Distortional failure and DSM design of cold-formed steel lipped channel beams under elevated temperatures

Thin-Walled Structures 98 (2016) 75–93

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Distortional failure and DSM design of cold-formed steel lipped channel beams under elevated temperatures Alexandre Landesmann a, Dinar Camotim b,n a b

Civil Engineering Program, COPPE, Federal University of Rio de Janeiro, Brazil ICIST, CEris, DECivil, Instituto Superior Técnico, Universidade de Lisboa, Portugal

art ic l e i nf o

a b s t r a c t

Article history: Received 3 February 2015 Received in revised form 9 June 2015 Accepted 10 June 2015 Available online 11 August 2015

The aim of this paper is to report an ongoing shell finite element investigation on the distortional postbuckling behaviour, ultimate strength and DSM design of cold-formed steel single-span lipped channel beams under elevated temperatures due to fire conditions. The beam ultimate strengths are computed by means of a steady-state loading strategy that consists of applying an increasing major-axis uniform bending moment to a beam under a uniform and constant (elevated) temperature distribution, until failure is reached. The steel constitutive law at elevated temperatures is simulated by means of the stress–strain–temperature models prescribed in EC3-1.2 for cold-formed steel. The materially and geometrically non-linear response of the cold-formed lipped channel beams is determined through ANSYS shell finite element analyses that incorporate critical-mode (distortional) initial geometrical imperfections. Finally, since there are no specific rules to predict the bending strength of cold-formed steel beams failing in distortional modes at elevated temperatures, the above failure moments and temperatures are used to establish preliminary guidelines for the design of cold-formed steel lipped channel beams under fire conditions. The approach followed is based on the Direct Strength Method (DSM) and has been already employed by other researchers – the currently available design/strength equations/curves, developed for ambient temperature, are modified to account for the appropriate Young’s modulus and yield stress reductions due to the temperature increase. The results presented in the paper provide solid numerical evidence that the currently codified DSM distortional strength curve is not appropriate to predict beam failure moments at elevated temperatures. This is because it either (i) does not account for the material non-linearity due to the elevated temperature (low-to-moderate slenderness range) or (ii) is already inadequate to predict room temperature failure moments (unsafe estimates obtained in the high slenderness range). & 2015 Elsevier Ltd. All rights reserved.

Keywords: Cold-formed steel beams Lipped channels Elevated temperatures Distortional buckling/failure Numerical (SFE) analysis Direct Strength Method (DSM) design

1. Introduction The use of cold-formed steel structures has grown steadily during the last few years, as they became extremely popular in different areas of the construction industry. The knowledge about the structural behaviour of cold-formed steel members at room temperature has advanced considerably in the last few years and, moreover, such advances have been incorporated into design specifications at a fairly rapid pace. Since it is well known that many cold-formed steel members are prone to distortional failure, the current design specifications include provisions dealing with this collapse mode. In particular, the Direct Strength Method (DSM –e.g., Schafer [1]), already incorporated into the current versions of n

Corresponding author. E-mail addresses: [email protected] (A. Landesmann), [email protected] (D. Camotim). http://dx.doi.org/10.1016/j.tws.2015.06.004 0263-8231/& 2015 Elsevier Ltd. All rights reserved.

the North-American [2], Australian/New Zealand [3] and Brazilian [4] specifications for cold-formed steel structures, includes specific provisions (strength curves) for the design of columns and beams against distortional failure – their application requires only knowledge about the steel yield stress and member distortional buckling load or moment. However, such provisions/curves were developed for cold-formed steel members at room temperature and it is still unknown whether they can also be adopted (with or without modifications) to estimate the ultimate strength of members subjected to elevated temperatures caused by fire conditions, which may alter considerably the steel constitutive law, namely its Young's modulus, yield strength and amount of constitutive non-linearity. The use of (i) high-strength steels and (ii) very slender crosssections is responsible for making cold-formed steel construction significantly vulnerable to fire hazards. However, the research activity devoted to cold-formed steel members under fire conditions was only initiated in this century and is still relatively scarce,

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even if the number of researchers drawn to this field of activity has consistently and fast grown in the last few years. Important work has been reported by Outinen et al. [5], Kaitila [6], Feng et al. [7–11], Lee et al. [12], Zhao et al. [13], Feng and Wang [14,15], Chen and Young [16–19], Ranawaka [20], Lim and Young [21], Ranawaka and Mahendran [22–24], Landesmann and Camotim [25–29], Shahbazian and Wang [30–34], Kankanamge and Mahendran [35], Chen et al. [36,37], Abreu and Schafer [38], Gunalan and Mahendran [39,40], Gunalan et al. [41], Ellobody [42] and Laím et al. [43,44]. Only a small fraction of these studies addresses failures associated with distortional buckling, an instability phenomenon often governing the structural response of lipped members with intermediate unrestrained lengths. Moreover, although some research has been conducted on the distortional behaviour and strength of cold-formed steel beams (e.g., [45,46] or [47]), to the authors’ best knowledge no such work has been reported for beams under at elevated temperatures. In order to contribute towards filling this gap, the authors recently initiated a shell finite element investigation on the distortional post-buckling behaviour, ultimate strength and DSM design of cold-formed steel singlespan lipped channel beams subjected to uniform elevated temperature distributions [48] – this research effort is continued and expanded in the present work, reporting all the available results on the topic. The paper begins by presenting the beam geometry selection, achieved by means of sequences of “trial-and-error” buckling analyses (Section 2). It aims at identifying beam cross-section dimensions and lengths leading to, as much as possible, “pure” distortional buckling and failure modes – i.e., the distortional critical buckling moments are significantly lower than their local and global counterparts. Then, Section 3 briefly describes the shell finite element (SFE) model employed to perform the ANSYS [49] geometrically and materially non-linear analyses at room and elevated temperatures. Next, Section 4 is devoted to room temperature analysis and starts with the validation of the ANSYS SFE model employed, through the replication of numerical simulations reported by Yu and Schafer [45]. Furthermore, (i) illustrative numerical results concerning the distortional post-buckling behaviour and ultimate strength of lipped channel beams are presented and discussed, and (ii) a parametric study is carried out, to assemble a fairly extensive lipped channel beam numerical ultimate strength “data bank”. The simply supported beams analysed (i) display three end support conditions, several room temperature yield stresses, covering a wide distortional slenderness range, and distortional initial geometrical imperfections with small amplitude (10% of the wall thickness t), and (ii) are submitted to an increasing uniform major-axis bending moment until failure for various uniform temperatures up to 800 ºC. After comparing the trends of

Table 1 End support conditions for the simply supported beams analysed. Support conditions

F

PF

P

Warping Minor-axis flexural rotations

Free Free

Prevented Free

Prevented Prevented

the numerical ultimate moments obtained, for beams under room temperature, with experimental and numerical values reported in the literature, their estimates provided by the current DSM distortional design curve are presented and discussed, making it possible to assess how the “quality” of the latter depends on the beam geometry and (mostly) end support conditions. Finally, Section 5 addresses the influence of the elevated temperature (according to the steel constitutive law dependence prescribed by EC3-1.2) on the elastic–plastic post-buckling and ultimate strength behaviour of the simply supported beams. Moreover, the beam failure moments obtained are also used (i) to assess how the current DSM distortional strength curve is able to cope with the constitutive law variation due to the temperature increase and (ii) to present some considerations on how to improve its performance.

2. Beam geometry selection – buckling behaviour The first task in this work consisted of carefully selecting the cross-section dimensions and lengths of the cold-formed steel single-span lipped channel beams to be analysed, which (i) are simply supported with respect to major-axis bending, (ii) have their end cross-section torsional rotations prevented and (iii) differ in the end cross-section warping (this designation covers here (i) differential longitudinal displacements and (ii) wall/local displacements and rotations) and minor-axis flexural rotation restraints. Three different situations are considered, namely (i) free warping and rotations, termed here “F” (the “fork-type” supports typically used in experimental lateral–torsional investigations), (ii) prevented warping and free rotations, termed here “PF” (physically speaking, they differ from “F” in the fact that rigid end plates are attached to the beam end cross-sections), and (iii) prevented warping and rotations, termed here “P” (they differ from “PF” ones in the fact that the beam supports rest on cylindrical hinges, instead of spherical ones) – Table 1 summarises the differences between these three support conditions. The selection procedure involved sequences of “trial-and-error” buckling analyses, performed by means of either (i) Generalised Beam Theory (GBT – code GBTUL, developed by Bebiano et al. [50,51]) or (ii) ANSYS SFE analyses (SFEA), aimed at satisfying the

Table 2 Selected lipped channel beam cross-section dimensions, area and major-axis elastic modulus. Cross-section

bw (mm)

bf (mm)

bl (mm)

t (mm)

Area (cm²)

S (cm3)

C120 C150 C160 C200 C210

120 150 160 200 210

75 120 100 100 70

10 10 10 10 9

3 3.5 2.2 2.5 2.5

8.7 14.4 8.4 10.5 9.2

37.24 69.18 47.69 71.19 59.46

A. Landesmann, D. Camotim / Thin-Walled Structures 98 (2016) 75–93

Table 3 Selected beam lengths, critical buckling moments and bifurcation-to-critical moment ratios. Beam F beams

