On the mechanics of local-distortional interaction in thin-walled lipped channel beams

Thin-Walled Structures xxx (xxxx) xxx–xxx

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On the mechanics of local-distortional interaction in thin-walled lipped channel beams ⁎

André Dias Martinsa, Dinar Camotima, , Rodrigo Gonçalvesb, Pedro Borges Dinisa a b

CERIS, ICIST, DECivil, Instituto Superior Técnico, Universidade de Lisboa, Portugal CERIS, ICIST and Universidade Nova de Lisboa, Portugal

A R T I C L E I N F O

A B S T R A C T

Keywords: Thin-walled lipped channel beams Local-distortional interaction True and secondary-bifurcation L-D interaction Generalised Beam Theory (GBT) Geometrically non-linear analysis

This paper investigates the elastic post-buckling behaviour of simply supported thin-walled lipped channel beams undergoing local-distortional (L-D) interaction. The beams are uniformly bent about the major-axis and experience flange-triggered local buckling (most common situation in practice). Three beam geometries are considered, each exhibiting a different type of L-D interaction, namely (i) “true L-D interaction” (close local and distortional critical buckling moments – two critical-mode initial geometrical imperfections are used, akin to the competing critical buckling modes), (ii) “secondary local-bifurcation L-D interaction” or (iii) “secondary distortional-bifurcation L-D interaction” (critical distortional-to-local buckling moment ratio well below or above 1.0, respectively, with “high enough” yield stresses). The results presented and discussed are obtained through elastic geometrically non-linear Generalised Beam Theory (GBT) analyses and provide the evolution, along given equilibrium paths, of the beam deformed configuration (expressed in GBT modal form), relevant displacement profiles and modal participation diagrams, making it possible to acquire in-depth knowledge on the beam L-D interaction mechanics. Particular attention is devoted to interpreting the quantitative and qualitative differences exhibited by the beam post-buckling behaviours associated with the three aforementioned L-D interaction types.

1. Introduction The complex thin-walled open cross-section shapes commonly exhibited by cold-formed steel members (e.g., with several intermediate and end stiffeners) makes them highly prone to several individual (local, distortional, global – L, D, G) and/or coupled (L-D, L-G, D-G or LD-G) buckling phenomena. Concerning the latter, the establishment of efficient design rules against interactive failures constitutes a considerable challenge, much more so than in hot-rolled steel members, which are “only” affected by L-G interaction, already adequately handled by steel design specifications around the world. Note that, as far as cold-formed steel columns are concerned, the authors proposed recently design/strength curves, based on the Direct Strength Method (DSM), able to handle L-D (Martins et al. [1]) and L-D-G (Dinis et al. [2], Dinis and Camotim [3]) interactive failures. Traditionally, the non-linear behaviour of prismatic thin-walled members could only be rigorously assessed by resorting to time-consuming and computation-intensive shell finite element simulations, which provide outputs that are not easy to apprehend/interpret. However, in the last few years Generalised Beam Theory (GBT) has emerged as a very promising alternative to obtain similarly accurate results in a more efficient and (mostly)

⁎

clarifying manner – only a few structurally meaningful d.o.f. are required. By performing GBT-based geometrically non-linear imperfect analyses (GNIA) of prismatic thin-walled members, it becomes possible to unveil and quantify the contributions of the various deformation modes to the member structural response under consideration. This feature makes it possible to acquire much deeper insight on that response, thus providing a much clearer picture of the mechanical aspects involved – i.e., GBT GNIA is an ideally suited numerical tool to investigate complex coupled instability problems, like the one addressed in this work. As far as thin-walled members affected by coupling phenomena are concerned, GBT post-buckling results are only available for thin-walled lipped channel (i) columns experiencing L-D (Silvestre and Camotim [4], Martins et al. [5–7]) and D-G (Martins et al. [8]) interaction, and (ii) beams affected by D-G interaction (Martins et al. [9]) – to the authors' best knowledge, no such results exist for thin-walled beams affected by L-D interaction. Thus, the aim of this work is to present and discuss GBT numerical results concerning the geometrically non-linear behaviour of simply supported lipped channel beams experiencing flange-triggered L-D interaction under uniform major-axis bending – these results shed fresh light on the mechanics underlying this coupling

Corresponding author. E-mail addresses: [email protected] (A.D. Martins), [email protected] (D. Camotim), [email protected] (R. Gonçalves), [email protected] (P. Borges Dinis).

http://dx.doi.org/10.1016/j.tws.2017.05.035 Received 25 February 2017; Received in revised form 29 May 2017; Accepted 31 May 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.

Please cite this article as: Dias Martins, A., Thin-Walled Structures (2017), http://dx.doi.org/10.1016/j.tws.2017.05.035

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2.1. Formulation, non-linear beam finite element and solution procedure – brief overview Consider the arbitrary open cross-section member depicted in Fig. 1, also showing the reference system x-s-z – coordinates along the member length, cross-section mid-line and wall thickness, respectively. As for u (x, s), v(x, s) and w(x, s), they denote the corresponding displacement fields, also indicated in Fig. 1. According to GBT, the member midsurface displacement field considers the variable separation:

Fig. 1. Arbitrary prismatic thin-walled member, local coordinate system and displacement components.

u (x , s ) = uk (s ) ϕk, x (x )

(1.1)

v (x , s ) = vk (s ) ϕk (x )

(1.2)

w (x , s ) = wk (s ) ϕk (x )

(1.3)

where (i) (.),x≡d(.)/dx, (ii) Einstein's summation convention applies to subscript k, (iii) ϕk(x) are modal displacement amplitude functions defined along the member length (member analysis unknowns) and (iv) functions uk(s), vk(s), wk(s) are the displacements profiles associated with deformation mode k, obtained from the cross-section analysis (briefly discussed in Section 2.2). The initial geometrical imperfections incorporated in the analysis are also expressed in GBT modal form – in this work, they are obtained from a preliminary GBT buckling analysis and subsequently normalised (as discussed in Section 4) – i.e., one has

phenomenon, already unveiled through shell finite element analyses (Dinis and Camotim [10]). Note that the beams analysed are highly prone to L-D interaction due to their end support conditions (Martins et al. [11]) – in fact, it is very difficult to avoid this coupling phenomenon in the short-to-intermediate length range. Three beam geometries are selected and analysed, each of them exhibiting a different type of L-D interaction, namely (i) “true L-D interaction” (close local and distortional critical buckling moments), (ii) “secondary local-bifurcation L-D interaction” and (iii) “secondary distortional-bifurcation L-D interaction” (e.g., [11]) – in order to assess the influence of the initial geometrical imperfection shape on the post-buckling behaviour of beams undergoing “true L-D interaction”, results concerning beams with local and distortional critical-mode initial imperfections are compared. The GBT elastic post-buckling results provide the evolution, along given equilibrium paths, of the beam deformed configuration (expressed in GBT modal form), displacement profiles and modal participations diagrams. Particular attention is paid to characterising the deformation patterns akin to the local and distortional critical buckling modes, as well as relevant longitudinal displacement profiles, thus making it possible to acquire in-depth knowledge on the mechanics of beam L-D interaction. This knowledge will certainly impact the development of rational design rules for cold-formed steel beams failing in LD interaction modes, currently under way [11]. This work begins with a brief review of the GBT elastic geometrically non-linear formulation, presented in Section 2. Then, Section 3 addresses the selection of beam geometries prone to L-D interaction, carried out by means of GBT buckling analyses, and Section 4 (the main one) presents and discusses the GBT GNIA results of lipped channel (LC) beams undergoing different types of L-D interaction. Finally, the paper closes with a few concluding remarks.

u (x , s ) = uk (s ) ϕk, x (x )

(2.1)

v (x , s ) = vk (s ) ϕk (x )

(2.2)

w (x , s ) = wk (s ) ϕk (x )

(2.3)

GBT geometrically non-linear formulations were previously developed by Miosga [13] and, much more recently, by Silvestre [14], Basaglia [15] and Silva [16]. These authors adopted a total Lagrangian kinematic description and an additive decomposition of the strain terms into Green-Lagrange membrane strains and small-strain bending. The main novelty of the present formulation is the consideration of the whole set of non-linear membrane strain terms, which is equivalent (but not similar) to the recent formulation of Gonçalves and Camotim [17]. The strain-displacement relationships adopted read

εxx = u, x − zw , xx +

εss = v , s − zw , ss +

1 2 1 (u, x + v ,2x + w ,2x ) − u, x + zw , xx − (u,2x + v ,2x + w ,2x ) 2 2 (3.1)

1 2 1 (u, s + v ,2s + w ,2s ) − v , s + zw , ss − (u,2s + v ,2s + w ,2s ) 2 2 (3.2)

γxs = u, s + v , x − 2zw , xs + u, x u, s + v , x v , s + w , x w , s − u, s − v , x + 2zw , xs − u, x u, s − v , x v , s − w , x w , s

2. GBT geometrically non-linear analysis

(3.3) The member strain energy is then expressed, in terms of the strain and stress components, as

The performance of a GBT structural analysis involves two (independent) main tasks, namely (i) a cross-section analysis, concerning the identification of the deformation modes and evaluation of the corresponding modal mechanical properties, and (ii) a member analysis (elastic buckling and post-buckling analyses, in the case of this paper), which consists of solving the appropriate differential equilibrium equation system. Next, a brief overview of the geometrically non-linear GBT formulation recently reported by the authors (Martins et al. [12]) is presented, followed by a brief description of the deformations modes most relevant to the analysis of the lipped channel beams affected by LD interaction.

