Numerical investigation and design of cold-formed steel built-up open section columns with longitudinal stiffeners

Thin-Walled Structures 89 (2015) 178–191

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Numerical investigation and design of cold-formed steel built-up open section columns with longitudinal stiffeners Jia-Hui Zhang, Ben Young n Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong

art ic l e i nf o

a b s t r a c t

Article history: Received 3 September 2014 Received in revised form 19 November 2014 Accepted 17 December 2014 Available online 22 January 2015

A built-up I-section with longitudinal stiffeners is expected to have better performance to resist against local and distortional buckling compared to conventional built-up I-section by simply connecting two plain channels back-to-back. This paper presents a non-linear finite element analysis to investigate the behaviour of cold-formed steel built-up open section columns with edge and web stiffeners. A finite element model was firstly developed and verified against tests of cold-formed steel built-up compression members, in which the initial geometric imperfections and material properties of the test specimens were included. Secondly, the verified finite element model was used for an extensive parametric study of fixed-ended cold-formed steel built-up open section columns. The parametric study was designed to investigate the effect of edge and web stiffeners in the built-up open sections. The finite element results together with the test results were compared with the design predictions calculated from the current design rules in the North American Specification and the Australian/New Zealand Standard. Furthermore, design rules of the current direct strength method were modified. It is shown that the design strengths predicted by the modified direct strength method are generally in good agreement with the ultimate loads of the built-up open section columns. In addition, the current design rules and the modified direct strength method were evaluated by reliability analysis. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Bucking Built-up sections Cold-formed steel Columns Distortional buckling Local buckling

1. Introduction Cold-formed steel built-up sections are being used as structural members in construction industry. Stone and LaBoube [1], Whittle and Ramseyer [2] and Reyes and Guzman [3] investigated the cold-formed steel built-up compression members. The investigations focused on typical built-up sections that were formed by lipped and plain channel sections without web stiffeners. Due to the high width-to-thickness ratio of the web, built-up compression members are easily failed by local buckling. A built-up section having edge and web stiffeners is expected to have better performance to resist against local and distortional buckling compared to sections without stiffeners. Young and Chen [4] investigated the cold-formed steel built-up closed sections with web stiffeners compression members and Zhang and Young [5] conducted a series of column tests on cold-formed steel built-up open sections with edge and web stiffeners. However, limited research has been reported on such kinds of structural members. Therefore, it is necessary to investigate the structural behaviour of cold-formed steel built-up compression members.

n

Corresponding author. Tel.: þ 852 2859 2674; fax: þ852 2559 5337. E-mail address: [email protected] (B. Young).

http://dx.doi.org/10.1016/j.tws.2014.12.011 0263-8231/& 2014 Elsevier Ltd. All rights reserved.

Finite element analysis (FEA) is a powerful tool for investigating the strength and complex behaviour of cold-formed steel structural members. Compared with an experimental investigation, a parametric study performed by FEA has the advantages of higher efficiency and lower cost. Finite element analysis has been used successfully for cold-formed steel open section columns by Yan and Young [6], Young [7] and Young and Ellobody [8]. The behaviour of lipped channel columns failed by interaction of local– distortional-overall buckling was numerically investigated by Dinis et al. [9]. It should be noted that an accurate and reliable finite element model (FEM) is the key of parametric studies, in which the geometric and material nonlinearities of the specimen should be included in the model. Therefore, a FEM was developed and verified against experimental results of cold-formed steel built-up open section compression members with longitudinal stiffeners in this study. The purpose of this study is firstly to investigate the behaviour of cold-formed steel built-up open sections with edge and web stiffeners subjected to axial compression. An accurate and reliable FEM was developed and used for an extensive parametric study. Six series of built-up open section compression members, having different dimension of edge and web stiffeners, were performed in the parametric study to further examine the complex behaviour of built-up open section columns. The effect of edge and web stiffeners

J.-H. Zhang, B. Young / Thin-Walled Structures 89 (2015) 178–191

Notation Ag bf bl bw Cp E e Fm fod fol fy K L ln Mm P Pcrd Pcre Pcrl PDSM-pro1

PDSM-pro2

PDSM-s

PDSM-2t

PEXP PFEA

gross cross-sectional area width of flange width of lip width of web correction factor in reliability analysis Young's modulus axial shortening mean value of fabrication factor elastic distortional buckling stress of cross section elastic local buckling stress of cross section yield stress is taken as the static 0.2% proof stress effective length factor length of column specimen side length of artificial stiffener mean value of material factor axial load critical elastic distortional column buckling load critical elastic column buckling load in flexural buckling critical elastic local column buckling load nominal axial strength calculated using the current direct strength method and rational design calculation for elastic buckling stress nominal axial strength calculated using the modified direct strength method and rational design calculation for elastic buckling stress nominal axial strength calculated using the direct strength method that assumes the built-up section as two independent single sections nominal axial strength calculated using the direct strength method that assumes the thickness of contact area to be 2t ultimate load obtained by experiment ultimate load obtained by finite element analysis

in the cold-formed steel built-up open sections was investigated. Secondly, the direct strength method was used for the design calculation of built-up open sections. Thirdly, the current direct strength method was modified for cold-formed steel built-up open section columns with longitudinal stiffeners. Finally, reliability analysis was conducted to evaluate the reliability of the current and modified design rules.

179

Pm Pn Pnd Pne Pnl Pu Py r

mean value of tested-to-predicted load ratios nominal axial strength nominal axial strength for distortional buckling nominal axial strength for flexural buckling nominal axial strength for local buckling column strength yield strength radius of gyration of full unreduced cross section about axis of buckling ri inside corner radius at flanges of specimen t nominal plate thickness of specimen VF coefficient of variation of fabrication factor VM coefficient of variation of material factor VP coefficient of variation of tested-to-predicted load ratios w1,w2, w3 dimensions of web stiffeners β reliability index (safety index) β1 reliability index determined using ϕc1 β2 reliability index determined using ϕc2 ε engineering strain εf elongation (tensile strain) after fracture εpltrue true plastic strain; θ angle between the inclined web stiffener and vertical axis ϕc resistance (capacity) factor for compression member ϕc1 resistance (capacity) factor for prequalified columns in the current DSM ϕc2 resistance (capacity) factor for other columns in the current DSM λc ; λd ; λl non-dimensional slenderness used in the direct strength method σ engineering stress σ0.2 static 0.2% tensile proof stress σtrue true stress σu static ultimate tensile strength

thickness and the specimen length can be identified clearly from the label. For example, the label “IT1.0L2000R” defines the specimen as follows, the first letter indicates the cross-section shape where the letter “I” refers to built-up I-shaped section. The second letter “T” refers to the nominal plate thickness and follows by the digits showing the plate thickness of 1.0 mm. The third letter “L” refers to the nominal length of the specimen and follows by the digits showing the length of the column. If a specimen was repeated, the letter “R” is included at the end of the label. The

2. Summary of experimental investigation

bf

2.1. General The test programme on cold-formed steel built-up open section columns with edge and web stiffeners has been reported by Zhang and Young [5]. The columns were tested between fixed ends with different column lengths ranged from 300 to 3200 mm. Two identical lipped channel sections with web stiffeners were connected back-to-back to form a built-up open section by using selftapping screws, as shown in Fig. 1. The nominal dimensions of the cross-section are listed in Table 1 and the measured cross-section dimensions are detailed in Zhang and Young [5]. The nominal screw spacing was 100 mm. The nominal plate thicknesses (t) of the specimens were 0.48, 1.0 and 1.2 mm. The test specimens were divided into three series and labelled as IT0.48, IT1.0 and IT1.2 accordingly. The specimens were labelled such that the plate

t

bl

w1

w2 w3

bw

θ

ri w1/2 Fig. 1. Definition of symbols for built-up open sections.