C120 C150 C160 C200 C210

PF and P beams

LD (cm) Mcr.D.20 (kN cm)

Mb1. L . T Mcr . D . T

Mb1. e . T Mcr . D . T

LD (cm) Mcr.D.20 (kN cm)

Mb1. L . T Mcr . D . T

Mb1. e . T Mcr . D . T

32 42 46 45 34

3.29 3.72 2.63 3.05 2.57

39.34 105.68 86.19 82.29 44.22

50 55 70 70 55

2.33 2.57 1.83 2.15 1.90

25.28 93.03 57.42 53.32 27.21

1788.1 1904.9 841.8 1393.0 1954.6

2580.6 2857.7 1236.5 2013.2 2751.1

following requirements: (i) Beams buckling in “pure” distortional modes and exhibiting also distortional collapses. This goal is achieved by ensuring that the critical buckling moment (i1) is clearly distortional and (i2) falls considerably below the lowest local and global bifurcation moments. (ii) Cross-section (lipped channel) dimensions associated with “pure” distortional failures for the three (F–PF–P) end support conditions dealt with here (only the lengths may differ). Although this requirement is not essential, it facilitates the performance of the parametric study. (iii) Beam lengths (iii1) associated with single half-wave buckling modes and (iii2) as close as possible to the values of the F beam minimum distortional critical buckling moments. (iv) Cross-section dimensions commonly used in practise and, if possible, distinct wall width proportions (e.g., web-to-flange width ratios. This requirement is intended to enable assessing whether such width proportions influence the beam distortional post-critical strength. The end product of the “trial-and-error” selection procedure are the 5 cross-section dimensions given in Table 2 – their web-toflange width ratios range from 1.25 to 3.0. On the other hand, Table 3 provides, for each beam cross-section geometry, (i) the length associated with critical distortional buckling (LD), (ii) the corresponding critical (distortional) buckling moment at room temperature (Mcr.D.20) and (iii) its ratios with respect to the lowest local (Mb1.L.T) and global (Mb1.e.T) bifurcation moments – all the buckling/bifurcation moments were calculated for E20 ¼210 GPa (steel Young’s modulus at room temperature) and ν ¼ 0.3 (Poisson’s ratio, assumed not to vary with the temperature). Note that the first “non-distortional” buckling moment is always local and that Mb1.L.T/Mcr.D.T varies between 2.57 and 3.72 (F beams) and 1.83

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and 2.57 (PF and P beams). The first global (lateral-torsional) buckling moment is much higher – Mb1.e.T/Mcr.D.T varies from 39.4 to 105.7 (F beams) and from 25.28 to 93.03 (PF and P beams). The curves depicted in Fig. 1(a)–(b) provide the variation of Mcr.T (critical buckling moment at temperature T) with the length L (logarithmic scale) and temperature T, for F and P beams with the C160 cross-section – (i) three temperatures are considered (20/ 100 ºC, 400 ºC and 600 ºC), (ii) the vertical scales are different in Fig. 1(a) and (b) and (iii) the EC3-1.2 [52] constitutive model, presented in Section 3, is adopted. Also shown are the critical (distortional) buckling modes of the F-beam with LD ¼4 cm and P-beam with LD ¼70 cm. Note that any given buckling curve can be obtained through a “vertical translation” of the top one, with a magnitude that depends exclusively on the Young’s modulus erosion due to the temperature rise. Moreover, the critical distortional moment Mcr.D.T corresponds to the same length (LD) for each temperature value.

3. Numerical model The beam distortional post-buckling equilibrium paths and ultimate strength values were determined through ANSYS [49] geometrically and materially non-linear SFEA. The beams were discretized into SHELL181 elements (ANSYS nomenclature – 4-node shear deformable thin-shell elements with six degrees of freedom per node and full integration). The analyses (i) were performed by means of an incremental-iterative technique combining NewtonRaphson’s method with an arc-length control or stabilisation strategies [53] and (ii) simulate the response of beams subjected to uniform constant temperature distributions (i.e., the beams are deemed engulfed in flames, thus sharing the surrounding air temperature, which remains unaltered  [54]) and subsequently acted by an increasing uniform major-axis bending moment up until failure  steady state structural analyses providing failure moments. Note that it has been shown [25,26] that the failure loads yielded by steady state analyses match the more realistic failure temperatures obtained through the “corresponding” transient analyses  axially compressed columns heated up to failure. As mentioned earlier (see Table 1), the simply supported beams analysed exhibit three end support conditions (F, PF and P beams) – both PF and P beams are modelled by attaching rigid plates to the beam end cross-sections. Uniform bending moment is achieved through the application of either (i) sets of concentrated forces acting on the nodes of both end cross-sections (F beams) or (ii) two concentrated moments acting on the rigid end-plates (PF and P beams). The force/moment application is made in small

Fig. 1. Variation of Mcr.T with L and T for (a) F and (b) PF-P C160 beams (EC3-1.2 model).

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Fig. 2. (a) Variation of the reduction factors ke, ky, kp with T r 800 °C and (b) cold-formed steel stress–strain curves sT/sy.20 vs. ε (ε r2%) for T ¼ 20/100–400–600–800 ºC – EC3-1.2 model.

increments, taking advantage of the ANSYS automatic “load stepping procedure”. All the beams contained initial geometrical imperfections with a critical-mode (distortional) shape and small amplitudes (10% of the wall thickness t). These initial imperfections involve inward compressed flange-lip motions, since they are most detrimental, in the sense of leading to lower post-buckling strengths [55]. Each critical buckling mode shape was determined by means of a preliminary ANSYS buckling analysis, performed with exactly the same shell finite element mesh employed to carry out the subsequent non-linear (post-buckling) analysis. It is still worth noting that no strain-hardening, residual stresses and/or corner strength effects were considered in this work.

3.1. Steel material behaviour The multi-linear stress–strain curve available in ANSYS is used to model the steel material behaviour for several yield stresses. Note that a fairly large fraction of these yield stresses are unrealistically high, leading to E20/sy.20 values well below the current DSM limit for pre-qualified beams (340) – they were selected to enable covering a wide beam distortional slenderness range. The steel constitutive law at elevated temperatures adopted to carry out the research work reported in this paper is defined by the analytical expression provided in Annex E of Part 1.2 of Eurocode 3 [52]. Fig. 2(a) compares the temperature variations of the cold-formed steel reduction factors applicable to the steel Young’s modulus (ke ¼ET/E20), nominal yield stress (ky ¼sy.T/sy.20) and proportionality limit stress (kp ¼sp.T/sp.20), which are tabulated in EC3-1.2. As for Fig. 2(b), it illustrates the qualitative differences between the steel stress–strain curves prescribed by the EC3-1.2 model for T ¼20/100 ºC (room temperature), T ¼400 ºC, T ¼600 ºC and T ¼800 ºC  sT/sy.20 vs. ε, where the applied stress at a given temperature, sT, is normalised with respect to the room temperature yield stress sy.20. Note that the steel stress–strain curve non-linearity increases substantially with the temperature (for T ¼20/100 ºC, the EC3-1.2 model prescribes a bi-linear constitutive law). The corresponding stress–strain curve, given by

⎧ ε⋅E for ε ≤ εp . T ⎪ T 0.5 ⎪ 2⎤ ⎡ 2 σT = ⎨ σp . T − c + (b/a )⎢a − εy . T − ε ⎥ for εp . T < ε < εy . T ⎣ ⎦ ⎪ ⎪ for εy . T ≤ ε ≤ εu . T ⎩ σy . T

(

)

2

a 2 = εy . T − εp . T εy . T − εp . T + c /ET , b = c εy . T − εp . T ET + c 2,

(

)(

)

(

)

2

c=

(

(σy.T − σp.T ) ) (

εy . T − εp . T E T − 2 σy . T − σp . T

)

(1)

is divided into three regions, associated with distinct strain ranges. Although the EC3-1.2 model further extends the strain range, to include additional strain-hardening for steel temperatures up to 400 °C (such strain-hardening is negligible for higher temperatures), this effect is not considered in this work. Notice that the stress–strain curve shape is considerably influenced by the temperature and proportionality limit strain (εp.T ¼sp.T /ET). The first part of the yield plateau of the T ¼20/100 °C curve is replaced by a strain-hardening region (more pronounced as the temperature rises). The stress–strain curve (i) is linear elastic, with slope ET (E20 ¼210 GPa), up to the proportional limit stress sp.T, then (ii) becomes elliptic in the transition between the elastic and plastic ranges, up to the effective yield stress sy.T (corresponding to εy.T ¼0.02), which accounts for (kinematic) strain-hardening and (iii) ends with a perfectly flat yield plateau up to a limit strain εu.T ¼ 0.15  in all cases, Prandtl–Reuss’s plasticity model (von Mises yield criterion and associated flow rule) is adopted. Finally, since the distortional post-buckling analyses involve large inelastic strains, the nominal (engineering) static stress–strain curve is replaced by a relation between the true stress and logarithmic plastic strain.

