Interaction of local-flexural buckling for cold-formed lean duplex stainless steel hollow columns

Thin-Walled Structures 112 (2017) 20–30

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Full length article

Interaction of local-flexural buckling for cold-formed lean duplex stainless steel hollow columns

MARK

⁎

M. Anbarasua, , M. Ashrafb a b

Department of Civil Engineering, Government College of Engineering, Salem 636011, Tamil Nadu, India School of Engineering and IT, The University of New South Wales, Canberra ACT 2610, Australia

A R T I C L E I N F O

A B S T R A C T

Keywords: Direct Strength Method Hollow columns Lean duplex stainless steel Local-flexural buckling interaction Numerical modelling

This paper investigates the structural behaviour and design of cold-formed lean duplex stainless steel (LDSS) hollow columns that primarily fail due to interaction of local and flexural buckling modes. Individual buckling modes observed in thin-walled metallic columns are relatively easy to deal with using available design rules, but interactions between buckling modes make it difficult to predict their structural response, especially with new material types such as LDSS. In the current study, geometric dimensions of the hollow column sections were chosen in such a way that they have almost equal local and flexural buckling stress. Finite element (FE) models for columns with pin-ended conditions were developed using ABAQUS [1], and numerically obtained results were validated using relevant test results available in literature. A total of 16 sections was used to produce 64 models offering various combination of geometry and material strength. Column resistances obtained from the parametric study were compared with those predicted using relevant American, Australian, European and DSM design techniques. Overall, it was observed that the current code guidelines fail to accurately capture the interaction of local and flexural buckling producing unconservative and unreliable predictions. Suitable modifications are proposed herein for AS/NZS standard [2], Eurocode 3 [3] and Direct Strength Method (DSM) [4] to account for the interaction of local and flexural buckling. Reliability analysis was also carried out to assess the performance of the existing as well as the proposed design rules for LDSS hollow columns.

1. Introduction Cold-formed stainless steel members are increasingly used now-adays as structural elements due to their corrosion resistance, aesthetic appearance, recyclability, better performance against fire, easy maintenance and durability. Stainless steel grades with low nickel content such as Lean Duplex Stainless Steel (LDSS) that offers high strength at a reasonable price, could be considered as a viable alternative to ordinary carbon steel. LDSS grades, especially EN 1.4162, has a low nickel content of ~1.5%, and an increased strength compared to conventional austenitic and ferritic stainless steels, making it a potentially attractive material for use in construction. Hollow sections are very common in thin-walled metallic construction due to their inherent efficiency against axial loading. Coupled instabilities mainly influence the behaviour and strength of hollow columns. Local plate buckling and flexural buckling modes can occur separately or interact with each other depending on the geometric dimensions of the cross-section, the geometric length of the column as well as its boundary conditions. Local buckling mode typically occurs at relatively short-wave length

⁎

due to the buckling of individual plate elements. Commonly used coldformed steel hollow section column members may possess similar local (L) and flexural (F) buckling resistances depending on the geometric dimensions, and their post-buckling behaviour, ultimate strengths and failure mechanisms could be strongly affected by coupling / interaction effects involving these two buckling modes [5–7]. Although column behaviour against individual buckling modes for hollow LDSS columns have recently been investigated, very little evidence is currently available on how the compression members would behave as a result of interaction between buckling modes. With the recent advancements in research to evaluate LDSS as a construction material, it is worth investigating this possible interaction between typical buckling modes in LDSS members for an appropriate understanding of the structural response of hollow columns. The current paper examines this interaction using nonlinear FE models developed using ABAQUS. A numerical parametric analysis was carried out, and the obtained results were compared against those predicted using available design codes. In accuracy and inconsistency observed in code predictions were dealt with by proposing appropriate modifications to the current design

Corresponding author. E-mail addresses: [email protected] (M. Anbarasu), [email protected] (M. Ashraf).

http://dx.doi.org/10.1016/j.tws.2016.12.006 Received 21 May 2016; Received in revised form 27 November 2016; Accepted 13 December 2016 0263-8231/ © 2016 Elsevier Ltd. All rights reserved.

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Nomenclature P*DSM A bf bl bw Eo Et L Le n Pcrl Pcrd Pcre PEXP PFEA Pnl Pnd Pne

cross-sectional area (mm2) breadth of flange of lipped channel section breadth of lip of lipped channel section breadth of web of lipped channel section initial Young's modulus tangent modulus length of specimen effective length of specimen exponent in Ramberg–Osgood expression critical elastic local buckling load (kN) critical elastic distortional buckling load (kN) critical elastic Euler's buckling load (kN) Experimental failure load (kN) ultimate load calculated from the FE analysis (kN) nominal member capacity of a member in compression for local buckling (kN) nominal member capacity of a member in compression for distortional buckling (kN) nominal member capacity of a member in compression for flexural, torsional and

P*EN P# AS/NZ P# DSM P# EN ri ro t β χ λl λd λc λl

Flexural/torsional buckling (kN)

λ0

Pý PASCE

λs λo σcr σ0.2 σu β φ

PAS/NZ PDSM PEN P* AS/NZ

yield capacity of the member in compression (kN) unfactored design strengths (nominal strength) by American specification unfactored design strengths (nominal strength) by AS/NZS Standard (kN) unfactored design strength calculated by Direct Strength Method(kN) unfactored design strengths by European code (kN) unfactored design strengths (nominal strength) by AS/NZS

Standard (kN) unfactored design strength calculated by Direct Strength Method (kN) unfactored design strengths by European code (kN) unfactored design strengths calculated using the proposed design rule for AS/NZS Standard (kN) unfactored design strengths calculated using the proposed design rule for Direct Strength Method (kN) unfactored design strengths calculated using the proposed design rule for European code (kN) inner corner radius of the specimen outer corner radius of specimen thickness of lipped channel section coefficient of buckling stress in Australian/New Zealand Standard reduction factor for members in compression in European Code local buckling slenderness distortional buckling slenderness overall buckling slenderness coefficient of buckling stress in Australian/New Zealand Standard for the non-dimensional slenderness to determine Pnl coefficient of buckling stress in Australian/New Zealand Standard cross-sectional slenderness overall slenderness elastic critical buckling stress of the section static 0.2% tensile proof stress static ultimate tensile strength reliability index resistance factor