U=

1 2

∫V σij εij dV = U1 + U2 + U3 − (U1 + U2 + U3)

(4)

where Ui and Ui (i = 1, 2, 3) are the strain energy terms associated with the total deformation and initial geometric imperfections (see [12]). The member analysis is performed by means of (one-dimensional) beam finite elements similar to those developed by Silvestre and Camotim [18] and Basaglia et al. [19]. The modal amplitude functions are approximated by linear combinations of (i) Lagrange cubic polynomial primitives ψiL (ξ ), i = 1, ..., 4 , for deformations modes involving only warping displacements (axial extension and shear modes), and (ii) Hermite cubic polynomials ψiH (ξ ), i = 1, ..., 4 , for the remaining conventional deformation modes and transverse extension modes – the various deformation mode families are addressed in Section 2.2. Such 2

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A.D. Martins et al.

polynomials have the form:

ψ1H

= Le

(ξ 3

−

2ξ 2

+ ξ)

(5.1)

ψ1,Lx =

1 (1 − ξ )(3ξ − 1)(3ξ − 2) 2

(5.2)

ψ2H

=

2ξ 3

(5.3)

ψ2,Lx

9 = ξ (ξ − 1)(3ξ − 2) 2

−

3ξ 2

+

+1

II 121 II 121 III 022 IV 112 + Djhi k ηαβ + Djih k ηβα + Ejhi k ηαβ + Ehij kαβη ) diβ djη

(5.4) (5.5)

9 ξ (1 − ξ )(3ξ − 1) 2

3

f hα =

(5.8)

where Le is the element length and ξ = x / Le . Therefore, the amplitude functions ϕk (x ) associated with the conventional (except axial extension) and transverse extension modes are approximated through

ϕk (x ) = ψ1H dk1 + ψ2H dk 2 + ψ3H dk3 + ψ4H dk 4

+

(6.1)

In these equations, it should be noted that (i) Subscripts h, i, j, k correspond to the deformations modes associated with the shape function subscripts α, β , η , μ , respectively. ab abc abcd , kαβη and kαβημ are defined by (ii) Coefficients kαβ

(6.2)

where dk1 = ϕk, x (0), dk 2 = ϕk, x (Le /3), dk3 = ϕk, x (2Le /3), dk 4 = ϕk, x (Le ) The finite element internal force vector is obtained by differentiating the strain energy Eq. (4), after the incorporation of Eqs. (6.1) and (6.2), to obtain 1

2

3

e 1 2 3 f hα = fhα + fhα + fhα − (f hα + f hα + f hα )

=

22 (Chi kαβ

+

00 Bhi kαβ

+

11 Dhi kαβ

+

20 Ehi kαβ

+

20 Eih kβα ) diβ

∫Le

∂aψα ∂bψβ ∂x a ∂x b

dx

(8.1)

1 I 1 I 0000 3 1111 II III = ⎜⎛ (Chijk + Bhijk + Ehijk + Bhijk fhα kαβημ ) kαβημ 2 2 ⎝ 1 I 1 I 1100 1100 III III + (Ehijk + Bhijk + (Ejkhi + Bjkhi ) kαβημ ) k ημαβ 2 2 1 I 1010 1010 1010 1010 I I I + (Dhijk + Djihk + Dkijh kαβημ + Dihjk kβαημ k ηβαμ kμβηα ) 2 1 II 2222 1 III 2211 1 III 2211 2121 II + Chijk kαβημ + Chijk kαβημ + Cjkhi k ημαβ + Dhijk kαβημ 2 2 2 1 II 2121 1 II 2121 III 2110 III 2110 + Dihjk kβαημ + Dkijh kμβηα + Dhijk kαβημ + Dihjk kβαημ 2 2 1 II 1122 1 II 1122 2110 III III 2110 + Djihk + Dkijh k ηβαμ kμβηα + Ehijk kαβημ + Ejkhi k ημαβ 2 2 1 III 1111 1 IV 0022 1 IV 0022 ⎞ Ejkhi k ημαβ + Ehijk kαβημ + Ejkhi k ημαβ ⎟ diβ djη dkμ 2 2 2 ⎠

22 00 11 20 20 f hα = (Chi kαβ + Bhi kαβ + Dhi kαβ + Ehi kαβ + Eih kβα ) diβ

∂aψα ∂bψβ ∂cψη ∂x a ∂x b ∂x c

dx

(8.2)

∫Le

∂aψα ∂bψβ ∂cψη ∂dψμ ∂x a ∂x b ∂x c ∂x d

dx

(8.3)

(iii) The components of the second-order tensors Chi, Bhi , Dhi , Ehi are modal mechanical properties characterising the cross-section linear behaviour. (iv) The third and fourth-order tensors, denoted by I II I II I II I II III IV I II Chij , Chij , Bhij , Bhij , Dhij , Dhij , Ehij , Ehij , Ehij , Ehij Chijk , Chijk , and IV III I II III I II III I II III Chijk , Bhijk , Bhijk , Bhijk , Dhijk , Dhijk , Dhijk , Ehijk , Ehijk , Ehijk , Ehijk , are associated with the member geometrically non-linear behaviour – note that tensors C, B, D stem from longitudinal extensions, transverse extensions and shear strains, respectively, while tensor E is associated with the coupling between longitudinal and transverse extensions (Poisson effects). The components of these tensors, obtained exclusively on the basis of the cross-section deformations modes, are given by1

(7.3)

1

abc kαβη =

abcd kαβημ =

1 III 022 1 IV 122 ⎞ III 022 IV 112 Ehij kαβη + Eihj kβαη + Ehij kαβη + Ejih k ηβα ⎟ diβ djη 2 2 ⎠

+

∫Le

(7.2)

1 I 211 1 I 000 2 211 000 110 I I I I 110 = ⎜⎛ Chij + Bhij + Dhij + Dihj fhα kαβη + Cihj kβαη kαβη + Bihj kβαη kαβη kβαη 2 2 ⎝ 1 I 200 1 II 110 200 011 I I II + Djih + Ehij + (Ehij + Bhij k ηβα kαβη + Eihj kβαη ) kαβη 2 2 3 II 222 011 II II II 121 II 121 II 121 + (Eihj + Bihj + Chij ) kβαη kαβη + Dhij kαβη + Dihj kβαη + Djih k ηβα 2 +

ab kαβ =

(7.1)

where 1 fhα

1 IV 0022 ⎞ E jkhi k ημαβ ⎟ diβ djη dkμ 2 ⎠

(7.7)

where dk1 = ϕk, x (0), dk 2 = ϕk (0), dk3 = ϕk, x (Le ), dk 4 = ϕk (Le ) and the amplitude functions ϕj, x (x ) associated with the deformation modes involving only warping are given by

ϕj, x (x ) = ψ1,Lx dj1 + ψ2,Lx dj2 + ψ3,Lx dj3 + ψ4,Lx dj 4

⎛⎜ 1 (C I + E III ) k 1111 + 1 C II k 2222 + 1 C III k 2211 hijk hijk αβημ hkji αμηβ hijk αβημ 2 2 ⎝2 +

(5.7)

1 ξ (3ξ − 1)(3ξ − 2) 2

(7.6)

1 III 2211 1 I 0000 1 II 1111 Cjkhi k ημαβ + Bhijk kαβημ + Bhkji kαμηβ 2 2 2 1 I III 1100 + (Ehijk + Bhijk ) kαβημ 2 1 I III 1100 I 1010 I 1010 II 2121 + (E jkhi + B jkhi ) k ημαβ + Dhijk kαβημ + Dihjk kβαημ + Dhijk kαβημ 2 II 2121 III 2110 III 2110 III 2110 III 2110 + Dihjk + Dhijk + Djkih kβαημ kαβημ + Dihjk kβαημ + Djkhi k ημαβ k ημβα 1 II 1122 1 II 1122 1 III 1111 1 IV 0022 + Ehijk kαβημ + E jkhi k ημαβ + E jkhi k ημαβ + Ehijk kαβημ 2 2 2 2

(5.6)

ψ4H = 3ξ 2 − 2ξ 3

ψ4,Lx =

1 IV 112 ⎞ 211 000 110 I I I Eijh kβηα ⎟ diβ djη + (Cjhi k ηαβ + Bjhi k ηαβ + Djhi k ηαβ 2 ⎠

110 200 011 I I II II II 222 + Djih k ηβα + Ejhi k ηαβ + (Ejhi + Bjhi ) k ηαβ + Chij kαβη

ψ3H = Le (ξ 3 − ξ 2)

ψ3,Lx =

1 I 211 1 I 000 1 I 200 2 110 I f hα = ⎜⎛ Chij kαβη + Bhij kαβη + Dhij kαβη + Ehij kαβη 2 2 2 ⎝ 1 II 1 II 222 1 III 022 011 II II 121 + (Ehij + Bhij ) kαβη + Chij kαβη + Dhij kαβη + Ehij kαβη 2 2 2

Chi = Q11

∫A (uh − zwh)(ui − zwi) dA

(9.1)

1 The determination of the fourth-order tensors can be (computationally) very costly if a large number of deformation modes is included in the analysis – the order of magnitude of the computations is p × NM 4, where NM is the deformation mode number and p is the number of the cross-section wall segments considered. In order to reduce drastically the computation time involved in this procedure, simplifications were made accounting for the (i) deformation mode nature (e.g., CIhijk is null if any subscript corresponds to a shear mode) and (ii) symmetry properties (e.g., CIhijk is symmetric with respect to (h, i) and (j, k)).

(7.4) (7.5) 3

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A.D. Martins et al.

