Effect of thickness on the behaviour of axially loaded back-to-back cold-formed steel built-up channel sections - Experimental and numerical investigation

Accepted Manuscript Effect of thickness on the behaviour of axially loaded back-toback cold-formed steel built-up channel sections - Experimental and numerical investigation

Krishanu Roy, Tina Chui Huon Ting, Hieng Ho Lau, James B.P. Lim PII: DOI: Reference:

S2352-0124(18)30111-5 doi:10.1016/j.istruc.2018.09.009 ISTRUC 332

To appear in:

Structures

Received date: Revised date: Accepted date:

28 December 2017 29 July 2018 21 September 2018

Please cite this article as: Krishanu Roy, Tina Chui Huon Ting, Hieng Ho Lau, James B.P. Lim , Effect of thickness on the behaviour of axially loaded back-to-back cold-formed steel built-up channel sections - Experimental and numerical investigation. Istruc (2018), doi:10.1016/j.istruc.2018.09.009

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ACCEPTED MANUSCRIPT Effect of thickness on the behaviour of axially loaded back-to-back coldformed steel built-up channel sections - Experimental and numerical investigation. Krishanu Roy a*, Tina Chui Huon Ting b, Hieng Ho Lau b and James B.P. Lim a Department of Civil and Environmental Engineering, The University of Auckland, New Zealand School of Engineering and Science, Curtin University Sarawak, Miri, Sarawak, Malaysia

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Corresponding Author Contact Details: Krishanu Roy

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b

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E: [email protected] , Building 906, Level 4, Room 413, Newmarket campus, University of Auckland

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Auckland-1010, New Zealand, T: +64 223917991, F: +64 9 373 7462

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Abstract: In cold-formed steel structures, such as trusses, wall frames and portal frames, the use of back-to-back built-up cold-formed steel channel sections are becoming increasingly popular. In such an arrangement, intermediate fasteners are required at discrete points along the length, preventing the

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channel-sections from buckling independently. Current guidance in the AISI &AS/NZS for such

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back-to-back built-up cold-formed steel channel sections requires the use of modified slenderness in order to take into account the spacing of the fasteners. Limited research has been done on back-toback built-up cold-formed steel columns to understand the effect of column thickness and slenderness’s on axial capacity. This issue is addressed herein. This paper presents the results of 60

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experimental tests performed on back-to-back built-up cold-formed steel channel sections under compression. Detailed observations on different failure modes and column strengths were made through varying thickness, length and cross section of columns. A non-linear finite element model was developed which includes material non-linearity, geometric imperfections and explicit modelling of web fasteners. The finite element model was validated against experimental results. A comprehensive parametric study consisting of 204 models has been carried out covering a wide range of thickness and slenderness for the considered back-to-back built-up columns. Axial capacities obtained from the numerical study were used to assess the performance of the current AISI& AS/NZS

ACCEPTED MANUSCRIPT standards when applied to cold-formed back-to-back built up columns; obtained comparisons showed that AISI& AS/NZS standards are un-conservative for stub and short columns which were failed by local buckling whereas standards were over-conservative for the strength of intermediate and slender columns which were failed mainly by overall member buckling. This paper has therefore proposed improved design rules and verified their accuracy using finite element analysis and test results of

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back-to-back built-up cold-formed channel sections, subjected to axial compression.

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Keywords: Cold-formed steel, Back-to-back sections, Built-up columns, Finite element modelling

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Buckling, Fasteners.

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Overall web length of section; Effective area of the section; Overall flange width of section; Overall lip width of section; Cold-formed steel; Thickness of section; Coefficient of variation; Young’s modulus of elasticity; Finite element; Finite element analysis; Critical buckling stress; Modified slenderness; Overall Slenderness; Compressive strength obtained from American Iron and Steel Institute; Compressive strength obtained from Experiment; Compressive strength obtained from the finite element analysis; Screw spacing; Non dimensional slenderness ratio;

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Notations A’ Ae B’ C’ CFS t COV E FE FEA Fn KL r ms KL r o PAISI PEXP PFEA S λc

ACCEPTED MANUSCRIPT 1 Introduction Use of back-to-back built-up cold-formed steel sections (see Fig. 1), as compression members are increasing because of its superior strength-to-self weight ratios and economic design. Cold-formed steel (CFS) structures are easy to construct compared to hot-rolled steel structures; the CFS industry is thus in search for more structurally efficient cross-sectional

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shapes. One of the most effective way to achieve this task is to connect two or more single members together to form a built-up section, e.g. simply connecting two channel sections back to back to form a built-up I-section. As a result, such a member with built-up sections can be utilized to carry more load and span more distance. Applications include struts in steel

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trusses and space frames, wall studs in wall frames and columns in portal frames.

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The American Iron and Steel Institute, AISI [1] and Australian and New Zealand Standards, AS/NZS [2] both prescribe the same modified slenderness approach to determine

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the axial strength of back-to-back built-up cold-formed steel columns. It should be noted that this modified slenderness approach has been adapted from design guidance for hot-rolled

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steel. The applicability of the modified slenderness method for cold-formed steel columns, should be justified.

In the literature, very limited research has been described to determine axial capacities

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of back-to-back built-up cold-formed steel channel-sections in the arrangement shown in Fig. 1, and specifically investigated the effect of thickness on axial strengths and failure modes i.e. local, distortional and overall buckling and buckling interactions. Ting et al. [3], investigated the effect of screw spacing on the behavior of back-to-back cold-formed steel under compression. Following this, Roy et al. [4], investigated the beneficial effect of gap on the axial strength of back-to-back gapped built-up CFS channels. Dabaon et al. [5] investigated built-up battened columns; and concluded that both AISI and the Eurocodes

ACCEPTED MANUSCRIPT were un-conservative for columns failed through local buckling and conservative when the built-up columns failed through flexural buckling. Piyawat et al. [6] investigated welded back-to-back channel-section. Zhang and Young [7] considered an opening in the back-toback built-up channel-sections. (See Fig. 2). Whittle et al. [8] investigated the axial strengths of built-up channel-sections which were welded to-to-toe. Stone and LaBoube [9] considered

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stiffened flange and track back-to-back channel-sections. On the other hand, Roy et al. [8] investigated the effect of screw spacing on the axial strength of back-to-back built-up CFS un-lipped channel sections and showed that AISI & AS/NZS can be un-conservative for stub and short columns. Other works include that of Fratamico et al. [11, 12] who considered

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wood-sheathed and screw-fastened back-to-back built-up cold-formed columns and investigated the buckling behaviour of such columns and Anbarasu et al. [13, 14] who

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investigated the behaviour of cold-formed steel web stiffened built-up battened columns. Roy et al. [15] recently presented an experimental and numerical investigation on the behaviour of

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face-to-face built-up CFS channel sections under axial compression. This paper presents 60 experimental tests and 204 non-linear finite element analyses

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results of back-to-back built-up cold-formed steel sections under axial compression. Tensile coupon tests were conducted to determine the material properties of the columns. Geometric imperfections were measured for all 60 specimens. Experimental failure loads, failure modes

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and deformed shapes were observed and discussed in detail for different length and cross section of columns. The finite element model includes modeling of web fasteners, material non-linearity and geometric imperfections. Finite element results compared well against experimental results. The test and finite element strengths are then compared against the design strengths calculated using AISI Specification and AS/NZS Standard for cold-formed steel structures. The validated finite element model is then used for a parametric study comprising 204 models to check the effect of thickness on strength of such columns. From

ACCEPTED MANUSCRIPT the parametric study results, it is shown that specifications are in general conservative for all columns failed by overall buckling but the standards are un-conservative for columns failed mainly by local buckling. Therefore, improved design rules are proposed for back-to-back built-up CFS channel sections subjected to axial compression.

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2. Design rules as per AISI Specification and AS/NZ Standard

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The un-factored design strengths calculated in accordance with the American Iron and Steel Institute specifications and the Australia/New Zealand standard, basically uses two methods for cold-formed steel members are: effective width method (EWM) and the direct strength method (DSM). However DSM does not include built-up sections, hence, effective

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width method is only available for cold-formed built-up members, which specifies the

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modified slenderness approach to calculate the axial strengths. For built-up sections, the unfactored design strength of axially loaded compression members calculated in accordance

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with AISI & AS/NZ standard are as follows: PAISI =Ae F n

(1)

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The critical buckling stress (Fn) can be calculated as follows: For, λ c ≤ 1.5: Fn = (0.658 λ c ) F y 2

 0.877   Fy 2  λ c 

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For, λ c >1.5: Fn  

(2)

(3)

The non-dimensional critical slenderness (λc) can be calculated as follows:

λc =

Fy Fe

(4)

All the calculations above were based on the modified slenderness ratio which is calculated as per the equation below:

ACCEPTED MANUSCRIPT 2  KL   KL   s =     +  r ms  r o  ryc

  

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  ; For which  s   0.5  KL     ryc 

(5)

 r o

3. Experimental investigation 3.1 Test specimens

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Channel-sections C75 and C90, considered in the experimental program, are shown in Fig. 1. The measured specimen dimensions are shown in Table 1. The experimental test program comprised 60 specimens, subdivided into four different column heights: 300 mm, 500 mm, 1000 mm and 2000 mm. The columns were tested with pin-ended conditions, apart

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from the stub column (300 mm). In Table 1, the specimens have been sub-divided into stub,

screw spacing were considered:

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short and intermediate and slender columns. In the experimental test program, the following

Columns of 300 mm height; screw spacing of 50 mm, 100 mm, and 200 mm

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Columns of 500 mm height; screw spacing of 100 mm, 200 mm, and 400 mm

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Columns of 1000 mm height; screw spacing of 225 mm, 450 mm, and 900 mm

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Columns of 2000 mm height; screw spacing of 475 mm, 950 mm, and 1900 mm

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•

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3.2 Material Properties

Material properties of the channel sections were determined, taking tensile coupons from the center of the web plate in the longitudinal direction. The tensile coupons were prepared and tested according to the British Standard for Testing and Materials [16]. Coupons were 12.5 mm wide with a gauge length of 50 mm. The coupons were tested in a MTS displacement controlled testing machine using friction grips. Two strain gauges and a calibrated extensometer of 50 mm gauge length were used to measure the longitudinal strain.

