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Experimental studies of hollow flange channel beams subject to combined bending and shear actions
Thin-Walled Structures 77 (2014) 129–140
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Experimental studies of hollow flange channel beams subject to combined bending and shear actions Poologanathan Keerthan, David Hughes, Mahen Mahendran n Science and Engineering Faculty, Queensland University of Technology, Brisbane, QLD 4000, Australia
art ic l e i nf o
a b s t r a c t
Article history: Received 11 November 2013 Received in revised form 3 December 2013 Accepted 3 December 2013 Available online 21 January 2014
This paper presents the details of an experimental study of a cold-formed steel hollow flange channel beam known as LiteSteel beam (LSB) subject to combined bending and shear actions. The LSB sections are produced by a patented manufacturing process involving simultaneous cold-forming and electric resistance welding. Due to the geometry of the LSB, as well as its unique residual stress characteristics and initial geometric imperfections resultant of manufacturing processes, much of the existing research for common cold-formed steel sections is not directly applicable to LSB. Experimental and numerical studies have been carried out to evaluate the behaviour and design of LSBs subject to pure bending actions and predominant shear actions. To date, however, no investigation has been conducted into the strength of LSB sections under combined bending and shear actions. Combined bending and shear is especially prevalent at the supports of continuous span and cantilever beams, where the interaction of high shear force and bending moment can reduce the capacity of a section to well below that for the same section subject only to pure shear or moment. Hence experimental studies were conducted to assess the combined bending and shear behaviour and strengths of LSBs. Eighteen tests were conducted and the results were compared with current AS/NZS 4600 and AS 4100 design rules. AS/NZS 4600 design rule based on circular interaction equation was shown to grossly underestimate the combined bending and shear capacities of LSBs and hence two lower bound design equations were proposed based on experimental results. Use of these equations will significantly improve the confidence and costeffectiveness of designing LSBs for combined bending and shear actions. & 2013 Elsevier Ltd. All rights reserved.
Keywords: LiteSteel beam Cold-formed steel structures Hollow flange Bending and shear Interaction Experimental study
1. Introduction Cold-formed steel members are used extensively in residential, commercial and industrial steel buildings due to the availability of advanced roll-forming processes and high strength (4550 MPa) and very thin ( o1 mm) steels, strict quality standards, and enhanced efficiency and economy over hot-rolled alternatives. Common cold-formed sections include Z- and C-sections, and the torsionally rigid hollow sections (SHS, RHS and CHS), for which a significant number of studies has been completed to develop suitable capacity equations. Since early 1990s, Australian companies such as OneSteel Australian Tube Mills [1] have produced innovative cold-formed hollow flange sections known as hollow flange beams (HFB) and LiteSteel beams (LSB) (see Fig. 1). The development of these sections was based on improving structural efficiency by adopting torsionally rigid hollow flanges, minimising local buckling of plate
n
Corresponding author. Tel.: þ 61 434070139; fax: þ 61 731381170. E-mail addresses: [email protected] (P. Keerthan), [email protected] (M. Mahendran). 0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.12.003
elements by eliminating free edges, distributing material away from the neutral axis to afford greater bending stiffness than conventional cold-formed sections, and optimising manufacturing efficiency. The HFB sections were produced from a single steel strip using a combined dual electric resistance welding and automated continuous roll-forming process [2]. Further developments in this unique manufacturing process and the need to facilitate easier connections between members led to the release of LSB, a hollow flange channel section in 2005, primarily for use as flexural members in residential and light commercial/industrial applications. Table 1 gives the nominal dimensions of LSB sections. The base steel used for LSB production has a yield strength of 380 MPa and a tensile strength of 490 MPa. However, due to cold-forming, the nominal yield strengths of the web and flange elements are 380 and 450 MPa, respectively [1]. The manufacturing process also introduces residual stresses and initial geometric imperfections which differ from those of common cold-formed and hot-rolled sections. Whilst the LSB is similar to the HFB in many respects, it is monosymmetric and possesses rectangular hollow flanges rather than triangular. Due to the geometry of the LSB, as well as its unique residual stress characteristics and initial geometric imperfections
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d1
Fig. 1. Hollow flange sections. (a) Hollow flange beam. (b) LiteSteel B eam.
Table 1 Nominal dimensions of LiteSteel beam sections (mm) [1]. LSB section
d
bf
df
t
ro
riw
300 75 3.0 300 75 2.5 300 60 2.0 250 75 3.0 250 75 2.5 250 60 2.0 200 60 2.5 200 60 2.0 200 45 1.6 150 45 2.0 150 45 1.6
300 300 300 250 250 250 200 200 200 150 150
75 75 60 75 75 60 60 60 45 45 45
25.0 25.0 20.0 25.0 25.0 20.0 20.0 20.0 15.0 15.0 15.0
3.0 2.5 2.0 3.0 2.5 2.0 2.5 2.0 1.6 2.0 1.6
6.00 5.00 4.00 6.00 5.00 4.00 5.00 4.00 3.20 4.00 3.20
3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
n
Refer Fig. 1(b) for notation.
resultant of manufacturing processes, much of the existing research for common cold-formed sections is not likely to be directly applicable to the LSB. Many studies have been conducted into the structural behaviour and design of LSBs. Both laboratory testing and finite element analysis have been completed to evaluate the behaviour and design of LSBs subject to pure bending [3–5] and shear [6,7], respectively. The above mentioned research has significantly improved the understanding of the structural behaviour of LSBs, particularly in flexural applications. To date, however, no investigation has been conducted into the strength of LSB sections under combined bending and shear actions. Combined bending and shear is especially prevalent at the supports of continuous spans and cantilever beams, where the interaction of high shear force and bending moment can reduce the capacity of a section to well below that for the same section subject only to pure shear or moment. The behaviour of steel beams in combined bending and shear has been investigated by numerous researchers, including LaBoube and Yu [8], Bleich [9], Evans [10], and Shahabian and Roberts [11], with provisions for the design of members subject to such loading included in both AS/NZS 4600 [12] and AS 4100 [13]. Design of LSBs is governed by the Australian cold-formed steel structures code, AS/NZS 4600 [12]. Due to the geometry of the LSB, as well as its unique residual stress characteristics and initial geometric imperfections, it is uncertain whether the design rules for combined bending and
shear outlined in AS/NZS 4600 are appropriate for use with the LSB. First, AS/NZS 4600 [12] suggests a circular interaction equation for the design of flexural members subject to combined bending and shear actions. LaBoube and Yu [8] performed an experimental investigation of the combined bending and shear behaviour of cold-formed channel sections without transverse stiffeners, and identified that a circular relationship for bending and shear interaction is quite conservative. The circular equation was originally derived for individual disjointed plates in combined bending and shear, and is not accurate when applied to webs that are restrained by flanges. This is likely to be especially true for LSB given the significant rigidity of its flanges. For the channel sections used in the study by LaBoube and Yu [8], it was shown that when the applied bending moment was less than approximately 50% of the moment capacity, the full shear capacity was developed. Likewise, little or no reduction in the moment capacity resulted until the applied shear was approximately 65% of the shear capacity [8]. It is expected that similar limits may also apply to LSBs, which would allow more economical design than a circular interaction equation. Second, most studies concerned with combined bending and shear have focused on hot-rolled plate girders or cold-formed members with “open” flanges rather than hollow. Whether the results of such research apply equally to cold-formed LSB sections with hollow flanges is not known. For example, Keerthan and Mahendran [6,7] have shown that the hollow flanges of the LSB provide nearly fixed restraint to the web, which has a significant effect on the shear buckling coefficient and shear buckling strength of LSB. AS/NZS 4600, however, currently considers only simply supported conditions at the edges of the web panel. Keerthan and Mahendran [6,7] have also shown that significant post-buckling strength is available for slender LSBs. Although such post-buckling capacity is ignored in AS/NZS 4600 provisions, it may affect the combined bending and shear behaviour of LSBs. An evaluation of LSBs subject to combined bending and shear is therefore considered important so that refinements to design rules can be made to more adequately reflect the behaviour of the innovative LSB sections. This paper presents the details of an experimental study on the behavior of LSBs subject to combined bending and shear actions and its results.
