distortional deformations occur. (iv) In columns containing pure distortional imperfections (h = 0° or 180°), the plastic strain evolution is qualitatively different from those described in the previous two items. Plasticity first appears at both lip free ends in the vicinity of the outer distortional half-wave crests. Collapse follows shortly after and is due to the full yielding of both webflange corners (see diagram II in Fig. 16(b3)). (v) The ultimate load ratios given in Table 1 show that the variation of Pu/Pcr with h is qualitatively similar for all column sets. Indeed, one observes that (v1) the h = 105° column always exhibits the lowest value and that (v2) there is very little variation within the 90° 6 h 6 120° interval (the maximum and minimum Pu/Pcr are never more than 1.4% apart) – this confirms that, for practical purposes, one may view the pure global initial imperfection (h = 90°) as the most detrimental one.18 (vi) The strength erosion stemming from the distortional/global mode interaction effects is quite considerable. For the h = 90° column, the ultimate strength corresponds to 29% (fy = 235 MPa), 21% (fy = 355 MPa) and 20% (fy > 355 MPa) drops with respect to the critical buckling load – it is interesting to notice there is no benefit in having a yield stress much larger than fy = 355 MPa, since the column collapse is totally governed by elastic distortional/global interaction effects. (vii) The features addressed in the previous two items are bound to have far-reaching implications in the design of coldformed steel columns experiencing D/G interaction, mostly because the (uncoupled) distortional and global post-buckling behaviours are looked upon as stable (even if only marginally) – to the authors’ best knowledge, no adverse mode interaction effects have ever been reported.

5. Concluding remarks This work dealt with a numerical (shell finite element) investigation on the elastic and elastic–plastic post-buckling behaviour and strength of simply supported cold-formed steel lipped channel columns affected by distortional/global mode interaction. The analyses, performed in the code ABAQUS, involved columns containing initial imperfections with shapes obtained by combining differently the two competing distortional (three half-waves) and global (one half-wave) buckling modes with amplitudes equal to (i) 10% of the wall thickness t (distortional mode) and (ii) L/1000 (global mode). 18

It is worth recalling that a pure outward distortional initial imperfection was found to be the most detrimental in the context of simply supported lipped channel columns affected by local-plate/distortional mode interaction [11].

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Initially, one addressed the lipped channel column (i) buckling behaviour and (ii) uncoupled distortional and global post-buckling behaviours, a task that (i) made it possible to select the most appropriate column length (i.e., that maximising the D/G interaction effects) and (ii) ended up disclosing a few unexpected (and surprising) features. Then, one presented and discussed several numerical results concerning the elastic post-buckling behaviour of lipped channel columns with the selected length (i.e., experiencing strong D/G mode interaction effects) and containing criticalmode initial imperfections with various configurations and the same amplitude (linear combinations of the two competing buckling mode shapes). These results consisted of (i) non-linear equilibrium paths and (ii) figures providing the evolution, along those paths, of column and cross-section deformed configurations. Finally, the paper closed with a few post-buckling results concerning elastic–plastic columns with (i) the same geometry and initial imperfection shapes and (ii) three yield stress values. Besides the non-linear equilibrium paths and deformed configuration evolution, one addressed also (i) issues related to the onset and spread of plasticity, as well as (ii) the variation of the column ultimate strength with the initial imperfection shape and yield stress value. Among the various conclusions drawn from this investigation, the following ones deserve to be specially mentioned: (i) A GBT analysis revealed that a large portion of the column critical buckling curve descending branch corresponds to distortional–flexural–torsional buckling modes with relevant anti-symmetric distortional components – this contradicts the general belief that such column lengths are associated with global buckling. Thus, the columns analysed in this work are affected by the interaction between (i1) three half-wave symmetric distortional and (i2) single half-wave (anti-symmetric) distortional–flexural–torsional modes. (ii) The participation of the anti-symmetric distortional mode just mentioned was shown to reduce the column post-critical strength and provided the explanation for the surprising unstable ‘‘global’’ post-buckling behaviour exhibited by the intermediate-to-long lipped channel columns. (iii) The equilibrium paths describing the post-buckling behaviours of the columns affected by distortional/global (‘‘global’’ means ‘‘distortional–flexural–torsional’’) mode interaction exhibit features that vary considerably with the initial imperfection shape. Those equilibrium paths can be grouped in three categories, depending on whether the initial imperfection shape is (iii1) pure distortional, (iii2) predominantly distortional with outward outer half-waves or (iii3) none of the above – this last category comprises the vast majority of the post-buckling paths and led to the identification of a ‘‘coupled buckling mode’’, defined by CG  0.45CD. (iv) A key aspect concerning the distortional/global mode interaction is the influence exerted by the outer half-wave flangelip motions (distortional feature) on the major axis flexure (global feature). Indeed, (iv1) while inward flange-lip motions reduce the cross-section major moment of inertia, thus facilitating the occurrence of the corresponding flexure, (iv2) the outward flange-lip motions have precisely the opposite effect.19 (v) The columns containing pure distortional initial imperfections are affected by a different type of distortional/global mode interaction phenomenon, which (v1) involves only minor axis flexure and (v2) is not caused by the closeness 19 This aspect is particularly relevant because the (symmetric) distortional buckling mode exhibits three half-waves (2 outward and 1 inward or vice-versa) - if the mode has an even or larger odd half-wave number, it should play a much lesser role (this issue is currently under investigation).