ACCEPTED MANUSCRIPT The average values of Young’s modulus and yield stress were 207 N/mm2 and 560 N/mm2, respectively. 3.3 Labelling Specimen labelling was used to represent the type of section, screw spacing, nominal

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length of specimen and specimen number. Fig.3 shows an example of the labelling used. The

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channel-sections are denoted by their web depth i.e. 90 in the label (see Fig. 3). The intermediate fastener spacing is denoted as s for the spacing. The column length is stated last in the label as L together with the nominal column length.

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3.4 Test-rig and testing procedure

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Fig. 4 shows a photograph of the test set-up for stub and intermediate columns. The external load cell was used at the base and two LVDTs were positioned at the web, and a

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third positioned at the top. LVDT positions are numbered as 1, 2, 3, 4, 5 and 6 in Fig. 4(a).Axial load was applied to the specimens via a 600 kN capacity GOTECH, GT-7001-

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LC60 Universal Testing Machine (UTM). The loading rate was kept below 25 kg/cm2/s for all test specimens.

3.5 Measurement of initial geometric imperfections

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Fig. 5(a) shows the geometric imperfection measurement equipment. An LVDT with 0.01 mm accuracy was used. The LVDT was able to record the readings at every 20 mm along the length of the sections at the center of the web, flanges, and edge of the lips. Position of LVDTs are shown in Fig. 5(b). A typical plot of the imperfections versus length is shown in Fig.5(c) for BU90-S200-L300-1. The maximum values of imperfections for the test specimens were 0.2 mm, 0.2 mm, 0.4 mm, and 0.6 mm for the 300 mm, 500 mm, 1000 mm and 2000 mm section lengths, respectively. These imperfections can be included in finite

ACCEPTED MANUSCRIPT element models to accurately predict the axial capacities of the built-up members. 3.6 Experimental results The dimensions of the test specimens and the experimental ultimate loads (PEXP) are shown in Table 1 (a) and (b) for BU75 and BU90. Table 1 also shows the strength of the

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back-to-back channel-sections in accordance with the AISI & AS/NZS standards. The non-

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dimensional slenderness of the channel-sections are also included in Table 1. As can be seen, AISI & AS/NZ Standards are un-conservative for the stub column tests and conservative for the other column tests.

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Load-axial shortening relationship for BU75-S50-L300-1 is plotted in Fig. 6. It is shown that the relationship was almost linear up to a load of 85 kN, which is approximately

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70.4% of the ultimate failure load for BU75-S50-L300-1. After that, non-linear behavior is

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continued until the failure load is reached, which is 120.7 kN. Different failure modes were observed with varying length of the columns. Also, screw

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spacing has significant effect in terms of strength and failure modes of built-up columns. Almost all stub columns of both BU75 and BU90 series, failed under local buckling. The failure modes for both the BU75-S100-L300 and BU90-S100-L300 test specimens are similar

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to the BU75-S50-L300 and BU90-S50-L300 test specimens. However, the individual channels tended to buckle separately between the intermediate fasteners as shown in Fig.7 (a). However, the individual channels pry apart at mid-length for both the BU75-S200-L300 and BU90-S200-L300 test specimens due to the lack of fasteners along the length of the column. It was observed that for the BU75-L500 test specimens, local buckling waves were formed during the initial phase of testing but global buckling dominated the final observed deformation as shown in Fig.7 (b).Flexural-torsional buckling was observed in some of the BU90-L1000 test specimens during testing, but for the intermediate columns global buckling

ACCEPTED MANUSCRIPT dominated the final and total deformation, as shown in Fig.7(c). Local and distortional buckling was not observed during the testing of the slender columns for the BU75-L2000 test specimens. Global buckling was noticeable immediately with a large curved deformation at mid-height. After the ultimate load was reached, localized deformation was visible near the

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mid-length of the compression side of the specimens.

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4. Numerical investigation 4.1 General

A non-linear elasto-plastic finite element model was developed using ABAQUS [17]. The modelling techniques described by Ying et al. [18-21] are applied. The model was based

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on the center line dimensions of the cross-sections. In the finite element model, the measured

4.2 Finite element idealization

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cross-sectional dimensions were used. Specific modelling issues are described below.

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The full geometry was modelled for the test specimens. Contact surfaces were defined between the web surfaces of the channel sections in the built-up section. In this study,

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surface-to-surface contact of the built-up columns is defined between the outside surfaces of the webs of the two-individual channel-sections. One of the web surfaces is defined as a master surface while the other as a slave surface. Axial load was applied through the center of

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gravity of the specimens. Displacement control load was used to apply the load. Static RIKS method was used to apply displacements on each ends of the built-up column. In total, 100 increments were used. Initial arc length increment was 1, whereas the minimum and maximum arc length increments were 1E-005 and 1E+036 respectively with an estimated total arc length of 1. 4.3 Geometry and material properties The full geometry was modelled for the test specimens and parametric study. The material non-linearity was incorporated in the finite element model by specifying ‘true’

ACCEPTED MANUSCRIPT values of stresses and strains. The ABAQUS classical metal plasticity model was adopted for all the analyses and for validation of the model. This model implements the von Mises yield surface to define isotropic yielding, associated plastic flow theory, and isotropic hardening behavior. For the parametric study, a simplified elastic perfect plastic stress–strain curve obeying Von Mises yield criterion was used. Yield stress of 560 MPa, ultimate stress of 690

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MPa, along with Young’s modulus of 207 GPa, and was used in numerical modelling. As per the ABAQUS manual, the engineering material curve is converted into a true material curve:

σtrue =σ (1+ε)

Where E is the Young’s Modulus,

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and

are the engineering stress and strain respectively

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in ABAQUS [17].



(7)

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σ true E

ε true(pl) =ln (1+ε)-

(6)

4.4 Element type and mesh sensitivity

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The channel sections were modelled using the linear 4-node quadrilateral thick shell

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element S4R5. An element size of 5 mm by 5 mm was found to be appropriate, based on the results of a convergence study. Along the length of the sections, the number of elements was chosen so that the aspect ratio of the elements was as close to one as possible. Mesh sensitivity analyses were performed to verify the number of elements. Fig. 8 shows a typical

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finite element mesh at failure.

4.5 Boundary conditions and load application The back-to-back built-up cold-formed steel columns investigated in this study were pin-ended columns other than the stub column which was fixed-ended. To define the upper and lower pin-end conditions, the displacements and rotations were applied to the upper and lower end plates through reference Points. The load was applied to the reference points of the upper end plate as shown in Fig.9. Cartesian basic connector elements available in the

ACCEPTED MANUSCRIPT ABAQUS library were used to model the screw connections. Connector elements were assigned a stress of 6210 MPa to validate the experimental results accurately. The load was applied in increments using the modified RIKS method available in the ABAQUS library. The RIKS method can predict post buckling behaviour. 4.6 Modelling of local and overall geometric imperfections

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Eigenvalue analyses were used to determine the contours for the global and local imperfections. The imperfections were scaled to the values determined in the experimental program. In addition, local imperfections of magnitude 0.5% of the section thickness were incorporated as recommended by Ellobody and Young [22]. Fig. 10 shows the local and

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overall buckling modes obtained for a typical stub column. 4.7 Modelling of residual stresses

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Residual stresses can be incorporated into the FE model as initial state using the ABAQUS (*INITIAL CONDITIONS, TYPE = STRESS) option. However, previous studies

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detailed in [5, 22–24] have shown that it has a negligible effect on the column strength,

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stiffness of the column, load-axial shortening behavior and failure modes. Therefore, residual stresses were not included in the model to avoid the complexity of the analysis. 4.8 Verification of finite element model

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Fig. 6 compares the experimental test results and the finite element results of stub column for both BU75 and BU90 series. Fig.11 shows the modes of failure for the stub, short and intermediate columns. As can be seen, the finite element results are close the experimental test results. Thus, the experimental and finite element results show good agreement for both the ultimate strength and the failure mode. Table 1(a) and (b) compares the failure load from the experimental tests with that of the finite element analysis. As can be seen, the mean value of the ratio PEXP /PFEA is 1.04, with a co-efficient of variation of 0.02 for stub column of BU75.