4. Distortional response at room temperature Besides the validation of the above ANSYS SFE model, this section includes also the presentation and discussion of numerical results concerning the influence of the cross-section dimensions and end support conditions on the beam distortional post-buckling behaviour and strength. Elastic and elastic–plastic numerical results are addressed separately. 4.1. Validation studies In order to validate the use of the ANSYS SFE model to assess the distortional post-buckling behaviour and strength of cold-formed steel beams, one begins by reproducing numerical simulations

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Table 4 Beam cross-section dimensions selected from [45]. Cross-section

bw (mm)

bf (mm)

bl (mm)

t (mm)

r (mm)

θl (deg)

8C097

204.216

52.832

14.097

2.4892

7.11

85.7

reported by Yu and Schafer [45], concerning cold-formed steel lipped channel beams that originated from experimental tests carried out by these authors. The cross-section dimensions considered are given in Table 4 and consist of mean values of the dimensions/angles adopted by the above authors (including many beams that were not experimentally tested). The numerical analyses performed by Yu and Schafer [45] were aimed at simulating four-point bending tests of beams formed by assemblages of two identical lipped channel members (i) with an overall length of 487.68 cm (16 ft), (ii) acted by two in-span point loads applied at the 1/3 points and (iii) connected back-to-back (to avoid torsion) by means of (rigid) hot-rolled tubes bolted to both webs and located at the end and loaded cross-sections (to avoid shear and web crippling problems). While the beam central 1/3span (length 162.56 cm and uniformly bent) was basically unrestrained, a through-fastened steel decking was attached (screwed) to stabilise the compression flange along the two outer spans  Fig. 3(a), taken from Yu and Schafer [45], provides an overall view of the experimental test set-up. In view of this experimental set-up, the two lipped channels “work almost independently” in the beam central span – the world “almost” stems from the fact that small angles (1¼  1¼  0.057 in.) were screwed to the tension flanges at 12 in. spacing (see Fig. 3(a)). Since Yu and Schafer wanted to simulate their tests, they modelled the whole experimental set-up: cold-formed steel beams, steel decking and hot-rolled tubes. They used ABAQUS and the 8C097 beam itself was discretised by means of 25.4 mm  25.4 mm meshes of S4R SFE (ABAQUS nomenclature  4-node isoparametric thin-shell elements with six degrees of freedom per node and reduced shear integration). The steel material properties were E ¼203.4 GPa (29500 ksi), ν ¼0.3 and sy.20 ¼ 227.53–303.37–386.80–428.85–506.08 MPa (33–44–56.1– 62.2–73.4 ksi) and the beams were analysed with critical-mode (distortional) initial geometrical imperfections with amplitude equal to 0.94 t (residual stresses and corner strength enhancement effects were neglected). An overall view of the finite element

Table 5 Beam ultimate bending moment obtained (Mu.obt) and reported by Yu and Schafer (Mu.YS). sy.20

Mu . YS

Mu . obt

Mu . YS − Mu . obt Mu . YS

(MPa)

(kN cm)

(kN cm)

(%)

227.53 303.37 386.80 428.85 506.08

1042.85 1333.22 1588.57 1675.57 1839.39

1010.60 1274.20 1527.00 1645.10 1849.35

3.1 4.4 3.9 1.8  0.5

model used by Yu and Schafer to simulate their experimental test set-up is shown in Fig. 3(b). The numerical analyses carried out in this work were performed in ANSYS, adopting the models described in Section 3  the structural models are much simpler because the end support conditions are well defined (unlike those adopted by Yu and Schafer, aimed at simulating realistically a complex experimental set-up). Indeed, the model adopted in this work merely consists of a single-span simply supported lipped channel beam (not beam pair) (i) with length L¼ 162.56 cm (like the central span of the three-span beams analysed by Yu and Schafer) and the crosssection dimensions given in Table 4, (ii) exhibiting an elasticperfectly plastic material behaviour with five yield stresses (also considered by Yu and Schafer  see Table 5), (iii) acted by equal end moments and (iv) containing critical-mode (distortional) initial geometrical imperfections with amplitude equal to 94% of the wall thickness (like the beam analysed by Yu and Schafer). Concerning the end support conditions, a P beam (see Section 2 and Table 1) is analysed, in order to replicate more closely the experimental set-up of Yu and Schafer. Moreover, the rounded corners were included in the cross-section model (radius given in Table 4)  the inclusion of the rounded corners is restricted to this validation study, since sharp corners are considered in all the remaining numerical results presented and discussed in this work,

Fig. 3. (a) Experimental test set-up and (b) associated finite element model [45].

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Fig. 4. 8C097 beam distortional (a) critical buckling and (b) failure mode shapes (sy.20 ¼386.80 MPa).

both at room and elevated temperatures. The ultimate moments obtained (Mu.obt) are compared with those reported by Yu and Schafer [45] (Mu.YS  halves of the values obtained for beams formed by two identical lipped channels) in Table 5, for beams with five yield stresses (sy.20 ¼ 227.53–303.37–386.80–428.85– 506.08 MPa)  Fig. 4(a)–(b) shows the distortional critical buckling and failure modes of the sy.20 ¼386.80 MPa beam. The percentage differences between Mu.obt and Mu.YS, also given in Table 5, never exceed 4.4% and, with a single exception, the values obtained are higher, probably because the model of Yu and Schafer does not ensure fully warping and minor-axis flexural fixity. This is confirmed by the difference between the critical (distortional) buckling moments of the beams with sy.20 ¼386.80 MPa: 3022.90 kN cm vs. 2915.01 kN cm  3.7% difference. In view of this comparison, it is fair to argue that the SFE model employed in this work may be deemed validated. 4.2. Elastic post-buckling behaviour Fig. 5(a)–(c) shows the elastic post-buckling equilibrium paths obtained for the F, PF and P C120-150-160-200-210 beams, relating the applied bending moment M, normalised with respect to Mcr.D.20, to the normalised displacement |δ|/t, where |δ| is the midspan (maximum) vertical displacement of the compressed flangestiffener corner and t is the wall thickness. The observation of these three sets of distortional post-buckling equilibrium paths prompts the following remarks: (i) The higher stiffness and strength of the P and PF beams is amply confirmed by comparing Fig. 5(a) and (b)–(c). Moreover, there is also a clear difference between the F and PF/P beam equilibrium paths: while the former exhibit a pronounced convexity, associated with a progressive stiffness degradation leading to elastic limit points (clearly visible in the C120–160–200–210 beams), the latter display a concavity, stemming from the stiffness increase due to the end cross-

section warping fixity, which precludes the occurrence of an elastic limit point (at least for not too high displacement values). (ii) Regarding the influence of the cross-section dimensions, note the similarity between the C120 and C160 beam stiffness and strength  they share the same bw/bf ¼1.60 value. For the PF and P beams, they are virtually identical for the whole displacement range displayed. For the F beams, on the other hand, the equilibrium paths coincide only up to |δ|/tE9, when the C120 beam one quickly evolves towards its limit point  conversely, the C160 beam equilibrium path exhibits much more ductility prior to failure. Although no sound mechanical reason was (yet) found for this behavioural difference, it should be noticed that the C120 beam is thicker (3.0 mm vs. 2.2 mm) and has “more effective” lips (higher bl/bf: 0.133 vs. 0.100). (iii) Both the F and PF/P C150 beams have clearly higher postcritical stiffness and strength than the remaining ones. This stems most certainly from their lower bw/bf ¼1.25 value and higher thickness, which implies that the elastic rotational restraint provided by the web to the compressed flange-lip assembly is significantly higher. (iv) In spite of its limited scope (15 beams analysed), this study makes it possible to anticipate that both the end support conditions and cross-section dimensions are bound to affect the characteristics of the beam elastic distortional postbuckling stiffness and strength. This may have some impact on the corresponding (elastic–plastic) ultimate strength and, therefore, also on its prediction by design methods [56].

4.3. Elastic-plastic post-buckling behaviour and strength The ANSYS SFE model validated in Section 4.1 is now employed to perform a parametric study aimed at assessing the elastic– plastic post-buckling and ultimate strength of F, PF and P simply

Fig. 5. Elastic equilibrium paths M /Mcr.D.20 vs. |δ|/t of the F, PF and P C120-150-160-200-210 beams.