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Table 1 Nominal dimension of test specimen. Series

Web

Flange

Lip

Thickness

Radius

Angle

IT0.48

bw (mm) 100

w1 (mm) 25

w2 (mm) 25

w3 (mm) 18

bf (mm) 30

bl (mm) 10

t (mm) 0.48

ri (mm) 3.0

θ (deg) 60

IT1.0

100

25

IT1.2

100

25

25

18

30

10

1.0

3.0

60

25

18

30

10

1.2

3.0

60

Note: 1 in. ¼25.4 mm.

Table 2 Comparison of test and finite element analysis results. Specimen

Test

FEA

Table 3 Nominal and measured material properties obtained from tensile coupon tests. Comparison

Test series

Nominal

Measured

σ0.2 (MPa)

E (GPa)

σ0.2 (MPa)

σu (MPa)

εf (%)

IT0.48

550

221

697

720

1.9

IT1.0

500

204

604

609

8.1

IT1.2

500

211

612

614

9.6

P EXP (kN) Failure mode P FEA (kN) Failure mode P EXP =P FEA Lþ D Lþ D Lþ D Lþ DþF Lþ DþF Lþ DþF Lþ DþF

72.4 72.9 67.9 66.7 53.8 48.3 39.6

L þD L þD L þD L þD L þDþ F L þDþ F L þDþ F

IT0.48L300 IT0.48L300R IT0.48L800 IT0.48L1400 IT0.48L2000 IT0.48L2600 IT0.48L3200 Mean, P m COV, V p

68.8 70.4 67.1 65.5 52.9 45.3 42.1

0.95 0.97 0.99 0.98 0.98 0.94 1.06 0.98 0.041

IT1.0L300 IT1.0L800 IT1.0L1400 IT1.0L2000 IT1.0L2000R IT1.0L2600 IT1.0L3200 Mean, P m COV, V p

201.9 199.3 191.2 150.6 157.6 127.0 71.1

D D DþF DþF DþF DþF F

211.4 208.9 204.8 156.8 163.9 125.8 76.4

D D Dþ F Dþ F Dþ F F F

0.96 0.95 0.93 0.96 0.96 1.01 0.93 0.96 0.027

IT1.2L300 IT1.2L300R IT1.2L800 IT1.2L1400 IT1.2L2000 IT1.2L2600 IT1.2L3200 Mean, P m COV, V p

261.1 262.3 249.5 230.5 178.7 149.6 105.3

D D D DþF DþF F F

266.4 267.9 262.0 247.9 179.8 144.9 103.2

D D D Dþ F Dþ F F F

0.98 0.98 0.95 0.93 0.99 1.03 1.02 0.98 0.036

Note: 1 kip¼ 4.45 kN; L¼ Local buckling; D¼distortional buckling; and F¼ Flexural buckling.

test rig and operation are detailed in Zhang and Young [5]. The experimental ultimate loads (PEXP) and the failure modes of the specimens are shown in Table 2. The failure modes of the tested specimens involve local buckling, distortional buckling, flexural buckling and the interaction of these buckling modes. 2.2. Material properties and geometric imperfections The test specimens were brake-pressed from high strength zinccoated grades G500 and G550 structural steel sheets having nominal yield stresses of 500 and 550 MPa, respectively, and the steel sheets specified in accordance with the Australian Standard AS 1397 [10]. Tensile coupon tests were conducted to determine the material properties of the specimens. The nominal and measured material properties are summarised in Table 3. A Mitutoyo digital gage was used to measure the initial local geometric imperfections of the test specimens for Series IT0.48, IT1.0 and IT1.2. The maximum absolute values of the initial local imperfections were 0.121, 0.196 and 0.318 mm for Series IT0.48, IT1.0 and IT1.2, respectively. The initial overall geometric imperfections about the minor axis were also measured prior to testing. The maximum initial overall geometric

Note: 1 ksi¼ 6.89 MPa.

imperfections were 1/783, 1/881 and 1/1243 of the specimen length for Series IT0.48, IT1.0 and IT1.2, respectively. The measured geometric imperfections are detailed in Zhang and Young [5].

3. Development of finite element model 3.1. General The finite element programme ABAQUS [11] was used to develop an FEM that can provide accurate and reliable predictions for cold-formed steel built-up open sections. The measured crosssection dimensions, material properties, and initial geometric imperfections of the tested specimens, as reported in Zhang and Young [5], were incorporated into the FEM. The FEA was divided into two steps. Firstly, an eigenvalue buckling analysis was conducted using the linear perturbation analysis. Secondly, a nonlinear static analysis was performed to obtain the ultimate loads and failure modes of cold-formed steel built-up compression members. 3.2. Type of element and finite element mesh The four-node doubly curved shell element with reduced integration and hourglass control (S4R) allows finite membrane strain and arbitrarily large rotations. Therefore, the S4R element was used to model the main components of the built-up open sections that may fail in elastic buckling. The eight-node linear brick element with reduced integration and hourglass control (C3D8R) was adopted in the FEM to model the self-tapping screws. The nominal diameter of the screw was 4.8 mm with the screw length of 12.5 mm, and the screw threads were not modelled in this study. To achieve an accurate and efficient FEM, the built-up open sections were meshed with the size of about 5 mm  5 mm at the flat portions and with a finer mesh size at the corner portions. In addition, the self-tapping screws were meshed with the mesh size around 1 mm  1 mm by C3D8R element. 3.3. Boundary condition and loading method The cold-formed steel built-up compression members were tested between fixed ends. Therefore, the FEA restrained all

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3.4. Material properties The material properties obtained from the flat tensile coupons were adopted in the FEM. In the stage of the non-linear static analysis, the nonlinearity of the material should be included in the FEM using the incremental plasticity model, as specified in ABAQUS [11]. The engineering stresses (σ) and strains (ε), obtained from the tensile coupon tests, should be converted to the true stresses (σtrue) and true plastic strains εpl true . As specified in ABAQUS [11], σ true ¼ σ ð1 þ εÞ and εpl ¼ lnð1 þ ε Þ  σ true =E are given, where true E is the Young's modulus of the static engineering stress–strain curve. The material properties of the self-tapping screw were also incorporated into the FEM, where Young's modulus and Poisson's ratio were 200 GPa and 0.3, respectively.

3.5. Initial geometric imperfections The structural behaviour of cold-formed steel compression members are susceptible to initial geometric imperfections. Therefore, both initial local and overall geometric imperfections were incorporated carefully into the FEM to ensure its accuracy for columns. The buckling modes were obtained by the use of an eigenvalue buckling analysis, in which the specimen was assumed to have a thinner plate of 0.1 mm to obtain a local buckling, and the specimen was also assumed to have a thicker plate of 10 mm to achieve an overall buckling. In the eigenvalue buckling analysis, the buckling modes are normalised to 1.0. Therefore, the achieved buckling modes were factored correspondingly by the measured magnitudes of initial local and overall geometric imperfections obtained from the tests. Finally, the superposition of the factored local buckling mode and overall buckling mode was imported into the FEM.