ACCEPTED MANUSCRIPT 5. Comparison with design standards Table 1(a) and (b) compares the experimental strengths with the design strengths calculated in accordance with AISI and AS/NZS. As can be seen, for both BU75 and BU90 series, the design strength calculated using the modified slenderness approach, proposed by AISI & AS/NZS are lower than experimental strengths by around 12% average for columns

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failed mainly by overall buckling. However, AISI & AS/NZS standard overestimated the strength of back-to-back built-up columns which were failed mainly by local buckling i.e. stub columns.

The effect of screw spacing was investigated with the length of the columns for both

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BU75 and BU90 series. For the case of the stub and slender columns, increasing the number

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of screws has negligible effect on axial strength of the sections but for short and intermediate columns, the strength of the section was dependent on the number of screws. In case of the

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short column, when the spacing of the screws was doubled, the strength of the section was reduced by around 5 % to 10%. When the screw spacing was doubled, the axial strength was

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decreased by around 10 % to 15 % for intermediate columns. Fig.12 (a) shows the experimental, finite element and design strengths of the BU75 columns plotted against modified slenderness. As can be seen, columns having modified

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slenderness ratio less than 32 failed mainly by local buckling, while columns having a modified slenderness greater than 53 failed through overall member buckling. On the other hand, Fig.12 (b) shows the plot of experimental, numerical and design strengths against modified slenderness for BU90 section. It is shown that, columns having modified slenderness ratio less than 29 failed mainly by local buckling, whereas columns having a modified slenderness greater than 48 failed through overall member buckling. As can be expected, for the slender columns, the increase in strength is less. Figs. 13(a) and 13(b)

ACCEPTED MANUSCRIPT compares the FEA strength and design strength calculated in accordance with AISI& AS/NZS for BU75 and BU90. For both BU75 and BU90 columns, standards were conservative for columns failed by overall buckling but for columns failed by local buckling, standards were un-conservative by around 10%.

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6. Parametric study

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A parametric study comprising 204 models was conducted using the verified finite element model. BU75 was considered for the parametric study, where back-to-back channels had dimension of 75×20×10 mm. Stub, short and intermediate and slender columns were investigated for thickness varying from 0.75 mm to 2.5 mm. Spacing between the screws

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were kept constant to investigate the effect of one parameter: thickness. Considered spacing

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for stub, short, intermediate and slender columns were 50 mm, 100 mm, 225 mm and 475 mm respectively. Table 2(a) to 2(d) has presented the comparison of FEA and design

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strengths when the thickness is varied from 0.75 to 2.5 mm for stub, short, intermediate and slender columns. Fig.14 shows the variation of strength against thickness for BU75 column.

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As can be seen, channel thickness was varied from 0.75 mm to 2.5 mm at an increment of 0.05 mm. Fig.14 also shows the strengths predicted by AISI & AS/NZS Standards. The effect of thickness was observed for different length of columns from the

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parametric study. For the case of the stub columns, when the thickness was increased from 0.75 mm to 2.5 mm, the difference in strength predicted from FEA and AISI&AS/NZS remains almost constant, while for the case of the short and intermediate columns, the strength difference of BU75 predicted from FEA and AISI&AS/NZS remains almost constant up to a thickness of 1.55 mm but above this thickness, AISI&AS/NZS overestimated the strength by around 32%. There was no significant difference in strength predicted between FEA and AISI&AS/NZS for the slender column up to a thickness of 1.15 mm, above which, standards underestimated the strength of back-to-back built-up cold formed steel column

ACCEPTED MANUSCRIPT significantly. The parametric study was further extended to investigate the axial strength of welded back-to-back channels with 0 mm spacing and compared results with screw spacing of 50 mm, 100 mm, 200 mm, 400 mm and 900 mm results. It was found that the axial strength of welded back-to-back channels with 0 mm spacing was 22% higher than the axial strength of

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built-up channels with 50 mm screw spacing. For 100 mm screw spacing, axial strength of built-up channels was 28% lower on average when compared with the axial strength of welded built-up channels. For 200, 400 and 900 mm fastener spacing, axial strengths of builtup back-to-back channels were 34%, 41% and 53% lower than the axial strength of welded

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back-to-back channels respectively.

7. Proposed design rules:

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There are mainly two regions in the design curve of AISI & AS/NZS. The first region is for stub columns (where modified slenderness is less than or equal to 32 or λ c ≤ 1.5) and the

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second region is for slender columns (where modified slenderness is greater than 53 or λ

c

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>1.5), which is defined by Eq. (2) and Eq. (3) in AISI & AS/NZS [1-2]. Tables 1(a) and 1(b) compares the experimental and FEA results with existing design equations for back-toback built-up CFS columns subjected to axial compression. As can be seen from Table 1,

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AISI & AS/NZS is over conservative for intermediate and slender columns which failed through flexural buckling. However, for stub columns, which failed though local buckling, AISI & AS/NZS are un-conservative by around 12%, which emphasize the need for improved design rules. Therefore this paper proposes a new set of equations as below: Fn = (0.61 λ c ) F y for, λ c ≤ 1.5

(8)

 0.84 Fn   1.5  λ c

(9)

2

  Fy for, λ c >1.5 

Fig. 15 shows a close agreement of test and FEA results with the proposed equations. Also

ACCEPTED MANUSCRIPT shown in Tables 1(a) and 1(b), the comparison of axial strength from FEA and design strengths calculated using eqns. (8) and (9) for BU75 and BU90 respectively. As can be seen from Table 1 and Fig. 15, design strengths are very close to FE strengths, when eqns. (8) and (9) were used.

8. Capacity reduction factor:

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The American cold-formed steel structures code [1] recommends a statistical model to determine the capacity reduction factors. This model accounts for the variations in material, fabrication and the loading effects. The capacity reduction factor 𝜙 is given by the following

𝜙 = 1.52𝑀𝑚 𝐹𝑚 𝑃𝑚 𝑒

2 +𝑉 2 +𝐶 𝑉 2 +𝑉 2 −𝛽0 √𝑉𝑚 𝑝 𝑝 𝑞 𝑓

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equation:

(10)

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where, 𝑀𝑚 and Vm are the mean and coefficient of variation of the material factor 1.1, 0.1; 𝐹𝑚 , Vf are the mean and coefficient of variation of the fabrication factor 1, 0.05; Vq is the

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coefficient of variation of load effect 0.21; β0 is the target reliability index 2.5; 𝐶𝑝 is the correction factor depending on the number of tests; 𝑃𝑚 is the mean value of the tested to

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predicted load ratio; Vp is the coefficient of variation of the tested to predicted load ratio. Vp and Pm values have to be determined from experiments or analyses. In this investigation ultimate loads obtained from FEA were considered. Hence Vp and Pm are the

standards.

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mean and coefficient of variation of the ratio of ultimate loads from FEA and design

The substitution of all the above values leads to the following equation. 𝜙 = 1.615𝑃𝑚 𝑒

−2.5√0.0566+𝐶𝑝 𝑉𝑝2

(11)

Eqn. 11 was used to determine the capacity reduction factors for the values obtained from the current AISI & AS/NZS [1-2] and the proposed design rules. AS/NZS [2] recommends a capacity reduction factor of 0.85 for compression members. Table 1 compares the tests

ACCEPTED MANUSCRIPT and FEA results with design strengths calculated from the existing design equations and proposed design equations for back-to-back built-up CFS columns subjected to axial compression. The capacity reduction factors according to the current AISI & AS/NZS [1-2], using Eqns. (2) and (3) are 0.80 and 0.92, for stub and slender columns respectively, which emphasize the need for improved design rules. The capacity reduction factor

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according to the proposed Eqns. (8) and (9) is 0.87. Therefore, Eqns. (8) and (9) are recommended for designing back-to-back built-up CFS columns under axial compression.

9. Conclusion and future work

This paper has presented the results of 60 experimental tests and 204 non-linear finite

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element analyses on back-to-back built-up cold-formed steel channels, subject to axial compression. Material properties and geometric imperfections were measured for all

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specimens. The buckling modes and deformed shapes at failure have been discussed in detail with varying thickness, length and cross section of columns. The non-linear finite element

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model includes explicit modeling of web fasteners, material non-linearity and geometric

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imperfections. Finite element results are compared against the experimental test results which showed good agreement.

The validated finite element models were then used to perform a parametric study to investigate the effect of thickness and slenderness on behavior of back-to-back cold-formed

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steel built-up columns under compression. The column strengths predicted from the experimental tests and finite element analyses were compared against the design strengths calculated in accordance with AISI and the AS/NZS Standards. From the comparison, it is shown that standards are is in general conservative for all columns failed by overall buckling but are un-conservative for columns failed mainly by local buckling by around 12%. Hence improved design rules were proposed for back-to-back built-up CFS channel sections subjected to axial compression.