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81

Fig. 6. Room temperature elastic–plastic distortional equilibrium paths (M /Mcr.D.20 vs. |δ|/t) of (a) F, (b) PF and (c) P C120 beams with λ¯ D.20 varying between 0.25 and 2.0.

supported beams buckling and failing in distortional modes. The numerical results presented and discussed concern 210 beams, combining (i) the 5 geometries defined in Tables 1 and 2, (ii) the 3 end support conditions dealt with (F, PF and P) and (iii) 14 room distortional slenderness values (λ¯ D.20 varying between 0.25 and 3.5 in 0.25 intervals)  recall that λ¯ D.20 ¼[My.20 /Mcr.D.20]0.5, where My.20 ¼S.sy.20 and S is the major-axis elastic section modulus (see Table 1). The first few rows of Tables A, included in Annex A, concern T ¼20/100 ºC and provide an illustrative sample of the results obtained: for each C200 beam, they display (i) the distortional buckling moment Mcr.D.20 and slenderness λ¯ D.20, (ii) the yield moment My.20, (iii) the numerical ultimate moment Mu.20 and associated (|δ|/t)lim value, (iv) the DSM ultimate moment prediction Mn.D.20 and (v) the ratios Mu.20/Mn.D.20 and Mn.D.20/My.20. Fig. 6(a)–(c) displays, for the 3 support conditions considered, a representative sample of the non-linear (geometrically and materially) equilibrium paths M/Mcr.D.20 vs. |δ|/t determined to obtain the ultimate moments Mu.20 (white circles)  these equilibrium paths concern C120 beams with room distortional slenderness λ¯ D.20 varying between 0.25 and 2.0. The corresponding elastic equilibrium paths, already shown in Fig. 5(a)–(c), are displayed again for comparison purposes  note that Fig. 6(a)–(c) only extend up to |δ|/ t¼12, while Fig. 5(a)–(c) extended up to |δ|/t¼24 (this is why the F and PF/P beam equilibrium paths appear to be much more similar in the former than in the latter). As for Fig. 7(a)–(c), they depict the F and PF/P beam and mid-span cross-section deformed configurations in the close vicinity of failure, for λ¯ D.20 ¼0.75 (sy.20 ¼ 270–385 MPa, respectively for F and PF/P beams)  the distortional nature of the beam collapse modes is clearly shown. The observation of the results shown in these figures, as well as the data obtained (an illustrative sample is provided in Tables A), leads to the following conclusions: (i) The F-beam elastic–plastic post-buckling behaviour and ultimate strength differ from their PF/P-beam counterparts, both qualitatively and quantitatively – in order to quantify this

statement, note that the C120 beam Mu.20/Mcr.D.20 values are (i1) quite similar for λ¯ D.20 r0.75 and (i2) ordered in the sequence F to PF to P, even if significant differences only occur in the high slenderness (yield stress) range. (ii) The failure bending moment ratio Mu.20 /Mcr.D.20 and associated (|δ|/t)lim values increase with the slenderness (yield stresses sy.20), regardless of the beam cross-section dimensions and end support conditions. Fig. 8(a)–(c) plots the Mu.20 /Mcr.D.20 vs. λ¯ D.20 curves for all the beams analysed and it is visible that those failing below the critical bending moment level (i.e., for which Mu.20 /Mcr.D.20 r1) exhibit a fairly small elastic–plastic strength reserve and very little ductility prior to failure  moreover, there are only small qualitative differences between the values concerning the F and PF/P beams. However, this assertion does not remain valid for the beams with Mu.20 /Mcr.D.20 4 1: while the F beams collapse almost immediately after the onset of yielding, the PF/P beams exhibit a much higher elastic–plastic strength reserve, which stems directly from the elastic post-buckling differences addressed earlier – indeed, note that the largest post-critical strength reserve is exhibited by the C150 P-beams. In this range, the differences between the the F, PF and P beam values are very clear: the last ones are consistently higher and the differences increase with λ¯ D.20. (iii) All the F, PF and P beams exhibit single half-wave distortional buckling and failure modes characterised by inward flange-lip motions (see Fig. 7(c)). It is worth noting that, almost 20 years ago, Schafer [57] reported a thorough numerical investigation on the moment capacity of laterally braced cold-formed steel Z-section beams failing in local or distortional modes. A series of 60 single span beams were analysed by means of ABAQUS SFEA  Fig. 9 provides an overall view of the end support and loading conditions adopted. The analyses (i) modelled the steel material behaviour through an elastic–plastic constitutive law with strain hardening, (ii) included

Fig. 7. Distortional failure modes of (a) F and (b) PF/P C120 beams and (c) associated (common) mid-span cross-section deformed configuration.

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Fig. 8. Plots of the (a) F, (b) PF and (c) P beams ultimate bending moment ratios Mu.20 /Mcr.D.20 against the distortional slenderness λ¯ D.20.

Fig. 9. End support and loading conditions of the Z-section beams numerically analysed by Schafer [57].

initial geometrical imperfections combining the local and distortional buckling modes, with magnitudes having a 75% probability of not being exceeded [58], and (iii) accounted for flexural residual stresses with maximum values (flange-web corners) equal to 30– 40% of the yield stress. The specimens tested had lengths corresponding to two or three (mostly) distortional half-wavelengths and were subjected to equal end moments causing a “horizontal” (normal to the web) neutral axis, thus leading to uniformly compressed top flanges – see Fig. 9. Note that the end support conditions adopted by Schafer [57] are quite similar to those termed “F” in this work (see Table 1). Out of the 60 beams analysed by Schafer [57], 31 collapsed in distortional modes, most of them exhibiting a small amount of yielding on the compressed lip. Such beams (i) had web-to-flange width ratios comprised between 2 and 8, (ii) covered a wide distortional slenderness range (0.88–3.28) and (iii) were found to exhibit Mu.20/My.20 ratios varying between 0.12 and 1.0. Fig. 10(a)–(c) plots Mu.20/My.20 against λ¯ D.20 for the 210 beams

Fig. 10. Plots Mu.20/My.20 vs. λ¯ D.20 for (a) F, (b) PF and (c) P beams (values from this work and the literature).

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Fig. 11. C120 (a) F, (b) PF and (c) P beam elastic and elastic–plastic equilibrium paths, deformed configurations (including the collapse mechanisms) and von Mises stress contours (λ¯ D.20 ¼ 1).

dealt with in this work and also (i) the 31 Z-section F beams reported by Schafer [57] and (ii) the 80 P beams numerically (50) or experimentally (24) analysed by Yu and Schafer [45], which concern 41 (36 þ9) Z-section and 29 (14 þ15) lipped channel beams, and were selected to cover a wide variety of cross-section geometries  e.g., the web-to-flange width ratio varies from 2 to 6. The observation of these plots shows that: (i) Naturally, the Mu.20/My.20 vs. λ¯ D.20 “clouds” follow the trend of “Winter-type” strength/design curves, with some “vertical dispersion” observed for the PF/P beams (unlike for the F beams), due to the distortional post-critical strength reserve differences addressed earlier  e.g., the C150 P-beam Mu.20/ My.20 values are considerably above the remaining ones. (ii) All the slender F-beam values are quite well aligned (with very little vertical dispersion) along the elastic buckling strength curve 1/(λ¯ D.20)2 (dashed curve) – a similar observation was made by Schafer [57], on the sole basis of the Z-section beams that he analysed. On the other hand, the PFbeam and P-beam values are above that curve for λ¯ D.20 4 1.5  the differences grow with the slenderness increases and

are more pronounced for the P-beams. (iii) The results of Yu and Schafer [45], which concern beams (iii1) formed by back-to-back lipped channel and Z-section profiles, and (iii2) exhibiting solely small-to-moderate distortional slenderness (0.59–1.53), “mingle” fairly well with the P-beam values obtained in this work. Finally, Fig. 11(a)–(c) concerns C120 F, PF and P beams with λ¯ D.20 ¼1 (sy.20 ¼480 and 700 MPa) and display their elastic and elastic–plastic equilibrium paths, and the evolution of their deformed configurations and von Mises stress (sVM.20) contours (before, at and beyond the peak load)  each set of 4 diagrams corresponds to the equilibrium states indicated on the associated equilibrium path. It is worth noting that (i) the deformed configurations are amplified 3 times, and (ii) state II always corresponds to the beam collapse (failure modes depicted). The observation of the results presented in these figures prompts the following remarks: (i) In all (F, PF and P) beams, yielding starts at the mid-span zone of the compressed lip free edge – see diagrams I. Collapse is

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Fig. 12. Comparison between the numerical and experimental Mu.20/Mn.D.20 values and their DSM estimates for the (a) F, (b) PF and (c) P beams.

also quite similar for all beams: it is associated with the full yielding of the web-flange corner at mid-span, leading to the formation of a “distortional plastic hinge” – see diagrams II, which also reveal that plasticity has already spread throughout the whole compressed lip mid-span zone – Dinis and Camotim [46] reported similar observations. (ii) The compressed flange yields gradually along the equilibrium path descending branch – see diagrams III and IV, corresponding to |δ|/t ¼4.5 and 6, respectively. However, the spread of plasticity rate, after the onset of yielding, is much higher in the F-beams than in their PF/P counterparts. Moreover, the stress diagrams IV indicate the occurrence of elastic unloading in the mid-span compressed flange regions of the F (mostly) and PF/P beams. The differences between the PF and P beam post-buckling behaviours are minute.