3.6. Interaction and constraint In this study, a built-up section was formed by connecting two open sections back-to-back with self-tapping screws. Therefore, the interaction relationship between the two open sections, and the constraint relationship between the screws and the open sections were defined carefully in the FEM. The surface-to-surface contact using the finite-sliding tracking method was used to define the interaction relationship occurred on the surfaces of two open sections as well as to define the contact between the two open sections and the screw heads. However, the large scale surface-tosurface contact in the FEM is very likely to lead to a convergence problem and to an early termination of the calculation. Thus, for the aim of obtaining a convergence and stable solution, an artificial damping was applied at the first increment of the non-linear static analysis, where a default value of 0.0002 was used for the dissipated energy fraction. Meanwhile, to keep the accuracy of the solution, the adaptive stabilisation was activated with the stabilisation-to-strain energy ratio of 0.005. In the test programme, the screw looseness was not found. Therefore, the constraint relationship between the screw shank and the open sections was modelled by tied constraint.

4. Verification of the finite element model The developed FEM was verified against the experimental results for the built-up open section compression members. For built-up open sections, the ultimate loads (PFEA) and the failure modes predicted by FEA were compared with the experimental results of Series IT0.48, IT1.0 and IT1.2, as shown in Table 2. It is shown that the ultimate loads (PFEA) are in good agreement with the test results (PEXP). The maximum percentages of difference between PFEA and PEXP are 6%, 7% and 7% for Series IT0.48, IT1.0 and IT1.2, respectively. Generally, the ultimate loads obtained from the tests are slightly lower than those predicted by the FEA. The mean values of PEXP/PFEA ratio are 0.98, 0.96 and 0.98 with the corresponding coefficients of variation (COV) of 0.041, 0.027 and 0.036 for Series IT0.48, IT1.0 and IT1.2, respectively. The failure modes at ultimate loads, which were obtained from the test and FEA specimens, are also shown in Table 2. The test specimens were failed by local buckling (L), distortional buckling (D), flexural buckling (F) and the interaction of these buckling modes. Good agreements were achieved between the test results and FEA results, except for the specimens IT0.48L1400 and IT1.0L2600. Fig. 2 shows the axial load versus axial shortening curves for specimens IT1.2L2600. The curves were obtained from the column test as well as the FEA prediction. It can be seen from the figure that the FEA curve follows the test curves closely, where the ultimate load from the tests is 149.6 kN, with the corresponding FEA result of 144.9 kN for specimen IT1.2L2600. The deformed shapes at the peak load of the test and FEA prediction for specimen IT1.0L2000 is compared in Fig. 3. As shown in Fig. 3(i), the test specimen IT1.0L2000 was failed by the interaction of distortional and flexural buckling and these failure modes were well predicted by the FEA, as shown in Fig. 3(ii) for the same column in different views. It is shown that the developed FEM can be used to predict the column strengths and failure modes of the test specimens. The comparisons between the test results and FEA predictions have led to the conclusion that the developed FEM is accurate and reliable for predicting the ultimate loads as well as the failure modes of cold-formed steel built-up open section compression members.

5. Parametric study 5.1. Design of the parametric study An extensive parametric study was performed to: (1) enhance the understanding of the complicated structural behaviour of coldformed steel built-up open section columns; (2) supply sufficient data on built-up open sections to evaluate current design rules as well as to propose modification to the direct strength equations.

200

Axial load, P (kN)

degrees of freedom at both ends of the specimen, except for the translational degree of freedom in the axial loading direction of one end of the column to simulate the fixed-ended boundary condition. The nodes other than the two ends were free to translate and rotate in any direction. The displacement loading control method was used in the FEA, which is identical to the loading method used in the test programme. Therefore, an axially compressive load was applied to an FEA specimen by specifying the axial displacement to each node of one end of a column.

181

150

100

50

Experimental FEA 0

0

1

2 3 4 Axial shortening, e (mm)

5

6

Fig. 2. Load versus axial shortening curves for specimen IT1.2L2600.

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The developed FEM has been used successfully to predict the structural behaviour of cold-formed steel built-up open section compression members, in which the column tests were reported by Zhang and Young [5]. Therefore, the verified model was used to perform the parametric study. In the parametric study, the influence of edge and web stiffeners of cold-formed steel builtup open sections was investigated. In total, 126 FEA specimens were considered in the parametric study of cold-formed steel built-up open section compression members. The FEA specimens were divided into six series according to the geometries of cross-section, as shown in Fig. 4. Series I-E16A60, I-E08A60, I-E24A60 had the same dimensions of crosssection, except for the edge stiffeners, whereas, Series I-E16A60, I-E16A45, I-E16A30 had the same dimensions of cross-section,

except for the web stiffeners. Series I-E16A00, formed by two lipped channel sections, was used as a reference to the other series that have web stiffeners. The dimensions of the cross-section for each series were designed so that the effects of the edge stiffeners and web stiffeners could be identified easily. The dimensions of the cross-section for each series are listed in Table 4, using the nomenclature defined in Fig. 1. Each series consisted of three subseries with plate thicknesses of 1.0, 1.5 and 2.4 mm. The FEA specimens were simulated between fixed-ended boundary condition with column lengths in the range of 450–4000 mm. The nominal screw spacing of the FEA specimens was 100 mm, and a smaller screw spacing was used at both ends of the specimens, as shown in Fig. 5. The specimens were labelled so that the series and the specimen length could be identified easily from the label. For example, the label “I-E16A60T1.0L450” defines the specimen as follows:

 The first letter indicates the cross-section shape, where the letter “I” refers to the built-up I-shaped open section.

 The second letter and the following digits “E16” refer to the edge stiffeners with the length of 16 mm.

 The third letter and the following digits “A60” refer to the web  

stiffeners inclined from the vertical line along its web with an angle of 601, as shown in Fig. 1. The fourth letter and the following digits “T1.0” indicate the nominal thickness of the specimen of 1.0 mm. The last letter and the following digits “L450” mean the column specimen length of 450 mm.

The material properties of the FEA specimens of the built-up open sections were obtained from those of Series IT1.0 that described in Zhang and Young [5], as shown in Table 3. The maximum initial local and overall imperfections were taken as 1/4 of the plate thicknesses and 1/2000 of the column lengths, respectively.

5.2. Results and discussion

(i) Experimental

(ii) different views of FEA

Fig. 3. Comparison of experimental and finite element analysis deformed shapes for specimen IT1.0L2000.

Different edge stiffeners

I-E24A60

I-E08A60

The parametric studies were used to investigate the effect of edge and web stiffeners on the column strengths of cold-formed steel built-up open section columns. The column strengths (PFEA) of the built-up open sections are shown in Table 5. The comparison was made between the column strengths (PFEA) with yield strengths (Py) to investigate the effect of edge and web stiffeners on loading capacity for the cold-formed steel built-up open sections, as shown in Figs. 6–8 and Tables 6–8. Series I-E16A60, I-E08A60 and I-E24A60 having different edge stiffeners, had similar efficiency to resist against axial compressive load because of their close values of PFEA/Py ratio. However, Series I-E08A60 generally performed the worst of these three series

Different web stiffeners

I-E16A60

I-E16A45

I-E16A30

I-E16A00

Fig. 4. Geometries of cross-section of built-up open sections in parametric study.