ACCEPTED MANUSCRIPT Authors are currently investigating the effect of different cross section and arrangement of screws for cold-formed steel built-up columns under axial and eccentric load to develop a new Direct Strength Method (DSM) for such built-up columns that will incorporate more accurate estimations of axial strength of column cross sections and screw spacing for different end conditions.

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Declaration of interest:

There is no conflict of interest between the authors or any third party regarding the publication of this article.

Funding:

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This research did not receive any specific grant from funding agencies in the public,

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commercial, or not-for-profit sectors.

ACCEPTED MANUSCRIPT References [1] American Iron and Steel Institute (AISI). North American Specification for the Design of Cold-formed Steel Structural Members, AISI S100-07; 2007. [2] Australia/New Zealand Standard (AS/NZS). Cold-Formed Steel Structures, AS/NZS

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4600:2005, Standards Australia/ Standards New Zealand, 2005.

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[3] Ting TCH, Roy K, Lau HH, Lim JBP. Effect of screw spacing on behavior of axialy loaded back-to-back cold-formed steel built-up channel sections, Adv. Struct. Eng. 2018; 21 (3) : 474-487.

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[4] Roy K, Ting TCH, Lau HH, Lim JBP. Nonlinear behaviour of back-to-back gapped builtup cold-formed steel channel sections under compression. J. Constr. Steel Res 2018; 147:

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257-276.

[5] Dabaon M, Ellobody E, Ramzy K. Nonlinear behavior of built-up cold-formed steel

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section battened columns, J Constr Steel R 2015; 110:16-28.

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[6] Piyawat K, Ramseyer C, Kang THK. Development of an axial load capacity equation for doubly symmetric built-up cold-formed sections, J Struct Eng ASCE 2013;139(12): 04013008-13.

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[7] Zhang JH, Young B. Compression tests of cold-formed steel I-shaped open sections with edge and web stiffeners, Thin-Walled Struct 2012; 52:1-11. [8] Whittle J, Ramseyer C. Buckling capacities of axially loaded, cold-formed, built-up channels, Thin-walled Struct 2009; 47:190-201. [9] Stone TA, LaBoube RA. Behaviour of cold-formed steel built-up I-sections, Thin-Walled Struct 2015; 43: 1805 – 1817.

ACCEPTED MANUSCRIPT [10] Roy K, Ting TCH, Lau HH, Lim JBP. Nonlinear behavior of axially loaded back-toback built-up cold-formed steel un-lipped channel sections. Steel and Composite Structures, an International Journal 2018; 28 (2): 233-250. [11] Fratamico DC, Schafer B. Numerical Studies on the Composite Action and Buckling

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Behavior of Built-Up Cold-Formed Steel Columns. Proc. of 22nd International Specialty

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Conference on Cold-Formed Steel Structures, St. Louis, Missouri, USA, November 5-6, 2014.

[12] Fratamico DC, Torabian S, Rasmussen KJR. Schafer B. Experimental Studies on the

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Compositeaction in Wood-sheathed and Screw-fastened Built-up Cold-formed Ssteel Columns. Proc. of the Annual Stability Conf., Structural Stability Res. Co., Orlando, FL

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2016.

[13] Anbarasu M, Kanagarasu K, Sukumar S. Investigation on the behaviour and strength of

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cold-formed steel web stiffened built-up battened, Materials and Struct 2015; 48(12):4029-

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4038.

[14] Anbarasu M, Kumar PB, Sukumar S. Study on the capacity of cold-formed steel built-up batten columns under axial compression, Latin American J Solids and Struct 2014; 11:2271-

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2283.

[15] Roy K, Mohammadjani C, Lim JBP.Experimental and numerical investigation into the behaviour of face-to-face built-up cold-formed steel channel sections under compression, Thin-

Walled Struct. (2018) revised. [16] BS EN. Tensile Testing of Metallic Materials Method of Test at Ambient Temperature, British Standards Institution, 2001. [17] ABAQUS Analysis User’s Manual-Version 6.14-2. ABAQUS Inc., USA; 2014.

ACCEPTED MANUSCRIPT [18] Lian Y, Uzzaman A, Lim JBP, Abdelal G, Nash D, Young B. Web crippling behaviour of cold-formed steel channel sections with web holes subjected to interior-one-flange loading condition-Part I: Experimental and numerical investigation, Thin-Walled Struct 2017; 111: 103–112. [19] Lian Y, Uzzaman A, Lim JBP, Abdelal G, Nash D, Young B. Web crippling behaviour

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of cold-formed steel channel sections with web holes subjected to interior-one-flange loading

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condition – Part II: parametric study and proposed design equations, Thin-Walled Struct 2017; 114: 92–106.

[20] Lian Y, Uzzaman A, Lim JBP, Abdelal G, Nash D, Young B. Effect of web holes on

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web crippling strength of cold-formed steel channel sections under end-one-flange loading condition – Part I: Tests and finite element analysis, Thin-Walled Struct 2016; 107 : 443–452.

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[21] Lian Y, Uzzaman A, Lim JBP, Abdelal G, Nash D, Young B. Effect of web holes on web crippling strength of cold-formed steel channel sections under end-one-flange loading

2016; 107: 489-501.

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condition - Part II: Parametric study and proposed design equations, Thin-Walled Struct

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[22] Ellobody E, Young B. Behavior of cold-formed steel plain angle columns, J Struct Eng ASCE 2005; 131(3): 457–466

[23] Young B, Ellobody E. Buckling analysis of cold-formed steel lipped angle columns,

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J Struct Eng ASCE USA 2005; 131(10): 1570–9. [24] Young B, Ellobody E. Design of cold-formed steel unequal angle compression members, Thin-Walled Struct 2007; 45(3): 330–8.

ACCEPTED MANUSCRIPT List of tables Table 1 Axial strength comparison from laboratory, finite element analyses and design standards. (a) BU75 (b) BU90 Table 2 Finite element strength and AISI&AS/NZS strength of BU75 with varying thickness

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MA

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Stub column Short column Intermediate column Slender column

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(a) (b) (c) (d)

ACCEPTED MANUSCRIPT Table 1 Axial strength comparison from laboratory, finite element analyses and design (a) BU75 Web

Flange

Lip

Length

Thickness

Spacing

Modified Slenderness

standards.

Test Results

Current AISI& AS/NZS Design Strengths

Specimen

Stub BU75-S50-L300-1 BU75-S50-L300-2 BU75-S50-L300-3 BU75-S100-L300-1 BU75-S100-L300-2 BU75-S100-L300-3 BU75-S200-L300-1 BU75-S200-L300-2 BU75-S200-L300-3 Mean COV Short BU75-S100-L500-1 BU75-S100-L500-3 BU75-S200-L500-1 BU75-S200-L500-2 BU75-S200-L500-3 BU75-S400-L500-1 BU75-S400-L500-2 Mean COV Intermediate BU75-S225-L1000-1 BU75-S225-L1000-2 BU75-S450-L1000-1 BU75-S450-L1000-2 BU75-S450-L1000-3 BU75-S900-L1000-1 BU75-S900-L1000-2 BU75-S900-L1000-3

A’

B’

C’

L

t

S

(KL/r)m

PEXP

(mm)

(mm)

(mm)

(mm)

(mm)

(mm)

-

(kN)

73.1 73.1 72.7 73.1 73.1 73.6 73.7 73.6 72.9

19.8 19.8 19.5 19.8 19.9 19.7 19.8 19.9 20.0

11.1 11.2 10.8 11.2 11.2 11.2 11.2 11.2 11.2

273 280 270 267 273 273 266.5 266 268

1.20 1.21 1.20 1.18 1.19 1.20 1.21 1.20 1.20

50.0 50.0 50.9 99.7 100.2 99.5 200.0 199.5 200.0

15.63 15.93 15.92 19.48 19.41 19.56 30.31 30.22 29.97

120.7 118.8 118.7 117.5 122.7 115.4 122.5 119.1 113.1

73.6 73.6 73.5 73.6 73.4 73.6 73.5

19.8 19.7 19.5 19.6 19.7 19.7 19.7

11.2 11.2 11.3 11.3 11.3 11.3 11.3

655.0 680.0 653.0 678.0 680.0 678.0 679.0

75.3 75.7 75.8 75.6 75.9 76.0 76.3 75.9

20.2 19.9 19.9 19.9 19.8 19.9 19.8 19.8

C A

E C

10.4 10.4 10.4 10.4 10.3 10.3 9.1 10.3

T P

1133 1131 1131 1133 1182 1131 1133 1183

D E

1.20 1.21 1.20 1.18 1.19 1.20 1.22

1.20 1.20 1.21 1.20 1.18 1.19 1.20 1.22

C S

A M

U N

T P

I R

PAISI&AS/NZS

FEA Results

Design Strengths Using eqns. 8 and 9

PFEA

PEXP/ PFEA

PAISI&AS/NZS (using eqns. 8 & 9)

(kN)