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of two buckling loads – it stems from the occurrence of horizontal shifts of the column cross-section effective centroids, due to the ‘‘weakening’’ (axial stiffness drop) of the flangelip assemblies brought about by high compressive normal stresses. (vi) Regardless of the initial imperfection shape, all elastic equilibrium paths exhibit limit points that take place below the critical applied load level. Concerning the elastic–plastic post-buckling behaviour, it was found that (vi1) the strength erosion due to distortional/global modal interaction effects is considerable, (vi2) there is virtually no elastic–plastic strength reserve and/or ductility (the onset of yielding often triggers the column failure) and, for yield stresses a bit larger than fy = 355 MPa, (vi3) the column collapse is fully governed by elastic distortional/global interaction. (vii) For practical purposes, pure global initial imperfections may be taken as the most detrimental ones, in the sense that they lead to the lowest ultimate loads, both in the elastic and elastic–plastic columns. (viii) Finally, a minor imperfection-sensitivity study carried out showed that the initial imperfection global amplitude has a relatively small impact on the column elastic post-buckling behaviour and ultimate strength – nevertheless, it is far more relevant than its distortional counterpart. Some of the features just described are bound to have far-reaching implications in the design of cold-formed steel columns experiencing distortional/global interaction, mostly because (i) the uncoupled distortional and global post-buckling behaviours are now looked upon as stable (even if only marginally) and (ii) no adverse mode interaction effects between them had ever been reported. The authors are currently investigating the influence of this mode interaction phenomenon in the post-buckling and ultimate strength behaviours of lipped channel columns with other cross-section dimensions and support conditions – the completion of this task should pave the way towards the development and validation of a direct strength approach to design this type of columns. References [1] Kwon Y, Hancock GJ. Postbuckling analysis of thin-walled channel sections undergoing local and distortional buckling. Comput Struct 1993;49(3):507–16. [2] Prola L, Camotim D. On the distortional post-buckling behavior of cold-formed lipped channel steel columns. In: Proceedings of structural stability research council annual stability conference (Seattle, 24–26/4); 2002. p. 571–90. [3] Camotim D, Silvestre N. GBT distortional post-buckling analysis of cold-formed steel lipped channel columns and beams. In: Program and book of abstracts of 17th asce engineering mechanics conference (Newark, 13–16/6); 2004. p. 38– 39 [full paper in Cd-Rom proceedings]. [4] Silvestre N, Camotim D. Local-plate and distortional post-buckling behavior of cold-formed steel lipped channel columns with intermediate stiffeners. J Struct Eng (ASCE) 2006;132(4):529–40. [5] Batista EM. Local-global buckling interaction procedures for the design of coldformed columns: effective width and direct method integrated approach. ThinWall Struct 2009;47(11):1218–31. [6] Schafer BW. Direct strength method design guide. Washington, DC: AISI (American Iron & Steel Institute); 2005. [7] Schafer BW. Review: the direct strength method of cold-formed steel member design. J Constr Steel Res 2008;64(7–8):766–78.

