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Nonlinear behaviour of back-to-back gapped built-up cold-formed steel channel sections under compression
Journal of Constructional Steel Research 147 (2018) 257–276
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Journal of Constructional Steel Research
Nonlinear behaviour of back-to-back gapped built-up cold-formed steel channel sections under compression Krishanu Roy a,⁎, Tina Chui Huon Ting b, Hieng Ho Lau b, James B.P. Lim a a b
Department of Civil and Environmental Engineering, The University of Auckland, New Zealand School of Engineering and Science, Curtin University Sarawak, Miri, Sarawak, Malaysia
a r t i c l e
i n f o
Article history: Received 18 January 2018 Received in revised form 29 March 2018 Accepted 10 April 2018 Available online xxxx Keywords: Gap Cold-formed steel Back-to-back gapped sections Buckling Direct strength method Axial strength
a b s t r a c t In cold-formed steel structures, such as trusses and portal frames, the use of back-to-back gapped built-up coldformed steel channel-sections for column members are becoming increasingly popular. In such an arrangement, the lowest flexural buckling mode may not necessarily be overall buckling of the whole column. In the literature, only three test results have been previously reported for such cold-formed steel columns, and limited to values of non-dimensional slenderness ranging from 1.08 to 1.16. This issue is considered herein. The results of 40 experimental tests are reported, conducted on back-to-back gapped built-up cold-formed steel channel-sections covering the range of non-dimensional slenderness from stub to slender columns. A nonlinear finite element model is then described that shows good agreement with the experimental results. The finite element model is then used for the purposes of a study comprising 84 models. Using the experimental and finite element results, it is shown that design in accordance with the American Iron and Steel Institute (AISI) and Australian and New Zealand Standards (AS/NZS) can be conservative by as much as 53%. However, use of a modification to the non-dimensional slenderness, that considers the gap, results in the design standards being within 5% conservative with respect to the experimental and finite element results. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction In this paper, the results of forty new experimental tests on back-toback gapped built-up cold-formed steel channel sections, with the sections acting as columns, are presented. Fig. 1 shows the details of gapped section investigated herein. As can be seen from Fig. 1, the gap is formed through a link-channel screwed between the webs of the back-to-back channel-sections. Such gaps are commonly introduced in
Abbreviations: A′, Overall web length of section; A1, Cross sectional area of single channel-section; Ae, Effective area of the section; B′, Overall flange width of section; C′, Overall lip width of section; CFS, Cold-formed steel; COV, Coefficient of variation; E, Young's modulus of elasticity; Fe, Least of the elastic flexural, torsional, and flexural torsional buckling stress; Fn, Nominal buckling stress as per AISI & AS/NZS; Fy, Yield stress which is equal to the 0.2% proof stress (σ0.2); FEA, Finite element analysis; I1, Moment of inertia of single channel-section; k, Effective length factor; l, Effective length of the builtup gapped section; L, Total length of the built-up gapped section; P, Applied axial load; PAISI& AS/NZS, Axial strength from AISI & AS/NZS; PDSM, Axial strength from Direct Strength Method; PEXP, Axial strength from experiments; PM-DSM, Axial strength from Modified Direct Strength Method; S, Longitudinal spacing of link-channels; w′, Gap between back-to-back channel-sections; λc, Non-dimensional slenderness ratio as per AISI & AS/NZS; λc,GAP, λc as for sections with gap; σ0.2, Static 0.2% proof stress. ⁎ Corresponding author at: Building 906, Level 4, Room 413, Newmarket campus, The University of Auckland, Auckland 1023, New Zealand. E-mail address: [email protected]. (K. Roy).
https://doi.org/10.1016/j.jcsr.2018.04.007 0143-974X/© 2017 Elsevier Ltd. All rights reserved.
struts in steel trusses and columns in portal frames, increasing the lateral stability of such columns. In the literature, only three such experimental results are available, as reported by Rondal and Niazi [1] in 1990; the values of non-dimensional slenderness in these tests ranged from 1.08 to 1.16. The forty new experimental tests reported herein have a value of non-dimensional slenderness ranging from 0.23 to 1.42, thus covering stub to slender columns. In current design standards, such as American Iron and Steel Institute [2] and Australian and New Zealand Standards AS/NZS 4600:2005 [3], the beneficial effect of the gap is ignored i.e. the design axial compressive strength is simply twice that of a single channel-section. This is the case regardless of whether the Effective Width Method (EWM) (reproduced in Section 2) or the Direct Strength Method (DSM) is used. It should be noted that the DSM does not include post-local-buckling capacity, however, Kumar and Kalyanaraman [4], modified the DSM equations, referred here as M-DSM, to include postlocal-buckling capacity. The axial strength calculated in accordance with EWM, DSM and M-DSM are all presented in this paper. Ting et al. [5] recently presented an experimental and numerical investigation on the behaviour of back-to-back built-up CFS channel sections under axial compression (see Fig. 2). The experimental tests reported herein, extends the work of Ting et al. [5]. As a result of the gap, for some combinations of column length and gap size, the lowest
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Fig. 2. Back-to-back built-up cold-formed steel channel columns without a gap as investigated by Ting et al. (2017).
(a) General arrangement (a) Mode A
(b) Cross section Fig. 1. Back-to-back gapped built-up cold-formed steel channel-sections.
(b) Mode B
Fig. 3. Overall flexural buckling modes of back-to-back gapped built-up cold-formed steel channel-sections.
K. Roy et al. / Journal of Constructional Steel Research 147 (2018) 257–276
(a) Built-up double Z members with a gap after Georgieva, Schueremans and Pyl (2012)
(b) Back to Back built-up cold-formed steel channel sections with an opening after Young & Zhang (2012) Fig. 4. Built-up cold-formed steel columns from literature.
Fig. 5. Nominal cross-sections of back-to-back gapped built-up cold-formed steel channel sections considered in this paper.
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flexural buckling mode may be as shown in Fig. 3a, as opposed to overall buckling of the whole column, taking into consideration of the gap, as shown in Fig. 3b. As mentioned previously, the design approach of the design standards ignores the beneficial presence of the gap. No other work in the literature has been reported for back-to-back gapped built-up CFS channel columns. While Georgieva, Schueremans and Pyl [6] considered built-up columns composed of zed-sections,
the gap considered was due to the zed-sections being connected toeto-toe (see Fig. 4a). Other work includes that of Zhang and Young [7] who considered back-to-back built-up channel-sections, but these were with an opening, not with a gap (see Fig. 4b). Dabaon et al. [8] investigated CFS built-up battened columns, while Stone and LaBoube [9] considered back-to-back channel-sections, which were flange stiffened and track sections. Whittle et al. [10] and Piyawat et al. [11] considered
Table 1 Measured specimen dimensions for built-up back-to-back gapped channel-sections. a) GBU75 Specimen
Web
Flange
Lip
Length
Spacing
Slenderness
Gap between columns
Experimental results
Finite element results
Comparison
Failure mode
A′
B′
C′
L
s
w′
λc
λC,GAP
PEXP
PFEA
PEXP/PFEA
–
(mm)
(mm)
(mm)
(mm)
(mm)
(mm)
–
–
(kN)
(kN)
–
–
Stub GBU75-S50-L300-1 GBU75-S50-L-300-2 GBU75-S50-L300-3 GBU75-S200-L300-1 GBU75-S200-L300-2 GBU75-S200-L300-3
73.2 73.6 73.6 73.6 73.6 72.3
19.8 19.9 19.8 19.7 19.7 18.5
11.2 11.1 11.1 11.1 11.2 10.5
271 270 268 268 271 263
50.0 50.1 50.2 201.0 200.0 199.0
73.2 73.6 73.6 73.6 73.6 72.3
0.38 0.39 0.38 0.42 0.42 0.41
0.26 0.27 0.26 0.29 0.29 0.28
122.6 120.5 121.9 114.7 114.3 117.9
126.3 124.2 126.8 119.2 119.9 119.2
1.03 1.03 1.04 1.04 1.05 1.01
Local Local Local Local Local Local
Short GBU75-S100-L500-1 GBU75-S100-L500-2 GBU75-S100-L500-3 GBU75-S400-L500-1 GBU75-S400-L500-2 GBU75-S400-L500-3
73.6 73.6 73.5 73.6 73.5 73.5
19.8 19.9 19.8 19.9 19.8 19.8
11.3 11.2 11.3 11.2 11.1 11.1
678 679 681 679 680 680
100.2 100.0 100.0 399.0 400.0 400.0
73.6 73.6 73.5 73.6 73.5 73.5
0.57 0.57 0.59 0.60 0.61 0.61
0.37 0.36 0.38 0.40 0.41 0.41
113.9 109.8 115.1 108.2 101.8 101.1
111.6 110.9 120.9 111.5 98.7 99.1
1.01 1.01 1.01 1.00 0.96 1.01
Mode A Mode A Mode A Local +Distortional Local +Distortional Local +Distortional
Intermediate GBU75-S225-L1000-1 GBU75-S225-L1000-2 GBU75-S225-L1000-3 GBU75-S900-L1000-1 GBU75-S900-L1000-2 GBU75-S900-L1000-3
76.1 76.2 75.8 75.9 76.0 76.0
19.8 20.3 19.8 19.8 19.9 19.8
10.4 10.4 10.4 10.4 10.3 10.3
1131 1132 1183 1133 1132 1182
225.0 225.0 224.8 900.0 897.5 900.0
76.1 76.2 75.8 75.9 76.0 76.0
0.89 0.88 0.92 0.98 0.99 1.00
0.75 0.74 0.77 0.83 0.85 0.86
77.1 76.2 73.2 65.3 61.5 61.2
79.4 77.7 74.6 63.3 60.3 59.3
1.01 1.00 0.96 1.01 1.01 1.01
Mode A Mode A Mode A Mode A Mode A Mode A
Slender GBU75-S475-L2000-1 GBU75-S475-L2000-2 GBU75-S475-L2000-3 GBU75-S1900-L2000-1 GBU75-S1900-L2000-2 GBU75-S1900-L2000-3
73.6 73.9 73.9 73.8 73.9 73.9
20.3 20.3 20.4 20.3 20.4 20.4
10.6 10.6 10.8 10.8 10.7 10.8
2183 2183 2184 2183 2183 2184
474.3 474.5 474.8 1901 1907 1902
73.6 73.9 73.9 73.8 73.9 73.9
1.36 1.36 1.36 1.41 1.42 1.42
1.23 1.23 1.23 1.26 1.27 1.26
25.6 25.2 25.7 21.3 20.9 21.6
24.6 24.5 24.7 20.9 20.3 20.8
1.04 1.03 1.04 1.02 1.03 1.04
Mode B Mode B Mode B Mode B Mode B Mode B
b) GBU90 Specimen
Web
Flange
Lip
Length
Spacing
Gap between columns
Slenderness
Experimental results
Finite element results
Comparison
Failure mode
A′
B′
C′4
L
S
w′
λc
λC,GAP
PEXP
PFEA
PEXP/PFEA
–
(mm)
(mm)
(mm)
(mm)
(mm)
(mm)
–
–
(kN)
(kN)
–
–
Stub GBU90-S50-L300-1 GBU90-S50-L300-2 GBU90-S200-L300-1 GBU90-S200-L300–2 GBU90-S200-L300-3
92.2 90.7 90.7 90.7 90.7
49.5 49.6 49.6 49.5 49.5
14.5 14.6 14.6 14.6 14.6
265 270 258 270 262
50.0 50.3 200.0 200.0 199.0
92.2 90.7 90.7 90.7 90.7
0.23 0.23 0.23 0.24 0.23
0.13 0.13 0.13 0.14 0.13
147.7 146.8 140.6 138.5 139.4
151.4 150.5 142.6 140.0 140.8
0.98 0.98 0.99 0.99 0.99
Local Local Local Local Local
Short GBU90-S100-L500-1 GBU90-S100-L500-2 GBU90-S100-L500-3 GBU90-S400-L500-1 GBU90-S400-L500-2 GBU90-S400-L500-3 GBU90-S400 L-500-4
90.5 90.6 88.9 90.5 90.5 90.3 90.6
49.4 49.4 49.4 49.5 49.5 49.5 49.4
14.6 14.6 15.5 13.4 14.6 14.7 14.6
680 680 680 680 680 680 680
100.5 100.5 100.3 400.0 400.0 400.0 401.0
90.5 90.6 88.9 90.5 90.5 90.3 90.6
0.41 0.41 0.41 0.42 0.45 0.45 0.45
0.23 0.23 0.24 0.26 0.27 0.28 0.27
132.5 133.7 133.2 129.7 130.4 129.5 130.6
135.9 135.9 135.1 130.1 131.8 131.7 132.9
0.97 0.98 0.99 1.00 0.99 0.98 0.98
Mode A Mode A Mode A Local +Distortional Local +Distortional Local +Distortional Local +Distortional
Intermediate GBU90-S225-L1000-1 GBU90-S225-L1000-2 GBU90-S900-L1000-1 GBU90-S900-L1000-2
90.5 89.8 90.6 90.6
49.7 48.5 49.6 49.7
14.4 13.7 14.4 14.4
1133 1183 1132 1183
225.3 224.8 892.0 900.0
90.5 89.8 90.6 90.6
0.85 0.86 0.86 0.87
0.64 0.65 0.66 0.67
86.5 87.5 82.9 83.3
84.3 84.5 80.2 81.8
1.03 1.04 1.03 1.02
Mode A Mode A Mode A Mode A
K. Roy et al. / Journal of Constructional Steel Research 147 (2018) 257–276
Fig. 6. Specimen labelling.