conducted a series of tests on lean duplex stainless steel SHS and RHS stub columns, and reported similar conservatism for LDSS stub columns. Patton and Singh [16,17] developed numerical models for LDSS columns with various cross-section types, such as square, L, T, and + shaped sections. Both American and Eurocode provisions were reported to be conservative for fixed ended stub columns of all shapes except for the L-shaped section. A forementioned major investigations reported in literature related to stainless steel hollow section columns clearly show that the currently available international design codes are mostly conservative in predicting compression resistances. Significant research has been reported to date suggesting modifications to the current design guidelines as well as proposing innovative design rules for accurate prediction of stainless steel member resistances. Gardner and Nethercot [18] devised a strain based design approach for stainless steel hollow sections to address the shortcomings of the international design codes. Their proposed design technique was later modified by Gardner and Ashraf [19] for nonlinear metallic materials. Ashraf et. al [20,21] reported a comprehensive study on FE modelling for different types of stainless steel cross sections and extended the design technique to be applied for both closed and open sections. Later, Gardner [22] generalized the design principle for all cross-section types and proposed Continuous Strength Method for stainless steel sections. Becque et. al [23] recently proposed a modified Direct Strength Method to predict the strength of ferritic and austenitic steel columns. Becque and Rasmussen [24] conducted a parametric study based on FE modelling approach to investigate the interaction of local and overall buckling modes for lipped channel sections produced from AISI304, AISI430 and 3Cr12 stainless steel grades. Numerically obtained results were compared to those predicted using American,

rules.

2. Literature survey Significant research has been conducted on stainless steel hollow section columns in the recent past. Young and Hartono [8] conducted experiments on fixed ended cold-formed stainless steel circular hollow columns produced from austenitic stainless steel grade EN 1.4301, and identified that the existing design rule predictions for compression resistance were mostly conservative. Liu and Young [9] presented the results of fixed-ended cold-formed stainless steel square hollow section (SHS) columns and also reported that American, Australian/New Zealand and European specifications for cold-formed stainless steel structures were significantly conservative. Gardner and Nethercot [10] conducted experiments on cold-formed austenitic stainless steel square, rectangular and circular hollow section beams and columns, and reported that predictions obtained using EC3 were significantly conservative. Ellobody and Young [11] developed a numerical model for high strength stainless steel compression members using ABAQUS and reported that existing design codes were inconsistent in predicting compression resistance; predicted resistances for long columns were mostly conservative but those for a number of stub columns were reported to be unconservative. Young and Lui [12,13] and Lui et al. [14] investigated the performance of SHS and RHS columns produced from cold-formed duplex and high strength austenitic stainless steel both experimentally and numerically. Significant conservatism was reported when test results were compared against those obtained using relevant American, Australian/New Zealand and European specifications for cold-formed stainless steel structures. Huang and Young [15] 21

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3.2. Boundary conditions and loading application

Australian/New Zealand and European design standards for stainless steel. Significant conservatism was observed for stocky sections but the interactions became more pronounced for relatively slender sections with less conservative predictions. Theofanous and Gardner [25] carried out experimental and numerical investigations on stub columns and pin-ended SHS and RHS columns produced fromlean duplex stainless steel LDSS (EN 1.4162 grade), and proposed new design rules based on the effective width approach for class 3 sections according to EC3. Huang and Young [26] recently proposed modifications to the AS/ NZS Standard, EC3 Code and DSM guidelines for cold-formed LDSS columns. However, their proposed equations do not explicitly account for the interaction of local and flexural buckling phenomenon, which is outlined in detail in Sections 7 and 8 of this paper. Anbarasu and Ashraf [27] recently carried out FE investigations for LDSS lipped channel columns and subsequently proposed DSM formulations to tackle local and global buckling for lipped channel columns. This paper, however, specifically looks into the interaction of local and flexural buckling modes; this type of failure is very likely to occur in high strength thinwalled column sections produced from LDSS but is yet to be investigated with sufficient details. Numerical results generated using the developed FE models were used to propose modifications to the currently available design guidelines to incorporate the effect of possible interaction between local and flexural buckling modes.

Fig. 2(a) illustrates the validation model used for RHS stub columns, where fixed end conditions were modelled by following the approach recently outlined in [27]. Fig. 2(b) shows the pin-ended boundary conditions used for the parametric analysis. Rigid body tie constraints were used to simulate pin ended conditions; details of the modelling technique employed in this study for pin ended columns are also reported in [27]. To reduce the computational time, half of the section was modelled for longer specimens, and appropriate symmetry boundary conditions were applied at the centre. Fig. 2(b) shows the developed FE model of a half-span column used in the parametric model. 3.3. Geometrical imperfections Initial local and overall geometric imperfections were both included in the parametric models. A combination of local and global buckling mode shapes was used as initial geometric imperfection patterns to perturb the geometry of the validation model. The local imperfection amplitude and the overall geometric imperfection magnitude were taken following the suggested guidelines reported in [25]. Initial geometric shape for the columns were obtained as a linear combination of the scaled buckled mode shapes for the chosen local and global modes in the parametric models to initiate nonlinear analysis.

3. FEM technique adopted in the current study

3.4. Nonlinear analysis

FE modelling technique is used in the current paper to investigate a special case of buckling interaction where the flexural resistance of a given column is almost equal to its cross-sectional resistance in compression. Fig. 1 shows a typical characteristic buckling curve obtained using CUFSM software [28] for an RHS 60×90×1.5 column. The variation of the critical buckling stress σcr with length L is presented in Fig. 1, which shows that for an overall column length of approx. 2275 mm, the flexural buckling resistance of the column will be almost equal to its local buckling resistance. When such a column is subjected to compression, its failure mechanism could be dominated by the interaction between local and flexural buckling modes. This is a unique case of buckling, which could significantly affect the overall resistance of thin-walled columns. This possible interaction should be checked for by comparing the cross-sectional resistance and the flexural buckling resistance of a column using available design rules such as DSM or by using readily available numerical tools such as CUFSM. If the cross-sectional resistance and the column resistance against flexural buckling are almost equal, appropriate recognition for possible interactions should be incorporated in design calculations; the remainder of the current paper presents relevant numerical investigations and suggests appropriate design modifications for accurate prediction of column resistance for such a case. Commercial general purpose FE program ABAQUS [1] was used in the current study to develop nonlinear numerical models to predict the ultimate resistance as well as the failure modes for cold-formed LDSS hollow columns. The following sections present required summary of adopted FE modelling technique as all relevant details were recently reported in [27].