Bhi = Q22

∫A (vh,s − zwh,ss)(vi,s − zwi,ss) dA

(9.2)

Dhi = Q66

∫A (uh,s + vh − 2zwh,s)(ui,s + vi − 2zwi,s) dA

(9.3)

Ehi = Q12

∫A (uh − zwh)(vi,s − zwi,ss) dA

(9.4)

I Chij = Q11

∫A uh (vi vj + wi wj) dA

(10.1)

∫A uh ui uj dA

(10.2)

I Bhij = Q22

∫A vh,s (vi,s vj,s + wi,s wj,s) dA

(10.3)

II Bhij = Q22

∫A vh,s ui,s uj,s dA

(10.4)

I Dhij = Q66

∫A (uh,s + vh)(vi vj,s + wi wj,s) dA

(10.5)

II Dhij = Q66

∫A (uh,s + vh) ui uj,s dA

(10.6)

I Ehij = Q12

∫A uh (vi,s vj,s + wi,s wj,s) dA

(10.7)

II Ehij = Q12

∫A vh,s (vi vj + wi wj) dA

(10.8)

III Ehij = Q12

∫A vh,s ui uj dA

(10.9)

IV Ehij = Q12

∫A uh,s ui,s uj dA

I Chijk = Q11

∫A (vh vi + wh wi)(vj vk + wj wk ) dA

(11.1)

II Chijk = Q11

∫A uh ui uj uk dA

(11.2)

∫A uh ui (vj vk + wj wk ) dA

(11.3)

I Bhijk = Q22

∫A (vh,s vi,s + wh,s wi,s)(vj,s vk,s + wj,s wk,s) dA

(11.4)

II Bhijk = Q22

∫A uh,s ui,s uj,s uk,s dA

(11.5)

III Bhijk = Q22

∫A uh,s ui,s (vj,s vk,s + wj,s wk,s) dA

(11.6)

I Dhijk = Q66

∫A (vh vi,s + wh wi,s)(vj vk,s + wj wk,s) dA

(11.7)

II Dhijk = Q66

∫A uh ui,s uj uk,s dA

(11.8)

III Dhijk = Q66

∫A uh ui,s (vj vk,s + wj wk,s) dA

(11.9)

I Ehijk = Q12

∫A (vh vi + wh wi)(vj,s vk,s + wj,s wk,s) dA

(11.10)

II Ehijk = Q12

∫A uh,s ui,s uj uk dA

(11.11)

III Ehijk = Q12

∫A uh,s ui,s (vj vk + wj wk ) dA

(11.12)

IV Ehijk = Q12

∫A (vh,s vi,s + wh,s wi,s) uj uk dA

II Chij

= Q11

III Chijk

= Q11

E , 1 − v2

1 22 00 11 20 20 Thiαβ = Chi kαβ + Bhi kαβ + Dhi kαβ + Ehi kαβ + Eih kβα

2 I 211 I 211 I 211 I 000 I 000 Thiαβ kαβη kβαη k ηαβ kαβη kβαη = (Chij + Cihj + Cjhi + Bhij + Bihj I 000 I 110 I 110 I 110 I 110 I 110 k ηαβ kαβη kαηβ kβαη + Djhi k ηαβ k ηβα + Bjhi + Dhij + Dhji + Dihj + Djih I 110 I 200 I 200 I 200 II II 011 ) kαβη kβηα + Ehij kαβη kβαη k ηαβ + Dijh + Eihj +Ejhi + (Ehij + Bhij II II 011 II II 011 II 222 II 121 ) kβαη ) k ηαβ kαβη + Dhij kαβη + (Eihj + Bihj + (Ejhi + Bjhi + 3Chij II 121 II 121 II 121 II 121 II 121 III 022 kαηβ + Dihj kβαη + Djhi k ηαβ + Djih k ηβα + Dijh kβηα + Ehij kαβη + Dhji III 022 III 022 IV 112 IV 112 IV 112 kβαη + Ejhi k ηαβ + Ehij kαβη + Ehji kαηβ + Ejih k ηβα ) djη + Eihj

(12.3)

⎛ 1 C I k 1111 + C I k 1111 + 1 B I k 0000 + B I k 0000 hijk αβημ hjik αηβμ hijk αβημ hjik αηβμ 2 ⎝2

3 = Thiαβ

⎜

1 I 1010 1010 1010 1010 I I I + Dhkji + Dihjk kαηβμ kαμηβ kβαημ (Dhijk kαβημ + Dhjik 2 1010 1010 1010 1010 I I I I + Djhik + Dkhji + Djihk + Dijhk k ηαβμ kμαηβ k ηβαμ kβηαμ +

1010 1010 1010 1010 I I I I + Djkhi + Dkijh + Dkjih + Dikjh k ημαβ kμβηα kμηβα kβμηα ) 1 I 1100 1 I 1100 1100 1100 I I + Ehijk kαβημ + Ehjik kαηβμ + Ejkhi k ημαβ + Eikhj kβμαη 2 2 3 II 2222 1 III 2211 1 III 2211 III 2211 + Chijk kαβημ + Chijk kαβημ + Chjik kαηβμ + Cjkhi k ημαβ 2 2 2 1 III 1100 1 III 1100 III 2211 III 1100 + Cikhj kβμαη + Bhijk kαβημ + Bhjik kαηβμ + Bjkhi k ημαβ 2 2 3 II 1111 2121 2121 III 1100 II II + Bikhj +Dhjik kβμαη + Bhijk kαβημ + 2Dhijk kαβημ kαηβμ 2 1 II 2121 1 II 2121 2121 2121 II II + Dihjk + Dkhji kβαημ kμαηβ+ Dkijh kμβηα + Dkjih kμηβα 2 2 III 2110 III 2110 III 2110 III 2110 + Dhijk kαβημ + Dhjik kαηβμ + Dhkji kαμηβ + Dihjk kβαημ

(10.10)

E v , 1 − v2

2110 2110 III III 2110 III III 2110 + Djhik + Dkhji + Dijhk k ηαβμ kμαηβ + Djihk k ηβαμ kβηαμ III 2110 2110 III 2110 III 2110 III + Djkhi + Dkijh k ημαβ kμβηα + Dkjih kμηβα + Dikjh kβμηα 1 II 1122 1 II 1122 II 1122 II 1122 + Ehijk kαβημ+Ehjik kαηβμ + Ejkhi k ημαβ + Eikhj kβμαη 2 2 1 III 1111 1111 III 1111 1 III III 1111 + Ehijk + Eikhj kαβημ + Ehjik kαηβμ+ Ejkhi k ημαβ kβμαη 2 2 1 IV 0022 1 IV 0022 IV 0022 IV 0022 + Ehijk kαβημ + Ehjik kαηβμ + Ejkhi k ημαβ + Eikhj kβμαη) djη dkμ 2 2 (12.4) 2 I 211 I 000 I 110 I 110 I 200 Thiαβ k ηαβ k ηαβ k ηαβ k ηβα k ηαβ = (Cjhi + Bjhi + Djhi + Djih + Ejhi II II 011 II 222 II 121 II 121 ) k ηαβ kαβη +Djhi k ηαβ + Djih k ηβα + (Ejhi + Bjhi + Chij III 022 IV 112 k ηαβ + Ehij kαβη ) djη + Ejhi

(12.5)

1 I 1 II 2222 1 III 2211 3 1111 III = ⎜⎛ (Chijk + Ehijk + Chkji Thiαβ ) kαβημ kαμηβ + Chijk kαβημ 2 2 2 ⎝ 1 III 2211 1 I 0000 1 II 0000 1 I 1100 III + Cjkhi k ημαβ + Bhijk kαβημ + Bhkji kαμηβ + (Ehijk + Bhijk ) kαβημ 2 2 2 2 1 I III 1100 I 1010 I 1010 II 2121 + (Ejkhi + Bjkhi ) k ημαβ + Dhijk + Dihjk + Dhijk kαβημ kβαημ kαβημ 2 II 2121 III 2110 III 2110 III 2110 III 2110 + Dihjk + Dhijk + Djkih kβαημ kαβημ + Dihjk kβαημ + Djkhi k ημαβ k ημβα 1 II 1122 1 II 1122 1 III 1111 1 IV 0022 + Ehijk kαβημ + Ejkhi k ημαβ + Ejkhi k ημαβ + Ehijk kαβημ 2 2 2 2

(11.13)

+

E 2(1 + v )

Q12 = Q66 = where Q11 = Q22 = (E and v – Young's modulus and Poisson's ratio). In order to perform/implement an incremental-iterative technique it is essential to evaluate the tangent stiffness matrix corresponding to a given deformed configuration, obtained by differentiating the internal e force vector f hα (Eq. (7.1)) with respect to a given d.o.f diβ – one has e 1 2 3 2 3 Thiαβ = Thiαβ + Thiαβ + Thiαβ − (Thiαβ + Thiαβ )

(12.2)

1 IV 0022 ⎞ Ejkhi k ημαβ ⎟ djη dkμ 2 ⎠ (12.6)

The incremental-iterative procedure was implemented based on (i) Newton-Raphson's method and (ii) a load or arc-length control strategy (both strategies were implemented). The numerical integration is performed using the Legendre-Gauss Quadrature with three-point integration along x (Eq. (8.1)–(8.3)) and y (Eqs. (9.1)–(11.13)) – the integration along z (though thickness) is done analytically.