built-up channel-sections, but these were welded connection. A numerical investigation on built-up columns with battens and stiches are presented by Crisan et al. [12], where they have checked the accuracy of European and American standards, while calculating compressive strength of such built-up batten columns. Other works includes that of Fratamico et al. [13] and Anbarasu et al. [14] who considered back-toback built-up CFS channels, without any gap under compression. Fratamico et al. [13] investigated built-up columns and investigated the effect of screw spacing for back-to-back channels, again without any gap. On the other hand, Anbarasu et al. [14] investigated the behaviour and strength of cold-formed steel web stiffened built-up batten columns, without any gap. Effect of thickness on the behaviour of axially loaded back-to-back built-up cold-formed steel channel sections were studied by Roy et al. [15]. Fig. 5 shows the nominal geometry of the two sections considered in this paper: GBU75 and GBU90. Forty experimental test results are reported. All test specimens were brake-pressed from G550 structural steel sheets. The experimental tests were conducted for different lengths in combination with different link-channel spacing. The effect of gap, link-channel spacing, load-axial shortening, load-axial strain behaviour and buckling failure modes for different cross sections and
261
lengths of back-to-back gapped built-up columns has been investigated in this paper, none of which is available in the literature. The experimental test results are compared against the tests results of back-toback built-up CFS column with no gap by Ting et al. [5]. A finite element model is also described. The finite element model was used for the purposes of a study comprising 84 models. Good correlation was obtained when the test and finite element strengths are compared. Using the experimental and finite element results, it is shown that design in accordance with the American Iron and Steel Institute (AISI) and Australian and New Zealand Standards (AS/NZS) can be conservative by as much as 53%. However, use of a modification to the non-dimensional slenderness, that considers the gap, results in the design standards being within 5% conservative with respect to the experimental and finite element results. 2. Design rules as per AISI specification and AS/NZ standard In accordance with AISI & AS/NZ standards, for built-up sections, the design axial compressive strength is simply twice that of a single channel-section, the un-factored design strength of a single channelsection is as follows: PAISI&AS=NZS ¼ Ae Fn
ð1Þ
The nominal buckling stress (Fn) can be calculated from: λc ≤1:5 : Fn ¼ 0:658 λc 2 Fy
Fig. 7. Drawing of loading test-rig.
ð2Þ
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where,
λc ¼
sffiffiffiffiffiffi Fy Fe
ð4Þ
3. Experimental investigation 3.1. Test sections Fig. 5 shows detailed cross-sections of the two-back-to-back gapped built-up CFS channel columns considered in this paper: GBU75 and GBU90. As indicated by the names, GBU75 and GBU90 are built-up from C75 and C90 channel-sections, respectively. In this paper, the link-channels through which the gap is formed using the same channel-section as that of the built-up section i.e. C75 and C90 channel-sections are used for the link-channels in GBU75 and GBU90, respectively. Table 1 summarises the measured column dimensions. The experimental test programme is comprised of 40 specimens, subdivided into four different column heights: 300 mm, 500 mm, 1000 mm, and 2000 mm. All columns with pin-ended, other than the 300 mm columns which were tested as fixed-ended columns. In Table 1, the test specimens have been sub-divided into stub, short, intermediate and slender columns. In the experimental test programme, the following longitudinal spacing of link-channels (S) were considered:
(a) Stub tests
• • • •
Column height of 300 mm; spacing of 50 mm and 200 mm Column height of 500 mm; spacing of 100 mm and 400 mm Column height of 1000 mm; spacing of 225 mm and 900 mm Column height of 2000 mm; spacing of 475 mm and 1900 mm
3.2. Material testing In order to determine the material properties of test sections, tensile coupon tests were conducted. The tensile coupons were prepared from the centre of the web plate, in accordance with British Standard for Testing and Materials [16]. Five of each coupons were obtained from longitudinal and transverse direction of the untested specimens. MTS test machine was used to test the coupons. To determine the tensile strain, two strain gauges and a calibrated extensometer of 50 mm gauge length were used. The average modulus of elasticity was 205GPa and yield strength was 565 MPa.
(b) Intermediate column tests
3.3. Section labelling The sections were labelled such that the type of section, longitudinal link channel spacing, nominal length of section and specimen number were indicated by the label. As shown in Fig. 6, the label “GBU75-S50L300-1” are explained as follows: • “GBU75” indicates back-to-back built-up channels with a gap of 75 mm web depth. • “S50” indicates the intermediate link-channel spacing as 50 mm. • “L300” indicates nominal length of specimen as 300 mm. • “1” indicates the specimen number as 1.
(c) Position of LVDT for shortening
3.4. Test-rig and testing procedure
Fig. 8. Photograph of loading test-rig.
λc N1:5; Fn ¼
0:877 λ2 c
Fy
ð3Þ
A schematic drawing of the test setup is shown in Fig. 7. The external load cell was placed at the top of the column. Six LVDTs were used, the position of LVDTs is shown in Fig. 8. To measure the strain, two longitudinal strain gauges along with three other strain gauges were used on the web of the columns. Strain gauge arrangements are shown in Fig. 9. A Universal Testing Machine (UTM) GT-7001-LC60, of 600 kN
K. Roy et al. / Journal of Constructional Steel Research 147 (2018) 257–276
Fig. 9. Position of strain gauges.
Fig. 10. Photograph of imperfection measurements setup.
capacity, was used to apply axial load to the columns. The displacement control was used to apply the axial force to the columns, which can include the post buckling behaviour of the columns. Displacement rate was kept as 0.03 mm/s for all test specimens. Strain values were recorded from longitudinal strain gauges near the top end plate and middle of the columns to verify that the load was applied through the centroid of the sections. In order to ensure, there is no gap between the two pin-ends and end plates of the specimen, all columns were loaded initially up to 25% of their expected failure load and then released. LVDT and strain gauge readings were recorded with each increment of loading. 3.5. Measurement of initial geometric imperfections Initial geometric imperfections were measured as shown in Fig. 10. A LVDT with an accuracy of 0.01 mm, was used to record the readings at
Fig. 11. Position of LVDTs to measure imperfections.
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every 20 mm along the length of the sections. The imperfections were measured at the centre of the web, flanges, and edge of the lips for all sections. LVDT positions are shown in Fig. 11. In Fig. 12, a typical length-imperfection plot is shown for GBU75-S100-L500-1. The maximum imperfections of the test specimens were 1.8 mm, 1.7 mm, 1.6 mm and 1.9 mm for the 300 mm, 500 mm, 1000 mm and 2000 mm column lengths, respectively.
3.6. Experimental results
Fig. 12. Typical imperfection profile (GBU75-S100-L500-1).
(i) GBU75-S50-L300
Column dimensions and the experimental failure loads (PEXP) are shown in Table 1. Also shown in Table 1, are the buckling modes. As can be seen, strength of the built-up sections were reduced significantly for all columns beyond 1000 mm length for both GBU75 and GBU90 with two intermediate link-channels. On the other hand, for GBU75 with no intermediate link-channel, significant reduction in axial strength was observed for column length higher than 500 mm. This is because the intermediate link-channels holds the individual back-toback channels together. Failure modes were different for stub, short, intermediate and slender columns. In total, 16 stub columns were tested (see Table 1). GBU75-S50L300 and GBU90-S50-L300 test specimens with three link-channels spaced at 50 mm, failed through local buckling. Back-to-back channels remain integral at failure, showing some plastic deformation near the bottom of the stub columns as shown in Fig. 13(a). The failure modes of GBU75-L300 series, were different from the BU75-L300 series (Ting et al. [5]) because the link-channels in GBU75-L300 provided sufficient restraint to prevent buckling at mid-height of the column. Unlike BU75-L300, for GBU90-L300 series columns, localized deformation was observed near the top or bottom end of the columns. GBU75-S100-L500 with three intermediate link-channels at 100 mm spacing buckled as Mode A, leading to a hinge-like angular buckling shape at about one-third height, near to the top of the column. The buckling shape is shown in Fig. 13(b). Both short (500 mm length) and intermediate (1000 mm length) columns showed similar buckling
(ii) GBU90-S50-L300 (a) Local buckling
Fig. 13. Photograph of columns at failure for different modes.