4.4. DSM design considerations This section addresses the merits of the currently codified DSM distortional design curve in predicting the failure moments of the simply supported lipped channel beams analysed in this work. It should be noted that the currently codified DSM distortional design curve was developed exclusively in the context of beams at room temperature (e.g., [1,2]). The nominal ultimate moment of a cold-formed steel beam failing in a distortional mode is estimated by

Mn . D = Mn . D.20 ⎧M ⎪ y.20 ⎡ ⎪ = ⎨ My.20⎣⎢1−0.22 Mcr . D.20/My.20 ⎪ 0.5 ⎪ ⎩ Mcr . D.20/My.20

(

(

)

for λ¯D.20 ≤ 0.673 0.5⎤

)

for λ¯D.20 > 0.673 ⎦⎥ (2)

where (i) Mcr.D.20 and My.20 are the beam distortional critical buckling and yield moments, and (ii) the beam distortional slenderness is defined as λ¯ D.20 ¼(My.20 /Mcr.D.20)0.5. Fig. 12(a)–(c) compares the numerical ultimate moment ratios displayed in Fig. 10(a)–(c) with their DSM estimates  the former are either obtained in this work (all concerning lipped channel beams) or reported by Schafer [57] (Z-section beams) and Yu and Schafer [45] (Z-section and lipped channel beams). Fig. 13(a)–(c), on the other hand, plot the ratios Mu.20/Mn.D.20 against λ¯ D.20, thus providing pictorial representations of the accuracy and safety of the DSM distortional ultimate moment estimates – the averages, standard deviations and maximum/minimum values of Mu.20/ Mn.D.20 are also given. The analysis of these six plots leads to the following comments: (i) Naturally, the DSM design curve provides accurate and mostly safe predictions of the numerical and experimental distortional failure moments reported by Yu and Schafer [45] – indeed, these failure moments were part of those used to develop/calibrate this design curve. However, it is also worth noting that, concerning the F-beam failure moments obtained by Schafer [57], the DSM curve predictions are (i1) mostly underestimations in the low-to-moderate slenderness range

Fig. 13. Mn.D.20/Mu.20 vs. λ¯ D.20 plots for the (a) F, (b) PF and (c) P beams.

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Fig. 14. Distortional post-buckling equilibrium paths and deformed configurations and von Mises stress contours at collapse for (a) F, (b) PF and (c) P C120 beams loaded at various elevated temperatures.

(λ¯ D.20 r1.5) and (i2) clear overestimations in the moderate-tohigh slenderness range (λ¯ D.20 41.5) – recall that all these Mu.20/Mn.D.20 values were shown to be almost perfectly aligned along the elastic buckling strength curve (dashed curve of Fig. 12(a)). (ii) Concerning the numerical ultimate moments obtained in this work, the DSM estimates are (ii1) mostly safe and accurate in the low-to-moderate slenderness range (λ¯ D.20 r1.25) and (ii2) clearly unsafe in the moderate-to-high slenderness range (λ¯ D.20 41.25)  the overestimation grows with λ¯ D.20, is particularly severe for the most slender F beams and can only be deemed “acceptable” for the P-beams. This is reflected in the average, standard deviation and maximum/minimum values of the Mu.20/Mn.D.20 ratios: 0.75, 0.25, 1.19/0.39 (F beams), 0.86, 0.17, 1.26/0.58 (PF beams) and 1.00, 0.12, 1.26/0.75 (P beams)  if only beams with λ¯ D.20 r 1.25 are considered, these indicators improve to 1.04, 0.07, 1.19/0.94 (F beams), 1.01, 0.14, 1.26/0.65 (PF beams) and 1.07, 0.09, 1.26/0.89 (P beams). Note that all Mu.20 values reported by Yu and Schafer [45] concern P

beams with slenderness between 0.68 and 1.53, which explains the adoption of the currently codified distortional design curve. On the other hand, the indicators of the F beams analysed by Schafer [57] are close to those of the F beams analysed in this work: 0.77, 0.24, 1.09/0.37. (iii) There is one exception to the content of the previous item: the C150 P-beam failure moments, which are underestimated by the DSM curve along the whole slenderness range, as quantified by the corresponding Mu.20/Mn.D.20 indicators: 1.16, 0.04, 1.26/1.10. Recall that these beams exhibit a much higher postcritical strength reserve than the remaining ones (see Fig. 5 (c)). (iv) The results of the limited parametric study carried out in this work provide evidence that the current DSM distortional strength/design curve overestimates, to a lesser or greater extent, the overwhelming majority of the numerical failure moments concerning beams with λ¯ D.20 41.25.

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5. Distortional response under elevated temperature 5.1. Elastic–plastic post-buckling behaviour The influence of the elevated temperature on the distortional elastic–plastic post-buckling and ultimate strength of cold-formed steel simply supported lipped channel beams is examined in this section. The investigation deals with F, PF and P C120–150–160– 200–210 beams (see Tables 2 and 3) and is based on the temperature-dependent steel constitutive law prescribed in EC3-1.2. Fig. 14(a)–(c) shows, for F-PF-P C120 beams with λ¯ D.20 E1.00 (sy.20 ¼480–690–690 MPa), (i) the non-linear equilibrium paths M/Mcr.D.20 vs. |δ|/t for temperatures T ¼20/100–200–300–400– 500–600–700–800 ºC (white circles identify the ultimate moments Mu.T and the room temperature curve is shown again for comparative purposes), and (ii) the deformed configurations and von Mises stress contours at collapse (M ¼Mu.T) for T ¼200–400– 600–800 ºC). The observation of these post-buckling results prompts the following remarks: (i) Obviously (and regardless of the end support conditions), the various beam equilibrium paths “move down” as the temperature T rises, thus leading to lower ultimate moments. (ii) Concerning the deformed configurations at collapse (failure modes), note that, since the thermal action effects are negligible (uniform temperature and free-to-deform beams), they are virtually identical in all beams sharing the same end

support conditions, i.e., do not depend on the temperature. The same is not true for the corresponding von Mises stress contours, as those concerning the F and PF þ P beams are not similar (the PF and P beam ones almost coincide). This fact stems directly from the variation of the stress–strain–temperature curve shape. In quantitative terms, the stresses obviously decrease as the temperature rises and continuously erodes the steel material behaviour. (iii) The TZ 600 °C curves are clearly below their T r500 °C counterparts for all beams, which reflects the heavy degradation of the steel material behaviour between 500 °C and 600 °C, namely via the proportionality limit strain and smoothness of the (elliptic) transition between the elastic and plastic ranges (see Fig. 2(b))  for T Z 600 °C, the stress– strain curve exhibits again a well-defined yield plateau. (iv) No clear trend was detected concerning the influence of the temperature, geometry and/or steel grade on the amount of beam elastic–plastic strength reserve and ductility prior to failure  all the computed |δ|/t values corresponding to the ultimate moments Mu.T are included in Tables A of Annex A for C200 beams. Indeed, regardless of the temperature, all beams exhibit quite similar post-collapse behaviours (equilibrium path descending branches). 5.2. Failure moments This section presents and discusses the results of the

Fig. 15. Numerical Mu.T/My.T vs. λ¯ D.T plots the C120–C150–C160–C200–C210 beams under temperatures T ¼20/100–200–300–400–500–600–700–800 ºC.

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parametric study carried out to gather ultimate strength data that will enable assessing the quality of the failure moment estimates provided by the DSM design curve modified to handle elevated temperatures. This parametric study involves a total of 1050 beams, corresponding to all possible combinations of (i) five lipped channel geometries (C120–C150–C160–C200–C210  see Tables 1 and 2), (ii) 3 end support conditions (F, PF and P), (iii) eight uniform temperatures (T ¼20/100–200–300–400–500–600– 700–800 ºC) and (iv) several yield stresses, covering wide distortional slenderness ranges: λ¯ D.T varies from 0.23 to 3.54. Note that the 210 F–PF–P-beam results at 20/100 ºC (room temperature) were already discussed in Section 4 and are presented here again for comparison purposes. Tables A (see Annex A) provide an illustrative sample (C200 beams) of the numerical (Ansys SFEA) beam failure moments obtained in this work. They concern C200 beams with different end support conditions (F–PF–P) and have information about the (i) yield stresses and temperatures considered, (ii) temperature-dependent yield moments My.T, distortional critical buckling moments Mcr.D.T and slenderness values λ¯ D.T, (iii) the computed failure/ultimate moments Mu.T and associated (|δ|/t)lim values, and (iv) the ultimate moment ratios Mu.T/My.T. Fig. 15 shows Mu.T/My.T vs. λ¯ D.T plots for the various temperatures considered  each of them presents together F, PF and P beam results, and includes, for comparative purposes, the corresponding elastic buckling strength curve. The joint observation of these plots makes it possible to draw the following conclusions:

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(i) Regardless of the temperature, there is a considerable amount of “vertical dispersion” in the Mu.T/My.T vs. λ¯ D.T “clouds” along the whole slenderness range considered, thus reflecting the influence of the end support conditions and/or cross-section geometry on the beam distortional post-critical strength reserve. Qualitatively speaking, the increase in temperature does not alter the traits detected/unveiled in Fig. 10(a)–(c): (i1) the Mu.T/My.T values move visibly upwards as one “travels” in the sense F - PF - P and (i2) the C150 beam Mu.T/My.T values are well above those concerning the remaining beam geometries. (ii) In the low slenderness range, all Mu.T/My.T values concerning the beams under elevated temperatures (T 4200 ºC) are well below their room or mildly elevated temperature (T ¼20– 200 ºC) counterparts  the differences become larger as the temperature increases (the elastic buckling strength curve provides a convenient reference to assess these differences). It is worth noting that only the room temperature Mu.T/My.T values form a kind of plateau for λ¯ D.T r0.8  for all the remaining (elevated) temperatures, these same values are aligned along fairly well defined descending curves that (ii1) exhibit continuity for λ¯ D.T 40.8 (a single curve extends throughout the whole slenderness range, with no initial plateau) and (ii2) originate at points that move downwards as the temperature grows. The above feature stems directly from the temperature-dependence of the steel constitutive law prescribe by the EC3-1.2 model: (ii1) considerable drop of the

Fig. 16. Comparison between the modified current DSM distortional curve and the numerical beam failure moments at temperatures T ¼ 20/100–200–300–400–500–600– 700–800 ºC.