(12.1)

where 4

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Fig. 2. GBT lipped channel (a) nodal discretisation and in-plane deformed configurations of the conventional deformation modes, (b) warping displacement profiles of the shear deformation modes, and (c) configurations of the transverse extension deformation modes.

intermediate, namely 3 in the top/compressed flange, 1 in the bottom/ tensioned flange and 11 in the web – the post-buckling behaviour determined with less web intermediate nodes was found to be excessively stiff (see Section 4.3).2 Such discretisation leads to (i) 23 conventional modes (4 global, 2 distortional and 17 local) – modes 1–23 (the most relevant are shown in Fig. 2), (ii) 20 shear modes (5 global and 15 local) − modes 24–43 (see Fig. 2), (iii) 20 linear transverse extension modes (1 global isotropic, 4 global deviatoric and 15 local) – modes 44–63 (see Fig. 2) and (iv) 20 “quadratic” transverse extension modes (modes 64–83 – see Fig. 2), totalling 83 (sequentially numbered) deformation modes.3 It should be noted that the number of deformation modes

2.2. Cross-section analysis The GBT cross-section analysis involves a lengthy set of fairly complex operations which has been reported in the literature, with emphasis on the recent works of Gonçalves et al. [20] and Bebiano et al. [21] – the interested reader can find detailed accounts in these references, which provide the fundamentals of the procedure implemented in version 2.0 of code GBTUL (Bebiano et al. [22]). The deformation modes obtained can be divided into three main sets/families, namely the (i) conventional (or Vlasov) modes, (ii) shear modes and (iii) transverse extensions modes. These deformation mode sets are also adopted in this work, together with an additional family of quadratic transverse extension modes, which plays an essential role in capturing the Poisson effects that develop in the deformed configurations occurring in the advanced post-buckling stages (involving heavy cross-section in-plane deformation) and, therefore, in precluding reaching overly stiff solutions (Gonçalves and Camotim [17]) – the above Poisson effects will be addressed in item (ii) of Section 4.1. Fig. 2 shows the GBT nodal discretisation adopted in all the beams analysed in this work, involving 21 nodes: 6 natural and 15

2 It may be possible to obtain accurate results with less web intermediate nodes if wall segments with unequal widths are adopted, making it possible to have more intermediate nodes in the web compressed region than in tensioned one. However, such a strategy is yet to be attempted. 3 Note that none of the previously developed GBT geometrically non-linear formulations employed the set of deformation modes considered in this work. This is because the cross-section analysis procedure employed here was not yet available – alternative (less sophisticated) cross-section analysis procedures were adopted.

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Fig. 3. (a) RDL = 1.02, (b) RDL = 1.60 and (c) RDL = 0.60 beam Mcr vs. L curves, selected lengths and critical mode shapes.

interaction “level”. The critical local and distortional buckling moments, as well as the corresponding buckling mode half-wave numbers (nL and nD) are also given in Table 1 (between brackets) – moreover, the local and distortional buckling mode shapes are also depicted in Fig. 3(a)-(c). In order to preclude the interaction with global (flexuraltorsional) buckling (L-D-G interaction), it was ensured that the critical global buckling moment (McrG) is much higher than its local and distortional counterparts (see Table 1).

Table 1 Selected simply supported lipped channel beam geometries and associated local and distortional critical buckling moments and buckling mode half-wave numbers (dimensions in mm and moments in kNcm). Beam

bw

bf

bl

t

L

McrD

McrL

McrG

RDL

B1 B2 B3

110 130 150

75 70 95

12.5 16.0 10.0

1.050 0.935 1.350

500 500 600

220.9(1) 331.2(1) 285.6(1)

217.0(7) 207.4(7) 473.6(6)

9796(1) 10,103(1) 21,223(1)

1.02 1.60 0.60

4. GBT post-buckling behaviour of beams experiencing L-D interaction considered in this work is almost twice that required to analyse lipped channel columns experiencing L-D interaction [7] – this is because of the lack of symmetry of the beam cross-section deformation patterns (unlike in the columns).

The selected simply supported (end cross-sections pinned and free to warp) beams are now used to investigate the influence of L-D interaction on their post-buckling behaviour, by means of GBT GNIA analyses, with the objective of shedding fresh light on the mechanics of the three aforementioned L-D interaction types – all post-buckling results are obtained adopting an arc-length control strategy. The results concerning beam B1, experiencing “true L-D interaction” (TI) are addressed first (Section 4.1), followed by those concerning beams B2 and B3, undergoing “secondary distortional-bifurcation L-D interaction” (SDI – Section 4.2) and “secondary local-bifurcation L-D interaction” (SLI – Section 4.3) – this sequence corresponds to a decreasing relevance of the L-D interaction effects [11]. All the beams analysed contain critical-mode initial imperfections with small amplitudes (10% of the wall thickness t), obtained from preliminary buckling analyses. All the GBT results are compared with values provided by shell finite element analyses (SFEA) carried out in ABAQUS [23], employing models previously adopted in the context of beam L-D interaction [11]. Note that, in order to achieve accurate post-buckling results, (i) fairly fine cross-section and axial discretisations must be adopted, and (ii) a quite high number of deformation modes must be included in the GNIA – these inputs are much more demanding than those needed to perform accurate buckling analyses.

3. GBT buckling behaviour – beam geometry selection In order to study the mechanics of the geometrically non-linear behaviour of lipped channel beams experiencing L-D interaction, it is indispensable to begin by selecting beams geometries (cross-section dimensions and lengths) prone to this coupling phenomenon. Based on previous works, this selection can be achieved on the sole basis of the closeness between the local (McrL) and distortional (McrD) critical buckling moments (a necessary condition4). Therefore, in order to shed new light on this interactive post-buckling behaviour, three simply supported beams are identified, namely (i) one experiencing “true L-D interaction”, with RDL = McrD /McrL ≈ 1.00, and (ii) two affected by “secondary bifurcation L-D interaction”, which can be either “local” (RDL ≈ 0.60) or “distortional” (RDL ≈ 1.60) − local buckling is always triggered by the compressed (top) flange. Table 1 provides the crosssection dimensions (bw, bf, bl, t – web-flange-lip widths and wall thickness) of the three steel (E = 210 GPa, v = 0.3) beams selected. On the other hand, Fig. 3(a)-(c) show curves depicting the variation of Mcr (critical buckling moment) with the beam length L (logarithmic scale) for the lipped channel simply supported beams – marked on each of these curves is the selected length adopted to ensure the desired

4.1. True L-D interaction Fig. 4(a1)-(a2) show the equilibrium paths M/Mcr vs. (v + v0)/t (v is the mid-span top flange-lip corner vertical displacement due to the applied moment M, i.e., excluding the initial value v0 = 0.1t), concerning the RDL = 1.02 beam (beam B1) with two critical-mode initial

4 As will be shown later, this condition is not sufficient in beams affected by “secondary (local or distortional) bifurcation L-D interaction”.

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Fig. 4. GBT (a) M/Mcr vs. (v + v0)/t post-buckling equilibrium paths of the RDL = 1.02 beam undergoing “true L-D interaction” and (b) modal amplitude diagrams – column containing (1) local and (2) distortional critical-mode initial geometrical imperfections.

respectively – Fig. 4(b1)-(b2) provide the corresponding modal participation diagrams. The modal participations factors provided by the GBT buckling analysis indicate major contributions from (i) local modes in the critical local buckling mode: 7–28.1%, 8–32.9%, 9–23.8%,

geometrical imperfections: (i) local with 7 half-waves (see Fig. 3(a)) and outward web curvature in the central one (Fig. 4(a1)) and (ii) distortional with a single half-wave, involving inward top flange-lip motions (Fig. 4(a2)), here termed RDL=1.02+L and RDL=1.02+D,

Fig. 5. Evolution of the RDL=1.02 beam mid-top flange transverse bending displacement profiles due to deformation modes (a) 5+6 (w5+6(x)), (b) 7–23 (w7–23(x)) and (c) 1–83 (w1–83(x)≡w(x)) at early loading stages for (1) RDL=1.02+L and (2) RDL=1.02+D beams.

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mid-top flange transverse displacement (w(x)) contributions from modes 5+6 (w5+6(x)), 7–23 (w7–23(x)) and 1–83 (w1–83(x)≡w(x)) – Fig. 5(a1)-(c1) and 5(a2)-(c2) concern the RDL=1.02+L and RDL=1.02+D beams, respectively. Figs. 6(a) and 7(a) show the evolution of the mid-top flange transverse displacement profiles (w(x)) due to modes 2 (w2(x)), 4 (w4(x)), 5+6 (w5+6(x)), 7–23 (w7–23(x)), 44–63 (w44–63(x)) and 1–83 (w1–83(x)≡w(x)), while Figs. 6(b) and 7(b) concern the top flange-web corner horizontal displacement profiles due to modes 3 (w3(x)), 44–63 (w44–63(x)), 1–83 (w1–83(x)≡w(x)) − RDL=1.02+L and RDL=1.02+D beams. The contributions from modes 4 and 5 + 6 (the latter is tiny) are not displayed, since their evolution is already depicted in Figs. 6(a) and 7(a). Moreover, the symbol (↓), appearing in Figs. 6 and 7, identifies an equilibrium state located in the equilibrium path descending branch (i.e., after the limit point). Finally, Fig. 8 shows beam deformed configurations at the loading stages M/Mcr = 0.50; 0.75; 1.00; 1.14 (peak moment) for both beams – the amplifications adopted for these configurations are indicated between brackets. The observation of all these post-buckling results prompts the following remarks:

10–2.6%, 11–2.4%, 12–3.3%, 13–1.2% (the remaining contributions are minute and concern other local modes and the two distortional modes – all modal amplitude functions are sinusoidal with 7 halfwaves) and (ii) distortional modes in the distortional critical bucking mode: 5–50.9%, 6–42.2%, 7–2.6%, 8–2.4% (the remaining contributions concern other local modes − all single half-waved). Moreover, to assess the relevance of the different deformation mode sets to the beam post-buckling behaviour, several “approximate analyses” were performed, including (i) conventional and linear transverse extension modes (1–23 + 44–63), (ii) conventional, shear and linear transverse extension modes (1–63) – both involving 8 finite elements – and (iii) conventional, shear and transverse extension (linear and quadratic) modes, considering 8 and 24 finite elements. On the other hand, Figs. 5(a)-(c), 6(a)-(b) and 7(a)-(b) are intended to show the relative importance of the contributions of the various deformation modes or mode sets to several relevant beam displacement longitudinal profiles. They display the evolution, along the most accurate equilibrium path in Fig. 4(a1)-(a2), of such displacements profiles due to those mode or mode sets – e.g., w7–23(x) stands for the longitudinal profile of displacement w(x) due to the sole contributions of modes 7–23. Fig. 5(a)(c) concern exclusively the initial post-buckling stages and show the

(i) The GBT equilibrium paths obtained with 24 finite elements and

Fig. 6. RDL=1.02+L beam displacement profiles: (a) mid-top flange transverse displacements (1) w2(x), (2) w4(x), (3) w5+6(x), (4) w7–23(x), (5) w44–63(x), (6) w1–83(x)≡w(x), and (b) top flange-web corner horizontal displacement (1) w3(x), (2) w44–63(x), (3) w1–83(x)≡w(x).