K. Roy et al. / Journal of Constructional Steel Research 147 (2018) 257–276
(i) GBU75-S100-L500
265
(ii) GBU75-S225-L1000 (b) Mode-A Fig. 13 (continued).
shapes, although 1000 mm long columns showed clearer deformed shapes (see Fig. 15(b)). Local and distortional buckling were not noticeable for slender columns. Overall buckling was observed for slender columns of both GBU75 and GBU90 series. Localized deformation was noticeable on the compression side of specimens near the mid-height of the column as shown in Fig. 13(c). For Mode B, GBU75-S475-L2000 uses four linkchannels while GBU75-S1900-L2000 uses two link-channels. Load-axial shortening behaviour for GBU75-S225-L1000-1 is plotted in Fig. 14. It is observed that the relationship was almost linear up to a load of 53 kN, which is approximately 69% of the ultimate failure load for GBU75-S225-L1000-1. After that, nonlinear behaviour is continued until the failure load is reached, which is 77.09 kN.
Five strain gauges were used to determine axial strain at mid-length and end of all columns. Load-axial strain relationships for GBU75-S50-L300-1 and GBU90-S50-L300-1 are plotted in Fig. 15 (a) and (b) respectively. S-E denotes the axial strain at the end of column, where S-M is the strain at the middle of the column (see Fig. 9). Maximum compressive micro-strain observed at failure load, as shown in Fig. 15(a), is 812 and 987 in S-E and S-M points respectively for GBU75-S50-L300-1 column. Similarly, as shown in Fig. 15(b), for GBU90-S50-L300-1, axial strain values obtained at failure load is 1050 and 1207 respectively for S-E and S-M points i.e. at the end and at middle height of columns. The effect of increase in the vertical spacing of link-channels, are investigated and shown in Table 1. As can be seen from Table 1(a), the
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(i) GBU75-S475-L2000
(ii) GBU75-S1900-L2000 (c) Mode B Fig. 13 (continued).
average strength of stub column of GBU75 series is decreased by 5%, when the vertical spacing between the link-channels is increased from 50 mm to 200 mm. For short columns, when the link channel spacing is increased from 100 mm to 400 mm, the axial strength of the gapped built-up columns is decreased by 5.8% on average. For intermediate columns, the average decrease in strength is by around 17% when increasing the spacing from 225 mm to 900 mm for
GBU75 columns. Reduction in strength for slender columns is approximately by 16% for GBU75, when the screw spacing is increased from 475 mm to 1900 mm. On the other hand, for GBU90 short columns, increasing the vertical spacing of link-channels from 100 mm to 400 mm, reduces the axial strength by 2.3%. For short GBU90 columns, the axial strength is reduced by 4.8%, when the spacing is increased from 100 mm to 400 mm.
K. Roy et al. / Journal of Constructional Steel Research 147 (2018) 257–276
267
140
90
80
120 70
100
50
Load(kN)
Load (kN)
60
40
80
60
30
40
20
10
TEST-GBU75-S225-L1000-1 20
FEA-GBU75-S225-L1000-1 0 0
1
2
3
4
5
6
7
8
9
10
Displacement (mm)
0 0
200
400
Fig. 14. Comparison of finite element and experimental results.
600
800
1000
1200
1400
Axial Strain
(a) GBU75S50L300-1
4. Numerical investigation
160
4.1. General 140
4.2. Analysis procedure
120
100
Load(kN)
A nonlinear elasto-plastic finite element model was developed using ABAQUS [17]. The finite element model was based on the centre line dimensions of the cross-sections. In the model, the measured crosssectional dimensions were used. Specific modelling techniques are described below.
80
60
The full geometry was modelled for the test specimens. Screws between the link-channels and the web of back-to-back channel sections were idealised by connector elements. Axial load was applied through the centre of gravity (CG) of the specimens. The columns were loaded in displacement control, with the displacement applied at a centroid node. The displacement control load was applied using the RIKS algorithm available in ABAQUS library.
40
20
Strain Strain Strain Strain
at at at at
the Edge(Exp) middle(Exp) the Edge(FEA) middle(FEA)
0 0
200
400
600
800
1000
1200
1400
Axial Strain
(b) GBU90S50L300-1
4.3. Geometry and material modelling The whole geometry of the built-up columns was modelled. Classical metal plasticity model available in ABAQUS library was used for all the finite element analyses. The classical metal plasticity model implements the von Mises yield surface to define isotropic yielding, associated plastic flow theory and isotropic hardening behaviour. Yield strength of 565 MPa and ultimate stress of 690 MPa, along with Young's modulus of 205 GPa, was used in numerical modelling. The mean mechanical properties conducted from the tensile tests were also used for engineering stress-strain curve. As per the ABAQUS manual [16], the engineering material curve is converted into a true material curve by following the equation below: σ true ¼ σ ð1 þ εÞ
εtrueðplÞ ¼ ln ð1 þ εÞ−
ð5Þ σ true E
ð6Þ
where E is the Young's modulus, σtrue is the true stress, σu is the tensile ultimate strength, σ and ε are the engineering stress and strain respectively in ABAQUS [16].
Fig. 15. Load versus axial-strain for GBU75 and GBU90 columns.
4.4. Type of element and finite element mesh The channel sections and the link-channels were modelled using the linear 4-node quadrilateral thick shell element S4R. The S4R element has six degrees of freedom per node. The upper and lower end plates were modelled using rigid quadrilateral shell elements (R3D4). An element size of 5 mm by 5 mm was found to be appropriate for back to back channels and link-channels, based on the results of a convergence study. The number of elements along the length of the sections were chosen, such that the aspect ratio of the elements are as close to one. A typical finite element mesh is shown in Fig. 16.
4.5. Boundary conditions and load application The back-to-back gapped built-up CFS columns investigated in this study were pin-ended columns other than the stub column which was fixed-fixed. In order to simulate the upper and lower pin-end supports, the displacements and rotations (boundary conditions) were assigned
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Fig. 16. Typical finite element mesh (GBU75-S100-L500).
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. 17. The distance between two reference points are the effective length of each specimen between the two hinged supports. The boundary conditions were assigned to both reference points. MPC beam connector element available in ABAQUS library was used to model the
screw connections between the link channel and web of back to back channels. Connector elements were assigned a stress of 62.10 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, in general, can predict unstable and nonlinear collapse of a structure such as post buckling behaviour.
Fig. 17. Applied boundary conditions (section GBU90S100L500).
K. Roy et al. / Journal of Constructional Steel Research 147 (2018) 257–276
4.6. Modelling of local and overall geometric imperfections Geometric imperfections are caused in columns as a result of fabrication and transportation processes. Hence, local and overall buckling modes are superimposed for accurate finite element analysis. Eigenvalue analyses of the built-up columns were performed with very small to large channel thickness to determine the contours for the local and global imperfections. The lowest buckling mode (Eigenmode 1) in ABAQUS, is used as the shape of local and overall buckling mode. This technique is used to model the initial local and overall geometric imperfections of the gapped built-up columns. The imperfections were scaled to the values given in the experimental Section 3.5. Besides, local imperfections of magnitude 0.5% of the channel thickness was incorporated as recommended by Ellobody and Young [18]. In Fig. 18, local and overall buckling modes obtained for GBU75-S100-L500, are shown. 4.7. Finite element model validation The comparison of failure loads from laboratory and the finite element analysis for GBU75 and GBU90 columns, are shown in Table 1. Close comparison is obtained between the tests and finite element results in terms of axial strength and failure modes. Fig. 19 shows Mode
269
A and Mode B obtained from the finite element analysis, which has good agreement with experimental failure modes (see Table 1). Fig. 14 compares the load-displacement curves from experimental tests and FEA for GBU75-S225-L1000-1. Also shown in Fig. 15(a) and (b), is the comparison of axial strain obtained numerically and experimentally for GBU75-S50-L300-1 and GBU90-S50-L300-1 respectively. As can be seen, good agreement is shown between experimental and numerical values of strain measured at the edge and middle of the columns. 5. Comparison with design standards Table 2 compares the experimental strengths with the design strengths calculated in accordance with AISI and AS/NZS, DSM and M-DSM. Fig. 20(a) shows the experimental and finite element strengths of the GBU75 columns plotted against length. For ease of reference, the experimental strengths of the back-to-back built-up channel-sections after Ting et al. [5] are also shown. As can be seen, the presence of the gap has increased the strength of the column by around 29%, when the column length is approximately 700 mm. As can be expected, for the stockier and slenderer sections, the increase in strength is less. Also shown in Fig. 20(a) are the design strengths calculated in accordance with AISI and AS/NZS. Two curves are shown. The lower curve is
(a) Local buckling
(b) Overall buckling Fig. 18. Initial imperfection contours (GBU75-S100-L500).
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the strength of the built-up back to back channel-section calculated using λc. As can be seen, this curve is very conservative to both the experimental and finite element strengths by as much as 53%. This can be expected since the beneficial effect of the gap is ignored i.e. the design axial compressive strength is simply twice that of a single channel-section. A theoretical equation for the elastic critical bucking load (P), that takes into account the gap, is as follows [19]: klsinkl þ
I −2 ð1−cosklÞ ¼ 0 I0
ð7Þ
where, 2
k ¼
P 2EI1
ð8Þ
(i) Experimental
Solving the above equations numerically for P, leads to a value for the non-dimensional slenderness, λc, GAP. In Fig. 20(a), the upper curve is calculated using the value of λc,GAP. As can be seen, use of λc,GAP, results in the AISI and AS/NZS being close to the experimental test results, being within 5% conservative to the design standards. For ease of reference, the DSM and M-DSM curves are also shown, again with use of the value of λc,GAP. As can be seen, both DSM and M-DSM are close to the AISI and AS/NZS curves, albeit slightly lower. The same trends can be seen in Fig. 20(b) for the GBU90 columns. Fig. 21(a) and (b) shows the same strength of the GBU75 and GBU90 columns, respectively, but for varying values of non-dimensional slenderness, λc. As mentioned previously, Rondal and Niazi [1], tested three gapped built-up columns having values of λc ranging from 1.08 to 1.16. The range λc covered in this paper is from 0.2 to 1.42, thus providing additional points to Rondal and Niazi [1]. In both Figs. 20 and 21, the value of λc where Mode A changes to Mode B is shown.
(ii) Numerical (a) Mode A (GBU75-S100-L500) Fig. 19. Failure mode of columns at failure from finite element analysis.
(iii) Close up
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(i) Experimental
(ii) Numerical
271
(iii) Close up
(b) Mode B (GBU75-S475-L2000) Fig. 19 (continued).
Fig. 22(a) and (b) show the experimental strengths and design strengths of the GBU75 and GBU90 columns, respectively, when the value of λc,GAP, as proposed by authors is used. As can be seen, good agreement is shown between the experimental strengths and the design strengths.