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Fig. 17. Mu.T/Mn.D.T ratios plotted against the distortional slenderness λ¯ D.T at temperatures T ¼20/100–200–300–400–500–600–700–800 ºC.

proportional limit stress and (ii2) replacement of the room temperature yield plateau by an increasingly pronounced elliptic strain-hardening region  see Fig. 2(b). (iii) These results provide promising indications about the possibility of developing an efficient (safe and accurate) DSM approach to estimate the distortional failure moments of simply supported beams under elevated temperatures. Nevertheless, it became also clear that the failure moment predictions for beams at room temperature and under elevated temperatures must be handled separately in the low slenderness range (at least when the EC3-1.2 temperature-dependence model is adopted). The DSM design, against distortional failures, of simply supported beams under elevated temperatures, is addressed in the next section.

the quality of the DSM ultimate moment estimates is affected by this temperature-dependence. The approach followed, which has already been (partially) [25–28] and other researchers (e.g., [16–18] or [23]), in the context of cold-formed steel columns, consists of modifying Eq. (2) in order to account for the influence of the temperature on Mcr.D and My. This influence is felt through Young’s modulus and “yield stress” (effectively, stress at εy.T ¼0.02) values, which progressively fall as the temperature (caused by fire conditions) increases. In other words, Mcr.D and My (or sy) are replaced by Mcr.D.T and My.T (or sy.T), thus implying that λ¯ D also varies with T (λ¯ D.T). In this framework, the nominal failure moment of cold-formed steel beams failing in distortional modes is given by

Mn . D . T 5.3. DSM design considerations This section begins by addressing the adequacy of the current DSM distortional strength/design curve (appropriately modified to reflect the temperature effects) to predict the failure moments of simply supported cold-formed steel lipped channel beams analysed in this work. In particular, it is intended to assess whether

⎧M ⎪ y.T ⎡ ⎪ = ⎨ My . T ⎣⎢1 − 0.22 Mcr . D . T /My . T ⎪ 0.5 ⎪ ⎩ Mcr . D . T /My . T

(

(

)

for λ¯D . T ≤ 0.673 0.5⎤

)

for λ¯D . T > 0.673 ⎦⎥ (3)

where (i) Mcr.D.T and My.T are the beam temperature-dependent distortional critical buckling and yield moments, and (ii) the beam distortional slenderness is given by λ¯ D.T ¼(My.T /Mcr.D.T)0.5. The plots shown in Fig. 16 concern temperatures T¼ 20/100– 200–300–400–500–600–700–800 ºC and make it possible to

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compare the numerical Mu.T/My.T values obtained for F, PF, P beams with C120–C150–C160–C200–C210 geometries and several distortional slenderness (yield stress) values, with the (i) modified current DSM distortional strength curves (solid lines) and (ii) elastic buckling strength curves (dashed lines). As for Fig. 17, it provides Mu.T/Mn.D.T vs. λ¯ D.T plots that enable a quick quantitative assessment of the quality (accuracy and safety) of the modified DSM predictions of the beam failure moments at elevated temperatures – the Mu.T/Mn.D.T ratio averages, standard deviations and maximum/minimum values are also given. The observation of the results presented in Figs. 16 and 17 prompts the following remarks (already anticipated on the basis of Figs. 12 and 15): (i) The DSM beam failure moment predictions are only consistently safe and accurate for (i1) the stockier beams (λ¯ D.T r1.0) under room or mildly elevated temperature (T¼20–200 ºC) and (i2) the C150 P beams, whose Mu.T/Mn.D.T values are clearly above all the remaining ones along the whole slenderness range due to their outstanding distortional post-critical strength reserve – indeed, the C150 P beams are exceptions to practically all the remarks made here. For the sake of simplicity, this exceptionality will not be further mentioned, i.e., “all beams” means “all beams except the C150 P ones”. This means that the failure moments of the moderate-to-slender beams (λ¯ D.T 41.0) are never adequately predicted by the current DSM design curve – not even for beams at room temperature. (ii) Practically all beams under elevated temperature have their distortional failure moments overestimated by the corresponding modified DSM design curves  the few exceptions are stocky beams at T¼ 200 ºC. The amount of overestimation (ii1) decreases slightly up to around λ¯ D.T ¼ 1.25 and then (ii2) grows at a fair pace with λ¯ D.T  see Fig. 17. Moreover, this overestimation is not significantly affected by either the crosssection geometry or the temperature value. Conversely, the amount of overestimation is clearly influenced by the end support conditions  it grows as one “travels” in the sense P - PF - F. (iii) It is quite interesting to notice that, for λ¯ D.T Z1.25 (approximately the intersection between the DSM and elastic buckling strength curves) and regardless of the temperature (surprisingly, this assertion applies also to room temperature), the elastic buckling strength curves provide (iii1) lower bounds for all the associated Mu.T/My.T values and also (iii2) quite accurate estimates of those concerning the F beams. (iv) For λ¯ D.T o1.25, the DSM design curves provide safe and fairly accurate failure moment estimates for (iv1) all the beams at room temperature and (iv2) most of the beams at T ¼200 ºC. For higher temperatures, virtually all the failure moments are overestimated.

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approach. (ii) For the PF and P beams at room temperature, the currently codified DSM design curve can still be retained for λ¯ D.T r1.25. For λ¯ D.T 41.25, the elastic distortional buckling strength curve falls now too below the Mu.T/My.T values  the differences are a bit larger for the P beams. It should be replaced by one (PF and P beams handled jointly) or two (PF and P beams handled separately) similar strength curves located above the buckling one. (iii) The contents of the two previous items remain valid for beams (iii1) at T ¼200 ºC, along the whole slenderness range, and (iii2) at higher temperatures (TZ 300 ºC), only for λ¯ D.T 41.25. (iv) For the stocky (λ¯ D.T r1.25) beams at high temperatures (T Z300 ºC), novel (qualitatively different) strength curves must be sought. Indeed, the current DSM strength curve, developed in the context of beams with linear elastic-perfectly plastic constitutive laws, is clearly not adequate. Instead, it is necessary to account for the fact that, according to the EC3-1.2 model, the non-linearity of the stress–strain curve at elevated temperatures begins well below the so-called yield stress sy.T  indeed, it starts at the proportional limit stress sp.T (about 65% of sy.T) and exhibits a very pronounced curvature immediately after. In the authors’ opinion, it is worth exploring the idea of treating the steel stress–strain curves at elevated temperatures similarly to those displayed by stainless steel grades at room temperature (e.g., [59]), but replacing sy.T by sp.T. If such an approach is adopted, the stress–strain curve non-linearity would be akin to strainhardening and inspiration could be drawn from available DSM-based approaches to design stainless steel thin-walled members against distortional failures (e.g., [60] or [61]).

6. Concluding remarks

On the basis of (i) the above remarks, which are exclusively based on the parametric study carried out in this work, and (ii) the findings reported in Section 4.4, for beams at room temperature, it is possible to establish a few preliminary guidelines for the search of an efficient DSM approach to design cold-formed steel lipped channel beams against distortional failures  these guidelines apply to failure moments of beams at both room and elevated temperatures, as the former are not adequately predicted by the currently codified DSM distortional strength curve. Indeed, it may be said that:

This paper reported the available results of an ongoing numerical (SFEA) investigation on the distortional post-buckling behaviour, ultimate strength and DSM design of cold-formed steel beams. These results concern 1050 simply supported single-span lipped channel beams exhibiting (i) five geometries, selected to ensure pure distortional buckling and failure modes, (ii) three end support conditions, differing mostly in the warping restraint, (iii) several room-temperature yield stresses, chosen to cover wide distortional slenderness ranges, (iv) various uniform temperatures (up to 800 ºC) supposedly caused by fire conditions and (v) temperature-dependent material properties according to the model prescribed in Annex E of EC3-1.2. This study made it possible (i) to acquire in-depth knowledge about the combined influence of the end support conditions and elevated temperatures on the distortional post-buckling and ultimate strength behaviours of cold-formed steel beams, (ii) to gather a fairly extensive beam failure moment data bank, which was subsequently used (iii) to provide a first contribution (establish preliminary guidelines) towards the development of an efficient DSM-based approach to design beams failing in distortional modes under elevated temperatures. Amongst the various findings of this work, the following ones deserve to be specially mentioned:

(i) For F beams at room temperature, using a combination of (i1) the currently codified DSM design curve, for λ¯ D.T r1.25, and (i2) the elastic distortional buckling strength curve, for λ¯ D.T 41.25, provides safe and accurate failure moment estimates, i.e., constitutes an efficient DSM-based design