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Fig. 7. RDL=1.02+D beam displacement profiles: (a) mid-top flange transverse displacements (1) w2(x), (2) w4(x), (3) w5+6(x), (4) w7–23(x), (5) w44–63(x), (6) w1–83(x)≡w(x), and (b) top flange-web corner horizontal displacement (1) w3(x), (2) w44–63(x), (3) w1–83(x)≡w(x).

(iv) The comparison between the modal participation diagrams in Fig. 4(b1)-(b2) shows significant differences for M/Mcr < 0.90, reflecting the diverse natures of the initial geometrical imperfections included in the GNIA. Except the dominance of mode 2 in the early loading stages, the contributions of the local and distortional modes are quite distinct. For instance, at M/Mcr ≅ 0.20 one has p5+6 = 20% (34%) and p7–23 = 20% (3%), for the RDL = 1.02 + L (RDL = 1.02 + D) beam (remaining contributions minute at this stage). As loading progresses, the participation of mode 2 is gradually replaced by (iv1) the increasingly relevant distortional modes, whose contribution reaches a peak at M/Mcr = 0.920 (RDL=1.02+L) or M/Mcr = 0.879 (RDL=1.02+D) and (iv2) a smooth contribution decrease (increase) of the local modes in the RDL=1.02+L (RDL=1.02+D) beam. At the p5+6 peak one has: p1 = 0.2% (0.1%), p2 = 6.2% (10.7%), p3 = 1.8% (0.9%), p4 = 1.4% (0.4%), p5+6 = 72.5% (78.3%), p7–23 = 14.1% (7.9%), p24–43 = 1.0% (0.5%), p44–63 = 2.4% (1.0%) and p64–83 = 0.4%

all conventional, shear and transverse extension modes are extremely close to those provided by the ABAQUS SFEA up to (v + v0)/ t = 45 (RDL=1.02+L) and (v + v0)/t = 35 (RDL=1.02+D). (ii) Fig. 4(a1)-(b1) show the relevance of the quadratic transverse extension modes in capturing accurately the beam post-buckling behaviour, thus confirming the findings reported in [17]. Indeed, when these modes are not included in the analysis (solutions 1–63 in Fig. 4(a1)-(a2)), accurate results are only up to fairly low loading stages (M/Mcr ≅ 0.75) – for higher applied moments, membrane locking effects lead to overly stiff solutions. (iii) The most accurate equilibrium paths shown in Fig. 4(a1)-(a2) exhibit a fairly rapid stiffness erosion until an elastic limit point is reached at M/Mcr ≅ 1.14 – "true L-D interaction" precludes reaching higher strengths (peak moment close to Mcr). Higher post-critical strengths are reached in the beams affected by the other L-D interaction types – see Figs. 9 and 14, concerning beams B2 (SDI) and B3 (SLI). 9

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Fig. 8. ABAQUS RDL=1.02+L (top) and RDL=1.02+D (bottom) beam deformed configurations at M/Mcr = 0.50; 0.75; 1.00; 1.14 (peak moment).

Fig. 9. GBT (a) M/McrL vs. (v + v0)/t equilibrium paths of the RDL=1.60 beam (“secondary distortional-bifurcation L-D interaction”) with a local initial geometrical imperfection and (b) corresponding modal participation diagram.

waves as loading progresses (see Fig. 5(b1)) – indeed, as it would be logical to expect, w7–23(x) is a mere amplification of the initial imperfection. On the other hand, w5+6(x) (Fig. 5(a1)) shows very clearly the gradual deformed configuration change from 7 halfwaves (akin to the initial imperfection – e.g., at M/Mcr = 0.094) to a single distortional half-wave (akin to the distortional critical buckling mode) – e.g., at M/Mcr = 0.402 this displacement profile is already fairly close to a sinusoid. Conversely, the w5+6(x) profile of the RDL=1.02+D beam retains the initial single half-wave as loading progresses (Fig. 5(a2)) while the w7–23(x) shape begins to change from 1 half-wave (e.g., at M/Mcr = 0.597 in Fig. 5(b2)) to 7 half-waves (e.g., at M/Mcr ≥ 0.879 in Fig. 5(b2)), akin to the critical local bucking mode shape. Moreover, at these early loading stages w(x) (Fig. 5(c1)-(c2)) reflects the combination of these two deformation sources (and also mode 2), providing clear evidence of L-D interaction – the gradual emergence, at early

(0.2%) in the RDL=1.02+L (RDL=1.02+D) beam. As loading increases, the participations of modes 3 and 4 (and also the shear and transverse extension modes) start to become more relevant and are responsible for the occurrence of limit points approximately at the same loading level (M/Mcr ≅ 1.14). These participations are “compensated” by the reduction in the participation of modes 2 and (mostly) 5+6 – at this stage, the various modal contributions are practically the same in both beams: 1 (0.7% vs. 0.7%), 2 (5.1% vs. 5.0%), 3 (6.9% vs. 6.5%), 4 (9.1% vs. 8.6%), 5+6 (56.1% vs. 57.1%), 7–23 (9.7% vs. 10.2%), 24–43 (3.2% vs. 3.0%), 44–63 (8.1% vs. 7.8%) and 64–83 (1.1% vs. 1.1%), thus explaining the near coincidence of the two equilibrium paths in the advanced post-buckling stages. (v) Turning the attention now to the beam early loading stages (Fig. 5(a1)-(c2)), the RDL=1.02+L beam mid-top flange transverse displacement profile w7–23(x) retains the initial (equal) 7 half-

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Fig. 10. Evolution of the RDL=1.60 beam top flange transverse bending displacement profiles due to deformation modes (a) 5+6 (w5+6(x)), (b) 7–23 (w7–23(x)) and (c) 1–83 (w1–83(x)≡w(x)) – early loading stages.

4.2. Secondary distortional-bifurcation L-D interaction

loading stages, of deformation patterns akin to the critical local and distortional buckling modes characterises “true L-D interaction”5 (see all the deformed configurations in Fig. 8). (vi) At higher loading stages (Figs. 6(a1)-(b3) and 7(a1)-(b3)) there is a strong qualitative resemblance between the displacements profiles of the two beams. Moreover, these profiles show that the dominant contribution to w(x) comes from the distortional modes 5+6 (Figs. 6(a3) and 7(a3)), as was already quantified in the modal participation diagrams of Fig. 4(b1)-(b2) – this is a feature typical of L-D interaction (regardless of the type, as will be shown in the following sections). It is interesting to notice that the local mode shape w7–23(x) (Figs. 6(a4) and 7(a4)) progressively moves away from the initial 7 equal-shaped half-waves – these half-waves become unequal and more pronounced (downwards) in the beam central region. This feature clearly illustrates the key role of the distortional deformations in the beam deformed configuration (particularly in the RDL=1.02+D beam), since the web outward curvature of w5+6(x) attracts the three central local half-waves to display similar deformations – only the two outer half-waves (RDL=1.02+L and RDL=1.02+D beams) retain the initial “sign”, since w5+6(x) is much smaller at those regions. Naturally, a local initial imperfection with an opposite “sign”, or involving an even half-wave number, will certainly entail diverse deformation patterns, but all sharing the same qualitative feature: distortional deformations govern the behaviour of beams affected by L-D interaction – indeed, initial imperfections with even half-wave numbers invariably lead to asymmetric deformation patterns.6 (vii) It is also worth noting the emergence of modes 3 (Figs. 6(b1) and 7(b1)) and 4 (Figs. 6(a2) and 7(a2)), which are neither directly caused by the loading (major-axis bending) nor akin to the beam global critical buckling mode. They stem from the stress redistribution taking place in the beam compressed (top) half at the advanced loading stages and are, most likely, responsible for the beam rapid stiffness erosion.7 Concerning the contributions of the transverse extensions modes to w(x) (Figs. 6(a5)+(b2) and 7(a5) +(b2)), they exhibit “irregular” single half-waves (see Figs. 6(a5) and 7(a5)), which are more pronounced in the beam central region (where the highest w5+6(x) values occur).