PEXP/ PAISI

PFEA/ PAISI using eqns. 8 and 9 -

(kN)

-

(kN)

-

126.6 126.7 124.8 125.1 125.4 124.8 119.0 119.0 119.3

0.95 0.94 0.95 0.94 0.98 0.92 1.03 1.00 0.95 0.96 0.04

116.7 114.9 114.4 113.6 117.4 112.6 120.8 116.4 108.4

1.09 1.03 1.04 1.03 1.05 1.02 1.01 1.02 1.04 1.04 0.02

113.3 112.6 113.3 109.2 114.0 111.5 118.4 114.1 105.2

1.03 1.02 1.01 1.04 1.03 1.01 1.02 1.02 1.03 1.02 0.01

100.0 100.5 195.0 195.0 200.5 400.0 401.0

69.11 72.16 73.36 75.58 75.61 88.74 89.00

83.0 74.1 86.2 88.9 93.6 74.8 80.6

78.8 78.3 79.9 81.4 86.7 72.4 74.3

1.05 0.95 1.08 1.09 1.08 1.03 1.08 1.05 0.05

79.5 78.4 80.3 82.7 88.1 74.6 76.3

1.04 0.95 1.07 1.07 1.06 1.00 1.06 1.04 0.05

76.4 74.0 77.2 80.3 84.7 72.4 74.8

1.04 1.06 1.04 1.03 1.04 1.03 1.02 1.04 0.01

225.3 225.3 447.0 450.0 450.0 900.0 900.0 901.0

121.36 123.71 133.91 135.07 140.52 171.43 178.06 176.55

47.0 46.3 50.4 45.0 41.8 39.9 33.7 31.5

42.3 41.0 38.9 38.1 34.6 33.2 30.2 28.9

1.11 1.13 1.29 1.18 1.21 1.20 1.11 1.09

45.7 44.9 42.4 40.1 35.8 34.2 31.5 29.6

1.03 1.03 1.19 1.12 1.17 1.17 1.07 1.06

44.4 43.2 42.0 39.3 34.1 33.9 30.3 29.0

1.03 1.04 1.01 1.02 1.05 1.01 1.04 1.02

ACCEPTED MANUSCRIPT Mean COV Slender BU75-S475-L2000-1 BU75-S475-L2000-2 BU75-S950-L2000-1 BU75-S950-L2000-2 BU75-S1900-L2000-1 BU75-S1900-L2000-2 Mean COV

1.17 0.07 73.9 73.9 73.9 73.9 73.9 73.9

20.3 20.2 20.3 20.2 20.3 20.4

10.7 10.8 10.8 10.8 10.9 10.7

2184 2183 2184 2184 2183 2184

1.20 1.20 1.18 1.17 1.18 1.19

474.5 462.0 949.5 950.0 1900.0 1901.0

231.20 231.61 255.17 256.21 334.82 333.86

10.9 10.8 8.8 8.6 7.6 7.5

10.2 10.2 8.4 8.4 7.3 7.3

T P

I R

C S

U N

D E

T P

C A

E C

A M

1.03 1.03 1.02 1.01 1.03 1.01 1.02 0.01

1.11 0.07 10.6 10.5 8.6 8.5 7.4 7.4

1.03 1.03 1.02 1.01 1.03 1.01 1.02 0.01

1.03 0.01 10.2 10.0 8.3 8.3 7.3 7.2

1.04 1.05 1.03 1.02 1.01 1.03 0.01 0.01

ACCEPTED MANUSCRIPT (b) BU90 Web

Flange

Lip

Length

Thickness

Spacing

Modified Slenderness

Test Results

Current AISI& AS/NZS Design Strengths

FEA Results

PFEA

T P

Specimen

Stub BU90-S50-L300-1 BU90-S50-L300-2 BU90-S50-L300-3 BU90-S100-L300-1 BU90-S100-L300-2 BU90-S200-L300-1 BU90-S200-L300-2 BU90-S200-L300-3 Mean COV Short BU90-S100-L500-1 BU90-S100-L500-2 BU90-S200-L500-1 BU90-S200-L500-2 BU90-S200-L500-3 BU90-S400-L500-1 BU90-S400-L500-2 Mean COV Intermediate BU90-S225-L1000-1 BU90-S225-L1000-2 BU90-S450-L1000-1 BU90-S450-L1000-2 BU90-S450-L1000-3 BU90-S900-L1000-1 BU90-S900-L1000-2 BU90-S900-L1000-3 Mean COV

PEXP/ PFEA

Design Strengths Using eqns. 8 and 9 PAISI & AS/NZS (using

PFEA/ PAISI using eqns. 8 and 9 -

A’

B’

C’

L

t

S

(KL/r)m

PEXP

PAISI&AS/NZS

PEXP / PAISI

(mm)

(mm)

(mm)

(mm)

(mm)

(mm)

-

(kN)

(kN)

-

(kN)

-

91.3 91.8 92.9 90.8 90.6 90.7 90.7 89.5

49.8 49.7 49.4 49.7 49.5 49.4 49.4 48.3

14.6 14.5 14.5 14.6 14.6 14.6 14.6 14.0

277.0 272.0 261.0 262.0 268.0 273.5 269.5 280.5

1.20 1.19 1.21 1.20 1.18 1.18 1.20 1.20

50.0 49.8 50.0 99.9 100.0 201.0 199.0 199.0

7.95 7.89 7.93 9.45 9.42 11.93 11.83 11.87

172.5 171.6 170.6 166.2 165.8 163.3 163.5 162.9

179.7 182.6 179.6 178.7 176.4 175.6 173.9 173.3

0.96 0.94 0.95 0.93 0.94 0.93 0.94 0.94 0.94 0.01

162.7 160.4 160.9 152.5 156.4 157.0 155.7 158.2

1.06 1.07 1.06 1.09 1.06 1.04 1.05 1.03 1.06 0.02

159.5 154.2 153.2 151.0 153.3 152.4 149.7 155.1

1.02 1.04 1.05 1.01 1.02 1.03 1.04 1.02 1.03 0.01

90.6 90.6 90.4 90.4 90.4 90.6 90.4

49.5 49.4 49.3 49.3 49.3 49.4 49.4

14.6 14.6 14.7 14.7 14.6 14.7 14.7

656.0 678.0 653.0 678.0 680.0 678.0 678.0

1.21 1.20 1.18 1.19 1.21 1.18 1.20

100.5 100.5 199.5 199.5 200.5 400.0 399.0

35.42 34.25 38.52 39.41 40.20 50.20 49.41

160.4 158.1 152.2 150.9 149.2 132.4 134.5

149.9 152.0 140.9 138.4 135.6 124.9 126.9

1.04 1.08 1.09 1.10 1.06 1.06 1.07 1.07 0.02

152.8 153.5 142.2 142.4 143.5 127.3 128.1

1.05 1.03 1.07 1.06 1.04 1.04 1.05 1.05 0.01

148.3 150.5 139.4 138.3 138.0 123.6 123.2

1.03 1.02 1.02 1.03 1.04 1.03 1.04 1.03 0.01

90.8 90.6 90.6 90.4 90.5 90.5 91.0 90.1

49.6 49.6 49.7 49.7 49.8 49.6 49.3 49.2

225.0 225.0 450.0 448.0 452.0 897.0 899.0 896.0

60.42 58.21 64.21 66.21 65.29 75.21 77.21 76.50

102.6 102.0 96.5 94.4 93.3 89.5 87.5 87.5

92.4 92.7 86.1 82.7 82.5 82.8 80.3 79.5

1.11 1.10 1.12 1.14 1.13 1.08 1.09 1.10 1.11 0.07

100.6 99.0 90.2 89.1 87.2 85.3 82.6 84.1

1.02 1.03 1.07 1.06 1.07 1.05 1.06 1.04 1.05 0.02

96.7 94.3 89.3 89.1 85.5 82.0 80.2 80.9

1.04 1.05 1.01 1.00 1.02 1.04 1.03 1.04 1.03 0.02

C A

E C

14.4 14.3 14.4 14.4 14.5 14.4 14.4 14.5

T P

1182 1132 1130 1182 1180 1131 1182 1129

D E

1.21 1.20 1.21 1.18 1.19 1.20 1.21 1.22

C S

U N

A M

I R

eqns. 8 & 9) (kN)

ACCEPTED MANUSCRIPT

Slender BU90-S475-L2000-1 BU90-S475-L2000-2 BU90-S950-L2000-1 BU90-S950-L2000-2 BU90-S1900-L2000-1 BU90-S1900-L2000-2 Mean COV

90.6 90.7 90.5 90.4 90.5 90.9

49.5 49.4 49.5 49.2 49.3 49.7

14.5 14.3 14.6 14.5 14.6 14.2

2164 2172 2169 2148 2158 2152

1.20 1.20 1.18 1.17 1.18 1.19

474.2 466.6 960.4 949.3 1902.4 1906.7

92.52 94.42 101.17 103.21 115.20 116.42

65.4 66.0 54.0 45.6 48.0 43.2

61.1 61.6 50.9 43.4 44.8 41.1

T P

I R

C S

U N

D E

T P

C A

E C

A M

1.07 1.07 1.06 1.05 1.07 1.05 1.06 0.02

62.9 62.9 52.5 44.7 48.5 43.2

1.04 1.05 1.03 1.02 0.99 1.00 1.02 0.02

62.3 61.7 50.5 43.4 47.5 41.1

1.01 1.02 1.04 1.03 1.02 1.05 1.03 0.01

ACCEPTED MANUSCRIPT

Table 2: Finite element strength and AISI&AS/NZS strength of BU75 with varying thickness (a) Stub column Web