6. Conclusions A detailed experimental investigation on the axial strength of backto-back gapped built-up CFS channel-sections is presented in this paper. A total of 40 tests are reported. The failure modes, load-axial shortening,
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Table 2 Comparison of experimental and design strengths. a) GBU75 Specimen
Experimental results AISI
Stub GBU75-S50-L300-13 GBU75-S50-L300-2 GBU75-S50-L300-3 GBU75-S200-L300-1 GBU75-S200-L300-2 GBU75-S200-L300-3 Mean COV Short GBU75-S100-L500-1 GBU75-S100-L500-2 GBU75-S100-L500-3 GBU75-S400-L500-1 GBU75-S400-L500-2 GBU75-S400-L500-3 Mean COV Intermediate GBU75-S225-L1000-1 GBU75-S225-L1000-2 GBU75-S225-L1000-3 GBU75-S900-L1000-1 GBU75-S900-L1000-2 GBU75-S900-L1000-3 Mean COV Slender GBU75-S475-L2000-1 GBU75-S475-L2000-2 GBU75-S475-L2000-3 GBU75-S1900-L2000-1 GBU75-S1900-L2000-2 GBU75-S1900-L2000-3 Mean COV
AISI &AS/NZS using eq-8 DSM using eq-8 MDSMusing eq-8 Comparison
PEXP (kN)
PAISI (kN) PAISI-eq-8 (kN)
PDSM (kN)
PM-DSM
PEXP/PAISI PEXP/PAISI-eq-8 PEXP/PDSM PEXP/PM-DSM
122.63 120.57 121.86 114.64 114.25 117.98
112.89 112.34 111.70 105.43 104.88 105.72
120.50 119.85 119.31 114.74 114.45 116.14
116.51 115.37 115.70 112.10 110.30 110.93
117.94 117.34 118.69 111.31 111.98 112.64
1.09 1.07 1.09 1.09 1.09 1.12 1.09 0.012
1.02 1.01 1.02 1.00 1.00 1.02 1.01 0.011
1.05 1.05 1.05 1.02 1.04 1.06 1.05 0.014
1.04 1.03 1.03 1.03 1.02 1.05 1.03 0.013
113.88 109.82 115.11 108.21 101.78 109.07
67.30 65.18 69.72 63.90 61.43 62.80
113.30 109.18 114.31 108.25 105.56 108.15
101.26 100.49 101.61 94.87 94.15 94.34
112.00 109.12 112.23 108.10 108.02 108.35
1.69 1.68 1.65 1.69 1.66 1.74 1.69 0.031
1.01 1.01 1.01 1.00 0.96 1.01 1.00 0.017
1.12 1.09 1.13 1.14 1.08 1.16 1.12 0.029
1.02 1.01 1.03 1.00 0.94 1.01 1.01 0.030
77.09 76.21 73.15 65.29 61.52 61.18
42.62 42.38 45.79 31.98 31.39 30.80
76.62 76.31 76.12 64.72 60.87 60.38
60.58 60.03 59.75 50.13 49.73 49.42
66.74 66.51 66.10 58.33 57.34 57.17
1.81 1.80 1.60 2.04 1.96 1.99 1.87 0.164
1.01 1.00 0.96 1.01 1.01 1.01 1.01 0.020
1.27 1.27 1.22 1.30 1.24 1.24 1.26 0.029
1.16 1.15 1.11 1.12 1.07 1.07 1.11 0.036
24.57 24.48 24.67 20.97 20.27 20.76
11.77 11.23 11.45 10.42 10.27 10.87
23.17 22.88 22.97 19.68 20.21 19.81
18.75 18.44 18.62 16.38 16.43 16.32
21.07 21.57 21.70 18.28 17.51 18.47
2.09 2.18 2.15 2.01 1.97 1.91 2.05 0.106
1.06 1.07 1.07 1.07 1.00 1.05 1.05 0.026
1.31 1.33 1.33 1.28 1.23 1.27 1.29 0.036
1.17 1.13 1.14 1.15 1.16 1.12 1.14 0.016
(b) GBU90 Specimen
Stub GBU90S50L300-1 146.84 140.56 138.47 139.0.42 GBU90S50L300-2 GBU90S200L300-1 GBU90S200L300-2 GBU90S200L300-3 Mean COV Short GBU90S100L500-1 GBU90S100L500-2 GBU90S100L500-3 GBU90S400L500-1 GBU90S400L500-2 GBU90S400L500-3 GBU90S400L500-4 Mean COV Intermediate GBU90S225L1000-1 GBU90S225L1000-2 GBU90S900L1000-1
Experimental Results
AISI
AISI& AS/NZS using eq-8
DSM using eq-8
M-DSM using eq-8
Comparison
PEXP (kN)
PAISI (kN)
PAISI-eq-8 (kN)
PM-DSM (kN)
PM-DSM (kN)
PEXP/PAISI
PEXP/PAISI-eq-8
PEXP/PDSM
PEXP/PM-DSM
145.20 137.88 129.01 129.89 130.71
140.34 144.33 137.24 137.93 138.02
142.12 139.79 132.68 131.98 132.21
1.07 141.18 135.50 134.94 135.06
1.02 1.06 1.09 1.07 1.07 1.07 0.011
1.05 1.02 1.02 1.00 1.01 1.01 0.008
1.04 1.05 1.06 1.05 1.05 1.05 0.004
1.04 1.04 1.03 1.03 1.03 0.006
132.48 133.68 133.24 129.73 130.40 129.52 130.61
107.34 107.90 107.87 105.44 105.96 104.77 105.12
130.62 130.84 130.10 127.96 128.08 127.20 127.77
123.07 122.22 122.51 118.82 120.63 120.18 120.43
125.97 126.16 128.54 126.00 126.50 125.70 127.20
1.23 1.24 1.24 1.23 1.23 1.24 1.24 1.24 0.004
1.01 1.02 1.02 1.01 1.02 1.02 1.02 1.02 0.004
1.08 1.09 1.09 1.09 1.08 1.08 1.08 1.08 0.007
1.05 1.06 1.04 1.03 1.03 1.03 1.03 1.04 0.013
86.54 87.45 82.92
59.21 60.33 57.10
81.40 82.97 78.92
67.85 68.01 63.87
74.62 75.98 70.45
1.46 1.45 1.45
1.06 1.05 1.05
1.28 1.29 1.30
1.16 1.15 1.18
147.66
138.31
K. Roy et al. / Journal of Constructional Steel Research 147 (2018) 257–276
273
Table 2 (continued) (b) GBU90 AISI
AISI& AS/NZS using eq-8
DSM using eq-8
M-DSM using eq-8
Comparison
PAISI (kN)
PAISI-eq-8 (kN)
PM-DSM (kN)
PM-DSM (kN)
PEXP/PAISI
PEXP/PAISI-eq-8
PEXP/PDSM
PEXP/PM-DSM
83.34
58.54
79.80
64.89
71.88
1.42 1.45 0.016
1.04 1.05 0.008
1.28 1.29 0.005
1.16 1.16 0.011
160
140
120
100
Strength, P (kN)
GBU90S900L1000-2 Mean COV
Experimental Results PEXP (kN)
Mode-A
Mode-B
80
60
40
20
0 0
200
400
600
800
1000
1200
1400
1600
1800
2000
Length, L (mm)
(a) GBU75 160
140
120
100
Strength, P (kN)
Specimen
Mode-A
Mode-B
80
60
40
20
0 0
200
400
600
800
1000
1200
Length, L (mm)
(b) GBU90 Fig. 20. Varition of strength against length.
1400
1600
1800
2000
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K. Roy et al. / Journal of Constructional Steel Research 147 (2018) 257–276 160
140
120
Strength, P (kN)
100
Mode-A
Mode-B
80
60
40
20
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Non-dimensional slenderness (λ c)
(a) GBU75 160
140
120
Mode A
Mode B
Strength, P (kN)
100
80
60
40
20
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Non-dimensional slenderness (λ c)
(b) GBU90 Fig. 21. Variation of strength against λc.
load-axial strain and deformed shapes at failure is discussed. Effect of the gap on axial strength of columns are investigated. Also investigated the effect of link-channel spacing on the axial strength of back-to-back gapped built-up columns. Different buckling modes at failure are discussed as well. The results are compared against current AISI and AS/NZS standard. A nonlinear finite element model is then described, which includes material nonlinearity and geometric imperfections. Finite element
model includes the modelling of link-channels and fasteners. Comparison of test results against numerical models have shown good agreement. The column strengths are compared against the design strengths calculated using the AISI & AS/NZS, Direct Strength Method and Modified Direct Strength Method. Both the test and FEA results were as much as 53% higher than the design standards when nondimensional slenderness (λc) was used to calculate design capacity
K. Roy et al. / Journal of Constructional Steel Research 147 (2018) 257–276
275
160
AISI & AS/NZ( using eq 1) Strength, P (kN)
140
120
100
80
60
40
20
AISI & AS/NZS /Test = 1 Test strength
0 0
20
40
60
80
100
120
140
160
Test Strength, P (kN)
(a) GBU75 160
AISI & AS/NZ( using eq 1) Strength, P (kN)
140
120
100
80
60
40
20 AISI & AS/NZS /Test strengths = 1 Test strengths 0 0
20
40
60
80
100
120
140
160
Test Strength, P (kN)
(b) GBU90 Fig. 22. Comparison of test strength and design strength (AISI & AS/NZS (using eq. 8)).
of such columns. However, the design standards were conservative by only 5% on average to the experimental and FEA strengths, when λc,GAP was used. Hence it is recommended to use λc,GAP while calculating the axial strength of back-to-back gapped built-up CFS columns.
References [1] J. Rondal, M. Niazi, Stability of built-up beams and columns with thin-walled members, J. Constr. Steel Res. 16 (1990) 329–335. [2] American Iron and Steel Institute (AISI), North American Specification for the Design of Cold-formed Steel Structural Members, AISI, 2007 S100–07.