2. Review of the behaviour of steel beams subject to shear, bending and combined bending and shear actions 2.1. Shear Timoshenko and Gere [14] developed Eq. (1) for the critical elastic buckling stress (fcr) of a rectangular plate in compression, bending or shear, which is a function of plate width (b), thickness (t), Poisson0 s ratio (ν), Young0 s modulus of elasticity (E), and plate buckling coefficient (k). 2 π2 E t f cr ¼ k ð1Þ 12ð1 ν2 Þ b Eq. (1) can be rewritten as an expression for the critical shear buckling stress of a rectangular web plate, replacing fcr with τcr, plate thickness (t) with web thickness (tw), plate width (b) with web panel depth (d1), and substituting the shear buckling coefficient (kv). 2 π2 E tw τcr ¼ kv ð2Þ 12ð1 ν2 Þ d1 The shear buckling coefficient itself is a function of the web panel aspect ratio (a/d1, where a is the shear panel length) and the
P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
degree of fixity at the plate edges. Lee and Yoo [15] showed that for plate girders the support condition is closer to fixed than simple. Similar comments have been made regarding the flangeweb support condition for LiteSteel beams [6,7]. In past research and design provisions, however, a simple flange-web support has often been conservatively assumed, which has led to an underestimation of shear buckling strength. Keerthan and Mahendran [6] developed suitable design equations for the shear capacity of LSBs (Vv) by including the available post-buckling strength and increased shear buckling coefficient (kv). Suitable shear design rules were also developed under the direct strength method (DSM) format. They are given in Appendix A of this paper.
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In the case of cold-formed steel sections, Hancock [18] notes that the ultimate moment capacity is typically less than the plastic moment capacity based on a fully effective cross-section since cold-formed sections are not usually compact. However, compact cold-formed sections generally have inelastic reserve capacity in bending. For example, Anapayan et al. [3] showed the presence of inelastic reserve capacity for LSB sections in bending using both experiments and numerical studies. However, AS/NZS 4600 permits members to be designed based on inelastic reserve capacity only if they satisfy a number of strict criteria, including limits on the shear force taken by the web. The section moment capacities of LSBs could be calculated based on AS/NZS 4600 [12]. The section moment capacity (Ms) is defined in AS/NZS 4600 as follows: M s ¼ f yf Z e
ð3Þ
2.2. Bending Members subject to pure bending generally fail due to lateral– torsional buckling, inward buckling of the compression flange, or flange yielding [16]. In the case of hollow flange beams, failure can also result from lateral–distortional buckling, which is characterised by simultaneous lateral deflection, twist and crosssectional change due to web distortion [3–5]. Discussion herein will focus only on members with full lateral restraint. Slender flange elements subject to uniform compression are prone to local buckling effects, with the critical buckling stress given by Eq. (1), using k¼ 4.0 for plate elements with both longitudinal edges simply supported, and k ¼0.425 for elements with one longitudinal edge simply supported and the other free. Following local buckling, plate elements in compression can develop significant post-buckling capacity, with failure occurring when the highly stressed areas reach yield. AS/NZS 4600 allows for local buckling of compression elements using an effective width concept. If the web of a flexural member is relatively slender, the portion of the web which is in compression is also prone to local buckling. The elastic critical buckling stress of a web in bending may be determined from Eq. (1), substituting tw for t, and d1 for b. The buckling stress (fcr) is then the compressive stress at the extreme plate fibres in pure bending to cause initial elastic buckling. For a web panel with a simply supported flangeweb junction, k¼ 23.8 whereas k ¼39.6 for a fixed flange-web junction [17].
The effective section modulus (Ze) is based on the initiation of yielding in the extreme compression fibre and therefore does not allow for the inelastic reserve capacity of the section [12]. The effects of local buckling are accounted for by using reduced widths (be) of slender elements in compression for the calculation of the effective section modulus (see Eq. (4)). be ¼
1 0:22=λ brb λ
where 1:052 b λ ¼ pffiffiffi k t
ð4Þ
sffiffiffiffiffi n f E
where k ¼local buckling coefficient, f n ¼applied stress. The section moment capacities of all the LSB sections could be calculated using the AS/NZS 4600 method described above, with local buckling coefficients (k) equal to 4 and 24 for the compression flange and web, respectively. The direct strength method (DSM) is an alternative to the traditional effective width method (EWM) and has been adopted as an alternative design procedure in AISI S100 [19] and AS/NZS 4600 [12]. The nominal section moment capacity at local buckling (Msl) can be determined from Section 7.2.2.3 of AS/NZS 4600 [12]. For, λl r 0:776
M sl ¼ M y
ð5Þ
Table 2 Section moment capacities of LSBs from Anapayan et al.0 s tests [3]. Test no.
LSBs
tfo (mm)
tfi (mm)
tw (mm)
df (mm)
bf (mm)
fyo (MPa)
My (kN m)
Test Mu (kN m)
Mol (kN m)
λ
Mu/My
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
300 75 3.0 300 75 2.5 300 60 2.0 250 75 3.0 250 75 2.5 250 60 2.0 200 60 2.5 200 60 2.0 200 45 1.6 150 45 2.0 150 45 1.6 125 45 2.0 125 45 1.6 300 75 3.0 300 60 2.0 250 75 2.5 250 60 2.0 200 45 1.6 150 45 2.0 150 45 1.6
3.18 2.87 2.15 3.08 2.79 2.09 2.58 2.03 1.56 2.11 1.60 1.98 1.62 3.22 2.22 2.90 2.18 1.79 2.22 1.77
3.18 2.87 2.15 3.08 2.79 2.09 2.58 2.03 1.56 2.11 1.60 1.98 1.62 3.13 2.02 2.60 2.02 1.66 2.02 1.63
2.84 2.51 1.98 2.77 2.48 1.96 2.34 1.85 1.48 1.89 1.60 1.98 1.62 3.09 1.98 2.54 1.95 1.61 1.97 1.58
75.31 75.24 60.28 76.35 75.98 60.47 60.23 60.15 45.05 44.95 45.12 45.10 45.07 74.60 60.00 75.00 60.40 45.50 45.40 45.20
25.17 25.05 19.97 25.22 24.92 20.12 19.95 20.31 14.98 14.73 14.89 14.93 14.95 24.80 19.80 25.50 20.40 15.20 14.80 14.80
528 511 568 506 525 580 496 473 478 498 540 503 549 498 558 552 523 537 538 558
94.27 82.10 59.94 68.25 64.05 46.45 35.92 26.96 17.54 16.51 13.75 12.14 10.96 89.77 57.89 66.35 41.88 21.63 17.93 14.92
103.90 85.80 52.40 77.89 71.49 47.33 52.47 31.80 17.36 19.63 14.94 14.38 12.95 93.00 53.36 70.68 42.12 20.88 20.20 16.18
179.20 126.70 46.75 203.50 148.30 53.63 121.40 59.87 21.22 64.38 34.41 53.77 29.44 209.40 45.51 149.20 52.84 27.44 68.72 35.69
0.73 0.80 1.13 0.58 0.66 0.93 0.54 0.67 0.91 0.51 0.63 0.48 0.61 0.65 1.13 0.67 0.89 0.89 0.51 0.65
1.10 1.05 0.87 1.14 1.12 1.02 1.46 1.18 0.99 1.19 1.09 1.18 1.18 1.04 0.92 1.07 1.01 0.97 1.13 1.08
Note: Refer Fig. 1(b) for bf and df while tw, tfo and tfi are the measured web, outside and inside flange thicknesses.