[8] Schafer BW, Peköz T. Laterally braced cold-formed steel members with edge stiffened flanges. J Struct Eng (ASCE) 1999;125(2):118–27. [9] Yang D, Hancock GJ. Compression tests of high strength steel columns with interaction between local and distortional buckling. J Struct Eng (ASCE) 2004;130(12):1954–63. [10] Ungureanu V, Dubina D. Recent research advances on ECBL approach (Part I: plastic–elastic interactive buckling of cold-formed steel sections. Thin-Wall Struct 2004;42(2):177–94. [11] Dinis PB, Camotim D, Silvestre N. FEM-based analysis of the local-plate/ distortional mode interaction in cold-formed steel lipped channel columns. Comput Struct 2007;85(19–20):1461–74. [12] Camotim D, Dinis PB, Silvestre N. Local/distortional mode interaction in lipped channel steel columns: post-buckling behaviour, strength and DSM design. In: Proceedings of fifth international conference on thin-walled structures (Brisbane, 18–20/6), vol. 1; 2008. p. 99–114. [13] Kwon YB, Kim BS, Hancock GJ. Compression tests of high strength cold-formed steel channels with buckling interaction. J Constr Steel Res 2009;65(2):278–89. [14] Silvestre N, Camotim D, Dinis PB. Direct strength prediction of lipped channel columns experiencing local-plate/distortional interaction. Adv Steel Constr – Int J 2009;5(1):45–67. [15] Dinis PB, Camotim D. Post-buckling analysis of cold-formed steel lipped channel columns affected by distortional/global mode interaction. In: Proceedings of SSRC annual stability conference (Nashville, 2–5/4); 2008. p. 405–31. [16] Rossi B. Mechanical properties, residual stresses and structural behavior of thin-walled stainless steel profiles. Ph.D. thesis, University of Liège; 2009. [17] Thomasson PO. Thin-walled C-shaped panels in axial compression. Swedish Council for Building Research, D1:1978. [18] Höglund T. Design of trapezoidal sheeting provided with stiffeners in the flanges and webs. Swedish Council for Building Research, D28:1980. [19] Desmond TP, Peköz T, Winter G. Edge stiffeners for thin-walled members. J Struct Eng (ASCE) 1981;107(2):329–53. [20] European Convention for Constructional Steelwork (ECCS). Manual on the stability of steel structures. 2nd ed. Brussels; 1976. [21] Schafer BW, Peköz T. Computational modelling of cold-formed steel: characterising geometric imperfections and residual stresses. J Constr Steel Res 1998;47(3):193–210. [22] Dubina D, Ungureanu V. Effect of imperfections on numerical simulation of instability behaviour of cold-formed steel members. Thin-Wall Struct 2002;40(3):239–62. [23] Maiorana E, Pellegrino C, Modena C. Imperfections in steel girder webs with or without perforations under patch loading. J Constr Steel Res 2009;65(5):1121–9. [24] Simulia Inc. Abaqus standard (vrs. 6.7-5); 2008. [25] Dinis PB, Camotim D. On the use of shell finite element analysis to assess the local buckling and post-buckling behaviour of cold-formed steel thin-walled members. In: Soares CAM et al., editors. Book of abstracts of III European conference on computational mechanics: solids, structures and coupled problems in engineering (Lisboa, 5–9/6); 2006. p. 689 [full paper in Cd-Rom proceedings]. [26] Schafer BW. Cufsm (version 2.6); 2003. . [27] Camotim D, Silvestre N, Dinis PB. Numerical analysis of cold-formed steel members. Int J Steel Struct 2005;5(1):63–78. [28] Camotim D, Silvestre N, Gonçalves R, Dinis PB. GBT analysis of thin-walled members: new formulations and applications. In: Loughlan J, editor. Thinwalled structures: recent advances and future trends in thin-walled structures technology (International workshop – Loughborough, 25/6); 2004. p. 137–68. [29] Camotim D, Silvestre N, Basaglia C, Bebiano R. GBT-based buckling analysis of thin-walled members with non-standard support conditions. Thin-Wall Struct 2008;46(7–9):800–15. [30] Bebiano R, Silvestre N, Camotim D. GBTUL 1.0b – code for buckling and vibration analysis of thin-walled members; 2008. . [31] Bebiano R, Silvestre N, Camotim D. GBTUL – a code for the buckling analysis of cold-formed steel members. In: LaBoube R, Yu W-W, editors. Proceedings of 19th international specialty conference on recent research and developments in cold-formed steel design and construction (St. Louis, 14–15/10); 2008. p. 61–79. [32] Young B, Rasmussen KJ. Shift of effective centroid in channel columns. J Struct Eng (ASCE) 1999;125(5):524–31.