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183

Table 4 Dimensions of cross-section for each series in parametric study. Series

I-E16A60 I-E08A60 I-E24A60 I-E16A45 I-E16A30 I-E16A00

Web

Flange

Lip

Radius

Angle

bw (mm)

w1 (mm)

w2 (mm)

w3 (mm)

bf (mm)

bl (mm)

ri (mm)

θ (deg)

150 150 150 150 150 150

37.5 37.5 37.5 37.5 37.5 N/A

35 35 35 25 20 N/A

40 40 40 40 40 N/A

35 35 35 35 35 35

16 8 24 16 16 16

3 3 3 3 3 3

60 60 60 45 30 N/A

450mm

25mm 100mm 100mm

End plate

100mm

Mid-length of the specimen

Column specimen

20mm 80mm 100mm

100mm 25mm

Screw

100mm 100mm

Weld

100mm 80mm 20mm

Fig. 5. Arrangement of screws for FEA specimens with screw spacing of 100 mm. (a) Column length of 450 mm and (b) column lengths of 1000, 1600, 2200, 2800, 3400 and 4000 mm.

when the column lengths were greater than or equal to 2200 mm. It was found from the results that the short edge stiffeners may weaken a built-up open section column to resist axial load. For Series I-E16A60, I-E16A45 and I-E16A30 having different web stiffeners, it is obvious that Series I-E16A60 performed the best due to its deepest web stiffeners. Series I-E16A00, formed by connecting two lipped channel sections, was found to have the lowest values of PFEA/Py ratio for all the series. It was also shown that the specimens of plate thicknesses of 1.5 and 2.4 mm have higher values of PFEA/Py ratio than those specimens with plate thickness of 1.0 mm. The comparisons for all series concluded that: (1) web stiffeners can be used to obtain a significant enhancement in the column strengths of cold-formed steel built-up open sections, and (2) edge stiffeners with proper lengths are able to resist against distortional buckling.

6. Design rules 6.1. General The direct strength method, proposed by Schafer and Peköz [12], has been adopted by the NAS Specification [13] and AS/NZS Standard [14] as an alternative design method for cold-formed steel structural members. Compared with the effective width method, the direct strength method has the advantages that it: (1) adopts the gross area for the ease of design calculation,

(2) considers the interactions between the individual compression elements of a whole section, and (3) makes it possible to predict the structural behaviour of almost any sectional configurations. However, the direct strength method does not cover the design of built-up sections. Therefore, its appropriateness for built-up open section compression members has been evaluated primarily with test results, as detailed in Zhang and Young [5]. In this study, the appropriateness of the direct strength method for built-up open section compression members was assessed further by FEA in the parametric study. The accuracy and reliability of the direct strength method rely on the elastic buckling stresses, which can be obtained by using a rational elastic buckling analysis, i.e., the finite strip method proposed by Papangelis and Hancock [15]. The rational elastic buckling analysis using the finite strip method requires the given section to be uniform along the longitudinal direction. Obviously, the built-up open sections connected by screws cannot meet this requirement. This leads to the result that a built-up open section cannot be used directly for computation in the finite strip method. Therefore, this section developed a modified model of the sectional configuration of a built-up open section for the rational elastic buckling analysis. There are no explicit and direct design rules for cold-formed steel built-up open section columns. In this phase of the study, the direct strength method was attempted to provide accurate and reliable predictions for cold-formed steel built-up open sections. Compared with the test and FEA results, this method is capable to

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Table 5 Comparison of numerical and experimental results with design strengths for cold-formed steel built-up open sections. Specimen

I-E16A60T1.0L450 I-E16A60T1.0L1000 I-E16A60T1.0L1600 I-E16A60T1.0L2200 I-E16A60T1.0L2800 I-E16A60T1.0L3400 I-E16A60T1.0L4000 I-E16A60T1.5L450 I-E16A60T1.5L1000 I-E16A60T1.5L1600 I-E16A60T1.5L2200 I-E16A60T1.5L2800 I-E16A60T1.5L3400 I-E16A60T1.5L4000 I-E16A60T2.4L450 I-E16A60T2.4L1000 I-E16A60T2.4L1600 I-E16A60T2.4L2200 I-E16A60T2.4L2800 I-E16A60T2.4L3400 I-E16A60T2.4L4000 I-E08A60T1.0L450 I-E08A60T1.0L1000 I-E08A60T1.0L1600 I-E08A60T1.0L2200 I-E08A60T1.0L2800 I-E08A60T1.0L3400 I-E08A60T1.0L4000 I-E08A60T1.5L450 I-E08A60T1.5L1000 I-E08A60T1.5L1600 I-E08A60T1.5L2200 I-E08A60T1.5L2800 I-E08A60T1.5L3400 I-E08A60T1.5L4000 I-E08A60T2.4L450 I-E08A60T2.4L1000 I-E08A60T2.4L1600 I-E08A60T2.4L2200 I-E08A60T2.4L2800 I-E08A60T2.4L3400 I-E08A60T2.4L4000 I-E24A60T1.0L450 I-E24A60T1.0L1000 I-E24A60T1.0L1600 I-E24A60T1.0L2200 I-E24A60T1.0L2800 I-E24A60T1.0L3400 I-E24A60T1.0L4000 I-E24A60T1.5L450 I-E24A60T1.5L1000 I-E24A60T1.5L1600 I-E24A60T1.5L2200 I-E24A60T1.5L2800 I-E24A60T1.5L3400 I-E24A60T1.5L4000 I-E24A60T2.4L450 I-E24A60T2.4L1000 I-E24A60T2.4L1600 I-E24A60T2.4L2200 I-E24A60T2.4L2800 I-E24A60T2.4L3400 I-E24A60T2.4L4000 I-E16A45T1.0L450 I-E16A45T1.0L1000 I-E16A45T1.0L1600 I-E16A45T1.0L2200 I-E16A45T1.0L2800 I-E16A45T1.0L3400 I-E16A45T1.0L4000 I-E16A45T1.5L450 I-E16A45T1.5L1000 I-E16A45T1.5L1600 I-E16A45T1.5L2200 I-E16A45T1.5L2800

FEA and test

Comparison

PFEA and PEXP(kN)

ðP FEA and P EXP Þ=P DSM  2t

ðP FEA and P EXP Þ=P DSM  s

ðP FEA and P EXP Þ=P DSM  pro1

ðP FEA and P EXP Þ=P DSM  pro2

245.7 245.2 233.9 215.0 175.6 137.9 104.7 441.5 433.5 414.5 353.6 285.1 216.5 160.8 763.7 749.4 697.5 608.8 479.6 354.0 264.6 240.5 238.2 222.2 193.9 158.6 118.0 89.1 423.8 421.3 376.0 326.0 249.4 186.1 139.3 723.8 716.0 640.6 542.8 422.4 311.1 230.5 258.6 244.8 225.2 201.0 167.1 138.1 111.7 462.3 447.3 423.3 382.4 312.0 239.9 184.6 806.4 770.7 711.9 636.6 512.9 399.0 307.4 237.6 228.5 212.9 188.2 137.7 103.7 72.8 416.8 401.5 359.1 290.2 215.2