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1.4 1.2
1.0
S
S1
Mu/My
1.0 0.8
DSM (Equations 5 and 6) Tests
0.6
Vult/Vyw
0.4
Shear Type Mechanism
0.2 Flange Criterion Controls
0.0 0.0
0.2
0.4
0.6
λl =
0.8
1.0
1.2
1.4
1.0
My Mol
M/Mp Fig. 3. Bending and shear interaction diagram [10].
Fig. 2. Comparison of experimental section moment capacities of LSBs from [3] with DSM based design equations.
"
λl 40:776
M ol M sl ¼ 1 0:15 My
0:4 # M ol 0:4 My My
Eq. 9
ð6Þ
Eq. 11 Eq. 10
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi My λl ¼ M ol My ¼ Z f f y where Mol is elastic local buckling moment of the section, M ol ¼ Z f f ol , Zf is elastic section modulus, fol is elastic local buckling stress of the section in bending. Anapayan et al. [3] conducted 20 section moment capacity tests to investigate the behaviour and strength of LSB flexural members. The ultimate section moment capacities from their tests (Mu) are presented in Table 2 along with the measured dimensions and outside flange yield stress (fyo) of tested LSBs used in calculating the first yield moments (My). Elastic local buckling moments of LSBs (Mol) were determined using the finite strip analysis program THIN-WALL, and are also given in Table 2. Fig. 2 compares the test ultimate moment capacities with the section moment capacities predicted by the direct strength method (DSM) using the nondimensional format of Mu/My versus (My/Mol)0.5. It can be seen that compact LSB sections have greater moment capacities than their first yield moments (My). Fig. 2 shows that the direct strength method based equations are able to predict the section moment capacities of LSB sections with reasonably good accuracy (conservative predictions). Anapayan et al. [3] also developed suitable finite element models of LSBs to predict their section moment capacities of LSBs, which agreed well with test results given in Table 2. Hence these LSB finite element models can also be used to accurately predict their section moment capacities. 2.3. Bending and shear interaction The interaction between shear force and bending moment can have a significant effect on the strength of a section, and is generally presented in the form of an interaction diagram (see Figs. 3 and 4). Bleich [9] originally proposed a circular interaction Eq. (7) is for the design of disjointed flat rectangular plates subject to combined bending and shear. 2 2 fb τ þ r 1:0 ð7Þ τcr f cr
Fig. 4. Bending and shear interaction diagram [8].
where fb is theapplied compressive bending stress, fcr is the theoretical buckling stress in pure bending, τ is the applied shear stress, and τcr is the theoretical buckling stress in pure shear. Whilst Eq. (7) remains the basis of AS/NZS 4600 design provisions for bending and shear interaction in unstiffened flexural members, numerous studies have been completed which show a circular interaction equation to be overly conservative. The circular interaction equation was originally derived for individual disjointed plates in combined bending and shear, and is not likely to be accurate when applied to webs restrained by flanges [8]. Evans [10] presented the interaction diagram shown in Fig. 3 for stiffened plate girders and proposed a somewhat complicated method to determine the relevant points on the diagram. From point S to S1, it was assumed that little reduction in shear capacity resulted due to applied moment, and the capacity of the section was that for pure shear. Gradual reductions in shear capacity due to bending moment were considered as shown in Fig. 3. Shahabian and Roberts [11] developed a simpler approach to bending and shear interaction in response to the complexity of Evans0 method. Based on investigations of plate girders with web panel aspect ratios between 1 and 2, and web slenderness ratios between 150 and 300, the following empirical equation was recommended. 4 4 M V þ r 1:0 ð8Þ Mu Vu where Vu and Mu are the pure shear and moment capacities, respectively, V and M are the applied shear and moment, respectively.
P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
LaBoube and Yu [8] performed an experimental investigation of the combined bending and shear behaviour of cold-formed steel channel sections without transverse stiffeners. For the channels used in their study, it was shown that when the applied bending moment was less than approximately 50% of the moment capacity, the full shear capacity was developed (Fig. 4). Likewise, little or no reduction in the moment capacity resulted until the applied shear was approximately 65% of the shear capacity [8]. Based on the results of the study, LaBoube and Yu [8] presented the following interaction equation. ( ) M 2 M V 1:077 0:732 ð9Þ þ 1:100 r 1:0 Mu Mu Vu
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Clause 5.12.3 of AS 4100 [13] presents suitable design rules for members subject to bending and shear interaction. It is assumed that no reduction in the shear capacity of a section will occur until the applied moment (M) is greater than 75% of the design section moment capacity. Conversely, reductions in the section moment capacity are assumed to occur only when the applied shear force is greater than 60% of the design shear capacity. This approach is similar to that adopted by AS/NZS 4600 [12] for beams with transverse stiffeners. The following equations give the shear capacity (Vvm) of hot-rolled sections in the presence of bending. V vm ¼ V v
for
M r 0:75M u
1:6M V vm ¼ V v 2:2 Mu
ð13Þ
for
0:75M u r M rM u
ð14Þ
Eq. (9) was simplified as: M V þ r 1:6 Mu V u
ð10Þ
Fig. 4 shows the experimental results from LaBoube and Yu0 s study, plotted against Eqs. (9) and (10). Despite these successful research outcomes for cold-formed steel channel sections, bending and shear interaction for beams with unstiffened webs is allowed for in Clause 3.3.5 of AS/NZS 4600 [12] by the circular interaction equation only. 2 2 M V þ r 1:0 ð11Þ Mu Vv where Mn is the design bending moment, Vn is the design shear force, Mu is the nominal section moment capacity, Vv is the nominal shear capacity of the web. For beams with transverse stiffeners, bending and shear interaction is only considered to take effect if the design moment (Mn) is greater than 50% of the design section moment capacity and the design shear force (Vn) is greater than 70% of the design shear capacity. Bending and shear interaction is then governed by the following equation. M V M V 0:6 þ r 1:3 where 40:5 and 40:7 ð12Þ Mu Vv Mu Vv
As discussed in this section, there are a number of bending and shear interaction equations that can be used for steel beams. However, their applicability to LiteSteel beams subject to combined bending and shear is not known. The following section presents the details of the experimental study undertaken for this purpose.