Flange

Lip

Thickness

Length

Spacing

AISI& AS/NZS Design Strength

FEA strength

A’

B’

C’

t

L

S

PAISI&AS/NZS

PFEA

mm

mm

mm

mm

mm

mm

kN

BU75-S50-L273-0.75 BU75-S50-L273-0.80 BU75-S50-L273-0.85 BU75-S50-L273-0.90 BU75-S50-L273-0.95 BU75-S50-L273-1.00 BU75-S50-L273-1.05

75.0 75.0 75.0 75.0 75.0 75.0 75.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0

10.0 10.0 10.0 10.0 10.0 10.0 10.0

0.75 0.80 0.85 0.90 0.95 1.00 1.05

273 273 273 273 273 273 273

50.0 50.0 50.0 50.0 50.0 50.0 50.0

67.77 75.17 81.40 88.21 94.16 101.1 107.5

BU75-S50-L273-1.10

75.0

20.0

10.0

1.10

273

50.0

BU75-S50-L273-1.15

75.0

20.0

10.0

1.15

273

50.0

BU75-S50-L273-1.20

75.0

20.0

10.0

1.20

273

BU75-S50-L273-1.25

75.0

20.0

10.0

1.25

273

BU75-S50-L273-1.30

75.0

20.0

10.0

1.30

273

BU75-S50-L273-1.35

75.0

20.0

10.0

1.35

BU75-S50-L273-1.40

75.0

20.0

10.0

1.40

BU75-S50-L273-1.45

75.0

20.0

10.0

1.45

BU75-S50-L273-1.50

75.0

20.0

10.0

BU75-S50-L273-1.55

75.0

20.0

10.0

BU75-S50-L273-1.60

75.0

20.0

BU75-S50-L273-1.65

75.0

20.0

BU75-S50-L273-1.70

75.0

20.0

BU75-S50-L273-1.75

75.0

BU75-S50-L273-1.80

75.0

BU75-S50-L273-1.85

75.0

BU75-S50-L273-1.90

75.0

BU75-S50-L273-1.95

Specimen

D E

kN

-

63.5 70.0 75.7 80.6 87.0 94.3 98.7

1.07 1.07 1.08 1.09 1.08 1.07 1.09

113.8

104.7

1.09

120.2

110.6

1.09

126.7

116.7

1.09

50.0

133.2

123.4

1.08

50.0

139.8

129.7

1.08

A M

U N

50.0

C S

I R

50.0

146.4

136.3

1.07

273

50.0

153.1

142.8

1.07

273

50.0

159.8

150.6

1.06

273

50.0

166.5

158.1

1.05

1.55

273

50.0

173.3

166.6

1.04

1.60

273

50.0

180.1

173.6

1.04

1.65

273

50.0

187.0

178.9

1.04

10.0

1.70

273

50.0

193.8

187.1

1.04

T P 1.50

273

T P

PAISI & AS/NZS / PFEA

20.0

E C 10.0

1.75

273

50.0

200.7

195.6

1.03

20.0

10.0

1.80

273

50.0

207.5

201.5

1.03

20.0

10.0

1.85

273

50.0

214.4

208.9

1.03

20.0

10.0

1.90

273

50.0

221.2

215.5

1.03

75.0

20.0

10.0

1.95

273

50.0

228.1

222.9

1.02

BU75-S50-L273-2.00

75.0

20.0

10.0

2.0

273

50.0

234.9

229.6

1.02

BU75-S50-L273-2.05

75.0

20.0

10.0

2.05

273

50.0

241.7

236.4

1.02

BU75-S50-L273-2.10

75.0

20.0

10.0

2.10

273

50.0

248.5

243.4

1.02

BU75-S50-L273-2.15

75.0

20.0

10.0

2.15

273

50.0

255.2

248.1

1.03

C A

10.0 10.0

ACCEPTED MANUSCRIPT BU75-S50-L273-2.20

75.0

20.0

10.0

2.20

273

50.0

261.9

255.5

1.03

BU75-S50-L273-2.25

75.0

20.0

10.0

2.25

273

50.0

268.5

263.2

1.02

BU75-S50-L273-2.30

75.0

20.0

10.0

2.30

273

50.0

275.1

269.2

1.02

BU75-S50-L273-2.35

75.0

20.0

10.0

2.35

273

50.0

281.7

275.5

1.02

BU75-S50-L273-2.40

75.0

20.0

10.0

2.40

273

50.0

288.1

283.2

1.02

BU75-S50-L273-2.45 BU75-S50-L273-2.50 Mean COV

75.0 75.0

20.0 20.0

10.0 10.0

2.45 2.50

273 273

50.0 50.0

294.5 299.4

289.5 295.1

1.02 1.01 1.05 0.03

I R

T P

C S

U N

D E

T P

C A

E C

A M

ACCEPTED MANUSCRIPT (b) Short column Web

Flange

Lip

Thickness

Length

Spacing

AISI& AS/NZS Design Strength

FEA Strength

A’

B’

C’

t

L

S

PAISI&AS/NZS

PFEA

mm

mm

mm

mm

mm

mm

kN

kN

BU75-S100-L655-0.75 BU75-S100-L655-0.80 BU75-S100-L655-0.85 BU75-S100-L655-0.90 BU75-S100-L655-0.95 BU75-S100-L655-1.00 BU75-S100-L655-1.05

75.0 75.0 75.0 75.0 75.0 75.0 75.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0

10.0 10.0 10.0 10.0 10.0 10.0 10.0

0.75 0.80 0.85 0.90 0.95 1.00 1.05

655 655 655 655 655 655 655

100.0 100.0 100.0 100.0 100.0 100.0 100.0

45.2 49.8 54.0 57.7 61.5 63.3 69.0

BU75-S100-L655-1.10

75.0

20.0

10.0

1.10

655

100.0

BU75-S100-L655-1.15

75.0

20.0

10.0

1.15

655

100.0

BU75-S100-L655-1.20

75.0

20.0

10.0

1.20

655

100.0

BU75-S100-L655-1.25

75.0

20.0

10.0

1.25

655

100.0

BU75-S100-L655-1.30

75.0

20.0

10.0

1.30

655

BU75-S100-L655-1.35

75.0

20.0

10.0

1.35

655

BU75-S100-L655-1.40

75.0

20.0

10.0

1.40

655

BU75-S100-L655-1.45

75.0

20.0

10.0

1.45

BU75-S100-L655-1.50

75.0

20.0

10.0

1.50

BU75-S100-L655-1.55

75.0

20.0

10.0

BU75-S100-L655-1.60

75.0

20.0

10.0

BU75-S100-L655-1.65

75.0

20.0

10.0

BU75-S100-L655-1.70

75.0

20.0

BU75-S100-L655-1.75

75.0

20.0

BU75-S100-L655-1.80

75.0

20.0

BU75-S100-L655-1.85

75.0

BU75-S100-L655-1.90

75.0

BU75-S100-L655-1.95

75.0

BU75-S100-L655-2.00

75.0

BU75-S100-L655-2.05

Specimen

T P

PAISI & AS/NZS / PFEA

-

36.5 47.5 51.1 55.0 58.8 62.8 66.8

1.24 1.05 1.06 1.05 1.05 1.04 1.03

72.8

70.8

1.03

76.6

75.2

1.02

80.4

79.5

1.01

84.2

83.9

1.00

88.0

88.4

1.00

100.0

91.7

92.8

0.99

100.0

95.4

97.1

0.98

100.0

99.1

101.1

0.98

655

100.0

102.8

105.5

0.97

655

100.0

106.4

109.7

0.97

655

100.0

109.9

113.8

0.97

1.65

655

100.0

113.4

117.9

0.96

1.70

655

100.0

116.8

122.0

0.96

1.75

655

100.0

120.1

126.1

0.95

10.0

1.80

655

100.0

123.3

130.2

0.95

D E

T P 1.55 1.60

655

A M

U N

100.0

C S

I R

20.0

E C 10.0

1.85

655

100.0

126.5

134.3

0.94

20.0

10.0

1.90

655

100.0

129.5

138.3

0.94

20.0

10.0

1.95

655

100.0

131.7

142.4

0.92

20.0

10.0

2.0

655

100.0

133.4

146.5

0.91

75.0

20.0

10.0

2.05

655

100.0

135.0

150.5

0.90

BU75-S100-L655-2.10

75.0

20.0

10.0

2.10

655

100.0

136.5

154.7

0.88

BU75-S100-L655-2.15

75.0

20.0

10.0

2.15

655

100.0

137.9

157.6

0.87

BU75-S100-L655-2.20

75.0

20.0

10.0

2.20

655

100.0

139.2

162.9

0.85

BU75-S100-L655-2.25

75.0

20.0

10.0

2.25

655

100.0

140.4

166.2

0.84

C A

10.0 10.0

ACCEPTED MANUSCRIPT BU75-S100-L655-2.30

75.0

20.0

10.0

2.30

655

100.0

141.5

170.3

0.83

BU75-S100-L655-2.35

75.0

20.0

10.0

2.35

655

100.0

142.6

174.4

0.82

BU75-S100-L655-2.40

75.0

20.0

10.0

2.40

655

100.0

143.5

178.0

0.81

BU75-S100-L655-2.45

75.0

20.0

10.0

2.45

655

100.0

144.4

182.1

0.79

BU75-S100-L655-2.50

75.0

20.0

10.0

2.50

655

100.0

145.1

185.7

0.78

T P

0.95

Mean COV

I R

C S

U N

D E

T P

C A

E C

A M

0.09

ACCEPTED MANUSCRIPT (c) Intermediate column Web

Flange

Lip

Thickness

Length

Spacing

AISI& AS/NZS Design Strength

FEA Strength

A’