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[3] Australia/New Zealand Standard (AS/NZS), Cold-Formed Steel Structures, AS/NZS 4600:2005, Standards Australia/Standards New Zealand, 2005. [4] M.V.A. Kumar, V. Kalyanaraman, Design strength of locally buckling stub lipped channel columns, J. Struct. Eng. ASCE 138 (2012) 1291–1299. [5] T.C.H. Ting, K. Roy, H.H. Lau, J.B.P. Lim, Effect of screw spacing on behavior of axialy loaded back-to-back cold-formed steel built-up channel sections, Adv. Struct. Eng. 21 (3) (2017) 474–487. [6] I. Georgieva I, L. Schueremans, L. Pyl, Composed columns from cold-formed steel Z-profiles:experiments and code based predictions of the overall compression capacity, Eng. Struct. 37 (2012) 125–134. [7] J.H. Zhang, B. Young, Compression tests of cold-formed steel I-shaped open sections with edge and web stiffeners, Thin-Walled Struct. 52 (2012) 1–11. [8] M. Dabaon, E. Ellobody, K. Ramzy, Nonlinear behavior of built-up cold-formed steel section battened columns, J. Constr. Steel Res. 110 (2015) 16–28. [9] T.A. Stone, R.A. LaBoube, Behaviour of cold-formed steel built-up I-sections, Thin-Walled Struct. 43 (2015) 1805–1817. [10] J. Whittle, C. Ramseyer, Buckling capacities of axially loaded, cold-formed, built-up channels, Thin-Walled Struct. 47 (2009) 190–201. [11] K. Piyawat, C. Ramseyer, T.H.K. Kang, Development of an axial load capacity equation for doubly symmetric built-up cold-formed sections, J. Struct. Eng. ASCE 139 (12) (2013) 04013008–04013013. [12] A. Crisan, V. Ungureanu, D. Dubina, Calibration of design formula for buckling strength of built-up back-to-back cold-formed steel members in compression,
[13]
[14]
[15]
[16] [17] [18] [19]
Proceedings of the ICTWS 2014 7th International Conference on Thin Walled Structurees, 28th September to 2nd October 2014, Busan, Korea, ICTWS, 2014. D.C. Fratamico, S. Torabian, K.J.R. Rasmussen, B. Schafer, Experimental studies on the 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, 2016. M. Anbarasu, K.P. Bharath, S. Sukumar, Study on the capacity of cold-formes steel built-up battened colums under axial compression, Lat. Am. J. Solids Struct. 11 (2014) (2271-1375). K. Roy, T.C.H Ting, H.H. Lau, J.B.P. Lim, Effect of thickness on the behaviour of axially loaded back-to-back cold-formed steel built-up channel sections. 12th International Conference on ‘Advances in Steel-Concrete Composite Structures’-ASCCS 2018, 27– 29 June, 2018,Valencia, Spain. (Accepted). BS EN, Tensile Testing of Metallic Materials Method of Test at Ambient Temperature, British Standards Institution, 2001. ABAQUS Analysis User's Manual-Version 6.14-2. , ABAQUS Inc., USA, 2014. E. Ellobody, B. Young, Behavior of cold formed steel plain angle columns, J. Struct. Eng. ASCE 131 (3) (2005) 457–466. B.G. Johnston, Spaced steel columns, J. Struct. Div., Proc. Am. Soc. Civ. Eng. 97 (1971) 1465–1479.
Contents lists available at ScienceDirect
Journal of Constructional Steel Research
Nonlinear behaviour of back-to-back gapped built-up cold-formed steel channel sections under compression Krishanu Roy a,⁎, Tina Chui Huon Ting b, Hieng Ho Lau b, James B.P. Lim a a b
Department of Civil and Environmental Engineering, The University of Auckland, New Zealand School of Engineering and Science, Curtin University Sarawak, Miri, Sarawak, Malaysia
a r t i c l e
i n f o
Article history: Received 18 January 2018 Received in revised form 29 March 2018 Accepted 10 April 2018 Available online xxxx Keywords: Gap Cold-formed steel Back-to-back gapped sections Buckling Direct strength method Axial strength
a b s t r a c t In cold-formed steel structures, such as trusses and portal frames, the use of back-to-back gapped built-up coldformed steel channel-sections for column members are becoming increasingly popular. In such an arrangement, the lowest flexural buckling mode may not necessarily be overall buckling of the whole column. In the literature, only three test results have been previously reported for such cold-formed steel columns, and limited to values of non-dimensional slenderness ranging from 1.08 to 1.16. This issue is considered herein. The results of 40 experimental tests are reported, conducted on back-to-back gapped built-up cold-formed steel channel-sections covering the range of non-dimensional slenderness from stub to slender columns. A nonlinear finite element model is then described that shows good agreement with the experimental results. The finite element model is then used for the purposes of a study comprising 84 models. Using the experimental and finite element results, it is shown that design in accordance with the American Iron and Steel Institute (AISI) and Australian and New Zealand Standards (AS/NZS) can be conservative by as much as 53%. However, use of a modification to the non-dimensional slenderness, that considers the gap, results in the design standards being within 5% conservative with respect to the experimental and finite element results. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction In this paper, the results of forty new experimental tests on back-toback gapped built-up cold-formed steel channel sections, with the sections acting as columns, are presented. Fig. 1 shows the details of gapped section investigated herein. As can be seen from Fig. 1, the gap is formed through a link-channel screwed between the webs of the back-to-back channel-sections. Such gaps are commonly introduced in
Abbreviations: A′, Overall web length of section; A1, Cross sectional area of single channel-section; Ae, Effective area of the section; B′, Overall flange width of section; C′, Overall lip width of section; CFS, Cold-formed steel; COV, Coefficient of variation; E, Young's modulus of elasticity; Fe, Least of the elastic flexural, torsional, and flexural torsional buckling stress; Fn, Nominal buckling stress as per AISI & AS/NZS; Fy, Yield stress which is equal to the 0.2% proof stress (σ0.2); FEA, Finite element analysis; I1, Moment of inertia of single channel-section; k, Effective length factor; l, Effective length of the builtup gapped section; L, Total length of the built-up gapped section; P, Applied axial load; PAISI& AS/NZS, Axial strength from AISI & AS/NZS; PDSM, Axial strength from Direct Strength Method; PEXP, Axial strength from experiments; PM-DSM, Axial strength from Modified Direct Strength Method; S, Longitudinal spacing of link-channels; w′, Gap between back-to-back channel-sections; λc, Non-dimensional slenderness ratio as per AISI & AS/NZS; λc,GAP, λc as for sections with gap; σ0.2, Static 0.2% proof stress. ⁎ Corresponding author at: Building 906, Level 4, Room 413, Newmarket campus, The University of Auckland, Auckland 1023, New Zealand. E-mail address: [email protected]. (K. Roy).
https://doi.org/10.1016/j.jcsr.2018.04.007 0143-974X/© 2017 Elsevier Ltd. All rights reserved.
struts in steel trusses and columns in portal frames, increasing the lateral stability of such columns. In the literature, only three such experimental results are available, as reported by Rondal and Niazi [1] in 1990; the values of non-dimensional slenderness in these tests ranged from 1.08 to 1.16. The forty new experimental tests reported herein have a value of non-dimensional slenderness ranging from 0.23 to 1.42, thus covering stub to slender columns. In current design standards, such as American Iron and Steel Institute [2] and Australian and New Zealand Standards AS/NZS 4600:2005 [3], the beneficial effect of the gap is ignored i.e. the design axial compressive strength is simply twice that of a single channel-section. This is the case regardless of whether the Effective Width Method (EWM) (reproduced in Section 2) or the Direct Strength Method (DSM) is used. It should be noted that the DSM does not include post-local-buckling capacity, however, Kumar and Kalyanaraman [4], modified the DSM equations, referred here as M-DSM, to include postlocal-buckling capacity. The axial strength calculated in accordance with EWM, DSM and M-DSM are all presented in this paper. Ting et al. [5] recently presented an experimental and numerical investigation on the behaviour of back-to-back built-up CFS channel sections under axial compression (see Fig. 2). The experimental tests reported herein, extends the work of Ting et al. [5]. As a result of the gap, for some combinations of column length and gap size, the lowest
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Fig. 2. Back-to-back built-up cold-formed steel channel columns without a gap as investigated by Ting et al. (2017).
(a) General arrangement (a) Mode A
(b) Cross section Fig. 1. Back-to-back gapped built-up cold-formed steel channel-sections.
(b) Mode B
Fig. 3. Overall flexural buckling modes of back-to-back gapped built-up cold-formed steel channel-sections.
K. Roy et al. / Journal of Constructional Steel Research 147 (2018) 257–276
(a) Built-up double Z members with a gap after Georgieva, Schueremans and Pyl (2012)
(b) Back to Back built-up cold-formed steel channel sections with an opening after Young & Zhang (2012) Fig. 4. Built-up cold-formed steel columns from literature.
Fig. 5. Nominal cross-sections of back-to-back gapped built-up cold-formed steel channel sections considered in this paper.