(i) The end support conditions and cross-section dimensions may significantly affect the distortional post-buckling response and failure moment of simply supported lipped channel beams. (ii) The currently codified DSM distortional strength curve

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overestimates the vast majority of the non-stocky beams (λ¯ D.20 41.25) at room temperature  the amount of overestimation depends on the end support conditions (mostly) and cross-section dimensions. (iii) For all the slender beams with free end section warping and lateral restraint (both at room and elevated temperatures), the failure moments are safely and accurately predicted by the elastic buckling strength curve, thus reflecting their small post-critical strength reserve  this strength reserve increases when the end section are restrained (particularly against warping). (iv) In the moderate-to-large slenderness range (λ¯ D.20 41.25), the quality of the DSM failure moment predictions is quite similar for the beams at room and elevated temperatures. (v) In the low-to-moderate slenderness range (λ¯ D.20 r1.25), the current DSM design curve is clearly inadequate to predict the failure moments of beams at elevated temperatures (T Z300 ºC), due to the pronounced stress–strain curve nonlinearity prescribed by the EC3-1.2 model. Finally, one last word to mention that the final goal of this research effort is the development, calibration and validation of DSM-based design approaches to provide high-quality failure moment predictions for cold-formed steel beams failing in distortional modes, (i) first at room temperature and (ii) subsequently under elevated temperatures  in the latter case, inspiration will certainly be drawn from available work on stainless steel thinwalled members. Naturally, the first step consists of achieving the above goal for lipped channel beams, which requires concluding the investigation initiated with this paper. However, it is also necessary to widen the scope of the investigation to cover (i) beams with other commonly used cross-section shapes (mostly) and/or end support conditions, and (ii) available experimentally-based temperature-dependent constitutive models, namely those due to Chen and Young [17], Ranawaka and Mahendran [22] or Kankanamge and Mahendran [62].

Table A1 Numerical ultimate loads and DSM data/estimates concerning the C200 F beams analysed. T (ºC) Mcr.D.T (kN cm)

λ¯ D.T

100

1393.0

200

Mu.T (kN cm)

(|δ|/t)lim Mn.D.T (kN cm)

Mu . T My . T

Mu . T Mn . D . T

0.23 71.2 0.51 355.9 0.75 783.0 1.01 1423.7 1.25 2171.2 1.52 3203.4 1.74 4235.6 2.02 5694.9 2.25 7047.4 2.53 8898.3 2.75 10500.0 3.03 12813.5 3.25 14699.9 3.54 17440.6

73.0 356.4 764.9 1209.2 1376.8 1444.8 1520.2 1611.2 1689.3 1782.3 1841.5 1910.4 1959.3 1998.0

0.26 0.17 2.11 1.67 3.72 4.26 7.07 9.37 10.04 11.64 13.10 13.81 14.75 16.14

71.2 355.9 737.9 1101.8 1432.6 1805.9 2122.6 2510.1 2826.7 3214.2 3518.0 3918.4 4218.7 4622.5

1.03 1.00 0.98 0.85 0.63 0.45 0.36 0.28 0.24 0.20 0.18 0.15 0.13 0.11

1.03 1.00 1.04 1.10 0.96 0.80 0.72 0.64 0.60 0.55 0.52 0.49 0.46 0.43

1253.7

0.50 1.01 1.51 2.01 2.51 3.02 3.52

316.8 1267.1 2851.0 5068.5 7919.5 11404.0 15522.1

292.0 1023.3 1283.9 1437.5 1587.1 1709.5 1794.5

0.00 1.22 3.99 8.76 11.13 14.11 16.65

316.8 984.6 1614.8 2245.0 2875.1 3505.3 4135.5

0.92 0.81 0.45 0.28 0.20 0.15 0.12

0.92 1.04 0.80 0.64 0.55 0.49 0.43

300

1114.4

0.50 277.6 1.00 1110.5 1.50 2498.6 2.00 4442.0 2.50 6940.7 2.99 9994.5 3.49 13603.7

232.3 816.6 1119.9 1249.7 1389.1 1499.1 1581.3

0.21 0.93 3.60 8.07 11.45 13.44 15.75

277.6 867.3 1423.5 1979.7 2535.9 3092.2 3648.4

0.84 0.74 0.45 0.28 0.20 0.15 0.12

0.84 0.94 0.79 0.63 0.55 0.48 0.43

400

975.1

0.49 0.97 1.46 1.95 2.44 2.92 3.41

231.4 925.4 2082.2 3701.7 5783.9 8328.8 11336.4

175.7 594.8 941.7 1058.1 1177.8 1285.0 1359.7

0.53 1.21 2.83 7.73 10.56 12.85 16.18

231.4 735.4 1210.4 1685.3 2160.3 2635.3 3110.2

0.76 0.64 0.45 0.29 0.20 0.15 0.12

0.76 0.81 0.78 0.63 0.55 0.49 0.44

500

835.8

0.48 0.95 1.43 1.90 2.38 2.85 3.33

188.6 754.6 1697.8 3018.3 4716.1 6791.2 9243.5

146.5 506.9 806.2 1058.1 1003.5 1094.5 1157.9

0.51 1.13 2.71 7.73 10.28 13.30 14.39

188.6 610.3 1007.3 1404.4 1801.5 2198.6 2595.6

0.78 0.67 0.47 0.35 0.21 0.16 0.13

0.78 0.83 0.80 0.75 0.56 0.50 0.45

600

431.8

0.50 0.99 1.49 1.99 2.49 2.98 3.48

106.8 427.1 961.0 1708.5 2669.5 3844.1 5232.2

78.4 258.9 414.7 470.4 523.0 570.5 604.4

0.51 1.42 2.74 7.63 10.24 13.32 15.40

106.8 334.5 549.2 763.9 978.7 1193.4 1408.1

0.73 0.61 0.43 0.28 0.20 0.15 0.12

0.73 0.77 0.76 0.62 0.53 0.48 0.43

700

181.1

0.51 1.01 1.52 2.02 2.53 3.03 3.54

46.3 185.1 416.4 740.3 1156.8 1665.8 2267.3

33.5 108.6 173.8 197.9 220.4 240.7 254.6

0.40 1.35 2.86 7.73 11.19 12.61 15.38

46.3 143.2 234.8 326.3 417.8 509.4 600.9

0.72 0.59 0.42 0.27 0.19 0.14 0.11

0.72 0.76 0.74 0.61 0.53 0.47 0.42

800

125.4

0.45 0.89 1.34 1.78 2.23 2.67 3.12

24.9 99.7 224.2 398.6 622.9 896.9 1220.8

19.9 70.8 118.7 133.9 147.6 160.1 171.1

0.46 1.08 2.15 6.28 9.28 11.64 14.13

24.9 84.2 140.1 196.0 251.9 307.8 363.6

0.80 0.71 0.53 0.34 0.24 0.18 0.14

0.80 0.84 0.85 0.68 0.59 0.52 0.47

Acknowledgements The authors gratefully acknowledge the financial support of CNPq (Brazilian National Counsel of Technological and Scientific Development), through research project “Numerical and Experimental Analyses Cold-Formed Steel Structural Members” (MEC/MCTI/ CAPES/CNPq/FAPs No. 61/2011), in the context of the “Science without Borders Program”.

Annex A: Data concerning C200 beams at room and elevated temperatures Tables A1–A3 provide the numerical and DSM data concerning the cold-formed steel lipped channel C200 beams analysed in this work (at both room and elevated temperature)  due to space limitations, the data concerning the other beams analysed are omitted here. Each table concerns a different end support condition (F, PF and P beams) and includes the following information: (i) temperature T, (ii) critical (distortional) buckling moment Mcr.D.T and distortional slenderness λ¯ D.T, (iii) yield moment My.T, (iv) numerical failure moment Mu.T and associated (|δ|/t)lim value, (v) DSM failure moment estimate Mn.D.T and (vi) Mu.T/My.T and Mu.T/Mn.D.T ratios.

My.T (kN cm)

A. Landesmann, D. Camotim / Thin-Walled Structures 98 (2016) 75–93

Table A2 Numerical ultimate loads and DSM data/estimates concerning the C200 PF beams analysed. T (ºC) Mcr.D.T (kN cm)

λ¯ D.T

100

2013.2

200

91

Table A3 Numerical ultimate loads and DSM data/estimates concerning the C200 P beams analysed.