B’

C’

t

L

S

PAISI&AS/NZS

PFEA

mm

mm

mm

mm

mm

mm

kN

kN

BU75-S225-L1133-0.75 BU75-S225-L1133-0.80 BU75-S225-L1133-0.85 BU75-S225-L1133-0.90 BU75-S225-L1133-0.95 BU75-S225-L1133-1.00 BU75-S225-L1133-1.05

75.0 75.0 75.0 75.0 75.0 75.0 75.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0

10.0 10.0 10.0 10.0 10.0 10.0 10.0

0.75 0.80 0.85 0.90 0.95 1.00 1.05

1133 1133 1133 1133 1133 1133 1133

225.0 225.0 225.0 225.0 225.0 225.0 225.0

20.3 21.9 23.5 25.0 26.6 28.1 29.6

BU75-S225-L1133-1.10

75.0

20.0

10.0

1.10

1133

225.0

BU75-S225-L1133-1.15

75.0

20.0

10.0

1.15

1133

225.0

BU75-S225-L1133-1.20

75.0

20.0

10.0

1.20

1133

225.0

BU75-S225-L1133-1.25

75.0

20.0

10.0

1.25

1133

225.0

BU75-S225-L1133-1.30

75.0

20.0

10.0

1.30

1133

BU75-S225-L1133-1.35

75.0

20.0

10.0

1.35

1133

BU75-S225-L1133-1.40

75.0

20.0

10.0

1.40

1133

BU75-S225-L1133-1.45

75.0

20.0

10.0

1.45

BU75-S225-L1133-1.50

75.0

20.0

10.0

1.50

BU75-S225-L1133-1.55

75.0

20.0

10.0

BU75-S225-L1133-1.60

75.0

20.0

10.0

BU75-S225-L1133-1.65

75.0

20.0

10.0

BU75-S225-L1133-1.70

75.0

20.0

BU75-S225-L1133-1.75

75.0

20.0

BU75-S225-L1133-1.80

75.0

20.0

BU75-S225-L1133-1.85

75.0

BU75-S225-L1133-1.90

75.0

BU75-S225-L1133-1.95

75.0

BU75-S225-L1133-2.00

75.0

BU75-S225-L1133-2.05

Specimen

T P

PAISI & AS/NZS / PFEA

-

23.4 25.4 27.4 29.4 31.3 33.1 34.9

0.87 0.86 0.86 0.85 0.85 0.85 0.85

31.1

36.7

0.85

32.6

38.4

0.85

34.0

40.3

0.85

35.3

42.1

0.84

36.3

43.8

0.84

225.0

37.1

45.6

0.83

225.0

37.9

47.4

0.81

225.0

38.6

49.2

0.80

1133

225.0

39.3

51.0

0.79

1133

225.0

40.0

52.7

0.77

1133

225.0

40.6

54.5

0.76

1.65

1133

225.0

41.2

56.3

0.75

1.70

1133

225.0

41.8

58.1

0.73

1.75

1133

225.0

42.3

59.9

0.72

10.0

1.80

1133

225.0

42.8

61.6

0.71

D E

T P 1.55 1.60

1133

A M

U N

225.0

C S

I R

20.0

E C 10.0

1.85

1133

225.0

43.3

63.4

0.70

20.0

10.0

1.90

1133

225.0

43.7

65.2

0.68

20.0

10.0

1.95

1133

225.0

44.2

67.0

0.67

20.0

10.0

2.0

1133

225.0

44.5

68.7

0.66

75.0

20.0

10.0

2.05

1133

225.0

44.9

70.5

0.65

BU75-S225-L1133-2.10

75.0

20.0

10.0

2.10

1133

225.0

45.2

72.3

0.64

BU75-S225-L1133-2.15

75.0

20.0

10.0

2.15

1133

225.0

45.5

74.1

0.63

BU75-S225-L1133-2.20

75.0

20.0

10.0

2.20

1133

225.0

45.8

75.8

0.61

BU75-S225-L1133-2.25

75.0

20.0

10.0

2.25

1133

225.0

46.0

77.6

0.60

C A

10.0 10.0

ACCEPTED MANUSCRIPT BU75-S225-L1133-2.30

75.0

20.0

10.0

2.30

1133

225.0

46.2

79.4

0.59

BU75-S225-L1133-2.35

75.0

20.0

10.0

2.35

1133

225.0

46.4

81.2

0.58

BU75-S225-L1133-2.40

75.0

20.0

10.0

2.40

1133

225.0

46.6

83.0

0.57

BU75-S225-L1133-2.45

75.0

20.0

10.0

2.45

1133

225.0

46.7

84.8

0.56

BU75-S225-L1133-2.50

75.0

20.0

10.0

2.50

1133

225.0

46.8

86.6

0.55

T P

0.73

Mean COV

I R

C S

U N

D E

T P

C A

E C

A M

0.11

ACCEPTED MANUSCRIPT

Flange

Lip

Thickness

Length

Spacing

AISI& AS/NZS Design Strength

FEA Strength

A’

B’

C’

t

L

S

PAISI&AS/NZS

PFEA

mm

mm

mm

mm

mm

mm

kN

kN

-

BU75-S474.5-L2184-0.75 BU75-S474.5-L2184-0.80 BU75-S474.5-L2184-0.85 BU75-S474.5-L2184-0.90 BU75-S474.5-L2184-0.95 BU75-S474.5-L2184-1.00 BU75-S474.5-L2184-1.05