3. Experimental study of LiteSteel beams subject to combined bending and shear 3.1. General An experimental study was conducted to evaluate the behavior and strength of LSB sections subject to combined bending and shear actions. The results obtained from the study were used for the purposes of developing suitable design capacity equations for combined bending and shear, and validation of subsequent finite element analyses. Eighteen full-scale tests were performed on five different LSB sections with web panel aspect ratios of 2.0, 2.5, 3.0, 3.5, 3.9 and 4.0 (see Table 3). They were based on testing of simply supported back to back LSBs subject to a mid-span load (see Figs. 5 and 6). This section gives the details of the mechanical and geometric properties of LSB test specimens, and discusses the test set-up and
Table 3 Measured geometric and mechanical properties of LSB test specimens. Test no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
LSB section
150 45 1.6 150 45 1.6 150 45 1.6 150 45 1.6 150 45 1.6 150 45 2.0 150 45 2.0 150 45 2.0 150 45 2.0 150 45 2.0 200 45 1.6 200 45 1.6 200 45 1.6 200 45 1.6 200 60 2.0 200 60 2.0 250 60 2.0 250 60 2.0
Aspect ratio a/d1
2.0 2.5 3.0 3.5 3.9 2.0 2.5 3.0 3.5 3.9 2.0 2.5 3.0 4.0 2.0 3.5 2.0 3.0
d1 (mm)
121.3 120.0 120.1 120.0 120.0 121.2 120.0 120.0 120.9 120.0 170.0 169.6 170.7 170.8 161.5 160.0 212.2 211.8
Yield stress (MPa)
Thickness (mm)
fyo
fyi
fyw
tfo
tfi
tw
558 558 558 558 558 538 538 538 538 538 537 537 537 537 521 521 523 523
488 488 488 488 488 492 492 492 492 492 491 491 491 491 471 471 473 473
454 454 454 454 454 423 423 423 423 423 452 452 452 452 440 440 452 452
1.77 1.76 1.78 1.77 1.77 2.25 2.25 2.24 2.26 2.25 1.66 1.66 1.68 1.67 2.12 2.10 2.19 2.20
1.66 1.66 1.65 1.66 1.67 2.08 2.09 2.08 2.07 2.08 1.61 1.60 1.61 1.60 2.02 2.01 2.04 2.03
1.60 1.58 1.58 1.58 1.58 1.99 1.97 1.98 1.97 1.97 1.58 1.58 1.58 1.59 1.96 1.97 1.97 1.99
bf (mm)
df (mm)
45.4 45.6 45.4 45.5 45.4 45.5 45.6 45.5 45.4 45.5 45.6 45.4 45.5 45.4 59.9 59.9 60.2 60.1
14.8 14.7 14.8 14.9 14.8 14.8 14.9 14.8 14.7 14.8 15.3 15.0 15.1 14.9 20.4 20.5 20.7 20.7
Note: Refer Fig. 1(b) for bf, df and d1 while tw, tfo and tfi are the measured web, outside and inside flange thicknesses. fyw, fyo and fyi are the measured web, outside and inside flange yield stresses. All thicknesses represent base metal thickness (BMT) exclusive of coating thickness (0.03 mm).
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Table 4 Predicted failure modes of tested LSB specimens. Test no.
Fig. 5. Details of combined bending and shear test arrangement. (a) Schematic diagram of test set-up. (b) Shear force and bending moment diagrams.
Applied Load
30 mm Gap Back to Back LSBs
Web Side Plates
T-Shaped Stiffener LVDTs
Roller Support
Fig. 6. Experimental set-up used for combined bending and shear tests.
procedure used in the combined bending and shear tests. Experimental results and observations are also described in this section. 3.2. Selection of test specimens In selecting LSB sections and spans for inclusion in the experimental study, it was noted that specimens should ideally represent the full range of shear failure modes (yielding, inelastic buckling and elastic buckling) and section slenderness classifications relevant to bending failure (compact, non-compact and slender section based on AS 4100). Moreover, test spans should give an indication of LSB behaviour over a wide range of applied shear force and bending moment combinations. Keerthan and Mahendran [6] studied the primarily shear behaviour of LSBs with web panel aspect ratios up to 1.5, which was regarded as the lower limit in this study on combined bending and shear. It is considered that LSB specimens with an aspect ratio greater than 1.5 will exhibit the combined effects of bending and shear. In deciding on an upper limit for the web panel aspect ratio,
LSB section
Aspect ratio (a/d1)
d1/tw Predicted shear failure mode
1
150 45 1.6 2.0
75.8 Inelastic buckling
2
150 45 1.6 2.5
75.9 Inelastic buckling
3
150 45 1.6 3.0
76.0 Inelastic buckling
4
150 45 1.6 3.5
75.9 Inelastic buckling
5
150 45 1.6 3.9
75.9 Inelastic buckling
6 7 8 9 10 11 12 13 14 15
150 45 2.0 150 45 2.0 150 45 2.0 150 45 2.0 150 45 2.0 200 45 1.6 200 45 1.6 200 45 1.6 200 45 1.6 200 60 2.0
16
200 60 2.0 3.5
17 18
250 60 2.0 2.0 250 60 2.0 3.0
2.0 2.5 3.0 3.5 3.9 2.0 2.5 3.0 4.0 2.0
60.9 60.9 60.6 61.4 60.9 107.6 107.3 108.0 107.4 82.4
Yielding Yielding Yielding Yielding Yielding Elastic buckling Elastic buckling Elastic buckling Elastic buckling Inelastic buckling
81.2 Inelastic buckling 107.7 Elastic buckling 106.4 Elastic buckling
Section slenderness Noncompact Noncompact Noncompact Noncompact Noncompact Compact Compact Compact Compact Compact Slender Slender Slender Slender Noncompact Noncompact Slender Slender
attention was given to the investigation by Anapayan et al. [3] into the experimental section moment capacities of LSBs using four point loading conditions. Anapayan et al. [3] identified that a 150 45 1.6 LSB displayed shear failure characteristics at spans of up to 1000 mm. However, bending failure was the governing mode when the span was increased to 1500 mm. The web panel aspect ratio for the latter arrangement was approximately equal to 4, which was therefore taken as the upper limit in this study. Table 3 lists the 18 LSB sections that were tested in this experimental study. For the purposes of this study, the length of a web panel (a) was taken to be the distance between the centrelines of the inside pairs of bolts (see Fig. 5a). For LSBs d1 is defined as the clear height of web instead of the depth of the flat portion of web measured along the plane of the web as defined in AS/NZS 4600 [12] for cold-formed channel sections. The reasons for this are given in Keerthan and Mahendran [6]. Table 4 shows the predicted failure modes of LiteSteel beam specimens. The section moment classification of LSBs in Table 4 was determined based on the Australian steel structures code AS 4100 [13]. It was based on the plate slenderness calculations for LSBs using the measured dimensions and yield stresses given in Table 3. Although AS 4100 design rules are not applicable to the cold-formed LSBs, the AS 4100 [13] section classification method was used for LSBs since AS/NZS 4600 [12] design rules do not allow any inelastic reserve bending strength for cold-formed steel beams in general, and limit their section moment capacities to their first yield moments. Shear failure mode of LiteSteel beams was predicted based on Keerthan and Mahendran [6]. 3.3. Geometric and mechanical properties of test specimens Table 3 presents the measured cross-sectional dimensions, thicknesses and yield strengths of LSB test specimens. All the LSB specimens tested as part of this study and those used in the experimental shear investigation by Keerthan and Mahendran [6] were from the same production run. The web and flange yield strengths of such specimens (fyw, fyo and fyi) were determined previously through tensile tests conducted by Keerthan and