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flexural buckling mode may be as shown in Fig. 3a, as opposed to overall buckling of the whole column, taking into consideration of the gap, as shown in Fig. 3b. As mentioned previously, the design approach of the design standards ignores the beneficial presence of the gap. No other work in the literature has been reported for back-to-back gapped built-up CFS channel columns. While Georgieva, Schueremans and Pyl [6] considered built-up columns composed of zed-sections,
the gap considered was due to the zed-sections being connected toeto-toe (see Fig. 4a). Other work includes that of Zhang and Young [7] who considered back-to-back built-up channel-sections, but these were with an opening, not with a gap (see Fig. 4b). Dabaon et al. [8] investigated CFS built-up battened columns, while Stone and LaBoube [9] considered back-to-back channel-sections, which were flange stiffened and track sections. Whittle et al. [10] and Piyawat et al. [11] considered
Table 1 Measured specimen dimensions for built-up back-to-back gapped channel-sections. a) GBU75 Specimen
Web
Flange
Lip
Length
Spacing
Slenderness
Gap between columns
Experimental results
Finite element results
Comparison
Failure mode
A′
B′
C′
L
s
w′
λc
λC,GAP
PEXP
PFEA
PEXP/PFEA
–
(mm)
(mm)
(mm)
(mm)
(mm)
(mm)
–
–
(kN)
(kN)
–
–
Stub GBU75-S50-L300-1 GBU75-S50-L-300-2 GBU75-S50-L300-3 GBU75-S200-L300-1 GBU75-S200-L300-2 GBU75-S200-L300-3
73.2 73.6 73.6 73.6 73.6 72.3
19.8 19.9 19.8 19.7 19.7 18.5
11.2 11.1 11.1 11.1 11.2 10.5
271 270 268 268 271 263
50.0 50.1 50.2 201.0 200.0 199.0
73.2 73.6 73.6 73.6 73.6 72.3
0.38 0.39 0.38 0.42 0.42 0.41
0.26 0.27 0.26 0.29 0.29 0.28
122.6 120.5 121.9 114.7 114.3 117.9
126.3 124.2 126.8 119.2 119.9 119.2
1.03 1.03 1.04 1.04 1.05 1.01
Local Local Local Local Local Local
Short GBU75-S100-L500-1 GBU75-S100-L500-2 GBU75-S100-L500-3 GBU75-S400-L500-1 GBU75-S400-L500-2 GBU75-S400-L500-3
73.6 73.6 73.5 73.6 73.5 73.5
19.8 19.9 19.8 19.9 19.8 19.8
11.3 11.2 11.3 11.2 11.1 11.1
678 679 681 679 680 680
100.2 100.0 100.0 399.0 400.0 400.0
73.6 73.6 73.5 73.6 73.5 73.5
0.57 0.57 0.59 0.60 0.61 0.61
0.37 0.36 0.38 0.40 0.41 0.41
113.9 109.8 115.1 108.2 101.8 101.1
111.6 110.9 120.9 111.5 98.7 99.1
1.01 1.01 1.01 1.00 0.96 1.01
Mode A Mode A Mode A Local +Distortional Local +Distortional Local +Distortional
Intermediate GBU75-S225-L1000-1 GBU75-S225-L1000-2 GBU75-S225-L1000-3 GBU75-S900-L1000-1 GBU75-S900-L1000-2 GBU75-S900-L1000-3
76.1 76.2 75.8 75.9 76.0 76.0
19.8 20.3 19.8 19.8 19.9 19.8
10.4 10.4 10.4 10.4 10.3 10.3
1131 1132 1183 1133 1132 1182
225.0 225.0 224.8 900.0 897.5 900.0
76.1 76.2 75.8 75.9 76.0 76.0
0.89 0.88 0.92 0.98 0.99 1.00
0.75 0.74 0.77 0.83 0.85 0.86
77.1 76.2 73.2 65.3 61.5 61.2
79.4 77.7 74.6 63.3 60.3 59.3
1.01 1.00 0.96 1.01 1.01 1.01
Mode A Mode A Mode A Mode A Mode A Mode A
Slender GBU75-S475-L2000-1 GBU75-S475-L2000-2 GBU75-S475-L2000-3 GBU75-S1900-L2000-1 GBU75-S1900-L2000-2 GBU75-S1900-L2000-3
73.6 73.9 73.9 73.8 73.9 73.9
20.3 20.3 20.4 20.3 20.4 20.4
10.6 10.6 10.8 10.8 10.7 10.8
2183 2183 2184 2183 2183 2184
474.3 474.5 474.8 1901 1907 1902
73.6 73.9 73.9 73.8 73.9 73.9
1.36 1.36 1.36 1.41 1.42 1.42
1.23 1.23 1.23 1.26 1.27 1.26
25.6 25.2 25.7 21.3 20.9 21.6
24.6 24.5 24.7 20.9 20.3 20.8
1.04 1.03 1.04 1.02 1.03 1.04
Mode B Mode B Mode B Mode B Mode B Mode B
b) GBU90 Specimen
Web
Flange
Lip
Length
Spacing
Gap between columns
Slenderness
Experimental results
Finite element results
Comparison
Failure mode
A′
B′
C′4
L
S
w′
λc
λC,GAP
PEXP
PFEA
PEXP/PFEA
–
(mm)
(mm)
(mm)
(mm)
(mm)
(mm)
–
–
(kN)
(kN)
–
–
Stub GBU90-S50-L300-1 GBU90-S50-L300-2 GBU90-S200-L300-1 GBU90-S200-L300–2 GBU90-S200-L300-3
92.2 90.7 90.7 90.7 90.7
49.5 49.6 49.6 49.5 49.5
14.5 14.6 14.6 14.6 14.6
265 270 258 270 262
50.0 50.3 200.0 200.0 199.0
92.2 90.7 90.7 90.7 90.7
0.23 0.23 0.23 0.24 0.23
0.13 0.13 0.13 0.14 0.13
147.7 146.8 140.6 138.5 139.4
151.4 150.5 142.6 140.0 140.8
0.98 0.98 0.99 0.99 0.99
Local Local Local Local Local
Short GBU90-S100-L500-1 GBU90-S100-L500-2 GBU90-S100-L500-3 GBU90-S400-L500-1 GBU90-S400-L500-2 GBU90-S400-L500-3 GBU90-S400 L-500-4
90.5 90.6 88.9 90.5 90.5 90.3 90.6
49.4 49.4 49.4 49.5 49.5 49.5 49.4
14.6 14.6 15.5 13.4 14.6 14.7 14.6
680 680 680 680 680 680 680
100.5 100.5 100.3 400.0 400.0 400.0 401.0
90.5 90.6 88.9 90.5 90.5 90.3 90.6
0.41 0.41 0.41 0.42 0.45 0.45 0.45
0.23 0.23 0.24 0.26 0.27 0.28 0.27
132.5 133.7 133.2 129.7 130.4 129.5 130.6
135.9 135.9 135.1 130.1 131.8 131.7 132.9
0.97 0.98 0.99 1.00 0.99 0.98 0.98
Mode A Mode A Mode A Local +Distortional Local +Distortional Local +Distortional Local +Distortional
Intermediate GBU90-S225-L1000-1 GBU90-S225-L1000-2 GBU90-S900-L1000-1 GBU90-S900-L1000-2
90.5 89.8 90.6 90.6
49.7 48.5 49.6 49.7
14.4 13.7 14.4 14.4
1133 1183 1132 1183
225.3 224.8 892.0 900.0
90.5 89.8 90.6 90.6
0.85 0.86 0.86 0.87
0.64 0.65 0.66 0.67
86.5 87.5 82.9 83.3
84.3 84.5 80.2 81.8
1.03 1.04 1.03 1.02
Mode A Mode A Mode A Mode A
K. Roy et al. / Journal of Constructional Steel Research 147 (2018) 257–276
Fig. 6. Specimen labelling.
built-up channel-sections, but these were welded connection. A numerical investigation on built-up columns with battens and stiches are presented by Crisan et al. [12], where they have checked the accuracy of European and American standards, while calculating compressive strength of such built-up batten columns. Other works includes that of Fratamico et al. [13] and Anbarasu et al. [14] who considered back-toback built-up CFS channels, without any gap under compression. Fratamico et al. [13] investigated built-up columns and investigated the effect of screw spacing for back-to-back channels, again without any gap. On the other hand, Anbarasu et al. [14] investigated the behaviour and strength of cold-formed steel web stiffened built-up batten columns, without any gap. Effect of thickness on the behaviour of axially loaded back-to-back built-up cold-formed steel channel sections were studied by Roy et al. [15]. Fig. 5 shows the nominal geometry of the two sections considered in this paper: GBU75 and GBU90. Forty experimental test results are reported. All test specimens were brake-pressed from G550 structural steel sheets. The experimental tests were conducted for different lengths in combination with different link-channel spacing. The effect of gap, link-channel spacing, load-axial shortening, load-axial strain behaviour and buckling failure modes for different cross sections and
261
lengths of back-to-back gapped built-up columns has been investigated in this paper, none of which is available in the literature. The experimental test results are compared against the tests results of back-toback built-up CFS column with no gap by Ting et al. [5]. A finite element model is also described. The finite element model was used for the purposes of a study comprising 84 models. Good correlation was obtained when the test and finite element strengths are compared. Using the experimental and finite element results, it is shown that design in accordance with the American Iron and Steel Institute (AISI) and Australian and New Zealand Standards (AS/NZS) can be conservative by as much as 53%. However, use of a modification to the non-dimensional slenderness, that considers the gap, results in the design standards being within 5% conservative with respect to the experimental and finite element results. 2. Design rules as per AISI specification and AS/NZ standard In accordance with AISI & AS/NZ standards, for built-up sections, the design axial compressive strength is simply twice that of a single channel-section, the un-factored design strength of a single channelsection is as follows: PAISI&AS=NZS ¼ Ae Fn
ð1Þ
The nominal buckling stress (Fn) can be calculated from: λc ≤1:5 : Fn ¼ 0:658 λc 2 Fy
Fig. 7. Drawing of loading test-rig.
ð2Þ
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where,
λc ¼
sffiffiffiffiffiffi Fy Fe
ð4Þ
3. Experimental investigation 3.1. Test sections Fig. 5 shows detailed cross-sections of the two-back-to-back gapped built-up CFS channel columns considered in this paper: GBU75 and GBU90. As indicated by the names, GBU75 and GBU90 are built-up from C75 and C90 channel-sections, respectively. In this paper, the link-channels through which the gap is formed using the same channel-section as that of the built-up section i.e. C75 and C90 channel-sections are used for the link-channels in GBU75 and GBU90, respectively. Table 1 summarises the measured column dimensions. The experimental test programme is comprised of 40 specimens, subdivided into four different column heights: 300 mm, 500 mm, 1000 mm, and 2000 mm. All columns with pin-ended, other than the 300 mm columns which were tested as fixed-ended columns. In Table 1, the test specimens have been sub-divided into stub, short, intermediate and slender columns. In the experimental test programme, the following longitudinal spacing of link-channels (S) were considered:
(a) Stub tests
• • • •
Column height of 300 mm; spacing of 50 mm and 200 mm Column height of 500 mm; spacing of 100 mm and 400 mm Column height of 1000 mm; spacing of 225 mm and 900 mm Column height of 2000 mm; spacing of 475 mm and 1900 mm
3.2. Material testing In order to determine the material properties of test sections, tensile coupon tests were conducted. The tensile coupons were prepared from the centre of the web plate, in accordance with British Standard for Testing and Materials [16]. Five of each coupons were obtained from longitudinal and transverse direction of the untested specimens. MTS test machine was used to test the coupons. To determine the tensile strain, two strain gauges and a calibrated extensometer of 50 mm gauge length were used. The average modulus of elasticity was 205GPa and yield strength was 565 MPa.
(b) Intermediate column tests
3.3. Section labelling The sections were labelled such that the type of section, longitudinal link channel spacing, nominal length of section and specimen number were indicated by the label. As shown in Fig. 6, the label “GBU75-S50L300-1” are explained as follows: • “GBU75” indicates back-to-back built-up channels with a gap of 75 mm web depth. • “S50” indicates the intermediate link-channel spacing as 50 mm. • “L300” indicates nominal length of specimen as 300 mm. • “1” indicates the specimen number as 1.
(c) Position of LVDT for shortening
3.4. Test-rig and testing procedure
Fig. 8. Photograph of loading test-rig.
λc N1:5; Fn ¼
0:877 λ2 c
Fy
ð3Þ
A schematic drawing of the test setup is shown in Fig. 7. The external load cell was placed at the top of the column. Six LVDTs were used, the position of LVDTs is shown in Fig. 8. To measure the strain, two longitudinal strain gauges along with three other strain gauges were used on the web of the columns. Strain gauge arrangements are shown in Fig. 9. A Universal Testing Machine (UTM) GT-7001-LC60, of 600 kN
K. Roy et al. / Journal of Constructional Steel Research 147 (2018) 257–276
Fig. 9. Position of strain gauges.
Fig. 10. Photograph of imperfection measurements setup.
capacity, was used to apply axial load to the columns. The displacement control was used to apply the axial force to the columns, which can include the post buckling behaviour of the columns. Displacement rate was kept as 0.03 mm/s for all test specimens. Strain values were recorded from longitudinal strain gauges near the top end plate and middle of the columns to verify that the load was applied through the centroid of the sections. In order to ensure, there is no gap between the two pin-ends and end plates of the specimen, all columns were loaded initially up to 25% of their expected failure load and then released. LVDT and strain gauge readings were recorded with each increment of loading. 3.5. Measurement of initial geometric imperfections Initial geometric imperfections were measured as shown in Fig. 10. A LVDT with an accuracy of 0.01 mm, was used to record the readings at
Fig. 11. Position of LVDTs to measure imperfections.
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every 20 mm along the length of the sections. The imperfections were measured at the centre of the web, flanges, and edge of the lips for all sections. LVDT positions are shown in Fig. 11. In Fig. 12, a typical length-imperfection plot is shown for GBU75-S100-L500-1. The maximum imperfections of the test specimens were 1.8 mm, 1.7 mm, 1.6 mm and 1.9 mm for the 300 mm, 500 mm, 1000 mm and 2000 mm column lengths, respectively.