75.0 75.0 75.0 75.0 75.0 75.0 75.0

20.0 20.0 20.0 20.0 20.0 20.0 20.0

10.0 10.0 10.0 10.0 10.0 10.0 10.0

0.75 0.80 0.85 0.90 0.95 1.00 1.05

2184 2184 2184 2184 2184 2184 2184

474.5 474.5 474.5 474.5 474.5 474.5 474.5

6.7 6.9 7.3 7.6 7.9 8.2 8.5

7.6 8.2 8.8 9.5 10.1 10.7 11.3

0.88 0.85 0.83 0.81 0.79 0.77 0.76

BU75-S474.5-L2184-1.10

75.0

20.0

10.0

1.10

2184

474.5

8.8

11.8

0.74

BU75-S474.5-L2184-1.15

75.0

20.0

10.0

1.15

2184

474.5

9.1

12.4

0.73

BU75-S474.5-L2184-1.20

75.0

20.0

10.0

1.20

2184

474.5

9.3

12.9

0.72

BU75-S474.5-L2184-1.25

75.0

20.0

10.0

1.25

2184

474.5

9.6

13.6

0.70

BU75-S474.5-L2184-1.30

75.0

20.0

10.0

1.30

2184

474.5

9.8

14.1

0.69

BU75-S474.5-L2184-1.35

75.0

20.0

10.0

1.35

2184

474.5

10.0

14.7

0.68

BU75-S474.5-L2184-1.40

75.0

20.0

10.0

1.40

2184

474.5

10.2

15.3

0.67

BU75-S474.5-L2184-1.45

75.0

20.0

10.0

1.45

2184

474.5

10.42

15.9

0.66

BU75-S474.5-L2184-1.50

75.0

20.0

10.0

1.50

2184

474.5

10.6

16.4

0.65

BU75-S474.5-L2184-1.55

75.0

20.0

10.0

1.55

2184

474.5

10.8

17.0

0.63

BU75-S474.5-L2184-1.60

75.0

20.0

10.0

1.60

2184

474.5

10.9

17.6

0.62

BU75-S474.5-L2184-1.65

75.0

20.0

10.0

1.65

2184

474.5

11.1

18.2

0.61

BU75-S474.5-L2184-1.70

75.0

20.0

10.0

1.70

2184

474.5

11.3

18.7

0.60

BU75-S474.5-L2184-1.75

75.0

20.0

10.0

1.75

2184

474.5

11.4

19.3

0.59

BU75-S474.5-L2184-1.80

75.0

20.0

10.0

1.80

2184

474.5

11.6

19.9

0.58

BU75-S474.5-L2184-1.85

75.0

20.0

10.0

1.85

2184

474.5

11.7

20.5

0.57

BU75-S474.5-L2184-1.90

75.0

20.0

10.0

1.90

2184

474.5

11.8

21.0

0.56

BU75-S474.5-L2184-1.95

75.0

20.0

10.0

1.95

2184

474.5

11.9

21.6

0.55

BU75-S474.5-L2184-2.00

75.0

20.0

10.0

2.0

2184

474.5

12.0

22.2

0.54

BU75-S474.5-L2184-2.05

75.0

20.0

10.0

2.05

2184

474.5

12.1

22.8

0.53

BU75-S474.5-L2184-2.10

75.0

20.0

10.0

2.10

2184

474.5

12.2

23.3

0.52

BU75-S474.5-L2184-2.15

75.0

20.0

10.0

2.15

2184

474.5

12.3

23.9

0.52

BU75-S474.5-L2184-2.20

75.0

20.0

10.0

2.20

2184

474.5

12.4

24.5

0.51

BU75-S474.5-L2184-2.25

75.0

20.0

10.0

2.25

2184

474.5

12.5

25.0

0.50

BU75-S474.5-L2184-2.30

75.0

20.0

10.0

2.30

2184

474.5

12.5

25.6

0.49

BU75-S474.5-L2184-2.35

75.0

20.0

10.0

2.35

2184

474.5

12.6

26.2

0.48

BU75-S474.5-L2184-2.40

75.0

20.0

10.0

2.40

2184

474.5

12.6

26.8

0.47

BU75-S474.5-L2184-2.45

75.0

20.0

10.0

2.45

2184

474.5

12.7

27.4

0.46

BU75-S474.5-L2184-2.50

75.0

20.0

10.0

2.50

474.5

12.7

27.9

0.45

2184

Mean

EP TE

D

COV

AC C

SC RI P

MA

Specimen

T

Web

NU

(d) Slender column PAISI & AS/NZS / PFEA

0.63 0.12

ACCEPTED MANUSCRIPT List of figures Fig. 1 Details of back-to-back built-up cold-formed steel channel-sections

(a) BU75 (b) BU90 Fig. 2 Back-to-back built-up cold-formed steel channel sections with an opening after Young & Zhang [6] Fig. 3 Specimen labelling Fig. 4 Photograph of test set-up

(a) Stub tests (b) Intermediate column tests Fig. 5 Details of imperfection measurements

(a) Photograph of imperfection measurements setup (b) LVDT measurement positions (c) Typical imperfection profile Fig. 6 Typical experimental and finite element results

T

Fig. 7 Failure modes observed during experimental tests

SC RI P

(a) Stub column

(b) Short column (c) Intermediate column (d) Slender column

Fig. 9 Boundary condition applied to the finite element model (BU75-S100-L500-1) Fig. 10 Initial imperfection contours (BU75-S100-L500-1)

MA

(a) Local buckling (b) Overall buckling

EP TE

D

Fig. 11 Failure modes of built-up sections

(a) Stub column (b) Short column (c) Intermediate column

NU

Fig. 8 Typical finite element mesh at failure (BU75-S100-L500-1)

(c) BU75 (d) BU90

AC C

Fig.12 Variation of strength against modified slenderness

Fig. 13 Comparison of FEA strength and design strength (AISI & AS/NZ Standards)

(a) BU75 (b) BU90

Fig.14 Comparison of FEA and AISI& AS/NZ strengths for built up column section BU75 with different thickness

ACCEPTED MANUSCRIPT

AC C

EP TE

D

MA

NU

SC RI P

T

(a)BU75

(b) BU90 All dimensions are in mm

Fig. 1: Details of back-to-back built-up cold-formed steel channel-sections

ACCEPTED MANUSCRIPT

AC C

EP TE

D

MA

NU

SC RI P

T

Fig.2: Back-to-back built-up cold-formed steel channel sections with an opening after Young & Zhang [6]

ACCEPTED MANUSCRIPT

AC C

EP TE

D

MA

NU

SC RI P

T

Fig. 3: Specimen labelling

ACCEPTED MANUSCRIPT

EP TE

D

MA

NU

SC RI P

T

(a) Stub column tests

(b) Intermediate column tests

AC C

Fig. 4: Photograph of test set-up

ACCEPTED MANUSCRIPT

SC RI P

T

(a) Photograph of imperfection measurement setup

AC C

EP TE

D

MA

NU

(b) LVDT measurement positions

(c) Typical imperfection profile for BU90-S200-L300-1 Fig. 5: Details of imperfection measurements

ACCEPTED MANUSCRIPT

200

180

160

140

100

80

60

Test BU 75-S50-L-300-1

40

FEA BU 75-S 50-L 300 1 20

T

Test BU 90-S50-L-300-1

SC RI P

FEA BU 90-S50-L-300-1

0 0

1

2

3

4

5

6

7

Displacement (mm)

EP TE

D

MA

NU

Fig. 6: Typical experimental and finite element results

AC C

Load (kN)

120

8

9

10

ACCEPTED MANUSCRIPT

BU90-S50-L300

BU75-S100-L300

BU90-S100-L300 (a) Stub column

BU75-S200-L300

AC C

EP TE

D

MA

NU

SC RI P

T

BU75-S50-L300

BU75-S100-L500

BU75-S200-L500 (b) Short column

BU75-S400-L500

BU90-S200-L300

BU75-S225-L1000

D

MA

NU

SC RI P

T

ACCEPTED MANUSCRIPT

AC C

EP TE

(c) Intermediate column

BU90-S225-L1000

AC C

EP TE

D

MA

NU

SC RI P

T

ACCEPTED MANUSCRIPT

BU75-S950-L2000

BU90-S950-L2000 (d) Slender column Fig. 7: Failure modes observed during experimental tests

ACCEPTED MANUSCRIPT

AC C

EP TE

D

MA

NU

SC RI P

T

Fig. 8: Typical finite element mesh at failure (BU75-S100-L500-1)

AC C

EP TE

D

MA

NU

SC RI P

T

ACCEPTED MANUSCRIPT

Fig. 9: Boundary condition applied to the finite element model (BU75-S100-L500-1)

SC RI P

T

ACCEPTED MANUSCRIPT

AC C

EP TE

D

MA

NU

(a) Local buckling (b) Overall buckling Fig. 10: Initial imperfection contours (BU75-S100-L500-1)

D

MA

NU

(i) Experimental (ii) FEA (i) Experimental (ii) FEA (b) Short column (c) Intermediate column Fig. 11: Failure modes of built-up sections

EP TE

(ii) FEA

AC C

(i) Experimental (a) Stub column

SC RI P

T

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

200

EXPERIMENTAL FEA

180

AISI & AS/NZS 160

Strength, P (kN)

140 120 100 80 60 40 20 0 50

100

150

200

250

300

350

T

0

400

SC RI P

Modified Slenderness(KL/r)m

(a) BU 75 200

180

NU

160

MA

120

D

100

EP TE

80

60

40

20

0 0

AC C

Strength , P(kN)

140

20

40

60

80

100

120

Modified Slenderness(KL/r)

(b) BU90 Fig.12: Variation of strength against modified slenderness

140

ACCEPTED MANUSCRIPT 200 180

AISI & AS/NZ Strength, P (kN)

160 140 120 100 80 60 40 20

AISI & AS/NZS /FEA = 1 FEA

0 0

20

40

60

80 100 120 FEA Strength, P (kN)

140

180 160

NU

140 120

MA

100 80

D

60

20 0 0

20

EP TE

40

40

AC C

AISI & AS/NZ Strength, P (kN)

200

SC RI P

200

180

T

(a) BU75

160

60

AISI & AS/NZS /FEA = 1 FEA 80 100 120 FEA Strength, P (kN)

140

160

180

(b) BU90

Fig. 13: Comparison of FEA strength and design strength (AISI & AS/NZ Standards)

200

ACCEPTED MANUSCRIPT

350.0

300.0

Strength, P(kN)

250.0

FEA, Length=273 AISI, Length=273 FEA, Length=655 AISI, Length=655 FEA, Length=1133 AISI, Length=1133 FEA, Length=2184 AISI, Length=2184

200.0

150.0

100.0

50.0

0.75

0.95

1.15

1.35

1.55

1.75

1.95

SC RI P

Thickness, t (mm)

T

0.0

2.15

2.35

AC C

EP TE

D

MA

NU

Fig.14: Comparison of FEA and AISI& AS/NZ strengths for built up column section BU75 with different thickness

ACCEPTED MANUSCRIPT

200

Proposed design rules AISI & AS/NZS Experimental FEA

180 160

120 100 80 60 40 20 0 50

100

150

200

250

Modified Slenderness(KL/r)m

300

350

T

0

EP TE

D

MA

NU

SC RI P

Fig. 15: Comparison of FEA strength and design strength (AISI & AS/NZ Standards)

AC C

Strength, P (kN)

140

400