Elastic buckling analyses were carried out to obtain the required buckling modes, which were subsequently used to generate an appropriate initial geometry for nonlinear analysis. The modified Riks method available in ABAQUS was employed to solve the geometrically and materially nonlinear column models; this allowed for the postultimate behaviour to be traced accurately. 4. Validation of the modelling technique The adopted FE modelling approach was validated using the experimental results reported in Theofanous and Gardner [25] for LDSS hollow columns. Tables 1 and 2, respectively, show the measured dimensions of the tested specimens and the material properties reported in [25]. Table 3 compares the obtained FE results for ultimate resistances for all considered columns to those reported from experiments. Fig. 3 compares the obtained deformed shapes for stub column 80×80×4SC2 and long column 80×80×4–1200 to those reported from experiments, whilst Fig. 4 compares their complete load-deformation behaviour. It is obvious from the comparisons that the FE models were able to predict the compression resistance with more than 98% accuracy and the predictions were very consistent with a less than 4% scatter. This paved the way for a reliable parametric analysis using the developed FE models to evaluate the interaction of local and flexural buckling modes Buckling stress, cr (MPa)

3000 3

3.1. Element type, mesh size and material model LDSS hollow section columns were modelled using four-node doubly curved shell elements with reduced integration (S4R). The element size was kept at 5 mm×5 mm in the flat regions of the FE model following a mesh convergence study. Two-stage RambergOsgood material model proposed in [19] was used in the current study to accurately incorporate the nonlinear material behaviour exhibited by LDSS.

2 2500 2 2000 1 1500 1 1000 500 0

10

100

Length, L (mm)

10000

10000

Fig. 1. Characteristic buckling curve for RHS 60×90×1.5 showing variation in critical buckling stress σcr with column effective length L.

22

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Table 2 Material properties of tested columns (Theofanous and Gardner [25]). E (N/mm2)

Specimen ID

80×80×4 60×60×3 80×40×4 100×100×4

Flat Corner Flat Corner Flat Corner Flat Corner

199,900 210,000 209,800 212,400 199,500 213,800 198,800 206,000

σ0.2 (N/ mm2)

679 731 755 885 734 831 586 811

σ1.0 (N/ mm2)

736 942 819 1024 785 959 632 912

σu (N/ mm2)

773 959 839 1026 817 962 761 917

Compound R-O Coefficients n

n'

6.5 5.6 6.0 6.3 10.1 4.4 9.0 6.3

4.2 3.7 4.3 4.0 3.4 4.0 2.8 4.1

Table 3 Comparison of experimental and FEA ultimate load.

Fig. 2. Loading and boundary conditions used in the FE models (a) Validation model (b) Parametric model. Table 1 Tested column dimensions (Theofanous and Gardner [25]). Specimen ID

Depth (H) (mm)

Width (B) (mm)

Thickness (t) (mm)

Inner radius (ri) (mm)

Length (L) (mm)

Area (A) mm2

80×80×4SC2 80×80×41200 80×80×42000 60×60×3800 60×60×31200 60×60×31600 60×60×32000 80×40×4SC2 80×40×4MI-800 80×40×4MI-1200 80×40×4MI-1600 100x100x4SC2

80.0

80.0

3.81

3.60

332.2

1125.0

79.6

79.3

3.72

3.80

1199.5

1091.0

79.6

79.5

3.80

3.40

1999.0

1116.7

60.0

60.0

2.4

3.13

799.0

690.8

60.0

60.0

2.4

3.13

1199.0

689.8

60.0

59.6

2.4

3.15

1599.0

692.4

60.0

60.0

2.7

3.13

1999.0

689.1

79.5

39.6

3.81

4.30

237.8

808.8

79.4

39.5

3.80

3.60

797.2

810.0

79.2

40.0

3.80

3.80

1199.0

811.3

79.2

39.0

3.80

4.30

1600.0

800.4

103.0

102.0

3.97

3.9

400.0

1524.7

Specimen ID

PEXP (kN)

PFEA (kN)

PEXP/PFEA

80×80×4-SC2 80×80×4-1200 80×80×4-2000 60×60×3-800 60×60×3-1200 60×60×3-1600 60×60×3-2000 80×40×4-SC2 80x40×4-MI-800 80×40×4-MI-1200 80×40×4-MI-1600 100×100×4-SC2 Mean Std. Dev

915.00 672.50 361.90 445.90 326.90 231.70 162.30 710.00 366.60 237.40 160.40 1037.00

906.78 678.26 385.52 450.28 344.30 245.36 167.25 693.43 377.52 252.27 156.13 1002.49

1.009 0.992 0.939 0.990 0.949 0.944 0.970 1.024 0.971 0.941 1.027 1.034 0.983 0.035

Fig. 3. Comparison of failure modes obtained from test and FE modelling (a)For 80×80×4-SC2 (b) For 80×80×4-1200.

observed in high strength LDSS hollow section columns.

stresses as explained in Fig. 1. The first minimum of the characteristic buckling curve shows the critical stress for local buckling for a given cross-section. This local buckling stress also becomes the critical stress for flexural buckling at a specific effective length triggering a possible interaction between local and flexural buckling modes. In such a case, the compression resistance and the post-buckling behaviour will be strongly affected by the interaction of local-flexural buckling. A total of

5. Selection of sections to investigate interaction between local and flexural buckling The cross-sectional dimensions and geometric lengths were selected through buckling analysis using CUFSM software [28] to ensure that the columns possess almost identical elastic local and flexural buckling 23

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Fig. 4. Comparison of complete load vs deflection response for LDSS hollow columns (a)For 80×80×4-SC2 (b) For 60×60×3-800 (c) For 60×60×3-2000.

7. Existing design standards and their performance

16 column cross-section geometries was carefully selected using CUFSM to investigate this possible interaction through a numerical parametric study. Table 4 shows the cross-sections selected for this study, all of which belong to Class 4 slender sections according to EC3 Part 1–4 [3].

7.1. American specification The clause 3.4 of the ASCE specification [31] for concentrically loaded compression members provides necessary details for calculating design strengths (PASCE) for columns by tangent modulus approach. However, an iterative approach is required to calculate the tangent modulus (Et) which is quite lengthy when compared against other direct approaches.

6. Parametric study Parametric study was conducted to investigate, primarily, the influence of both cross-section slenderness and the interaction of local-flexural buckling phenomenon on column resistance. Carefully chosen 16 hollow sections were considered in this parametric study, and the nominal centreline dimensions, based on Table 1, were used in the developed FE models. Considered column sections had pin boundary conditions with geometric lengths in the range of 2007–12,450 mm. According to EN10088-4 [29], the minimum specified material properties of LDSS Grade EN1.4162 are: 0.2% proof stress σ0.2 of 530 MPa and ultimate stress σu of 700–900 MPa. As part of the current parametric study, theoretical and numerical analyses were carried out for the selected cross-sections by varying the 0.2% proof stress values of the flat material, as shown in Table 5, following the results for LDSS reported in [30]. A total of 64 columns with identical local and flexural buckling stresses were analysed numerically, and their resistances were also predicted using available design guidelines. Unfactored design resistances were calculated using the American code ASCE 8-02 [31], the Australian/New Zealand Standard (AS/NZS 4673:2001) for cold-formed stainless steel structures [2], Eurocode 3 Part 1.4: Design of steel structures – Part 1.4: General rules -Supplementary rules for stainless steels [3], and the Direct Strength Method (AISI-S100:2007) for cold-formed steel structures [4].