0.85 0.88 0.91 0.94 0.92 0.95 0.99 0.88 0.92 0.98 0.99 1.00 1.00 1.02 0.95 0.99 1.03 1.07 1.05 1.02 1.04 0.93 0.92 0.93 0.92 0.95 0.97 1.01 0.94 0.95 0.97 1.02 1.00 1.02 1.05 0.96 1.01 1.03 1.06 1.06 1.06 1.09 1.05 1.03 1.01 1.00 0.95 0.93 0.92 0.96 0.96 0.98 0.98 0.98 0.97 1.00 0.95 0.96 0.98 1.02 1.01 1.01 1.04 0.88 0.89 0.92 0.98 0.97 1.05 1.02 0.90 0.94 0.99 1.00 1.01

1.05 1.05 1.14 1.49 1.95 2.26 2.37 1.07 1.05 1.27 1.63 2.11 2.36 2.43 0.99 1.09 1.34 1.75 2.22 2.41 2.50 1.15 1.14 1.17 1.52 2.01 2.21 2.31 1.12 1.12 1.27 1.70 2.11 2.32 2.40 1.01 1.12 1.35 1.77 2.23 2.42 2.49 1.09 1.13 1.23 1.42 1.67 2.03 2.28 1.01 1.05 1.20 1.60 2.08 2.35 2.51 0.99 1.05 1.27 1.66 2.14 2.45 2.61 1.06 1.02 1.18 1.69 2.00 2.22 2.16 1.09 1.05 1.31 1.73 2.08

1.08 1.07 1.02 0.96 0.92 0.95 0.99 1.08 1.07 1.02 0.99 1.00 1.00 1.02 1.00 0.99 1.03 1.07 1.05 1.02 1.04 1.14 1.12 1.05 0.95 0.95 0.97 1.01 1.11 1.11 0.99 1.02 1.00 1.02 1.05 1.00 1.01 1.03 1.06 1.06 1.06 1.09 1.08 1.06 1.04 1.03 0.98 0.95 0.94 0.98 0.98 1.00 1.00 0.98 0.97 1.00 0.98 0.96 0.98 1.02 1.01 1.01 1.04 1.08 1.03 0.96 0.99 0.97 1.05 1.02 1.07 1.03 0.99 1.00 1.01

1.02 1.02 1.01 1.04 0.99 0.95 0.99 1.01 0.99 0.98 0.99 1.00 1.00 1.02 0.95 0.99 1.03 1.07 1.05 1.02 1.04 1.08 1.07 1.04 1.03 1.00 0.97 1.01 1.04 1.03 0.97 1.02 1.00 1.02 1.05 0.96 1.01 1.03 1.06 1.06 1.06 1.09 1.21 1.18 1.16 1.13 1.07 1.04 1.02 1.07 1.07 1.08 1.09 1.02 0.97 1.00 0.95 0.96 0.98 1.02 1.01 1.01 1.04 1.02 1.00 1.03 1.07 0.97 1.05 1.02 0.99 0.96 0.99 1.00 1.01

J.-H. Zhang, B. Young / Thin-Walled Structures 89 (2015) 178–191

185

Table 5 (continued ) Specimen

I-E16A45T1.5L3400 I-E16A45T1.5L4000 I-E16A45T2.4L450 I-E16A45T2.4L1000 I-E16A45T2.4L1600 I-E16A45T2.4L2200 I-E16A45T2.4L2800 I-E16A45T2.4L3400 I-E16A45T2.4L4000 I-E16A30T1.0L450 I-E16A30T1.0L1000 I-E16A30T1.0L1600 I-E16A30T1.0L2200 I-E16A30T1.0L2800 I-E16A30T1.0L3400 I-E16A30T1.0L4000 I-E16A30T1.5L450 I-E16A30T1.5L1000 I-E16A30T1.5L1600 I-E16A30T1.5L2200 I-E16A30T1.5L2800 I-E16A30T1.5L3400 I-E16A30T1.5L4000 I-E16A30T2.4L450 I-E16A30T2.4L1000 I-E16A30T2.4L1600 I-E16A30T2.4L2200 I-E16A30T2.4L2800 I-E16A30T2.4L3400 I-E16A30T2.4L4000 I-E16A00T1.0L450 I-E16A00T1.0L1000 I-E16A00T1.0L1600 I-E16A00T1.0L2200 I-E16A00T1.0L2800 I-E16A00T1.0L3400 I-E16A00T1.0L4000 I-E16A00T1.5L450 I-E16A00T1.5L1000 I-E16A00T1.5L1600 I-E16A00T1.5L2200 I-E16A00T1.5L2800 I-E16A00T1.5L3400 I-E16A00T1.5L4000 I-E16A00T2.4L450 I-E16A00T2.4L1000 I-E16A00T2.4L1600 I-E16A00T2.4L2200 I-E16A00T2.4L2800 I-E16A00T2.4L3400 I-E16A00T2.4L4000 IT0.48L300 IT0.48L300R IT0.48L800 IT0.48L1400 IT0.48L2000 IT0.48L2600 IT0.48L3200 IT1.0L300 IT1.0L800 IT1.0L1400 IT1.0L2000 IT1.0L2000R IT1.0L2600 IT1.0L3200 IT1.2L300 IT1.2L300R IT1.2L800 IT1.2L1400 IT1.2L2000 IT1.2L2600 IT1.2L3200 Mean, P m COV, V p Reliability index, β1 Resistance factor, ϕc1 Reliability index, β2 Resistance factor, ϕc2

FEA and test

Comparison

PFEA and PEXP(kN)

ðP FEA and P EXP Þ=P DSM  2t

ðP FEA and P EXP Þ=P DSM  s

ðP FEA and P EXP Þ=P DSM  pro1

ðP FEA and P EXP Þ=P DSM  pro2

152.5 112.8 709.8 684.7 606.3 486.2 358.8 256.5 188.4 231.2 213.2 189.7 157.3 120.7 85.7 64.0 401.0 375.8 345.0 271.2 199.7 135.8 99.4 673.8 656.5 583.4 455.8 320.2 230.6 164.8 140.9 139.1 119.7 111.0 96.5 74.7 56.7 258.6 257.5 215.6 196.2 161.9 120.4 88.1 483.2 478.5 429.2 344.3 275.8 190.5 146.9 68.8 70.4 67.1 65.5 52.9 45.3 42.1 201.9 199.3 191.2 150.6 157.6 127.0 71.1 261.1 262.3 249.5 230.5 178.7 149.6 105.3

1.03 1.06 0.96 1.00 1.04 1.05 1.05 1.08 1.10 1.04 0.96 0.89 0.90 0.97 1.01 1.05 1.02 0.96 1.01 1.03 1.07 1.07 1.08 0.94 1.01 1.07 1.09 1.08 1.14 1.12 0.93 0.94 0.91 1.01 1.12 1.12 1.06 0.90 0.89 0.83 0.91 0.96 1.02 1.03 0.81 0.80 0.82 0.86 0.99 1.01 1.08 0.83 0.85 0.84 0.91 0.86 0.91 1.10 0.91 0.89 0.97 0.95 0.99 1.08 0.88 0.91 0.92 0.90 0.95 0.92 1.03 1.05 0.98 0.073 2.63 0.85 2.88 0.80

2.18 2.23 1.02 1.11 1.38 1.81 2.16 2.28 2.32 1.19 1.09 1.08 1.42 1.77 1.85 1.91 1.22 1.15 1.28 1.63 1.95 1.95 1.98 1.12 1.10 1.36 1.72 1.96 2.08 2.05 1.42 1.50 1.54 1.84 2.16 2.15 2.02 1.35 1.38 1.38 1.63 1.83 1.76 1.60 1.16 1.15 1.24 1.30 1.45 1.48 1.57 1.01 1.03 0.98 1.16 1.34 1.64 2.27 1.13 1.11 1.26 1.64 1.72 2.34 1.99 1.10 1.10 1.05 1.23 1.57 2.22 2.37 1.62 0.299 3.02 0.85 3.17 0.80