Since the results addressed in this section, which concern beam B2, are qualitatively similar to those reported in Section 4.1 (for beams B1), their presentation and discussion are abbreviated as much as possible – the same will happen in Section 4.3 (results concerning beam B3). Fig. 9(a) shows M/McrL vs. (v + v0)/t equilibrium paths, obtained again with an arc-length control strategy and concerning a beam containing a 7 half-wave local initial geometrical imperfection (akin to the critical buckling mode) with the central half-wave involving outward web curvature. A preliminary GBT buckling analysis showed that the most relevant contributions to the 7 half-wave beam critical buckling mode are from local deformation modes: 7–27.1%, 8–26.5%, 9–24.0%, 10–5.6%, 11–1.9%, 12–6.4%, 13–4.9% (participations similar to the beam B1 ones – the remaining ones are minute and concern again other local modes and the two distortional modes). Once more, the several equilibrium paths are determined with various deformation mode sets: (i) conventional and linear transverse extension modes (1–23 + 44–63) and (ii) conventional, shear and linear transverse extension modes (1–63), with 8 finite elements in both cases, and (iii) conventional, shear and (linear and quadratic) transverse extension modes, with 8 or 30 finite elements – Fig. 9(b) shows the modal participation diagram associated with the most accurate equilibrium path (30 finite elements). Figs. 10(a)-(c) and 11(a1)-(b3) are the counterparts of Figs. 5(a)-(c) and 6(a1)-(b3) (or 7(a1)-(b3)) and display the evolutions of relevant displacement profiles along that same most accurate equilibrium path – the first ones cover only the early loading stages. Finally, Fig. 12 shows the beam deformed configurations at M/McrL = 0.75; 1.00; 1.25; 1.38 (peak moment). The observation of the results presented in these figures leads to the following conclusions: (i) The general comments made in items (i) and (ii) of Section 4.1 remain valid. However, it should be noted that, due to a more refined longitudinal discretisation (30 finite elements, instead of 24), the GBT and SFEA results are now practically coincident. Moreover, the gradual stiffness erosion is a bit smaller in beam B2, leading to the occurrence of an elastic limit point at a slightly higher applied loading level (M/McrL = 1.38). (ii) Generally speaking, the modal participation diagram of Fig. 9(b) is qualitatively similar to that shown in Fig. 4(b1) for the RDL=1.02+L beam. The main difference between them is the (expected) higher dominance of the local modes in the beam structural response. As before, mode 2 plays a key role in the beam deformed configuration at the early loading stages and, then, is gradually reduced, due to the surprising appearance modes 5 + 6 (mostly mode 5), which is discussed in detail in item (iii) – the local mode participations remain practically constant up to M/McrL

5 Note that the shape of w5+6(x) in the RDL=1.02+L beam becomes a sinusoid much sooner (lower loading stage) than, in the RDL=1.02+D beam, the shape of w7–23(x) changes to exhibit 7 half-waves – this behavioural feature will be discussed in Section 4.2. 6 This same behavioural feature also occurs in lipped channel beams undergoing distortional-global interaction – see [9]. 7 This same behavioural feature also occurs, although to a smaller extent (smaller rigidbody motions) in beams B2 and B3, addressed in Sections 4.2 and 4.3 – this fact probably explains the slower stiffness erosion, making it possible to achieve higher post-critical strengths.

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Fig. 11. RDL=1.60 beam displacement profiles: (a) mid-top flange transverse displacements (1) w2(x), (2) w4(x), (3) w5+6(x), (4) w7–23(x), (5) w44–63(x), (6) w1–83(x)≡w(x), and (b) top flange-web corner horizontal displacement (1) w3(x), (2) w44–63(x), (3) w1–83(x)≡w(x).

Fig. 12. ABAQUS RDL=1.60 beam deformed configurations at M/McrL = 0.75; 1.00; 1.25; 1.38 (peak moment).

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Fig. 13. RDL=1.60 beam mid-top flange transverse displacement profiles (1) w5(x), (2) w6(x), (3) w5+6(x): (a) early and (b) advanced loading stages.

the dominance of the major-axis flexural deformations8 (downwards with a single half-wave) in the early loading stages (p2 > 65% – see Fig. 9(b)). In order to search for the root of this behavioural feature, Fig. 13(a1)-(b3) show the individual and combined contributions from modes 5 and 6 (w5(x), w6(x), w5+6(x)) to the mid-top flange transverse displacement profiles at early (Fig. 13(a1)-(a3)) and advanced (Fig. 13(b1)-(b3)) loading stages. The observation of these figures makes it possible to conclude that: (iii.1) The contributions from modes 5 and 6 are clearly distinct in the early and advanced loading stages. In the early loading stages (Fig. 13(a1)-(a3)) w5(x), which involves inward top flangelip motions, strongly prevails over w6(x), which also involves inward top flange-lip motions – note that w5+6(x) does not differ too much from w5(x). In addition, w5(x) switches from 7 halfwaves to 1 half-wave much “sooner” than w6(x). In the authors’ opinion, the early emergence of single half-wave contributions from modes 5 and 6 stems from flange curling [24–26], a phenomenon occurring in beams with wide and thin flanges and characterised by inward flange-lip motions.9 This provides a

= 0.473. At this loading level, one has p2 = 39.4%, p4 = 1.8%, p5 = 18.5%, p6 = 6.7%, p7–23 = 30.0%, p24–43 = 0.8% and p44–63 = 2.2%. For higher loadings (M/McrL > 0.473) the participations of modes 5 and 6 grow steadily (mode 5 always prevalent, although such prevalence starts to diminish after M/McrL = 0.863), matching the local mode contribution at M/McrL = 0.800 and reaching a combined peak of 64% at M/McrL = 1.312 – this growth is compensated by the drastic reduction of the contributions from modes 2 (mostly) and 7–23. At M/McrL = 1.312, one has p2 = 5.2%, p3 = 2.7%, p4 = 1.9%, p5 = 33.9%, p6 = 30.0%, p7–23 = 20.0%, p24–43 = 1.5%, p44–63 = 3.8% and p64–83 = 0.6%. From here on the participations of modes 5 and 6 start to decrease, due to the “sudden” increase in relevance of the minor-axis flexure (3), torsion (4), shear and linear transverse extension modes – at M/ McrL = 1.345↓ the contributions from these modes reach 6.1%, 6.2%, 3.0%, 8.1%, respectively, while the participations from modes 2, 5+6, and 7–23 decrease to 4%, 58% and 12%, respectively. (iii) As expected, in the early loading stages w(x) (Fig. 10(c)) has dominant contributions from (iii1) the local modes 7–23 (Fig. 10(b)) and (iii2) major-axis flexure (mode 2 – Fig. 11(a1)). Like in beam B1, Fig. 10(a) shows clearly the w5+6(x) shape switch from 7 half-waves (akin to the initial imperfection – e.g., at M/McrL = 0.094) to a single half-wave (akin to the distortional critical buckling mode – e.g., at M/McrL = 0.613, w5+6(x) is already close to a perfect sinusoid). This early switch is quite unexpected, in view of how far apart the local and distortional critical buckling moments are. Indeed, the w5+6(x) profiles of beams B2 (RDL=1.60) and B1 (RDL=1.02) are fairly similar, even if the halfwave number switch takes place slightly “sooner” in the latter. It is also worth noting that such similarity was not observed in lipped channel columns experiencing TI and SDI (e.g., [7]), which means that there must be some mechanical reason causing/precipitating this “unexpected” behavioural feature, certainly associated with

8 Another effect due to the predominance of major-axis flexure in the early loading stages was recently observed in [9]: the equilibrium path of beams with a small-amplitude single half-wave distortional initial geometrical imperfection involving outward top flange-lip motions undergo, in the early loading stages, opposite distortional deformations (inward top flange-lip motions – see a complete discussion on this topic in [9]). This also explains the adoption of the initial imperfection shape in the RDL=1.02+D beam (and also employed in beam B3 – see Section 4.3): involving inward top flange-lip motions – beams with outward initial top flange-lip motions only exhibit the expected behaviour for initial geometrical imperfection amplitudes considerably higher than 0.1t. 9 The deformation pattern associated with flange curling is similar to deformation mode 5, in the sense that both flange-lip assemblies move inward. The presence of mode 6 reflects the asymmetry between the motions of the compressed and tensioned flange-lip assembly motions – since the former is larger than the latter, mode 6 performs the appropriate “reinforcing” and “alleviating” tasks.

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Fig. 14. nD = 2 RDL=1.60 lipped channel beam with a local initial geometrical imperfection: (a) M/McrL vs. (v + v0)/t equilibrium paths and (b) deformed configurations at M/McrL = 1.25; 1.50; 1.25↓ (amplified twice).

logical explanation for both the single half-wave (like ϕ2(x)) and the prevalence of mode 5 over mode 6. (iii.2) In the advanced loading stages (Fig. 13(b1)-(b3)), the picture changes significantly: both w5(x) and w6(x) exhibit single half-wave sinusoids with amplitudes of the same order of magnitude, even the w5(x) ones are a bit larger – now w5+6(x) is almost twice w5(x). The relation between the single half-wave contributions from modes 5 and 6 is now very similar to that exhibited by the beam critical distortional buckling mode (modal participations 5–48.5% and 6–41.7%) – the exact p5 /p6 ratio is achieved for M/McrL > 1.244. It seems fair to argue that this distortional deformation pattern stems from secondary distortional-bifurcation L-D interaction. (iii.3) Summarising the contents of the two previous items, the contributions of modes 5 and 6 to the beam deformed configuration have two sources along the beam equilibrium path (besides the small contribution associated with the initial geometrical imperfection, present from the beginning of loading), both involving single half-wave inward top flange-lip motions: (iii1) flange curling, starting at the early loading stages, involving predominantly mode 5 and fading as the loading progresses into the intermediate loading stages, and (iii2) secondary distortional-bifurcation L-D interaction, starting only at advanced loading stages (close vicinity of the critical distortional buckling moment level). (iii.4) The above explanation applies also to the sudden change in the w5+6(x) displacement profiles in the RDL = 1.02 + L, when compared with those of the RDL = 1.02 + D beam (see footnote 5 in Section 4.1). Moreover, it becomes also clear why L-D interaction in beams with nD = 1 and susceptible to flange curling always involves inward distortional top flange-lip motions: the “attractive power” of the deformation pattern associated with flange curling. (iii.5) The fact that the both sources mentioned in the previous item cause single half-wave flange-lip motions clouds significantly this issue, because it becomes very difficult to know what originates from each source. In an attempt to shed new light on the topic, a second RDL = 1.60 lipped channel beam (bw = 130, bf = 80, bl = 20, t = 0.975, L = 1500 mm) is analysed – it differs from the first one in the fact that its critical distortional buckling mode has two half-waves. Fig. 14(a) shows the M/McrL