7.2. Australian/New Zealand standard The clause 3.4.2 of the AS/NZS Standard [2] provides guidelines for a direct approach to calculate the design strengths (PAS/NZS) for columns. It is worth noting that the current AS/NZS Standard does not cover LDSS type EN 1.4162 grade. Buckling stresses (fn) for the considered LDSS columns were determined using the values given for duplex grade i.e. parameters α, β, λ0 and λ1were taken as 1.16, 0.13, 0.65 and 0.42 respectively. 7.3. European code Column design resistances following Euro code (PEN) were calculated based on the clause 5.4.1 of the EC3 Part1.4 [3]. The reduction factor χ was calculated using an imperfection factor α=0.49 and a limiting slenderness λo=0.4, which are suggested for hollow sections failing due to flexural buckling. Effective widths (be) for the considered Class 4 slender sections were calculated based on the reduction factor ρ as suggested in clause 5.2.3 of the EC3 Part 1.4. 24

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Table 4 Section dimensions and critical buckling stress for material-M1. Specimen designation

60×90×1 60×90×1.5 80×80×1 80×80×1.5 80×120×1 80×120×1.5 80×120×2.5 80×120×3 80×160×1 80×160×1.5 80×160×2.5 80×240×1 80×240×1.5 80×240×2.5 80×240×3 80×240×4

Section dimensions (mm) Depth (H)

Width (B)

Radius (r)

Thickness (t)

90 90 80 80 120 120 120 120 160 160 160 240 240 240 240 240

60 60 80 80 80 80 80 80 80 80 80 80 80 80 80 80

1.50 2.25 1.50 2.25 1.50 2.25 3.75 4.50 1.50 2.25 3.75 1.50 2.25 3.75 4.50 6.00

1.0 1.5 1.0 1.5 1.0 1.5 2.5 3.0 1.0 1.5 2.5 1.0 1.5 2.5 3.0 4.0

SHS 100×100×4 M1 RHS 80×40×4 M2 SHS 80×80×4 M3 SHS 60×60×3 M4

E (N/mm2)

σ0.2 (N/ mm2)

σ1.0 (N/ mm2)

σu (N/ mm2)

109.07 245.45 111.52 250.48 61.37 138.05 383.53 552.67 36.26 81.56 226.71 16.79 37.81 105.44 151.93 270.75

3430 2275 4310 2870 6105 4060 2420 2007 8205 5448 3251 12,450 8290 4950 4115 3062

Table 6 compares numerically obtained compression resistances (PFEA) against those predicted using international codes (PASCE, PAS/NZ, PEN and PDSM), whilst Table 7 compares FE results against those determined using the recent modifications proposed by Huang and Young [26] for LDSS columns (P*AS/NZ, P*EN and P*DSM). Obtained results were used to determine the reliability of the current code provisions and the design modifications proposed by Huang and Young [26] for LDSS columns subjected to simultaneous local and flexural buckling following the procedure outlined in the Commentary of the ASCE Specifications [31]. A target reliability index (β0) of 2.5 for stainless steel structural members was considered as a lower limit in this study. Load combinations of (1.2DL+1.6LL), (1.25DL+1.5LL) and (1.35DL+1.5LL) with resistance factors (ϕ) of 0.85, 0.90, and 0.91 for concentrically loaded compression members were used in calculating the reliability index β0 for ASCE [31], AS/NZS [2], and EC3 [3] respectively. A load combination of (1.2DL+1.6LL) was used for the Direct Strength Method (DSM) as presented in AISI-S100-2007 Specification [4]. Table 6 clearly shows that the design strengths predicted by ASCE 802 [31] and DSM specifications [4] are considerably unconservative, with mean predictions of 0.90 and 0.93 respectively, for the considered 64 LDSS columns subjected to simultaneous local and flexural buckling. AS/NZS [2] and EC3 [3] guidelines produced fairly accurate mean of 0.99 and 1.03 respectively but the predictions were very scattered producing almost 14% and 18% standard deviations. Most importantly, all international design codes failed to meet the target value of 2.50 for reliability index β0, which clearly shows that the possible interaction of local and flexural buckling is not appropriately taken care of by the current design guidelines. Huang and Young [26] recently conducted numerical investigations on the structural performance of cold-formed LDSS columns, and proposed some modifications to AS/NZS [2], EC3 [3] and DSM [4] guidelines to tackle the observed discrepancies in predicting compression resistance. Huang and Young's [26] proposed formulations were used in the current study to determine resistances for the considered columns where interaction between local and flexural buckling dominates structural response. But their suggested modifications [26] for EC3 failed to capture this phenomenon by a significant margin producing a mean of 0.90 with a scatter of 14%, and the predictions were significantly unreliable giving a β0 index of 1.80. Performance of the modified AS/NZS design rules was comparatively better with a mean of 0.96 and a standard deviation of 0.11, but still failed to achieve the desired level of reliability with β0=2.10. Huang and Young's

Compound RO coefficients n

n'

Flat

198,238

560

642

–

8.3

2.6

Corner Flat

206,000 203,964

811 607

912 734

917 –

6.3 4.6

4.1 2.9

Corner Flat

213,850 197,185

831 657

959 770

962 –

4.4 4.7

4.0 2.6

Corner Flat

210,000 206,430

731 711

942 845

959 –

5.6 5.0

3.7 2.7

Corner

212,400

885

1024

1026

6.3

4.0

7.4. Direct Strength Method The direct strength method (DSM) [4] was originally proposed for the design of cold-formed carbon steel members. The elastic buckling load for a given cross-section can be determined using the CUFSM software. In DSM, the slenderness ratio of a full cross-section is used instead of the most slender constituent elements, and hence this method recognizes the inherent element interactions within a thin-walled crosssection.