1.03 1.06 0.99 1.00 1.04 1.05 1.05 1.08 1.10 1.16 1.07 0.95 0.90 0.97 1.01 1.05 1.11 1.04 1.01 1.03 1.07 1.07 1.08 0.99 1.01 1.07 1.09 1.08 1.14 1.12 1.03 1.04 1.00 1.11 1.22 1.22 1.15 1.03 1.03 0.97 1.06 1.11 1.07 1.03 0.93 0.92 0.89 0.86 0.99 1.01 1.08 1.00 1.02 0.98 0.95 0.86 0.90 1.09 1.12 1.11 1.06 0.95 0.99 1.08 0.88 1.10 1.10 1.05 0.97 0.92 1.03 1.05 1.03 0.061 2.85 0.85 3.10 0.80

1.03 1.06 0.96 1.00 1.04 1.05 1.05 1.08 1.10 1.11 1.02 0.98 0.97 0.97 1.01 1.05 1.03 0.97 1.01 1.03 1.07 1.07 1.08 0.94 1.01 1.07 1.09 1.08 1.14 1.12 1.18 1.23 1.18 1.28 1.37 1.34 1.25 1.11 1.17 1.09 1.17 1.21 1.16 1.07 0.90 0.95 0.97 0.94 0.99 1.01 1.08 0.97 0.99 0.95 1.02 0.95 0.99 1.18 1.05 1.04 1.00 0.95 0.99 1.08 0.88 1.02 1.02 0.97 0.95 0.92 1.03 1.05 1.04 0.075 2.86 0.85 3.11 0.80

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J.-H. Zhang, B. Young / Thin-Walled Structures 89 (2015) 178–191

0.8

0.6

PFEA /Py

Table 7 Comparison of column strengths and yielding strengths of FEA specimens (t ¼1.5 mm).

I-E16A60 I-E08A60 I-E24A60 I-E16A45 I-E16A30 I-E16A00

Column length (mm)

0.4

0.2

0.0 0

500

1000

1500

2000

2500

Effective length L/2 (mm)

450 1000 1600 2200 2800 3400 4000

Comparison (PFEA/Py) IE16A60

IE08A60

IE24A60

IE16A45

IE16A30

IE16A00

0.86 0.85 0.81 0.69 0.56 0.42 0.31

0.88 0.87 0.78 0.67 0.52 0.38 0.29

0.85 0.83 0.78 0.71 0.58 0.44 0.34

0.87 0.84 0.75 0.61 0.45 0.32 0.24

0.87 0.81 0.75 0.59 0.43 0.29 0.22

0.57 0.57 0.48 0.43 0.36 0.27 0.20

Fig. 6. PFEA/Py versus effective length for FEA specimens with 1.0 mm plate thickness.

1.0

0.8

PFEA /Py

Table 8 Comparison of column strengths and yielding strengths of FEA specimens (t ¼2.4 mm).

I-E16A60 I-E08A60 I-E24A60 I-E16A45 I-E16A30 I-E16A00

Column length (mm)

0.5

0.3

0.0 0

500

1000

1500

2000

2500

450 1000 1600 2200 2800 3400 4000

Comparison (PFEA/Py) IE16A60

IE08A60

IE24A60

IE16A45

IE16A30

IE16A00

0.93 0.91 0.85 0.74 0.58 0.43 0.32

0.94 0.93 0.83 0.70 0.55 0.40 0.30

0.93 0.89 0.82 0.74 0.59 0.46 0.35

0.93 0.89 0.79 0.64 0.47 0.34 0.25

0.91 0.89 0.79 0.62 0.43 0.31 0.22

0.67 0.66 0.59 0.48 0.38 0.26 0.20

Effective length L/2 (mm) Fig. 7. PFEA/Py versus effective length for FEA specimens with 1.5 mm plate thickness.

1.0

I-E16A60 I-E08A60 I-E24A60 I-E16A45 I-E16A30 I-E16A00

PFEA /Py

0.8

0.5

be used for built-up compression members but with some limitations. However, from the view of an engineer, its limitations may prevent it from being a favourable choice for design. Therefore, providing direct and relatively simple design rules for built-up open section columns is one of the major tasks in this study. Based on both the test and FEA results, the current direct strength method was modified. The exponent and coefficient values in the direct strength equations were modified by calibrating against the test and FEA results. It is shown that the modified direct strength method is generally conservative and reliable for coldformed steel built-up open section columns.

0.3

6.2. Design calculation using elastic buckling stresses 0.0 0

500

1000

1500

2000

2500

Effective length L/2 (mm) Fig. 8. PFEA/Py versus effective length for FEA specimens with 2.4 mm plate thickness.

Table 6 Comparison of column strengths and yielding strengths of FEA specimens (t¼ 1.0 mm). Column length (mm)

450 1000 1600 2200 2800 3400 4000

Comparison (PFEA/Py) IE16A60

IE08A60

IE24A60

IE16A45

IE16A30

IE16A00

0.72 0.72 0.68 0.63 0.51 0.40 0.31

0.75 0.74 0.69 0.60 0.49 0.37 0.28

0.72 0.68 0.62 0.56 0.46 0.38 0.31

0.75 0.72 0.67 0.59 0.43 0.33 0.23

0.75 0.69 0.62 0.51 0.39 0.28 0.21

0.47 0.46 0.40 0.37 0.32 0.25 0.19

The determination of the buckling modes and the corresponding elastic buckling stresses is the first step for design calculation using the direct strength method. As mentioned earlier, a built-up section cannot be used directly in the finite strip method. Therefore, a modification of the sectional configuration was used, i.e. a single open section with solid stiffeners, as shown in Fig. 9. The solid stiffeners were added at the locations of the self-tapping screws. The solid stiffener was assumed to be a solid square with n its length (ln) to be l ¼ 6ðtÞ0:5 , where t is the plate thickness of the section. In the rational elastic buckling analysis, the solid stiffeners were used to take the role of the self-tapping screw to constrain the behaviour of the sections. Thus, the modification made it possible to perform elastic buckling analysis for built-up open sections. 6.3. Current direct strength method The direct strength method (DSM) has the advantage of simplicity in design calculation for the structural members with complicated sectional configurations. The design rules of the direct strength method in the North American Specification [13] for

J.-H. Zhang, B. Young / Thin-Walled Structures 89 (2015) 178–191

187

1000

DSM-2t

Column strength Pu (kN)

DSM-s 800

DSM-pro1 DSM-pro2 FEA

600

400

200

0 0

Solid stiffeners

2t

500

1000

1500

2000

2500

Effective length L/2 (mm) Fig. 12. Comparison of FEA and design strengths for I-E16A60T2.4.

DSM-s

DSM-pro1 DSM-pro2

300

DSM-2t

Fig. 9. Cross-section assumptions in the calculation of elastic buckling stresses for built-up open sections.