P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
Mahendran [6], respectively (see Table 3 for results). The yield strengths in Table 3 were adopted for specimens tested in this study. The thicknesses and cross-sectional dimensions of all LSB specimens were determined prior to testing, based on averages of measurements recorded at a minimum of three locations. Element thicknesses were measured to an accuracy of onehundredth of a millimetre using a micrometer screw gauge, whilst all cross-sectional dimensions were typically measured to the nearest five-hundredths of a millimetre using vernier calipers. LSBs are protected by an aluminium–zinc alloy coating which is not considered to provide any strength [1]. Keerthan and Mahendran [6] measured the thicknesses of coated LSB coupons before and after being immersed in a hydrochloric acid solution, which effectively removes the aluminium–zinc coating. On average, the coating thickness was found to be 0.015 mm on each face. In Table 3, base metal thicknesses (BMT) are given, which were determined by subtracting a coating thickness of 0.03 mm from the measured total coated thickness (TCT). Combined bending and shear tests were conducted on pairs of LSBs connected back-toback and hence the dimensions given in Table 3 are the average of those recorded for each specimen in the assembled pair. 3.4. Experimental test set-up and procedure All LSB specimens were tested using the Tinius Olsen machine in the Structures Laboratory. The experimental set-up was based on the arrangement shown in Figs. 5 and 6. LSB sections were tested in pairs bolted back-to-back using T-shaped stiffeners at the supports and loading point, with a 30 mm gap between them to ensure independent behaviour. A concentrated load was applied through the T-shaped stiffener at mid-span. As loads were applied directly to the beam webs and close to the shear centre, bearing failure (including web crippling) and torsional loading were avoided. High strength steel bolts (M16 8.8/S) were used to avoid any bolt failure during testing. Specimens were simply supported at each end on machine ground and lubricated half-rounds placed upon ball bearings, which ensured free rotation at the supports (see Fig. 6). Ten millimeter thick web side plates were used at the supports and loading point to prevent lateral flange movement, with the web side plate height being approximately equal to the clear height of the web. Assembled pairs of LSB sections were positioned in the testing rig, with care taken to align the T-shaped loading stiffener with the
135
centre of the testing machine cross-head. Longer specimen spans were accommodated by placing the half-round supports on a hotrolled universal column section which was set across the Tinius Olsen machine and propped at one end. The hot-rolled section was deemed to be sufficiently stiff so as not to introduce any significant error in deflection measurements. To measure mid-span vertical deflections in each test, 25 mm linear variable displacement transducers (LVDT) were positioned beneath the bottom flange of LSB specimens. An EDCAR data acquisition system was employed to record all load and deflection data from the commencement of testing until failure. At the commencement of testing, a small load was applied to allow the loading and support systems to settle evenly on the bearings. The measuring system was subsequently zeroed and loading initiated. Loading was applied at a constant rate of 1 mm/min until specimen failure. Testing was terminated shortly after reaching the ultimate load to avoid uncontrolled specimen collapse and potential damage to test equipment. 3.5. Experimental results As identified earlier, the experimental study was designed for the purposes of collecting section strength data and observing failure behavior of LSBs subject to combined bending and shear actions. Such results are critical for the validation of finite element analyses and form an important part of the development of suitable design equations. Table 5 summarizes the ultimate loads (Pu) applied in each of the combined bending and shear tests. Also included are the applied maximum shear forces (V) and bending moments (M) in each section at the ultimate load. The applied shear force in each LSB was taken to be Pu/4, assuming half of the load transferred through the T-shaped stiffener to be resisted by a single LSB section. The moment applied at mid-span to each section was calculated as the shear force multiplied by the shear span, a (see Fig. 5a). Specimens were typically observed to fail in a combined bending and shear mode, which was characterised by a combination of shear web buckling/yielding and some degree of flange buckling/yielding. In all members subject to combined bending and shear failures, local buckling/yielding of the compression flange typically occurred near the mid-span loading point (that is, the location of maximum bending moment). It should be noted that no signs of lateral buckling were observed during
Table 5 Summary of experimental results. Test no.
LSB section
Aspect ratio
Ultimate load (Pu) Applied shear force at (kN) failure V (kN)
Applied moment at failure M Shear capacity Vu Moment capacity Mu (kN m) (kN) (kN m)
V/ Vu
M/ Mu
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
150 45 1.6 150 45 1.6 150 45 1.6 150 45 1.6 150 45 1.6 150 45 2.0 150 45 2.0 150 45 2.0 150 45 2.0 150 45 2.0 200 45 1.6 200 45 1.6 200 45 1.6 200 45 1.6 200 60 2.0 200 60 2.0 250 60 2.0 250 60 2.0
2.0 2.5 3.0 3.5 3.9 2.0 2.5 3.0 3.5 3.9 2.0 2.5 3.0 4.0 2.0 3.5 2.0 3.0
178.6 163.5 151.6 133.5 124.4 233.9 207.9 196.4 175.2 157.5 185.9 170.2 155.0 129.6 267.1 208.9 277.3 230.4
10.83 12.26 13.66 14.02 14.55 14.18 15.59 17.68 18.53 18.43 15.80 18.04 19.84 22.14 21.57 29.24 29.42 36.59
0.94 0.90 0.84 0.74 0.70 0.96 0.87 0.81 0.73 0.66 0.96 0.90 0.83 0.70 0.94 0.75 0.92 0.78
0.72 0.81 0.90 0.93 0.96 0.75 0.83 0.94 0.98 0.98 0.77 0.88 0.97 1.09 0.69 0.94 0.72 0.90
44.65 40.88 37.90 33.38 31.10 58.48 51.98 49.10 43.80 39.38 46.47 42.56 38.75 32.40 66.77 52.21 69.32 57.59
47.30 45.49 45.11 44.86 44.72 61.16 59.94 60.25 60.39 59.94 48.58 47.41 46.65 46.59 71.28 69.74 75.49 74.23
15.10 15.10 15.10 15.10 15.10 18.90 18.90 18.90 18.90 18.90 20.40 20.40 20.40 20.40 31.10 31.10 40.80 40.80
136
P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
Bending
Shear
Fig. 7. Failure mode of 250 60 2.0 LSB (aspect ratio¼2.0). (a) Shear buckling in web. (b) Local flange buckling.
Local Flange Buckling
Shear Yielding Fig. 8. Shear yielding of 150 45 2.0 LSB (aspect ratio¼2.0).
Shear Buckling
Local Flange Buckling
Shear Buckling Fig. 10. Failure mode of 250 60 2.0 LSB (aspect ratio¼ 3.0). (a) Local flange buckling. (b) Shear buckling in web.
Fig. 9. Failure mode of 150 45 1.6 LSB (aspect ratio¼ 3.0).
experimental tests. All specimens displayed significant ductility at failure. Figs. 7–10 show the failure modes of LSBs while Figs. 11–14 show the plots of applied load versus vertical deflection for tested LSBs.