3.6. Experimental results
Fig. 12. Typical imperfection profile (GBU75-S100-L500-1).
(i) GBU75-S50-L300
Column dimensions and the experimental failure loads (PEXP) are shown in Table 1. Also shown in Table 1, are the buckling modes. As can be seen, strength of the built-up sections were reduced significantly for all columns beyond 1000 mm length for both GBU75 and GBU90 with two intermediate link-channels. On the other hand, for GBU75 with no intermediate link-channel, significant reduction in axial strength was observed for column length higher than 500 mm. This is because the intermediate link-channels holds the individual back-toback channels together. Failure modes were different for stub, short, intermediate and slender columns. In total, 16 stub columns were tested (see Table 1). GBU75-S50L300 and GBU90-S50-L300 test specimens with three link-channels spaced at 50 mm, failed through local buckling. Back-to-back channels remain integral at failure, showing some plastic deformation near the bottom of the stub columns as shown in Fig. 13(a). The failure modes of GBU75-L300 series, were different from the BU75-L300 series (Ting et al. [5]) because the link-channels in GBU75-L300 provided sufficient restraint to prevent buckling at mid-height of the column. Unlike BU75-L300, for GBU90-L300 series columns, localized deformation was observed near the top or bottom end of the columns. GBU75-S100-L500 with three intermediate link-channels at 100 mm spacing buckled as Mode A, leading to a hinge-like angular buckling shape at about one-third height, near to the top of the column. The buckling shape is shown in Fig. 13(b). Both short (500 mm length) and intermediate (1000 mm length) columns showed similar buckling
(ii) GBU90-S50-L300 (a) Local buckling
Fig. 13. Photograph of columns at failure for different modes.
K. Roy et al. / Journal of Constructional Steel Research 147 (2018) 257–276
(i) GBU75-S100-L500
265
(ii) GBU75-S225-L1000 (b) Mode-A Fig. 13 (continued).
shapes, although 1000 mm long columns showed clearer deformed shapes (see Fig. 15(b)). Local and distortional buckling were not noticeable for slender columns. Overall buckling was observed for slender columns of both GBU75 and GBU90 series. Localized deformation was noticeable on the compression side of specimens near the mid-height of the column as shown in Fig. 13(c). For Mode B, GBU75-S475-L2000 uses four linkchannels while GBU75-S1900-L2000 uses two link-channels. Load-axial shortening behaviour for GBU75-S225-L1000-1 is plotted in Fig. 14. It is observed that the relationship was almost linear up to a load of 53 kN, which is approximately 69% of the ultimate failure load for GBU75-S225-L1000-1. After that, nonlinear behaviour is continued until the failure load is reached, which is 77.09 kN.
Five strain gauges were used to determine axial strain at mid-length and end of all columns. Load-axial strain relationships for GBU75-S50-L300-1 and GBU90-S50-L300-1 are plotted in Fig. 15 (a) and (b) respectively. S-E denotes the axial strain at the end of column, where S-M is the strain at the middle of the column (see Fig. 9). Maximum compressive micro-strain observed at failure load, as shown in Fig. 15(a), is 812 and 987 in S-E and S-M points respectively for GBU75-S50-L300-1 column. Similarly, as shown in Fig. 15(b), for GBU90-S50-L300-1, axial strain values obtained at failure load is 1050 and 1207 respectively for S-E and S-M points i.e. at the end and at middle height of columns. The effect of increase in the vertical spacing of link-channels, are investigated and shown in Table 1. As can be seen from Table 1(a), the
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(i) GBU75-S475-L2000
(ii) GBU75-S1900-L2000 (c) Mode B Fig. 13 (continued).
average strength of stub column of GBU75 series is decreased by 5%, when the vertical spacing between the link-channels is increased from 50 mm to 200 mm. For short columns, when the link channel spacing is increased from 100 mm to 400 mm, the axial strength of the gapped built-up columns is decreased by 5.8% on average. For intermediate columns, the average decrease in strength is by around 17% when increasing the spacing from 225 mm to 900 mm for
GBU75 columns. Reduction in strength for slender columns is approximately by 16% for GBU75, when the screw spacing is increased from 475 mm to 1900 mm. On the other hand, for GBU90 short columns, increasing the vertical spacing of link-channels from 100 mm to 400 mm, reduces the axial strength by 2.3%. For short GBU90 columns, the axial strength is reduced by 4.8%, when the spacing is increased from 100 mm to 400 mm.
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267
140
90
80
120 70
100
50
Load(kN)
Load (kN)
60
40
80
60
30
40
20
10
TEST-GBU75-S225-L1000-1 20
FEA-GBU75-S225-L1000-1 0 0
1
2
3
4
5
6
7
8
9
10
Displacement (mm)
0 0
200
400
Fig. 14. Comparison of finite element and experimental results.
600
800
1000
1200
1400
Axial Strain
(a) GBU75S50L300-1
4. Numerical investigation
160
4.1. General 140
4.2. Analysis procedure
120
100
Load(kN)
A nonlinear elasto-plastic finite element model was developed using ABAQUS [17]. The finite element model was based on the centre line dimensions of the cross-sections. In the model, the measured crosssectional dimensions were used. Specific modelling techniques are described below.
80
60
The full geometry was modelled for the test specimens. Screws between the link-channels and the web of back-to-back channel sections were idealised by connector elements. Axial load was applied through the centre of gravity (CG) of the specimens. The columns were loaded in displacement control, with the displacement applied at a centroid node. The displacement control load was applied using the RIKS algorithm available in ABAQUS library.
40
20
Strain Strain Strain Strain
at at at at
the Edge(Exp) middle(Exp) the Edge(FEA) middle(FEA)
0 0
200
400
600
800
1000
1200
1400
Axial Strain
(b) GBU90S50L300-1
4.3. Geometry and material modelling The whole geometry of the built-up columns was modelled. Classical metal plasticity model available in ABAQUS library was used for all the finite element analyses. The classical metal plasticity model implements the von Mises yield surface to define isotropic yielding, associated plastic flow theory and isotropic hardening behaviour. Yield strength of 565 MPa and ultimate stress of 690 MPa, along with Young's modulus of 205 GPa, was used in numerical modelling. The mean mechanical properties conducted from the tensile tests were also used for engineering stress-strain curve. As per the ABAQUS manual [16], the engineering material curve is converted into a true material curve by following the equation below: σ true ¼ σ ð1 þ εÞ
εtrueðplÞ ¼ ln ð1 þ εÞ−
ð5Þ σ true E
ð6Þ
where E is the Young's modulus, σtrue is the true stress, σu is the tensile ultimate strength, σ and ε are the engineering stress and strain respectively in ABAQUS [16].
Fig. 15. Load versus axial-strain for GBU75 and GBU90 columns.
4.4. Type of element and finite element mesh The channel sections and the link-channels were modelled using the linear 4-node quadrilateral thick shell element S4R. The S4R element has six degrees of freedom per node. The upper and lower end plates were modelled using rigid quadrilateral shell elements (R3D4). An element size of 5 mm by 5 mm was found to be appropriate for back to back channels and link-channels, based on the results of a convergence study. The number of elements along the length of the sections were chosen, such that the aspect ratio of the elements are as close to one. A typical finite element mesh is shown in Fig. 16.
4.5. Boundary conditions and load application The back-to-back gapped built-up CFS columns investigated in this study were pin-ended columns other than the stub column which was fixed-fixed. In order to simulate the upper and lower pin-end supports, the displacements and rotations (boundary conditions) were assigned
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Fig. 16. Typical finite element mesh (GBU75-S100-L500).
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. 17. The distance between two reference points are the effective length of each specimen between the two hinged supports. The boundary conditions were assigned to both reference points. MPC beam connector element available in ABAQUS library was used to model the
screw connections between the link channel and web of back to back channels. Connector elements were assigned a stress of 62.10 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, in general, can predict unstable and nonlinear collapse of a structure such as post buckling behaviour.
Fig. 17. Applied boundary conditions (section GBU90S100L500).
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4.6. Modelling of local and overall geometric imperfections Geometric imperfections are caused in columns as a result of fabrication and transportation processes. Hence, local and overall buckling modes are superimposed for accurate finite element analysis. Eigenvalue analyses of the built-up columns were performed with very small to large channel thickness to determine the contours for the local and global imperfections. The lowest buckling mode (Eigenmode 1) in ABAQUS, is used as the shape of local and overall buckling mode. This technique is used to model the initial local and overall geometric imperfections of the gapped built-up columns. The imperfections were scaled to the values given in the experimental Section 3.5. Besides, local imperfections of magnitude 0.5% of the channel thickness was incorporated as recommended by Ellobody and Young [18]. In Fig. 18, local and overall buckling modes obtained for GBU75-S100-L500, are shown. 4.7. Finite element model validation The comparison of failure loads from laboratory and the finite element analysis for GBU75 and GBU90 columns, are shown in Table 1. Close comparison is obtained between the tests and finite element results in terms of axial strength and failure modes. Fig. 19 shows Mode
269
A and Mode B obtained from the finite element analysis, which has good agreement with experimental failure modes (see Table 1). Fig. 14 compares the load-displacement curves from experimental tests and FEA for GBU75-S225-L1000-1. Also shown in Fig. 15(a) and (b), is the comparison of axial strain obtained numerically and experimentally for GBU75-S50-L300-1 and GBU90-S50-L300-1 respectively. As can be seen, good agreement is shown between experimental and numerical values of strain measured at the edge and middle of the columns. 5. Comparison with design standards Table 2 compares the experimental strengths with the design strengths calculated in accordance with AISI and AS/NZS, DSM and M-DSM. Fig. 20(a) shows the experimental and finite element strengths of the GBU75 columns plotted against length. For ease of reference, the experimental strengths of the back-to-back built-up channel-sections after Ting et al. [5] are also shown. As can be seen, the presence of the gap has increased the strength of the column by around 29%, when the column length is approximately 700 mm. As can be expected, for the stockier and slenderer sections, the increase in strength is less. Also shown in Fig. 20(a) are the design strengths calculated in accordance with AISI and AS/NZS. Two curves are shown. The lower curve is
(a) Local buckling
(b) Overall buckling Fig. 18. Initial imperfection contours (GBU75-S100-L500).