350

DSM-2t DSM-s

Column strength Pu (kN)

300

DSM-pro1 250

DSM-s

250

Column strength Pu (kN)

DSM-2t

DSM-pro1 DSM-pro2

200

FEA 150 100 50

DSM-pro2 FEA

200

0 0

150

500

1000

1500

2000

2500

Effective length L/2 (mm)

100

Fig. 13. Comparison of FEA and design strengths for I-E08A60T1.0.

50 0 0

500

1000

1500

2000

500

2500

DSM-2t

Effective length L/2 (mm)

Column strength Pu (kN)

DSM-s

Fig. 10. Comparison of FEA and design strengths for I-E16A60T1.0.

600

DSM-2t DSM-s

Column strength Pu (kN)

500

DSM-pro1 DSM-pro2

400

400

DSM-pro1 DSM-pro2 FEA

300

200

100

FEA 0

300

0

500

1000

1500

2000

2500

Effective length L/2 (mm)

200

Fig. 14. Comparison of FEA and design strengths for I-E08A60T1.5.

100 0 0

500

1000

1500

2000

2500

P nl ¼

Effective length L/2 (mm)

8 P ne > <

if λl r 0:776

8 Py > <

if λd r 0:561

 0:4  0:4 P crl P crl P ne > P ne : 1  0:15 P ne

Fig. 11. Comparison of FEA and design strengths for I-E16A60T1.5.

columns are follows: P n ¼ minðP ne ; P nl ; P nd Þ 8 λ2 > > < ð0:658 c ÞP y if λc r1:5
 P ne ¼ 0:877 > P y if λc 4 1:5 > 2 : λc

P nd ¼ ð1Þ

ð2Þ

 0:6  0:6 P crd P crd Py > : 1 0:25 Py Py

if λl 4 0:776

if λd 4 0:561

ð3Þ

ð4Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where λc ¼ P y =P cre ; P y ¼ Ag f y ; λl ¼ P ne =P crl ; P crl ¼ Ag f ol ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λd ¼ P y =P crd ; P crd ¼ Ag f od . The Ag is the gross cross-section area and fy is the 0.2% proof stress (yield stress) of the material. The elastic flexural buckling load (Pcre) in accordance with Section

188

J.-H. Zhang, B. Young / Thin-Walled Structures 89 (2015) 178–191

1000

1000

DSM-2t

DSM-2t

800

DSM-s

Column strength Pu (kN)

Column strength Pu (kN)

DSM-s DSM-pro1 DSM-pro2 FEA

600

400

200

800

DSM-pro1 DSM-pro2 FEA

600

400

200

0

0 0

500

1000

1500

2000

0

2500

500

1000

1500

2000

2500

Effective length ley (mm)

Effective length L/2 (mm)

Fig. 18. Comparison of FEA and design strengths for I-E24A60T2.4.

Fig. 15. Comparison of FEA and design strengths for I-E08A60T2.4.

300

DSM-2t

DSM-2t

DSM-pro1 DSM-pro2

200

DSM-s

250

Column strength Pu (kN)

Column strength Pu (kN)

300

DSM-s

250

FEA 150 100 50

DSM-pro1 DSM-pro2

200

FEA 150 100 50

0 0

500

1000

1500

2000

2500

0 0

Effective length L/2 (mm)

500

1000

1500

2000

2500

Effective length L/2 (mm)

Fig. 16. Comparison of FEA and design strengths for I-E24A60T1.0.

Fig. 19. Comparison of FEA and design strengths for I-E16A45T1.0.

600

DSM-2t

DSM-2t DSM-s

DSM-pro1

Column strength Pu (kN)

Column strength Pu (kN)

500

DSM-s

500

DSM-pro2

400

FEA 300 200 100

400

DSM-pro1 DSM-pro2 FEA

300

200

100

0 0

500

1000

1500

2000

2500

Effective length L/2 (mm) Fig. 17. Comparison of FEA and design strengths for I-E24A60T1.5.

0 0

500

1000

1500

2000

2500

Effective length L/2 (mm) Fig. 20. Comparison of FEA and design strengths for I-E16A45T1.5.

C4.1.1 of the North American Specification [13] is calculated as P cre ¼ Ag π 2 E=ðKL=rÞ2 . In the design calculation, the effective length (KL) for minor axis flexural buckling was assumed to be half of the column length for the fixed-ended compression member. Furthermore, the overall slenderness ratio KL/r was replaced by the modified slenderness ratio (KL/r)m in accordance with Section D1.2 of North American Specification [13]. 6.4. Modified direct strength method The current direct strength method was modified for coldformed steel built-up open section columns. The local buckling curve proposed by Yap and Hancock [16] provides a relative

accurate prediction than the current DSM for cold-formed steel channel sections with web stiffeners due to the consideration of the local instability occurred on multiple compression elements. Therefore, the local bucking curve (Eq. (10) in Ref. [16]) proposed by Yap and Hancock [16] was used for cold-formed steel built-up open sections with edge and web stiffeners in this study. The distortional buckling curve of the current DSM was also modified for a relative accurate prediction. The modification to the current direct strength method is as follows: 8 if λl r0:673 P ne > <  0:5  0:5 P nl ¼ ð5Þ P crl P crl P ne if λl 40:673 > P ne : 1  0:22 P ne

J.-H. Zhang, B. Young / Thin-Walled Structures 89 (2015) 178–191

1000

189

1000

DSM-2t

DSM-2t

800

DSM-s

Column strength Pu (kN)

Column strength Pu (kN)

DSM-s DSM-pro1 DSM-pro2 FEA

600

400

200

0

800

DSM-pro1 DSM-pro2 FEA

600

400

200

0 0

500

1000

1500

2000

2500

0

500

2000

2500

Fig. 24. Comparison of FEA and design strengths for I-E16A30T2.4.

250

200

200

DSM-2t

DSM-2t

DSM-s

DSM-s

DSM-pro1 DSM-pro2 FEA

150

100

DSM-pro1

150

DSM-pro2 FEA 100

50

50

0 0

500

1000

1500

2000

0

2500

500

1000

1500

2000

2500

Effective length L/2 (mm)

Effective length L/2 (mm)

Fig. 25. Comparison of FEA and design strengths for I-E16A00T1.0.

Fig. 22. Comparison of FEA and design strengths for I-E16A30T1.0.

500

400

400

DSM-2t

DSM-2t

DSM-s

DSM-s

DSM-pro1

Column strength Pu (kN)

Column strength Pu (kN)

1500

Fig. 21. Comparison of FEA and design strengths for I-E16A45T2.4.

0

DSM-pro2 FEA

300

200

100

0

DSM-pro1

300

DSM-pro2 FEA 200

100

0 0

500

1000

1500

2000

2500

0

500

Effective length L/2 (mm)

8 Py > < > : 1  0:2



P crd Py

0:6 

P crd Py

0:6

if λd 40:761

1500

2000

2500

Fig. 26. Comparison of FEA and design strengths for I-E16A00T1.5.

if λd r0:761 Py

1000

Effective length L/2 (mm)

Fig. 23. Comparison of FEA and design strengths for I-E16A30T1.5.

P nd ¼

1000

Effective length L/2 (mm)

Column strength Pu (kN)

Column strength Pu (kN)

Effective length L/2 (mm)

ð6Þ

The design equations for the nominal axial strength (Pn) and nominal axial strength for flexural buckling (Pne) are identical to those of the current direct strength method. The exponent value of 0.4 and coefficient value of 0.15 in Eq. (3) were modified to 0.5 and 0.22 as shown in Eq. (5). The coefficient value of 0.25 in Eq. (4) was modified to 0.2 in Eq. (6). The non-dimensional slenderness (λl ) and (λd ) were also modified in order to have a smooth transition between the elastic and inelastic buckling curves for local and distortional buckling.