Mu.T (kN cm)

(|δ|/t)lim Mn.D.T (kN cm)

Mu . T My . T

Mu . T Mn . D . T

T (ºC) Mcr.D.T (kN  cm)

λ¯ D.T

0.23 106.8 0.50 498.3 0.74 1103.4 1.03 2135.6 1.25 3132.2 1.52 4627.1 1.75 6157.6 2.02 8186.4 2.25 10179.6 2.52 12813.5 2.75 15198.3 3.00 18152.5 3.25 21249.1 3.52 24915.2

119.2 525.8 1083.5 1662.6 1879.8 2094.5 2360.9 2793.2 3161.2 3560.1 3890.7 4263.2 4600.0 4928.3

1.01 0.81 1.77 3.72 5.27 6.79 14.04 15.53 17.10 19.48 21.19 22.66 23.75 25.21

106.8 498.3 1047.5 1630.6 2068.2 2609.2 3078.0 3616.8 4084.1 4636.1 5088.6 5602.4 6097.7 6639.5

1.12 1.06 0.98 0.78 0.60 0.45 0.38 0.34 0.31 0.28 0.26 0.23 0.22 0.20

1.12 1.06 1.03 1.02 0.91 0.80 0.77 0.77 0.77 0.77 0.76 0.76 0.75 0.74

100

2013.2

0.23 106.8 0.50 498.3 0.74 1103.4 1.03 2135.6 1.25 3132.2 1.52 4627.1 1.75 6157.6 2.02 8186.4 2.25 10179.6 2.52 12813.5 2.75 15198.3 3.00 18152.5 3.25 21249.1 3.52 24915.2

119.5 537.3 1117.0 1793.4 2108.2 2517.6 2992.3 3510.3 3921.0 4418.8 4811.0 4958.9 4958.9 4958.9

1.25 1.46 2.06 3.72 5.24 8.69 9.89 11.06 12.42 14.01 15.11 14.20 13.99 13.89

1811.9

0.49 1.02 1.51 2.01 2.51 2.99 3.50

443.5 1900.7 4118.1 7285.9 11404.0 16155.7 22174.5

435.0 1440.8 1846.6 2468.6 3159.4 3786.9 4387.7

0.89 3.38 6.54 15.86 19.32 22.57 25.15

443.5 1457.1 2333.0 3234.8 4147.1 5011.8 5940.0

0.98 0.76 0.45 0.34 0.28 0.23 0.20

0.98 0.99 0.79 0.76 0.76 0.76 0.74

200

1811.9

0.49 1.02 1.51 2.01 2.51 2.99 3.50

443.5 1900.7 4118.1 7285.9 11404.0 16155.7 22174.5

443.5 1532.4 2186.1 3061.4 3873.5 4607.0 4958.9

300

1610.6

0.49 388.7 1.02 1665.8 1.50 3609.1 1.99 6385.4 2.49 9994.5 2.96 14158.9 3.47 19433.8

345.8 1191.8 1590.7 2143.8 2757.6 3311.9 3846.3

1.18 2.86 6.12 15.88 19.32 22.50 25.05

388.7 1283.6 2056.7 2852.6 3657.8 4421.1 5240.3

0.89 0.72 0.44 0.34 0.28 0.23 0.20

0.89 0.93 0.77 0.75 0.75 0.75 0.73

300

1610.6

0.49 388.7 1.02 1665.8 1.50 3609.1 1.99 6385.4 2.49 9994.5 2.96 14158.9 3.47 19433.8

400

1409.3

0.48 0.99 1.46 1.94 2.43 2.89 3.39

323.9 1388.1 3007.6 5321.2 8328.8 11799.1 16194.9

256.8 894.1 1315.1 1786.6 2316.8 2793.0 3262.1

1.62 2.81 5.26 15.65 19.07 22.13 25.04

323.9 1088.6 1748.7 2428.4 3116.0 3767.7 4467.3

0.79 0.64 0.44 0.34 0.28 0.24 0.20

0.79 0.82 0.75 0.74 0.74 0.74 0.73

400

1409.3

0.48 0.99 1.46 1.94 2.43 2.89 3.39

500

1207.9

0.47 264.1 0.97 1131.9 1.42 2452.4 1.90 4338.8 2.37 6791.2 2.82 9620.8 3.31 13205.0

215.8 761.2 1122.4 1496.9 1938.0 2340.8 2738.7

1.55 2.66 5.12 15.39 18.61 21.76 24.25

264.1 903.5 1455.4 2023.6 2598.4 3143.3 3728.1

0.82 0.67 0.46 0.34 0.29 0.24 0.21

0.82 0.84 0.77 0.74 0.75 0.74 0.73

500

1207.9

600

624.1

0.49 1.01 1.49 1.98 2.48 2.95 3.46

149.5 640.7 1388.1 2455.9 3844.1 5445.7 7474.6

113.6 390.7 580.6 804.4 1042.5 1257.0 1464.8

1.68 2.95 5.44 15.96 19.51 22.48 25.17

149.5 495.0 793.5 1100.7 1411.6 1706.3 2022.5

0.76 0.61 0.42 0.33 0.27 0.23 0.20

0.76 0.79 0.73 0.73 0.74 0.74 0.72

600

700

261.7

0.50 1.03 1.52 2.02 2.52 3.00 3.52

64.8 277.6 601.5 1064.2 1665.8 2359.8 3239.0

48.1 164.1 244.1 342.0 444.0 534.1 620.1

1.68 3.02 5.50 16.16 19.62 22.67 25.83

64.8 212.0 339.2 470.2 602.7 728.3 863.1

0.74 0.59 0.41 0.32 0.27 0.23 0.19

0.74 0.77 0.72 0.73 0.74 0.73 0.72

800

181.2

0.44 0.91 1.34 1.78 2.22 2.65 3.10

34.9 149.5 323.9 573.0 896.9 1270.7 1744.1

29.5 107.4 163.8 208.9 271.9 328.9 387.4

1.39 2.49 4.64 14.70 17.40 20.39 23.11

34.9 124.7 202.4 282.4 363.3 440.0 522.3

0.85 0.72 0.51 0.36 0.30 0.26 0.22

0.85 0.86 0.81 0.74 0.75 0.75 0.74

My.T (kN cm)

My.T Mu.T (|δ|/t)lim Mn.D.T (kN  cm) (kN  cm) (kN  cm)

Mu . T My . T

Mu . T Mn . D . T

106.8 498.3 1047.5 1630.6 2068.2 2609.2 3078.0 3616.8 4084.1 4636.1 5088.6 5602.4 6097.7 6639.5

1.12 1.08 1.01 0.84 0.67 0.54 0.49 0.43 0.39 0.34 0.32 0.27 0.23 0.20

1.12 1.08 1.07 1.10 1.02 0.96 0.97 0.97 0.96 0.95 0.95 0.89 0.81 0.75

1.83 3.46 8.66 10.93 13.92 16.22 15.96

443.5 1457.1 2333.0 3234.8 4147.1 5011.8 5940.0

1.00 0.81 0.53 0.42 0.34 0.29 0.22

1.00 1.05 0.94 0.95 0.93 0.92 0.83

351.1 1249.8 1860.5 2612.5 3320.8 3974.5 4610.6

2.57 3.25 8.76 10.80 13.74 16.29 18.60

388.7 1283.6 2056.7 2852.6 3657.8 4421.1 5240.3

0.90 0.75 0.52 0.41 0.33 0.28 0.24

0.90 0.97 0.90 0.92 0.91 0.90 0.88

323.9 1388.1 3007.6 5321.2 8328.8 11799.1 16194.9

261.0 934.3 1498.7 2127.8 2728.6 3297.5 3860.2

3.95 3.70 8.69 10.48 13.74 16.36 18.51

323.9 1088.6 1748.7 2428.4 3116.0 3767.7 4467.3

0.81 0.67 0.50 0.40 0.33 0.28 0.24

0.81 0.86 0.86 0.88 0.88 0.88 0.86

0.47 264.1 0.97 1131.9 1.42 2452.4 1.90 4338.8 2.37 6791.2 2.82 9620.8 3.31 13205.0

219.4 795.2 1257.8 1792.5 2295.9 2772.4 3248.0

3.49 3.40 8.49 10.34 13.20 15.66 18.02

264.1 903.5 1455.4 2023.6 2598.4 3143.3 3728.1

0.83 0.70 0.51 0.41 0.34 0.29 0.25

0.83 0.88 0.86 0.89 0.88 0.88 0.87

624.1

0.49 1.01 1.49 1.98 2.48 2.95 3.46

149.5 640.7 1388.1 2455.9 3844.1 5445.7 7474.6

116.3 409.0 668.5 948.0 1220.5 1477.3 1729.3

3.80 3.98 8.83 10.85 14.16 16.67 19.00

149.5 495.0 793.5 1100.7 1411.6 1706.3 2022.5

0.78 0.64 0.48 0.39 0.32 0.27 0.23

0.78 0.83 0.84 0.86 0.86 0.87 0.85

700

261.7

0.50 1.03 1.52 2.02 2.52 3.00 3.52

64.8 277.6 601.5 1064.2 1665.8 2359.8 3239.0

49.4 172.0 283.4 402.0 518.0 627.2 733.9

4.33 3.85 8.89 10.99 14.47 16.88 19.28

64.8 212.0 339.2 470.2 602.7 728.3 863.1

0.76 0.62 0.47 0.38 0.31 0.27 0.23

0.76 0.81 0.84 0.86 0.86 0.86 0.85

800

181.2

0.44 0.91 1.34 1.78 2.22 2.65 3.10

34.9 149.5 323.9 573.0 896.9 1270.7 1744.1

30.3 111.5 180.4 252.5 324.3 391.6 460.2

2.64 3.03 4.90 9.84 12.00 14.54 16.90

34.9 124.7 202.4 282.4 363.3 440.0 522.3

0.87 0.75 0.56 0.44 0.36 0.31 0.26

0.87 0.89 0.89 0.89 0.89 0.89 0.88

92

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