PDSM=min(Pne,Pnl)

Length of the column (L) (mm)

8. Discussion on the obtained results

Table 5 Material properties for parametric study (Theofanous and Gardner [30]). Specimen ID and material type

Critical Buckling Stress (σcr) (N/mm2)

(1)

Table 6 compares the numerical results obtained from the parametric analysis to those predicted using ASCE 8-02 [31], AS/NZS 4673 [2], EN 1993-1-4 [3], and DSM [4], whilst Table 7 presents the performance of the modified design rules for LDSS columns recently proposed by Huang and Young [17] as well as those suggested as part of the current study. All specimens considered in the parametric study failed in combined local and flexural buckling as expected, and typical deformed shapes obtained from FE analysis for two sections are shown in Fig. 5. Typical load displacement responses observed for 80×120×2.5 columns with a geometric length of 2420 mm are also shown in Fig. 6. 25

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suggested DSM modifications performed the best with a mean of 0.94 with β0=2.46 but there were considerable number of unconservative

predictions. All these comparisons clearly show that there are still rooms for improvement to tackle this unique but likely case of

Table 6 Comparison of compression resistances obtained from FE and those predicted using current design codes. Specimen ID

60×90×1

60×90×1.5

80×80×1

80×80×1.5

80×120×1

80×120×1.5

80×120×2.5

80×120×3

80×160×1

80×160×1.5

80×160×2.5

80×240×1

80×240×1.5

80×240×2.5

80×240×3

80×240×4

Material type

M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4

Mean Std. Dev. Capacity reduction factor (φ) Reliability index (β0)

Interaction length (mm)

3430 3430 3430 3430 2275 2280 2280 2275 4310 4310 4315 4310 2870 2870 2870 2870 6105 6105 6105 6105 4060 4060 4060 4060 2420 2420 2420 2420 2007 2008 2008 2007 8205 8185 8185 8185 5448 5448 5448 5448 3251 3252 3252 3251 12,450 12,450 12,450 12,450 8290 8290 8290 8290 4950 4950 4950 4950 4115 4110 4115 4110 3062 3063 3063 3062

PFEA (kN)

21.35 23.80 22.92 23.82 84.66 85.95 83.31 87.16 26.14 27.10 26.01 27.16 95.77 96.34 94.37 98.74 16.61 17.11 16.44 17.15 60.85 62.51 60.13 62.70 297.94 284.84 289.53 307.30 444.52 425.15 438.04 474.01 11.09 11.41 10.96 11.43 41.88 42.19 40.70 42.45 209.12 213.66 207.05 216.84 6.13 6.30 5.96 6.20 23.46 24.06 23.13 24.12 120.98 124.34 119.90 125.07 203.01 224.81 216.64 226.24 546.46 551.46 536.47 562.18

ASCE

AS/NZS

EN

DSM

PASCE (kN)

PFEA/PASCE

PAS/NZS (kN)

PFEA/PAS/NZ

PEN (kN)

PFEA/PEN

PDSM (kN)

PFEA/PDSM

27.37 28.00 26.96 28.36 93.03 89.53 88.63 94.74 28.73 29.14 28.27 29.85 96.74 94.45 93.44 98.74 20.26 20.87 20.30 21.17 69.15 70.24 68.33 72.07 294.99 276.54 281.10 301.27 427.42 412.77 425.28 460.20 14.04 14.44 14.05 14.65 48.14 49.06 47.33 49.94 222.47 218.02 215.68 228.25 8.40 8.63 8.28 8.73 28.27 28.99 28.21 29.41 132.95 135.15 131.76 137.44 230.69 234.18 228.04 238.15 541.05 520.25 520.84 551.16

0.78 0.85 0.85 0.84 0.91 0.96 0.94 0.92 0.91 0.93 0.92 0.91 0.99 1.02 1.01 1.00 0.82 0.82 0.81 0.81 0.88 0.89 0.88 0.87 1.01 1.03 1.03 1.02 1.04 1.03 1.03 1.03 0.79 0.79 0.78 0.78 0.87 0.86 0.86 0.85 0.94 0.98 0.96 0.95 0.73 0.73 0.72 0.71 0.83 0.83 0.82 0.82 0.91 0.92 0.91 0.91 0.88 0.96 0.95 0.95 1.01 1.06 1.03 1.02 0.90 0.09 0.85 2.27

25.12 26.15 25.47 26.76 76.96 79.58 78.59 83.01 26.67 27.37 26.81 28.00 82.56 86.02 84.26 88.95 19.54 20.13 19.34 20.42 62.73 64.44 63.29 66.00 238.35 249.86 249.59 262.65 358.48 379.60 384.25 405.14 13.69 14.09 13.53 14.29 44.55 46.36 44.73 47.17 186.71 194.24 191.71 200.78 8.17 8.51 8.16 8.61 27.28 28.31 27.54 28.71 122.20 126.88 122.35 128.94 205.06 212.08 208.31 217.54 447.92 467.34 462.47 484.64

0.85 0.91 0.90 0.89 1.10 1.08 1.06 1.05 0.98 0.99 0.97 0.97 1.16 1.12 1.12 1.11 0.85 0.85 0.85 0.84 0.97 0.97 0.95 0.95 1.25 1.14 1.16 1.17 1.24 1.12 1.14 1.17 0.81 0.81 0.81 0.80 0.94 0.91 0.91 0.90 1.12 1.10 1.08 1.08 0.75 0.74 0.73 0.72 0.86 0.85 0.84 0.84 0.99 0.98 0.98 0.97 0.99 1.06 1.04 1.04 1.22 1.18 1.16 1.16 0.99 0.14 0.90 2.14

23.21 24.04 23.63 24.81 69.39 71.63 71.21 75.14 24.90 25.57 25.25 26.63 73.11 76.46 75.50 79.63 18.46 19.01 18.68 19.49 56.87 58.97 57.82 60.87 217.47 226.06 226.20 238.22 336.76 354.29 356.13 376.20 13.69 14.26 13.87 14.47 43.63 44.88 44.24 46.14 175.73 182.62 180.04 190.21 9.01 9.26 9.03 9.39 28.96 30.08 29.28 30.53 123.45 128.19 124.90 131.65 205.06 212.08 208.31 219.65 437.17 455.75 450.82 476.42

0.92 0.99 0.97 0.96 1.22 1.20 1.17 1.16 1.05 1.06 1.03 1.02 1.31 1.26 1.25 1.24 0.90 0.90 0.88 0.88 1.07 1.06 1.04 1.03 1.37 1.26 1.28 1.29 1.32 1.20 1.23 1.26 0.81 0.80 0.79 0.79 0.96 0.94 0.92 0.92 1.19 1.17 1.15 1.14 0.68 0.68 0.66 0.66 0.81 0.80 0.79 0.79 0.98 0.97 0.96 0.95 0.99 1.06 1.04 1.03 1.25 1.21 1.19 1.18 1.03 0.18 0.91 2.13