7. Reliability analysis A reliability analysis was performed to assess the reliability of the current direct strength method and the modified direct strength method on cold-formed steel built-up open section columns. A target reliability index (β) of 2.5 for structural members in the North American Specification (NAS) [13] was used as a lower limit. The resistance factor (ϕc1) of 0.85 for the prequalified columns and the resistance factor (ϕc2) of 0.8 for other columns are used in the NAS Specification [13] and AS/NZS Standard [14], and both resistance factors are also used for coldformed steel built-up open section columns in this study. The load combination of 1.2DL þ1.6LL was used in the analysis as shown in

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1.5

700

DSM-2t

Tests

DSM-pro1 500

DSM-pro2

1.0

FEA

400

Pu / Py

Column strength Pu (kN)

FEA Modified DSM

DSM-s

600

300

Current DSM 0.5

200 100 0.0

0 0

500

1000

1500

2000

2500

0.0

0.5

1.0

1.5

2.0

2.5

3.0

λd

Effective length L/2 (mm)

Fig. 30. Comparison of FEA and test results with design strength curves (PDSM and PDSM-pro2) for built-up open sections failed by distortional buckling.

Fig. 27. Comparison of FEA and design strengths for I-E16A00T2.4.

1.5

reliability index (β2) was calculated using the resistance factor ϕc2. Reliability analysis is detailed in the commentary of the NAS Specification [13].

FEA Tests Current DSM

Pu / Py

1.0

8. Comparison of numerical and experimental results with design predictions

0.5

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

λc Fig. 28. Comparison of FEA and test results with design strength curve (PDSM) for built-up open sections failed by flexural buckling.

1.5

FEA Tests

1.0

Pu / Pne

Current DSM

0.5

Modified DSM (Yap and Hancock [16]) 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

λl Fig. 29. Comparison of FEA and test results with design strength curves (PDSM and PDSM-pro2) for built-up open sections failed by local buckling.

the American Society of Civil Engineers (ASCE) Standard [17], where DL is the dead load and LL is the live load. The statistical parameters Mm, Fm, VM and VF are the mean values and coefficients of variation for material properties and fabrication factors. These values were taken from Table F1 of the NAS Specification [13] for concentrically loaded compression members, where Mm ¼1.10, Fm ¼1.00, VM ¼ 0.10 and VF ¼0.05. The statistical parameters Pm and VP are the mean value and coefficient of variation of tested and FEA-to-predicted ratios, respectively, as shown in Table 5. A correction factor CP was used in the reliability analysis considering the influence due to a small number of tests. The reliability index (β1) was calculated using the resistance factor ϕc1, whereas the

The column strengths obtained from test specimens (PEXP) as well as the FEA specimens (PFEA) are compared with the nominal (unfactored) design strengths PDSM-2t, PDSM-s, PDSM-pro1 and PDSMpro2, as shown in Table 5 and Figs. 10–27. The design strengths PDSM-2t, PDSM-s and PDSM-pro1 were obtained by using the current direct strength method, whereas the design strength PDSM-pro2 was calculated by the modified direct strength method. The design strengths obtained the elastic buckling stress using different crosssection assumptions, as shown in Fig. 9. The design strengths PDSM-2t obtained by the current direct strength method, assuming the built-up section as one rigidly connected section, are generally conservative or slightly unconservative for intermediate and long columns but unconservative for short columns, as shown in Table 5 and Figs. 10–27. The mean value of the (PFEA and PEXP)/PDSM-2t ratio is 0.98, with the coefficient of variation (COV) of 0.073, and the corresponding values of β1 ¼2.63 and β2 ¼ 2.88. The design strengths PDSM-s obtained by the current DSM and regarding the built-up section as two single separated open sections are conservative for short columns and very conservative for intermediate and long columns. The mean value of the (PFEA and PEXP)/ PDSM-s ratio is 1.62, with the coefficient of variation (COV) of 0.299, and the corresponding values of β1 ¼3.02 and β2 ¼3.17. The design strengths PDSM-pro1 obtained by the current DSM as well as the modified elastic buckling analysis are generally conservative or slightly unconservative for all the specimens. The mean value of the (PFEA and PEXP)/PDSM-pro1 ratio is 1.03, with the coefficient of variation (COV) of 0.061, and the corresponding reliability values of β1 ¼2.85 and β2 ¼ 3.10. The design strengths PDSM-pro2 were calculated using the modified DSM and rational design calculation for elastic buckling stress. The PDSM-pro2 predictions are more accurate compared with the PDSM-pro1 predications for short columns of Series I-E16A60, I-E08A60, I-E16A45 and I-E16A30. Generally, PDSM-pro2 conservatively predicted the design strengths for Series I-E24A60 and IE16A00. The mean value of the (PFEA and PEXP)/PDSM-pro2 ratio is 1.04, with the coefficient of variation (COV) of 0.075, and the corresponding reliability values of β1 ¼ 2.86 and β2 ¼3.11. The reliability analysis indicated that the design strengths of PDSM-pro1 and PDSM-pro2 are reliable for all series. The reliability indices (β1

J.-H. Zhang, B. Young / Thin-Walled Structures 89 (2015) 178–191

and β2) are greater than the target value of 2.5 for the current DSM and modified DSM. The column strengths obtained from the test specimens (PEXP) as well as the FEA specimens (PFEA) were also compared with the design strength curves for the current direct strength method and the modified direct strength method. The comparison is shown in Figs. 28–30 for specimens failed by flexural, local and distortional buckling, respectively. It is shown that the current DSM provides accurate predictions for specimens failed by flexural buckling, whereas the modified DSM provides more reasonable design curves for local and distortional buckling, as shown in Figs. 29 and 30.

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longitudinal stiffeners can be used as a prequalified section for column design in the current DSM.

Acknowledgements The research work described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project no. HKU719711E).

References 9. Conclusions The strength and behaviour of cold-formed steel built-up open section columns with longitudinal stiffeners have been investigated. Finite element model (FEM) was developed by including initial geometric imperfections and non-linear material properties of the built-up open section columns. The FEM was verified against the test results. It has been shown that the column strengths and failure modes predicted by finite element analysis are generally in good agreement with the test results. Therefore, the verified FEM was used to perform extensive parametric studies to investigate the effect of edge and web stiffeners on the column strengths and behaviour of cold-formed steel built-up open sections. The column strengths obtained from the finite element analysis and the tests were compared with the nominal strengths calculated by the current DSM and modified DSM. The finite strip method was used to obtain elastic buckling stresses for built-up compression members. Different sectional configuration assumptions were used to achieve a rational buckling analysis for the cold-formed steel built-up open sections. It is shown that the design strengths PDSM-2t are unconservative for short columns and PDSM-s are generally very conservative for all series. The design strengths PDSM-pro1 and PDSM-pro2, combining with the rational buckling analysis, are generally conservative and reliable, whereas PDSM-pro2 provided more accurate predictions for Series I-E16A60, I-E08A60, I-E16A45 and I-E16A30. Therefore, the current direct strength method and the modified direct strength method, using the rational buckling analysis, can be used for the design of coldformed steel built-up open section columns. In addition, it is also recommended that the built-up open section columns with

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