vs. (v + v0)/t equilibrium path, obtained by means of an ABAQUS SFEA,10 while Fig. 14(b) displays deformed configurations at M/ McrL = 1.25; 1.50; 1.25↓ – equilibrium states “I”, “II” and “III” indicated in the equilibrium path. It is possible to observe clearly that the deformed configurations corresponding to states “I” and “II” exhibit a single distortional half-wave (due to flange curling), while the deformed configuration associated with state “III” exhibits two distortional half-waves (due to L-D interaction). (iv) As pointed out earlier, the distortional deformations stemming from L-D interaction, which emerge at M/McrL ≈ 1.244, rapidly become the dominant source of the cross-section in plane deformation (see Fig. 11(a3)-(a6)). Concerning the contributions from the remaining modes, they follow the same general trends of those obtained for beams B1 (RDL=1.02+L and RDL=1.02+D) at advanced loading stages. Nevertheless, there are a few differences: (iv1) the two local half-waves adjacent to the central one reverse sign at higher loading levels11 (e.g., M/McrL = 1.364↓ – see Fig. 11(a4)), due to the added relevance of the local mode contribution in this beam (due to the larger RDL value, there is more “room” for the local deformations to develop before the dominant distortional deformations emerge), and (iv2) the contributions from the global (2, 3, 4) and transverse extension modes are now slightly more irregular/wavy, although they retain similar patterns. (v) Based on the above findings, it may be concluded that, in beams, the SDI effects are quite close to the TI ones (i.e., more severe than anticipated). This does not change when RDL increases (obviously, the strength also increases), which means that there is a significant strength erosion stemming from this type of L-D interaction – the current DSM local and distortional design curves are unable to predict adequately the failure moments of such beams [11].

4.3. Secondary local-bifurcation L-D interaction Fig. 15(a)-(b) shows the RDL = 0.60 beam (beam B3) equilibrium paths M/McrD vs. (v + v0)/t, obtained with a critical-mode initial geometrical imperfection (single distortional half-wave with inward top flange-lip motions – see Fig. 15(a)) and the corresponding modal 10 Due to current (temporary) memory limitations of the software developed to perform the GBT GNIA, it is not possible to present here GBT-based results for this beam. 11 Naturally, the difference becomes more pronounced as RDL increases.

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Fig. 15. (a) M/McrD vs. (v + v0)/t post-buckling equilibrium paths of the beam B3 (“secondary local-bifurcation L-D interaction”) with a distortional initial geometrical imperfection and (b) modal participation diagram corresponding to path F.

loading progresses, mode 2 is gradually replaced by modes 5+6. For instance, at M/McrD = 0.503 the contribution of mode 2 is reduced from 64% to 28% while that of the distortional modes 5+6 increases from 33% to 65% – the remaining modal contributions are minute at this stage: p4 = 1%, p7–23 = 3% and p44–63 = 1%. Progressing along the equilibrium path (M/McrD > 0.503), the participation of modes 5+6 gradually increases up to a peak value of 86% at M/McrD = 0.976, which corresponds to the emergence of relevant contributions from modes 3 and 4. Beyond M/McrD = 0.976, the contribution of modes 5+6 is partly replaced by that of the remaining modes or mode sets: at M/McrD = 1.500 one has p1 = 1%, p2 = 4%, p3 = 7%, p4 = 9%, p5+6 = 61%, p7–23 = 7% (mostly modes 7–10: p7 = 1.1%, p8 = 2.6%, p9 = 2.2% and p10 = 0.6%), p24–43 = 3% (mostly mode 25: p25 = 1%), p44–63 = 8% (mostly modes 44–48: p44 = 1.9%, p45 = 0.8%, p46 = 0.3%, p47 = 1.0% and p48 = 2.7%), and p64–83 = 1%. (iii) Fig. 15(a) clearly shows a significant stiffness drop of the beam B3 equilibrium path at M/McrD ≈ 1.20, the applied loading level corresponding to the emergence of local deformations (see Fig. 16(a4)). (iv) For M/McrD < 1.20, the beam post-buckling behaviour is clearly distortional, i.e., w(x) exhibits a single half-wave (see Fig. 16(a6)) combining a dominant contribution from the distortional modes 5+6 (Fig. 16(a3)) with contributions from mode 2 (major-axis flexure – Fig. 16(a1)) and, to a smaller extent, local (7–23 – Fig. 16(a4)) and transverse extension (44–63 – Fig. 16(a5)) modes − there are also minute contributions from modes 3 (minor-axis flexure pointing towards de web – Fig. 16(b1)) and 4 (torsion – Fig. 16(b2)), “flattened” in the beam central region. Qualitatively speaking, the displacement profiles evolve similarly to those reported in [11], concerning a beam exhibiting a distortional postbuckling behaviour (McrD < McrL ≪ McrG). This post-buckling behaviour is clearly illustrated by the first two deformed configurations in Fig. 17, concerning loading levels M/McrD = 0.75 and M/McrD = 1.00. (v) When the applied loading increases beyond M/McrD > 1.20 the w (x) shape gradually switches from a single half-wave to seven (unequal) half-waves, due to the emergence and development of local deformations akin to the second local buckling mode (see the evolution of w7–23(x) in Fig. 16(a4)). The (apparently) surprising emergence of local deformations not akin to the critical local buckling mode, which contains six half-waves (see Table 1), is due

participation diagram. The GBT buckling analysis showed that the critical buckling mode combines major contributions from modes 5–52.3% and 6–43.0% – the remaining (minute) contributions are from local modes (7-1.4%, 8-1.7%, 9-1.0%) and also exhibit a single halfwave. Unlike for beams B1 and B2, all equilibrium paths depicted in Fig. 15(a) were determined by including all the deformation modes available in the GNIA, while considering different cross-section and/or longitudinal discretisations, i.e., differences only in the numbers of intermediate nodes and/or finite elements (FEs): (i) 3 flange + 3 web nodes and 8 FEs (“A”), (ii) 3 flange + 5 web nodes and 8 FEs (“B”), (iii) 3 flange + 7 web nodes and 8 FEs (“C”), (iv) 3 top flange + 1 bottom flange + 9 web nodes and 8 FEs (“D”), (v) 3 top flange + 1 bottom flange + 11 web nodes and 20 FEs (“E”) and (vi) 3 top flange + 1 bottom flange + 11 web nodes and 26 FEs (“F”). Moreover, Fig. 16(a)(b) display displacement profiles similar to those shown in previous sections: Fig. 16(a) depicts the mid-top flange transverse displacement profiles due to the contributions of modes 2, 4, 5+6, 7–23, 44–63 and 1–83 (w2(x), w4(x), w5+6(x), w7–23(x), w44–63(x), w1–83(x)≡w(x)) while Fig. 16(b) displays the top flange-web corner horizontal displacement profiles due to the contributions of modes 3, 44–63 and 1–83 (w3(x), w44–63(x), w1–83(x)≡w(x)). Finally, Fig. 17 shows the beam deformed configurations at four post-buckling stages (M/McrD = 0.75; 1.00; 1.25; 1.45). These results make it possible to draw the following conclusions: (i) Fig. 15(a) provides clear confirmation of the key role played by the number of web intermediate nodes in the accuracy of the results obtained, particularly after the emergence and development of L-D interaction (as discussed below). Indeed, the accuracy of the GBT equilibrium path continuously improves with the number of web intermediate nodes (between solutions “A” and “E”).12 For the most refined web discretisation, the inclusion of all deformation modes available (conventional, shear and transverse extension – linear and quadratic) and the consideration of 26 FEs provide a virtually perfect solution up to M/McrD = 1.50 – solution “F”. (ii) The modal participation diagram of Fig. 15(b) shows that, at early loading stages, there are mainly contributions of modes 2 and 5 + 6 – the latter are already included in the initial imperfection. As

12 In columns affected by L-D interaction (e.g., [7]), it was found that accurate equilibrium paths can be obtained with only 3 web intermediate nodes – the fact that web deformation is now much more complex (double curvature and lack of symmetry) explains the need for a (much) more refined web discretisation.

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Fig. 16. Beam B3 displacement profiles: (a) mid-top flange transverse displacements (1) w2(x), (2) w4(x), (3) w5+6(x), (4) w7–23(x), (5) w44–63(x), (6) w1–83(x)≡w(x), and (b) top flange-web corner horizontal displacement (1) w3(x), (2) w44–63(x), (3) w1–83(x)≡w(x).

to the fact that (v1) the first and second local buckling moments are practically coincident (473.58 vs. 473.89 kN cm)13 and (v2) the latter is symmetric, like the (single half-wave) critical distortional buckling mode – this shared symmetry favours the interaction between these two buckling modes.14 This is confirmed by the growing visibility of a seven half-wave local contribution to w (x), as the applied loading increases beyond M/McrD = 1.20 – see also the deformed configurations at M/McrD = 1.25 and M/McrD = 1.45 in Fig. 17. Note that the amplitudes of the local half-waves are unequal, which stems from the dominant role of the distortional modes. Indeed, the three central half-waves exhibit outward web curvature (like the distortional modes) and only the outer half-waves (located near the beam ends) have opposite curvature –

the role of the distortional modes is much smaller. The emergence of a deformation pattern akin to the critical or non-critical15 local buckling mode, at fairly advanced loading stages, characterises the so-called “secondary local-bifurcation L-D interaction” (SLI). Like in columns, the beam SLI effects are less pronounced than the TI and SDI ones. Moreover, in beams with higher RDL values, it is necessary to progress further along the equilibrium path (i.e., reach higher loadings) to observe the emergence and development of SLI interaction – naturally, if RDL is high enough, this interaction does not occur and the beam exhibits a “pure” distortional post-buckling behaviour [11]. (vi) Concerning the other modal contributions to the evolution of w(x), the configurations of w2(x), w3(x) and w4(x) become progressively more “flattened” in the beam central region and their configurations remain pratically unchanged as loading progresses (only the amplitudes increase) – recall that modes 3+4 emerge due to the stress redistribution caused by the distortional deformations in the

13 Most likely because of this extreme buckling moment closeness, numerical difficulties were encountered in the vicinity of M/McrD = 1.20. In order to prevent “jumps” to neighbouring equilibrium states (involving 5 local half-waves – the odd half-wave number associated with the second lowest local buckling moment), it was necessary to adopt very small arc-length values (to stay in the correct equilibrium path). 14 Similar observations have been reported in other buckling phenomena (e.g., columns [8] and beams [9] under D-G interaction).