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the strength of the built-up back to back channel-section calculated using λc. As can be seen, this curve is very conservative to both the experimental and finite element strengths by as much as 53%. This can be expected since the beneficial effect of the gap is ignored i.e. the design axial compressive strength is simply twice that of a single channel-section. A theoretical equation for the elastic critical bucking load (P), that takes into account the gap, is as follows [19]: klsinkl þ
I −2 ð1−cosklÞ ¼ 0 I0
ð7Þ
where, 2
k ¼
P 2EI1
ð8Þ
(i) Experimental
Solving the above equations numerically for P, leads to a value for the non-dimensional slenderness, λc, GAP. In Fig. 20(a), the upper curve is calculated using the value of λc,GAP. As can be seen, use of λc,GAP, results in the AISI and AS/NZS being close to the experimental test results, being within 5% conservative to the design standards. For ease of reference, the DSM and M-DSM curves are also shown, again with use of the value of λc,GAP. As can be seen, both DSM and M-DSM are close to the AISI and AS/NZS curves, albeit slightly lower. The same trends can be seen in Fig. 20(b) for the GBU90 columns. Fig. 21(a) and (b) shows the same strength of the GBU75 and GBU90 columns, respectively, but for varying values of non-dimensional slenderness, λc. As mentioned previously, Rondal and Niazi [1], tested three gapped built-up columns having values of λc ranging from 1.08 to 1.16. The range λc covered in this paper is from 0.2 to 1.42, thus providing additional points to Rondal and Niazi [1]. In both Figs. 20 and 21, the value of λc where Mode A changes to Mode B is shown.
(ii) Numerical (a) Mode A (GBU75-S100-L500) Fig. 19. Failure mode of columns at failure from finite element analysis.
(iii) Close up
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(i) Experimental
(ii) Numerical
271
(iii) Close up
(b) Mode B (GBU75-S475-L2000) Fig. 19 (continued).
Fig. 22(a) and (b) show the experimental strengths and design strengths of the GBU75 and GBU90 columns, respectively, when the value of λc,GAP, as proposed by authors is used. As can be seen, good agreement is shown between the experimental strengths and the design strengths.
6. Conclusions A detailed experimental investigation on the axial strength of backto-back gapped built-up CFS channel-sections is presented in this paper. A total of 40 tests are reported. The failure modes, load-axial shortening,
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Table 2 Comparison of experimental and design strengths. a) GBU75 Specimen
Experimental results AISI
Stub GBU75-S50-L300-13 GBU75-S50-L300-2 GBU75-S50-L300-3 GBU75-S200-L300-1 GBU75-S200-L300-2 GBU75-S200-L300-3 Mean COV Short GBU75-S100-L500-1 GBU75-S100-L500-2 GBU75-S100-L500-3 GBU75-S400-L500-1 GBU75-S400-L500-2 GBU75-S400-L500-3 Mean COV Intermediate GBU75-S225-L1000-1 GBU75-S225-L1000-2 GBU75-S225-L1000-3 GBU75-S900-L1000-1 GBU75-S900-L1000-2 GBU75-S900-L1000-3 Mean COV Slender GBU75-S475-L2000-1 GBU75-S475-L2000-2 GBU75-S475-L2000-3 GBU75-S1900-L2000-1 GBU75-S1900-L2000-2 GBU75-S1900-L2000-3 Mean COV
AISI &AS/NZS using eq-8 DSM using eq-8 MDSMusing eq-8 Comparison
PEXP (kN)
PAISI (kN) PAISI-eq-8 (kN)
PDSM (kN)
PM-DSM
PEXP/PAISI PEXP/PAISI-eq-8 PEXP/PDSM PEXP/PM-DSM
122.63 120.57 121.86 114.64 114.25 117.98
112.89 112.34 111.70 105.43 104.88 105.72
120.50 119.85 119.31 114.74 114.45 116.14
116.51 115.37 115.70 112.10 110.30 110.93
117.94 117.34 118.69 111.31 111.98 112.64
1.09 1.07 1.09 1.09 1.09 1.12 1.09 0.012
1.02 1.01 1.02 1.00 1.00 1.02 1.01 0.011
1.05 1.05 1.05 1.02 1.04 1.06 1.05 0.014
1.04 1.03 1.03 1.03 1.02 1.05 1.03 0.013
113.88 109.82 115.11 108.21 101.78 109.07
67.30 65.18 69.72 63.90 61.43 62.80
113.30 109.18 114.31 108.25 105.56 108.15
101.26 100.49 101.61 94.87 94.15 94.34
112.00 109.12 112.23 108.10 108.02 108.35
1.69 1.68 1.65 1.69 1.66 1.74 1.69 0.031
1.01 1.01 1.01 1.00 0.96 1.01 1.00 0.017
1.12 1.09 1.13 1.14 1.08 1.16 1.12 0.029
1.02 1.01 1.03 1.00 0.94 1.01 1.01 0.030
77.09 76.21 73.15 65.29 61.52 61.18
42.62 42.38 45.79 31.98 31.39 30.80
76.62 76.31 76.12 64.72 60.87 60.38
60.58 60.03 59.75 50.13 49.73 49.42
66.74 66.51 66.10 58.33 57.34 57.17
1.81 1.80 1.60 2.04 1.96 1.99 1.87 0.164
1.01 1.00 0.96 1.01 1.01 1.01 1.01 0.020
1.27 1.27 1.22 1.30 1.24 1.24 1.26 0.029
1.16 1.15 1.11 1.12 1.07 1.07 1.11 0.036
24.57 24.48 24.67 20.97 20.27 20.76
11.77 11.23 11.45 10.42 10.27 10.87
23.17 22.88 22.97 19.68 20.21 19.81
18.75 18.44 18.62 16.38 16.43 16.32
21.07 21.57 21.70 18.28 17.51 18.47
2.09 2.18 2.15 2.01 1.97 1.91 2.05 0.106
1.06 1.07 1.07 1.07 1.00 1.05 1.05 0.026
1.31 1.33 1.33 1.28 1.23 1.27 1.29 0.036
1.17 1.13 1.14 1.15 1.16 1.12 1.14 0.016
(b) GBU90 Specimen
Stub GBU90S50L300-1 146.84 140.56 138.47 139.0.42 GBU90S50L300-2 GBU90S200L300-1 GBU90S200L300-2 GBU90S200L300-3 Mean COV Short GBU90S100L500-1 GBU90S100L500-2 GBU90S100L500-3 GBU90S400L500-1 GBU90S400L500-2 GBU90S400L500-3 GBU90S400L500-4 Mean COV Intermediate GBU90S225L1000-1 GBU90S225L1000-2 GBU90S900L1000-1
Experimental Results
AISI
AISI& AS/NZS using eq-8
DSM using eq-8
M-DSM using eq-8
Comparison
PEXP (kN)
PAISI (kN)
PAISI-eq-8 (kN)
PM-DSM (kN)
PM-DSM (kN)
PEXP/PAISI
PEXP/PAISI-eq-8
PEXP/PDSM
PEXP/PM-DSM
145.20 137.88 129.01 129.89 130.71
140.34 144.33 137.24 137.93 138.02
142.12 139.79 132.68 131.98 132.21
1.07 141.18 135.50 134.94 135.06
1.02 1.06 1.09 1.07 1.07 1.07 0.011
1.05 1.02 1.02 1.00 1.01 1.01 0.008
1.04 1.05 1.06 1.05 1.05 1.05 0.004
1.04 1.04 1.03 1.03 1.03 0.006
132.48 133.68 133.24 129.73 130.40 129.52 130.61
107.34 107.90 107.87 105.44 105.96 104.77 105.12
130.62 130.84 130.10 127.96 128.08 127.20 127.77
123.07 122.22 122.51 118.82 120.63 120.18 120.43
125.97 126.16 128.54 126.00 126.50 125.70 127.20
1.23 1.24 1.24 1.23 1.23 1.24 1.24 1.24 0.004
1.01 1.02 1.02 1.01 1.02 1.02 1.02 1.02 0.004
1.08 1.09 1.09 1.09 1.08 1.08 1.08 1.08 0.007
1.05 1.06 1.04 1.03 1.03 1.03 1.03 1.04 0.013
86.54 87.45 82.92
59.21 60.33 57.10
81.40 82.97 78.92
67.85 68.01 63.87
74.62 75.98 70.45
1.46 1.45 1.45
1.06 1.05 1.05
1.28 1.29 1.30
1.16 1.15 1.18
147.66
138.31
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Table 2 (continued) (b) GBU90 AISI
AISI& AS/NZS using eq-8
DSM using eq-8
M-DSM using eq-8
Comparison
PAISI (kN)
PAISI-eq-8 (kN)
PM-DSM (kN)
PM-DSM (kN)
PEXP/PAISI
PEXP/PAISI-eq-8
PEXP/PDSM
PEXP/PM-DSM
83.34
58.54
79.80
64.89
71.88
1.42 1.45 0.016
1.04 1.05 0.008
1.28 1.29 0.005
1.16 1.16 0.011
160
140
120
100
Strength, P (kN)
GBU90S900L1000-2 Mean COV
Experimental Results PEXP (kN)
Mode-A
Mode-B
80
60
40
20
0 0
200
400
600
800
1000
1200
1400
1600
1800
2000
Length, L (mm)
(a) GBU75 160
140
120
100
Strength, P (kN)
Specimen
Mode-A
Mode-B
80
60
40
20
0 0
200
400
600
800
1000
1200
Length, L (mm)
(b) GBU90 Fig. 20. Varition of strength against length.
1400
1600
1800
2000
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K. Roy et al. / Journal of Constructional Steel Research 147 (2018) 257–276 160
140
120
Strength, P (kN)
100
Mode-A
Mode-B
80
60
40
20
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Non-dimensional slenderness (λ c)
(a) GBU75 160
140
120
Mode A
Mode B
Strength, P (kN)
100
80
60
40
20
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Non-dimensional slenderness (λ c)
(b) GBU90 Fig. 21. Variation of strength against λc.
load-axial strain and deformed shapes at failure is discussed. Effect of the gap on axial strength of columns are investigated. Also investigated the effect of link-channel spacing on the axial strength of back-to-back gapped built-up columns. Different buckling modes at failure are discussed as well. The results are compared against current AISI and AS/NZS standard. A nonlinear finite element model is then described, which includes material nonlinearity and geometric imperfections. Finite element
model includes the modelling of link-channels and fasteners. Comparison of test results against numerical models have shown good agreement. The column strengths are compared against the design strengths calculated using the AISI & AS/NZS, Direct Strength Method and Modified Direct Strength Method. Both the test and FEA results were as much as 53% higher than the design standards when nondimensional slenderness (λc) was used to calculate design capacity
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275
160
AISI & AS/NZ( using eq 1) Strength, P (kN)
140
120
100
80
60
40
20
AISI & AS/NZS /Test = 1 Test strength
0 0
20
40
60
80
100
120
140
160
Test Strength, P (kN)
(a) GBU75 160
AISI & AS/NZ( using eq 1) Strength, P (kN)
140
120
100
80
60
40
20 AISI & AS/NZS /Test strengths = 1 Test strengths 0 0
20
40
60
80
100
120
140
160
Test Strength, P (kN)
(b) GBU90 Fig. 22. Comparison of test strength and design strength (AISI & AS/NZS (using eq. 8)).
of such columns. However, the design standards were conservative by only 5% on average to the experimental and FEA strengths, when λc,GAP was used. Hence it is recommended to use λc,GAP while calculating the axial strength of back-to-back gapped built-up CFS columns.
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