3.5.1. Test specimens failed primarily due to shear (aspect ratio 2.0) Specimens with aspect ratios of 2.0 failed primarily due to shear, with only minor evidence of flange buckling. The failure mode observed for the 250 60 2.0 LSB with an aspect ratio of 2.0 is typical of that for all sections which failed primarily due to shear (Fig. 7). The load–deflection plot from this test is given in
Fig. 12, which shows near linear load–deflection behaviour right up to the point of ultimate load. This was followed by unloading upon failure of the web, with the ultimate load coinciding with relatively low values of mid-span deflection. Diagonal shear web buckling was seen to occur at loads close to the ultimate load. However, it was not possible to accurately determine the web buckling load and hence post-buckling shear capacity during testing. It shows that some post-buckling shear capacity was clearly developed through tension field action. Local compression flange buckling was also initiated close to the ultimate load. Evidence of web shear buckling is shown in Fig. 7a. Shear buckling was characterised by the appearance of web buckles substantially before the ultimate load. Fig. 7b shows local flange buckling in the 250 60 2.0 LSB with an aspect ratio of 2.0.
P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
250
300
Applied Load (kN)
250 Applied load (kN)
137
200 150
LSB 1
100
LSB 2
200 150 LSB 1 LSB 2
100 50
50 0 0
2
4
6
8
0
10
0
2
Vertical Deflection (mm)
6
8
10
Fig. 12. Applied mid-span load versus vertical deflection for 150 45 1.6 LSB (aspect ratio¼ 3.0).
250
250
200
200
Applied load (kN)
Applied Load (kN)
Fig. 11. Applied mid-span load versus vertical deflection for 250 60 2.0 LSB (aspect ratio¼ 2.0).
150 LSB 1 LSB 2
100
4
Vertical Deflection (mm)
150 LSB 1 LSB 2
100 50
50
0
0 0
2
4
6
8
10
Vertical Deflection (mm)
0
2
4
6
8
10
Vertical Deflection (mm)
Fig. 13. Applied mid-span load versus vertical deflection for 150 45 2.0 LSB (aspect ratio¼ 3.0).
Fig. 14. Applied mid-span load versus vertical deflection for 250 60 2.0 LSB (aspect ratio¼ 3.0).
Failure patterns similar to those observed for the 250 60 2.0 LSB with aspect ratio of 2.0 were also observed in the tests on the 200 60 2.0 LSB and the 200 45 1.6 LSB (also with aspect ratios of 2.0). The exception was only extremely slight evidence of flange buckling in the 200 60 2.0 LSB. The failure mode for the 150 45 2.0 LSB with an aspect ratio of 2.0 is depicted in Fig. 8. The specimen displayed strong diagonal shear yielding patterns in the web, but did not exhibit any signs of buckling or yielding in the flanges. The 150 45 1.6 LSB with an aspect ratio of 2.0 displayed a failure mode, which closely matched that of the 150 45 2.0 LSB, with the exception of shear buckling as opposed to yielding. Again, signs of flange buckling were absent.
4. Combined bending and shear interaction diagram and proposed equations
3.5.2. Test specimens failed primarily due to bending (aspect ratios 3.0, 3.5 and 4.0) Specimens with web panel aspect ratios of 3.0, 3.5 and 4.0 were observed to fail primarily due to bending, and generally exhibited linear load–deflection behaviour up to approximately 80% of the ultimate load. Load–deflection plots typically became non-linear beyond this point and rose gradually to the peak ultimate load. This is evident in the load–deflection plot for the 250 60 2.0 LSB with an aspect ratio of 3.0 (see Fig. 14). Web yielding occurred locally at mid-span in all members subject to primary bending failure, which confirms that inelastic reserve capacity was mobilised at the ultimate load. Local buckling/yielding of the compression flange was much more pronounced in the tests on specimens with web panel aspect ratios of 3.0, 3.5 and 4.0, which tended to fail primarily due to bending. Photographs from tests on the 150 45 1.6 LSB and the 250 60 2.0 LSB (both with aspect ratios of 3.0) are given in Figs. 9 and 10. Well defined local flange yielding, which was the primary cause of failure, was seen to occur in both tests at the centre of the span. Shear buckling was also observed in both tests, which is evidence that shear was a contributing factor in the failure mechanism.
Keerthan and Mahendran [6] developed suitable design equations for the shear capacity of LSBs (Vv) by including the available post-buckling strength and the additional level of fixity at the web-flange juncture. Shear capacities of LSBs reported in Table 5 were calculated based on Keerthan and Mahendran0 s improved design rules [6] (see Appendix A). Similarly, the section moment capacities (Mu) of LSBs tested in this study were based on the developed finite element models in [3–5] that were validated against section moment capacity test results [3]. It was found that these finite element models predicted the ultimate section moment capacities of LSBs with good accuracy. Table 5 shows the ratios of V/Vu and M/Mu for each test calculated using the applied shear force (V) and moments (M) at failure and the corresponding shear and section moment capacities (Vu and Mu). Figs. 15 and 16 show the interaction diagram of V/Vu versus M/Mu for LSBs subject to combined bending and shear actions. Ultimate capacity results for members with aspect ratios of 2.0, 2.5, 3.0, 3.5, and 4.0 are included in these figures. Shear capacity results from the experimental study by Keerthan and Mahendran [6] on LSBs with aspect ratios of 1.0 and 1.5 are also included in these figures. Fig. 15 includes the circular interaction diagram based on AS/NZS 4600 to allow a comparison with test results. This figure shows that the current Australian/New Zealand design code provisions based on Eq. (11) are inadequate for predicting the strengths of LSBs subject to combined bending and shear. The overconservative nature of the circular interaction equation in AS/NZS 4600 [12] is clearly evident from Fig. 15. Test results are also compared with current AS 4100 [13] design provisions in Fig. 15. AS 4100 [13] predictions based on Eqs. (13) and (14) compared reasonably well with test results, which reinforces that the behaviour of the LSB is more in line with that of hot-rolled sections. This is likely to be the resultant of the stiff hollow flanges
P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
1.4
1.4
1.2
1.2
1.0
1.0
0.8
0.8
V/Vu
V/Vu
138
0.6
Equation 15
Test_Shear
Equation 17
Equation 12
Equation 11
0.6 (V/V
0.4
0.4
Test_Combined
0.2
0.2
0.0
0.0 0
0.2
0.4
0.6
0.8
1
1.2
0.0
Fig. 15. Interaction diagrams for LSBs subject to combined bending and shear actions. (Mu based on FEA and Vu based on shear design rules with post-buckling from [6]).
which contribute to a large proportion of the bending capacity much the same as in hot-rolled beams. In this study, Eq. (15) was proposed as a means of more accurately predicting the capacities of LSBs subject to combined bending and shear actions. Fig. 16 shows the proposed combined bending and shear interaction diagram for LSBs. 4 4 M V þ r1 ð15Þ Mu Vu Eqs. (16) and (17) were also proposed as a linear alternative: for
M o 0:65M u
V ¼ V V ½1:65 ðM=M u Þ
for
ð16Þ 0:65M u r M r M u
ð17Þ
With reference to Fig. 16, Eqs. (15) and (17) form lower bound solutions to the experimental results. M and V can be replaced by Mn and Vn, and likewise Mu and VV can be replaced by ϕbMs and ϕvVv to produce equations which are consistent with AS/NZS 4600 provisions. By adopting a lower bound solution, Mn rϕbMs and Vn rϕvVv holds true for all points on the interaction diagram. Note that capacity reduction factors are already included in the calculation of design bending and shear capacities. Hence, there is no need to include any further capacity reduction factor in the lower bound solutions given above. The interaction diagrams shown in Figs. 15 and 16 assume that section moment capacities (Mu) are based on those from finite element analyses (FEA), which include inelastic reserve capacity. Whilst the section moment capacities determined from finite element analysis are greater than those which would be calculated using the current AS/NZS 4600 provisions, the use of such values is justifiable given that they lead to more conservative capacity predictions. Moreover, this approach ensures that Eqs. (15) and (17) are compatible with any future revisions to AS/NZS 4600 which allow for inelastic reserve capacity in LSBs. It is interesting to note that Eq. (15) is identical to that proposed by Shahabian and Roberts [11] for plate girders. Again, this tends to indicate that LSB behaviour is more consistent with that of hot-rolled sections rather than typical cold-formed sections. Fig. 16 also includes the interaction diagram based on Eq. (12) although AS/NZS 4600 states that this equation is for beams with transverse stiffeners. Comparison with test results shows that Eq. (12) is unable to predict the strengths of LSBs subject to combined
0.2
0.4
0.6
0.8
1.0
1.2
M/Mu
M/Mu
V ¼ VV
+ (M/M ) = 1
Fig. 16. Proposed interaction diagrams based on Eqs. (15) and (17) for LSBs subject to combined bending and shear actions. (Mu based on FEA and Vu based on shear design rules with post-buckling from [6]).