26.36 26.15 25.19 26.47 85.52 86.82 85.01 88.94 27.52 28.23 27.38 28.59 92.98 95.39 92.52 96.80 19.09 19.44 18.90 19.71 64.05 65.80 63.97 66.70 275.87 287.72 283.85 301.27 419.36 442.86 446.98 469.32 13.36 13.91 13.37 14.11 45.52 46.88 45.22 47.17 211.23 215.82 209.14 219.03 8.28 8.63 8.28 8.73 28.27 28.99 27.87 29.41 130.09 133.70 130.33 135.95 225.57 231.76 225.67 235.67 535.75 551.46 536.47 562.18

0.81 0.91 0.91 0.90 0.99 0.99 0.98 0.98 0.95 0.96 0.95 0.95 1.03 1.01 1.02 1.02 0.87 0.88 0.87 0.87 0.95 0.95 0.94 0.94 1.08 0.99 1.02 1.02 1.06 0.96 0.98 1.01 0.83 0.82 0.82 0.81 0.92 0.90 0.90 0.90 0.99 0.99 0.99 0.99 0.74 0.73 0.72 0.71 0.83 0.83 0.83 0.82 0.93 0.93 0.92 0.92 0.90 0.97 0.96 0.96 1.02 1.00 1.00 1.00 0.93 0.08 0.85 2.42

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Table 7 Performance of the modified design rules proposed by Huang and Young [17] and those suggested in the current paper. Specimen ID

60×90×1

60×90×1.5

80×80×1

80×80×1.5

80×120×1

80×120×1.5

80×120×2.5

80×120×3

80×160×1

80×160×1.5

80×160×2.5

80×240×1

80×240×1.5

80×240×2.5

80×240×3

80×240×4

Material type

M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4

Mean Std. Dev. Capacity reduction factor (φ) Reliability index (β0)

PFEA (kN)

21.35 23.80 22.92 23.82 84.66 85.95 83.31 87.16 26.14 27.10 26.01 27.16 95.77 96.34 94.37 98.74 16.61 17.11 16.44 17.15 60.85 62.51 60.13 62.70 297.94 284.84 289.53 307.30 444.52 425.15 438.04 474.01 11.09 11.41 10.96 11.43 41.88 42.19 40.70 42.45 209.12 213.66 207.05 216.84 6.13 6.30 5.96 6.20 23.46 24.06 23.13 24.12 120.98 124.34 119.90 125.07 203.01 224.81 216.64 226.24 546.46 551.46 536.47 562.18

Column resistances predicted using modified rules Huang and Young [17]

Column resistances predicted using the equations proposed in this study

P*AS/NZS (kN)

PFEA/ P*AS/NZ

P*EN (kN)

PFEA/ P*EN

P*DSM (kN)

PFEA/ P*DSM

P# AS/NZS (kN)

PFEA/ P#AS/NZ

P# EN (kN)

PFEA/ P#EN

P# DSM (kN)

PFEA/ P#DSM

25.72 26.74 25.75 27.07 80.63 83.45 82.49 87.16 27.23 27.94 27.09 28.59 86.28 89.20 87.38 92.28 19.77 20.37 19.57 20.66 64.73 66.50 64.66 68.15 256.84 268.72 265.62 281.93 393.38 412.77 417.18 438.90 13.69 14.26 13.70 14.47 45.52 46.88 45.73 47.70 195.44 203.49 199.09 210.52 8.28 8.51 8.28 8.61 27.60 28.64 27.54 29.06 124.72 129.52 124.90 131.65 211.47 218.26 212.39 224.00 475.18 492.38 483.31 506.47

0.83 0.89 0.89 0.88 1.05 1.03 1.01 1.00 0.96 0.97 0.96 0.95 1.11 1.08 1.08 1.07 0.84 0.84 0.84 0.83 0.94 0.94 0.93 0.92 1.16 1.06 1.09 1.09 1.13 1.03 1.05 1.08 0.81 0.80 0.80 0.79 0.92 0.90 0.89 0.89 1.07 1.05 1.04 1.03 0.74 0.74 0.72 0.72 0.85 0.84 0.84 0.83 0.97 0.96 0.96 0.95 0.96 1.03 1.02 1.01 1.15 1.12 1.11 1.11 0.96 0.12 0.90 2.10

26.36 27.36 26.65 28.02 80.63 83.45 82.49 87.16 28.41 29.46 28.58 30.18 85.51 89.20 87.38 92.28 20.76 21.39 20.81 21.71 65.43 67.95 66.08 69.67 254.65 266.21 265.62 279.36 396.89 416.81 417.18 443.00 15.19 15.63 15.22 15.88 49.27 50.83 49.04 51.77 205.02 213.66 209.14 219.03 9.73 10.00 9.61 10.16 31.70 32.96 31.68 33.50 139.06 144.58 141.06 147.14 233.34 241.73 238.07 248.62 505.98 530.25 525.95 551.16

0.81 0.87 0.86 0.85 1.05 1.03 1.01 1.00 0.92 0.92 0.91 0.90 1.12 1.08 1.08 1.07 0.80 0.80 0.79 0.79 0.93 0.92 0.91 0.90 1.17 1.07 1.09 1.10 1.12 1.02 1.05 1.07 0.73 0.73 0.72 0.72 0.85 0.83 0.83 0.82 1.02 1.00 0.99 0.99 0.63 0.63 0.62 0.61 0.74 0.73 0.73 0.72 0.87 0.86 0.85 0.85 0.87 0.93 0.91 0.91 1.08 1.04 1.02 1.02 0.90 0.14 0.91 1.80

26.04 25.59 24.91 25.89 84.66 85.95 83.31 88.04 27.23 27.94 27.09 28.29 91.21 94.45 90.74 94.94 18.66 19.22 18.68 19.49 63.39 65.11 62.64 66.00 273.34 284.84 281.10 295.48 415.44 438.30 438.04 464.72 13.20 13.75 13.20 13.94 45.03 46.36 44.73 46.65 207.05 213.66 207.05 216.84 8.17 8.51 8.16 8.61 27.93 28.64 27.54 29.06 128.70 132.28 133.22 133.05 223.09 229.40 221.06 233.24 530.54 546.00 531.16 556.61

0.82 0.93 0.92 0.92 1.00 1.00 1.00 0.99 0.96 0.97 0.96 0.96 1.05 1.02 1.04 1.04 0.89 0.89 0.88 0.88 0.96 0.96 0.96 0.95 1.09 1.00 1.03 1.04 1.07 0.97 1.00 1.02 0.84 0.83 0.83 0.82 0.93 0.91 0.91 0.91 1.01 1.00 1.00 1.00 0.75 0.74 0.73 0.72 0.84 0.84 0.84 0.83 0.94 0.94 0.90 0.94 0.91 0.98 0.98 0.97 1.03 1.01 1.01 1.01 0.94 0.09 0.85 2.46