15 Depending on whether the critical local half-wave number is odd or even, respectively.

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Fig. 17. ABAQUS beam B3 deformed configurations at M/McrD = 0.75; 1.00; 1.25; 1.45.

distortional critical buckling modes are already present at quite early loading stages, and the two beam deformed configurations practically coincide at advanced loading stages. The L-D interaction effects increase along the equilibrium path, causing significant beam strength erosion – “true L-D interaction” (close critical local and distortional buckling moments) is the most detrimental type of interaction. Since these beams reach limit points at loading levels less than 15% above Mcr ≈ McrL ≈ McrD, there a fairly low “cap” on the strength benefit obtained from increasing the yield stress (i.e., the use of high-strength steels may be only marginally beneficial). (iii) In beam B2, undergoing “secondary distortional-bifurcation L-D interaction”, the deformation pattern akin to the local critical buckling mode, expected to remain unchanged until the advanced loading stages (note that RDL = 1.60), was altered at a loading level much lower than McrD due to the occurrence of flange curling, which leads to the appearance of a deformation pattern combining the distortional deformation modes 5 (dominant) and 6 but not akin to the beam critical distortional buckling mode – subsequently, it was found that flange curling also occurs in beams B1 (it remained undetected until the analysis of beam B2). The effect of flange curling fades as the loading increases and, in the vicinity of the critical distortional buckling moment level, is “replaced” by distortional deformations akin to the critical distortional buckling mode, which corresponds to the occurrence of “secondary distortional-bifurcation L-D interaction”. The findings unveiled from the analysis of beam B2 were confirmed (and even “reinforced”) by the analysis of a second beam undergoing “secondary distortional-bifurcation L-D interaction” (with the same RDL value), but exhibiting a two half-wave critical distortional buckling mode. In the end, it was found that the post-buckling behaviours of the B1 and B2 beams were qualitatively fairly similar (much more than it had been anticipated). (iv) In beam B3, undergoing “secondary local-bifurcation L-D interaction”, the deformation pattern akin to the critical distortional buckling mode remains practically unchanged (i.e., the beam exhibits a typical distortional post-buckling behaviour) until well above the critical buckling moment level (M/McrD ≈ 1.20). Beyond this loading level, local deformation emerge and L-D interaction effects cause a sudden stiffness drop. However, it should be noted that the local deformation pattern is not akin to the beam critical local buckling mode, which has six half-waves – instead, it is akin to the local buckling mode, which has seven half-waves. This behavioural feature, which is not restricted to beams undergoing “secondary local-bifurcation L-D interaction”, is due to (iv1) the shared longitudinal symmetry of second local and critical distortional buckling modes and (iv2) the virtual coincidence of the first two local buckling moments. Since the amount of L-D interaction taking place in beam B3 is much smaller than in beams B1 and B2, the strength erosion is also smaller and, thus, no limit point occurs prior to M/McrD = 1.50 (highest loading applied). (v) The mechanics of the three types of L-D interaction (true and secondary local/distortional-bifurcation L-D interaction) are strongly influenced by the distortional deformations (core of the

top half cross-section. The profiles w44–63(x) (Fig. 16(a5) and (b2)) pratically retain their shapes and exhibit larger values in the beam central region (like the profiles w5+6(x)). (vii) As it would be logical to expect, the mechanics of “secondary local bifurcation L-D interaction” are also strongly influenced by the distortional deformations (core of the beam deformed configuration). 5. Conclusion A GBT-based numerical investigation concerning the geometrically non-linear (elastic) post-buckling behaviour of simply supported lipped channel beams under uniform bending and affected by three types of LD interaction, namely (i) “true L-D interaction”, (ii) “secondary distortional-bifurcation L-D interaction” and (iii) “secondary local-bifurcation L-D interaction”, was reported. Particular attention was paid to the structural interpretation of the behavioural features stemming from the L-D interaction effects, aimed at shedding fresh light on the mechanics of this coupling phenomenon. The results presented and discussed consisted of equilibrium paths, modal participation diagrams and plots providing the evolution, along those equilibrium paths, of the beam (i) relevant displacement profiles and (ii) deformed configurations. They were validated through the comparison with values yielded by ABAQUS shell finite element analysis. After presenting the main steps involved in the development of the GBT formulation to analyse the elastic non-linear behaviour of thinwalled members, GBT buckling analyses were employed to select the geometries of simply supported lipped channel beams experiencing the three types of L-D interaction considered in this work – local buckling was triggered by the compressed flange in all of them, which corresponds to the case most commonly encountered in practice. Then, the results obtained by means of GBT geometrically non-linear imperfect analyses of beams undergoing distinct types of L-D interaction were presented and discussed, taking advantage of the GBT modal features – the compressed flange transverse bending behaviour was the main vehicle used to acquire in-depth knowledge on the mechanics of L-D interaction. Among the various findings of this investigation, the following ones deserve to be specially mentioned: (i) In spite of the distinct mechanics exhibited by all the beams analysed, experiencing different types of L-D interaction (true and secondary local/distortional-bifurcation L-D interaction), all of them share a common trait: occurrence of deformation patterns akin to both local and distortional critical (or non-critical) buckling modes. The coupling of these modes leads to a post-buckling strength erosion whose relevance depends on the distortional-tolocal critical buckling moment ratio (L-D interaction nature) and applied loading level. Thus, the ultimate strength of elastic-plastic beams with these geometries will be eroded by the occurrence of LD interaction – naturally, the amount of erosion depends also on the yield stress value. (ii) In beams B1 (RDL=1.02+L and RDL=1.02+D), undergoing “true L-D interaction”, deformation patterns akin to the local and 17

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deformation patterns related to this coupling phenomenon), even for “secondary distortional-bifurcation L-D interaction” (McrL < McrD or McrL ≪ McrD) – local buckling plays a secondary role. Regardless of the L-D interaction type and initial geometrical imperfection nature, distortional deformations are always more relevant than local ones, as is clearly attested (and quantified) by all the modal participation diagrams presented, as well as by several of the displacements profile evolutions displayed.

[8]

[9]

Finally, one last word to mention that the authors are currently extending the investigation reported in this work to cover beams subjected to non-uniform bending, a loading case much more often encountered than uniform bending and, therefore, of greater practical interest.

[10] [11]

[12]

Acknowledgments [13]

The first author gratefully acknowledges the financial support of FCT (Fundação para a Ciência e a Tecnologia – Portugal), through the doctoral scholarship SFRH/BD/87746/2012.

[14]

[15]

References

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[1] A.D. Martins, D. Camotim, P.B. Dinis, On the direct strength method approach for cold-formed steel columns failing in local-distortional interactive modes, in: R. LaBoube, W.-W. Yu (Eds.), Proceedings of Wei-Wen Yu International Specialty Conference on Cold-Formed Steel Structures (Ccfss 2016 – Baltimore, 9-10/9), 2016, pp. 135–154. [2] P.B. Dinis, B. Young, D. Camotim, Local-distortional-global interaction in coldformed steel lipped channel columns: behaviour, strength and DSM design, in: Website Proceedings of the SSRC Annual Stability Conference (Orlando, 12-15/4), 2016. [3] P.B. Dinis, D. Camotim, Behaviour and DSM design of hat, zed and rack columns experiencing local-distortional-global interaction, in: B. Young, Y. Cai (Eds.), USB Key Drive Proceedings of the 8th International Conference on Steel and Aluminium Structures (Icsas 2016 – Hong Kong, 5-7/12), Paper 60, 2016. [4] N. Silvestre, D. Camotim, GBT-based analysis of the local-plate/distortional buckling mode interaction in lipped channel columns, in: M. Pignataro, J. Rondal, V. Gioncu (Eds.), Proceedings of the 4th International Conference on Coupled Instabilities in Metal Structures (Cims 04 – Rome, 27-29/09), Editura Orizonturi Universitare (Timisoara), 2004, pp. 449–462. [5] A.D. Martins, D. Camotim, P.B. Dinis, P. Providência, On the mechanics of localdistortional interaction in lipped channel thin-walled columns, in: Website Proceedings of the SSRC Annual Stability Conference (Nashville, 24-27/3), 2015. [6] A.D. Martins, D. Camotim, P.B. Dinis, Mechanics of the local-distortional interaction in fixed-ended lipped channel columns, in: D. Dubina, V. Ungureanu (Eds.), Proceedings of the International Colloquium on Stability and Ductility of Steel Structures (Sdss 2016 – Timisoara, 30/05-01/6), Wiley-Ernst & Sohn (Mem Martins), 2016, pp. 345–352. [7] A.D. Martins, D. Camotim, P.B. Dinis, GBT-based assessment of the mechanics of

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