bending and shear compared to the similar linear alternative given by Eq. (17). Hence Eq. (12) is not considered suitable for LSBs. Fig. 17(a) shows the interaction diagram of M/Mu and V/Vu where Mu is calculated based on the direct strength method (DSM) and Vu is calculated based on Keerthan and Mahendran0 s [6] design rules with post-buckling strength. This figure shows that the section moment capacities predicted by the DSM are also reasonably accurate. Hence direct strength method (DSM) equations can also be used to determine the section moment capacities of LSBs instead of using finite element analyses of Anapayan and Mahendran [5]. Fig. 17(b) and (c) shows the interaction diagram of M/Mu and V/Vu where Mu is calculated based on finite element analyses and direct strength method (DSM), respectively and Vu is calculated based on Keerthan and Mahendran0 s [6] design rules without post-buckling strength. Since these shear design rules did not include the available post-buckling strength in LSBs, the ratios of V/Vu exceed 1 for test results. It shows that the shear capacities predicted by Keerthan and Mahendran0 s [6] design rules without post-buckling strength are conservative. Fig. 17(d) and (e) shows the interaction diagram of M/Mu and V/Vu where Mu is calculated based on finite element analyses and direct strength method (DSM), respectively and Vu is calculated based on AS/NZS 4600 [12] design rules. AS/NZS 4600 design rules assume that the web panel of lipped channel beam is simply supported at the web-flange juncture and hence the shear buckling coefficient of coldformed steel beams is only 5.34. Since AS/NZS 4600 design rules neither include the available post-buckling strength in LSBs nor the additional fixity at the web-flange juncture, the ratios of V/Vu vary from 0.8 to 2.07 for test results. It shows that the shear capacities predicted by AS/NZS 4600 design rules are very conservative.
5. Conclusions This paper has presented the details of an experimental investigation into the structural behaviour and design of LSBs subject to combined bending and shear actions. Eighteen tests were undertaken using a three point loading arrangement. It was found that noticeable reductions in shear capacity were observed when applied bending moments exceeded about 65% of the section moment capacity. Likewise, noticeable reductions in bending capacity were observed when applied shear forces exceeded about 65% of the shear capacity. LSBs which failed in a primary shear mode displayed well defined web yield zones and a diagonal post-buckling tension field mechanism. Minor local buckling of the
P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
1.4
1.4
1.2
1.2
1.0
1.0
0.8
V/Vu
V/Vu
139
0.6
0.8 0.6
0.4
0.4
0.2
0.2 0.0
0.0 0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
M/Mu
0.6
1
0.8
1.2
M/Mu
2.5
1.4
2.0
1.2 1.0 V/Vu
V/Vu
1.5 0.8
1.0
0.6 0.4
0.5 0.2 0.0
0.0 0
0.2
0.4
0.6
0.8
1
0
1.2
0.2
0.4
M/Mu
0.6
0.8
1
1.2
M/Mu
2.5 2.0
V/Vu
1.5 1.0 0.5 0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
M/Mu
Fig. 17. Comparison of test results with proposed interaction diagrams. (a) Mu based on DSM and Vu based on shear design rules with post-buckling from [6]. (b) Mu based on FEA and Vu based on shear design rules without post-buckling from [6]. (c) Mu based on DSM and Vu based on shear design rules without post-buckling from [6]. (d) Mu based on FEA and Vu based on AS/NZS 4600 design rules [12]. (e) Mu based on DSM and Vu based on AS/NZS 4600 design rules [12].
compression flange was also observed for such specimens. LSBs which failed in a primary bending mode displayed local buckling of the compression flange and yielding. AS 4100 [13] predictions compared reasonably well with the experimental results, which indicates that the behaviour of LSB is more in line with that of hot-rolled steel sections. Experimental results were compared with existing AS/NZS 4600 [12] design provisions based on a circular interaction equation for combined bending and shear actions. Such provisions were found to be overly conservative and to inadequately reflect the behaviour of the LSB. Two lower bound equations were proposed as a means of more accurately predicting the capacities of LSBs subject to combined bending and shear actions. It was found that current direct strength method (DSM) based equations can be used to determine the section moment capacities of LSBs.
Acknowledgements The authors would like to thank Australian Research Council for their financial support and the Queensland University of Technology
for providing the necessary facilities and support to conduct this research project.