22.01 22.67 22.25 23.35 70.55 72.84 70.60 74.50 23.76 24.41 23.86 25.15 76.62 78.97 77.35 80.93 17.30 17.82 17.31 18.24 54.82 56.83 55.17 58.06 242.23 249.86 241.28 253.97 435.80 442.86 421.19 443.00 12.19 12.68 12.31 12.99 39.51 40.57 39.51 41.62 171.41 176.58 171.12 180.70 7.66 7.88 7.64 8.05 24.69 25.60 24.87 25.94 108.02 111.02 109.00 113.70 182.89 188.92 183.59 193.37 423.61 434.22 422.42 446.17

0.97 1.05 1.03 1.02 1.20 1.18 1.18 1.17 1.10 1.11 1.09 1.08 1.25 1.22 1.22 1.22 0.96 0.96 0.95 0.94 1.11 1.10 1.09 1.08 1.23 1.14 1.20 1.21 1.02 0.96 1.04 1.07 0.91 0.90 0.89 0.88 1.06 1.04 1.03 1.02 1.22 1.21 1.21 1.20 0.80 0.80 0.78 0.77 0.95 0.94 0.93 0.93 1.12 1.12 1.10 1.10 1.11 1.19 1.18 1.17 1.29 1.27 1.27 1.26 1.08 0.13 0.90 2.50

20.93 21.83 21.22 22.26 65.63 67.68 66.12 69.73 22.73 23.36 22.82 24.04 69.91 72.44 70.95 74.80 16.61 17.11 16.61 17.50 52.01 53.89 52.75 55.49 215.90 224.28 219.34 231.05 352.79 366.51 362.02 379.21 12.19 12.68 12.31 12.99 39.14 40.57 39.51 41.21 164.66 170.93 166.98 174.87 8.07 8.29 8.05 8.49 25.78 26.73 25.99 27.41 110.99 115.13 112.06 117.99 186.25 192.15 188.38 198.46 413.98 430.83 422.42 446.17

1.02 1.09 1.08 1.07 1.29 1.27 1.26 1.25 1.15 1.16 1.14 1.13 1.37 1.33 1.33 1.32 1.00 1.00 0.99 0.98 1.17 1.16 1.14 1.13 1.38 1.27 1.32 1.33 1.26 1.16 1.21 1.25 0.91 0.90 0.89 0.88 1.07 1.04 1.03 1.03 1.27 1.25 1.24 1.24 0.76 0.76 0.74 0.73 0.91 0.90 0.89 0.88 1.09 1.08 1.07 1.06 1.09 1.17 1.15 1.14 1.32 1.28 1.27 1.26 1.11 0.17 0.91 2.50

24.26 24.04 23.15 24.31 79.12 80.33 78.59 82.23 25.63 26.31 25.25 26.63 85.51 88.39 85.02 88.95 17.48 18.01 17.49 18.24 59.08 60.69 58.95 61.47 256.84 266.21 263.21 279.36 393.38 412.77 417.18 438.90 12.32 12.82 12.45 12.99 41.88 43.05 41.96 43.76 193.63 199.68 193.50 202.65 7.66 7.97 7.64 8.05 26.07 26.73 25.99 27.10 120.98 124.34 123.61 123.83 209.29 216.16 208.31 217.54 492.31 510.61 496.73 515.76

0.88 0.99 0.99 0.98 1.07 1.07 1.06 1.06 1.02 1.03 1.03 1.02 1.12 1.09 1.11 1.11 0.95 0.95 0.94 0.94 1.03 1.03 1.02 1.02 1.16 1.07 1.10 1.10 1.13 1.03 1.05 1.08 0.90 0.89 0.88 0.88 1.00 0.98 0.97 0.97 1.08 1.07 1.07 1.07 0.80 0.79 0.78 0.77 0.90 0.90 0.89 0.89 1.00 1.00 0.97 1.01 0.97 1.04 1.04 1.04 1.11 1.08 1.08 1.09 1.00 0.090 0.85 2.71

27

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Fig. 5. Von-Misses stress contours at ultimate load (a) 80×120×3-M1 (b) 80×240×4-M3.

350 300

Axial Load (kN)

λo=0.82, and λl=0.47, are proposed herein for LDSS type EN 1.4162 hollow columns subjected to local-flexural buckling. Table 7 shows that when these coefficients are used in AS/NZS guideline [2] for coldformed stainless steel structures, the overall prediction (PFEA/P#AS/NZ) for the considered LDSS columns become safe with a mean of 1.08. Adoption of these new coefficients also make the AS/NZS formulations to produce an acceptable reliability index of 2.50; this clearly shows notable improvement in predicting resistances for columns failing due to simultaneous local and flexural buckling.

M1 M2 M3 M4

250 200 150 100 50 0 0

1

2

3

4

5

9.2. Proposed modifications for Euro code 3 Part 1.4 [3]

6

Axial Shortening (mm)

Predictions obtained using the current design rules inEC3 Part 1.4 [3] (PEN) produced a seemingly good average with a mean of 1.03 but the predicted results were very scattered (standard deviation 0.18) and unreliable (β0=2.13). Design modifications proposed for EC3 by Huang and Young [26] (P*EN) actually made things worse (mean 0.90, standard deviation=0.14 and β0=1.80) for the considered failure scenario where local and flexural buckling occurs simultaneously. It is worth noting that Huang and Young [26] carried out parametric analysis on all types of columns and their proposed techniques were shown to perform well from a holistic point of view, but failed to capture the unique buckling interaction case considered in this paper. Numerical constants required in EC3 were modified in the current study, and proposals are made herein for α=0.96 and λo=0.66 for LDSS hollow columns subjected to simultaneous local and flexural buckling. Table 7 shows that when column resistances were determined using the suggested constants, mean for PFEA/P#EN was 1.11 with a standard deviation of 0.17. Although the reliability index was at the target level of 2.5, the predictions were quite conservative as well as considerably scattered to an extent similar to those obtained using the original EC3 guidelines [3] and those proposed by Huang and Young [26]; these outcomes raise some concern on the suitability of the PerryRobertson type buckling curves in predicting buckling interactions.

Fig. 6. Load Vs axial shortening plot for 80×120×2.5 series.

simultaneous local-flexural buckling in thin-walled columns. 9. Proposed modifications to the design rules ASCE Specification [31] requires an iterative process, which is relatively lengthy when compared against other design provisions. As part of the current study, design modifications are proposed for AS/ NZS, EC3 and DSM specifications to appropriately account for the interaction of local and flexural buckling observed in high strength LDSS columns. 9.1. Proposed modifications for AS/NZS 4673 [2] Coefficients α, β, λo and λl are required to determine the critical buckling stress using AS/NZS standard [2]. In absence of any specified values for LDSS, those currently proposed for DSS were adopted in this paper for predicting resistances following the current AS/NZS Specifications (PAS/NZS) and the deign recommendations proposed by Huang and Young [26] (P*AS/NZ). Predicted PAS/NZS and P*AS/NZS values were in good agreement with FE results producing a mean of 0.99 and 0.96 respectively, but the results were significantly scattered with a standard deviation of 0.14. More importantly, both PAS/NZS and P*AS/ NZS values failed to achieve the required level of reliability. Hence, a new set of magnitudes for the coefficients i.e. α=1.25, β=0.82,

9.3. Proposed modifications for the Direct Strength Method The current DSM (PDSM) and the modifications proposed Huang and Young [26] (P*DSM) for LDSS hollow columns produced relatively 28

Thin-Walled Structures 112 (2017) 20–30

M. Anbarasu, M. Ashraf

consistent (standard deviation of 0.08) but unconservative results with mean predictions of 0.93 and 0.94 respectively. DSM formulations also failed to achieve the required target for reliability producing β0=2.42 and 2.46 respectively. Therefore, the current study proposes a new design equation to calculate the nominal axial strength for local buckling which accounts for the L-F buckling interaction in LDSS hollow columns. The proposed Eq. (2) is valid only when the elastic local and flexural buckling stresses are nearly equal.

• •

⎧ ⎡ ⎛ P ⎞0.55⎤ ⎛ P ⎞0.55 ⎪ Pnl= ⎨ Pne forλl≤0. 776/ ⎢1 − 0. 22 ⎜ crl ⎟ ⎥ ⎜ crl ⎟ Pne forλl > 0.776 ⎪ ⎢ ⎝ Pne ⎠ ⎥⎦ ⎝ Pne ⎠ ⎣ ⎩ (2) Table 7 clearly shows the nominal member capacities determined using the proposed DSM technique (P# DSM) produced the most accurate predictions with a mean of 1.00, and also achieved a high level of reliability with β0=2.71.

•

10. Conclusions A numerical investigation on the buckling behaviour, ultimate resistance and design of pin ended cold-formed hollow section columns produced from high strength lean duplex stainless steel subjected to simultaneous local-flexural buckling has been reported in the current study. Nonlinear FE models were validated using results available in the literature, and the developed modelling technique was subsequently used to carry out a parametric analysis on carefully chosen 64 specimens to investigate interaction between local and flexural buckling modes. The sections were selected based on the limits of compression member as stated in the AS/NZS Specifications for cold-formed stainless steel structures, whilst the overall column lengths were obtained using CUFSM to achieve identical critical stresses for local and flexural buckling. Results obtained from the parametric study were compared against those computed using ASCE 8-02 [31], AS/NZ-4673 [2], Eurocode 3 [3], Direct Strength Method [4], and the modified design equations proposed by Huang and Young [17]. Following are the primary observations and recommendations from the current study –

•

•

•

1.11 with β0 =2.50. DSM formulations were also modified and obtained predictions were in very good agreement producing a mean of 1.00 with a reliability index β0=2.71. Although all suggested techniques gave reasonably good agreement with FE results, EC3 formulations were highly scattered (standard deviation was 0.168); this raises some concern on the suitability of Perry-Robertson type formulation in capturing the interaction between buckling modes. It is worth noting that the findings and thus the recommended design modifications are limited to the unique buckling interaction considered in the current study i.e. simultaneous occurrence of local and flexural buckling in LDSS columns. The prospective users are advised to check their considered buckling length with the L-F buckling interaction length for the chosen geometry by elastic buckling analysis for the applicability of the current proposals to their chosen sections. Additional experimental investigations are required to understand this numerically observed behaviour. This study highlights the significance to check for any possible interactions between buckling modes prior to using currently available design guidelines; CUFSM could be used as a handy tool for this purpose.

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The design strengths predicted by the current ASCE 8-02 and DSM formulations were mostly unconservative with mean for predictions being 0.90 and 0.93 respectively. AS/NZS and EC3 specifications produced better mean predictions with 0.99 and 1.03 respectively. However, all codes failed to reach the target reliability index of 2.50. Despite producing better mean predictions, reliability indices for EC3 and AS/NZS were only 2.14 and 2.13 respectively. Reliability of the DSM was 2.42, which is close to the target value of 2.50, but obtained results were mostly unconservative (mean=0.93). Recent modifications proposed by Huang and Young [26] for LDSS columns were also investigated in the case of simultaneous local and flexural buckling. For AS/NZS guidelines, the mean showed some improvement with a value of 0.96 but the reliability deteriorated with β0=2.10. EC3 guidelines also performed very poorly with the suggested modifications giving a highly unconservative mean of 0.90 accompanied by a significant lack of reliability with β0=1.80. The performance of DSM formulations remained almost similar with a mean of 0.94 and β0 =2.46. This clearly shows that modifications suggested by Huang and Young [26] are not capable of tackling this special buckling interaction although their suggested modifications were shown to perform well from a holistic view on column resistance. Observed discrepancies eventually led to making appropriate modifications to current design formulations. For AS/NZS a new set of magnitudes were proposed (α=1.25, β=0.82, λo=0.82, and λl=0.47), and the obtained mean was 1.08 with a reliability index of 2.50. For EC3, new values for design coefficients α=0.96 and λo=0.66 were proposed, which resulted in a conservative mean of 29

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Baltimore, MD 21218, 2007 〈http://www.ce.jhu.edu/bschafer〉. [29] EN 10088-4. Stainless steels part 4: technical delivery conditions for sheet/plate andastrip of corrosion resisting steels for general purposes, CEN, 2009. [30] M. Theofanous, L. Gardner, Experimental and numerical studies of lean duplex stainless steel beams, J. Constr. Steel Res. 66 (6) (2010) 816–825. [31] ASCE, Specification for the design of cold-formed stainless steel structural members, SEI/ASCE8-02, American Society of Civil Engineers, Reston, VA, 2002.

stainless steel hollow section columns, Eng. Struct. 31 (12) (2009) 3047–3058. [26] Y. Huang, B. Young, Structural performance of cold-formed lean duplex stainless steel columns, Thin-Walled Struct. 83 (2014) 59–69. [27] M. Anbarasu, M. Ashraf, Behaviour and design of cold-formed lean duplex stainless steel lipped channel columns, Thin-Walled Struct. 104 (2016) 106–115. [28] B.W. Schafer, CUFSM 3.12 - Finite strip Buckling Analysis of Thin-Walled Members, Department of Civil Engineering, Johns Hopkins University, 208 Latrobe Hall

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