Appendix A. Proposed DSM design equations for the shear capacity of LSBs (a) Shear capacity of LiteSteel beams without post-buckling Vv ¼1 V yw
λ r 0:815
Vv 0:815 ¼ λ V yw Vv 1 ¼ V yw λ2
0:815 o λ r 1:23
λ 4 1:23
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s f yw V yw d1 ¼ 0:815 λ¼ V cr tw EkLSB
ðA1Þ
ðA2Þ
ðA3Þ
ðA4Þ
140
P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
V yw ¼ 0:6f yw d1 t w
ðA5Þ
kLSB π 2 Et 3w V cr ¼ 12ð1 ν2 Þd1 For LSBs kss ¼ 4 þ
kLSB ¼ kss þ 0:87ðksf kss Þ 5:34
for
ða=d1 Þ2
kss ¼ 5:34 þ ksf ¼
ðA6Þ
4
5:34 ða=d1 Þ2
ksf ¼ 8:98 þ
þ
a o1 d1
For
ða=d1 Þ2
ðA8Þ
a Z1 d1
ðA9Þ
2:31 3:44 þ 8:39ða=d1 Þ ða=d1 Þ
5:61 1
ða=d Þ2
ðA7Þ
1:99 1
ða=d Þ3
For
For
a o1 d1
a Z1 d1
ðA10Þ
ðA11Þ
(b) Shear capacity of LiteSteel beams with post-buckling Option 1 Vv ¼1 V yw
λ r 0:815
ðA12Þ
Vv 0:815 0:815 þ 0:25 1 ¼ λ λ V yw Vv 1 1 ¼ 2 þ 0:25 1 2 V yw λ λ
0:815 oλ r 1:23
λ 4 1:23
ðA13Þ ðA14Þ
Option 2 Vv ¼1 V yw
λ r 0:815
Vv 1 0:15 ¼ 2 V yw λ λ
λ 40:815
ðA15Þ ðA16Þ
Eq. (A16) can also be written as follows. " # Vv V cr 0:5 V cr 0:5 ¼ 1 0:15 λ 40:815 V yw V yw V yw
ðA17Þ
References [1] OneSteel Australian Tube Mills, (OATM) (2008), Design of LiteSteel Beams, Brisbane, Australia. [2] Dempsey, RI. (1990), Structural behaviour and design of hollow flange beams. In: Proceedings of the second national structural engineering conference, Adelaide, Australia. [3] Anapayan T, Mahendran M, Mahaarachchi D. Section moment capacity tests of LiteSteel beams. Thin-Walled Struct 2011;49:502–12. [4] Anapayan T, Mahendran M. Improved design rules for hollow flange sections subject to lateral distortional buckling. Thin-Walled Struct 2012;50:128–40. [5] Anapayan T, Mahendran M. Numerical modelling and design of LiteSteel beams subject to lateral buckling. J Construct Steel Res 2012;70:51–64. [6] Keerthan P, Mahendran M. New design rules for the shear strength of LiteSteel beams. J Construct Steel Res 2011;67:1050–63. [7] Keerthan P, Mahendran M. Experimental studies on the shear behaviour and strength of LiteSteel beams. J Eng Struct 2010;32:3235–47. [8] LaBoube, RA, Yu, WW. (1978), Cold-formed steel web elements under combined bending and shear. In: Proceedings of the fourth international specialty conference on cold-formed steel structures, University of MissouriRolla, St Louis, Missouri, USA. [9] Bleich F. Buckling strength of metal structures. New York, USA: McGraw Hill; 1952. [10] Evans HR. Longitudinally and transversely reinforced plate girders. In: Plated structures, stability and strength. London, UK: Applied Science Publishers; 1983. [11] Shahabian F, Roberts TM. Behaviour of plate girders subjected to combined bending and shear loading. Sci Iran 2008;15:16–20. [12] Standards Australia/Standards New Zealand (SA) (2005), Australia/New Zealand Standard AS/NZS 4600 Cold-Formed Steel Structures, Sydney, Australia. [13] Standards Australia (SA) (1998). Australian Standard AS 4100 Steel Structures, Sydney, Australia. [14] Timoshenko SP, Gere JM. Theory of elastic stability. New York, USA: McGrawHill Book Co. Inc; 1961. [15] Lee SC, Yoo CH. Strength of plate girder web panels under pure shear. J Struct Eng, ASCE 1998;124:184–94. [16] Galambos TV. Guide to stability design criteria for metal structures. 5th ed.. New York, USA: John Wiley & Sons; 1998. [17] Bulson PS. The stability of flat plates. London, UK: Chatto and Windus; 1970. [18] Hancock GJ. Design of cold-formed steel structures to AS/NZS 4600:1996. 3rd ed.. Sydney, Australia: Australian Institute of Steel Construction; 1998. [19] American Iron and Steel Institute (AISI) (2007), North American specification for the design of cold-formed steel structural members, AISI, Washington, DC, USA.
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Experimental studies of hollow flange channel beams subject to combined bending and shear actions Poologanathan Keerthan, David Hughes, Mahen Mahendran n Science and Engineering Faculty, Queensland University of Technology, Brisbane, QLD 4000, Australia
art ic l e i nf o
a b s t r a c t
Article history: Received 11 November 2013 Received in revised form 3 December 2013 Accepted 3 December 2013 Available online 21 January 2014
This paper presents the details of an experimental study of a cold-formed steel hollow flange channel beam known as LiteSteel beam (LSB) subject to combined bending and shear actions. The LSB sections are produced by a patented manufacturing process involving simultaneous cold-forming and electric resistance welding. Due to the geometry of the LSB, as well as its unique residual stress characteristics and initial geometric imperfections resultant of manufacturing processes, much of the existing research for common cold-formed steel sections is not directly applicable to LSB. Experimental and numerical studies have been carried out to evaluate the behaviour and design of LSBs subject to pure bending actions and predominant shear actions. To date, however, no investigation has been conducted into the strength of LSB sections under combined bending and shear actions. Combined bending and shear is especially prevalent at the supports of continuous span and cantilever beams, where the interaction of high shear force and bending moment can reduce the capacity of a section to well below that for the same section subject only to pure shear or moment. Hence experimental studies were conducted to assess the combined bending and shear behaviour and strengths of LSBs. Eighteen tests were conducted and the results were compared with current AS/NZS 4600 and AS 4100 design rules. AS/NZS 4600 design rule based on circular interaction equation was shown to grossly underestimate the combined bending and shear capacities of LSBs and hence two lower bound design equations were proposed based on experimental results. Use of these equations will significantly improve the confidence and costeffectiveness of designing LSBs for combined bending and shear actions. & 2013 Elsevier Ltd. All rights reserved.
Keywords: LiteSteel beam Cold-formed steel structures Hollow flange Bending and shear Interaction Experimental study
1. Introduction Cold-formed steel members are used extensively in residential, commercial and industrial steel buildings due to the availability of advanced roll-forming processes and high strength (4550 MPa) and very thin ( o1 mm) steels, strict quality standards, and enhanced efficiency and economy over hot-rolled alternatives. Common cold-formed sections include Z- and C-sections, and the torsionally rigid hollow sections (SHS, RHS and CHS), for which a significant number of studies has been completed to develop suitable capacity equations. Since early 1990s, Australian companies such as OneSteel Australian Tube Mills [1] have produced innovative cold-formed hollow flange sections known as hollow flange beams (HFB) and LiteSteel beams (LSB) (see Fig. 1). The development of these sections was based on improving structural efficiency by adopting torsionally rigid hollow flanges, minimising local buckling of plate
n
Corresponding author. Tel.: þ 61 434070139; fax: þ 61 731381170. E-mail addresses: [email protected] (P. Keerthan), [email protected] (M. Mahendran). 0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.12.003
elements by eliminating free edges, distributing material away from the neutral axis to afford greater bending stiffness than conventional cold-formed sections, and optimising manufacturing efficiency. The HFB sections were produced from a single steel strip using a combined dual electric resistance welding and automated continuous roll-forming process [2]. Further developments in this unique manufacturing process and the need to facilitate easier connections between members led to the release of LSB, a hollow flange channel section in 2005, primarily for use as flexural members in residential and light commercial/industrial applications. Table 1 gives the nominal dimensions of LSB sections. The base steel used for LSB production has a yield strength of 380 MPa and a tensile strength of 490 MPa. However, due to cold-forming, the nominal yield strengths of the web and flange elements are 380 and 450 MPa, respectively [1]. The manufacturing process also introduces residual stresses and initial geometric imperfections which differ from those of common cold-formed and hot-rolled sections. Whilst the LSB is similar to the HFB in many respects, it is monosymmetric and possesses rectangular hollow flanges rather than triangular. Due to the geometry of the LSB, as well as its unique residual stress characteristics and initial geometric imperfections
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d1
Fig. 1. Hollow flange sections. (a) Hollow flange beam. (b) LiteSteel B eam.
Table 1 Nominal dimensions of LiteSteel beam sections (mm) [1]. LSB section
d
bf
df
t
ro
riw
300 75 3.0 300 75 2.5 300 60 2.0 250 75 3.0 250 75 2.5 250 60 2.0 200 60 2.5 200 60 2.0 200 45 1.6 150 45 2.0 150 45 1.6
300 300 300 250 250 250 200 200 200 150 150
75 75 60 75 75 60 60 60 45 45 45
25.0 25.0 20.0 25.0 25.0 20.0 20.0 20.0 15.0 15.0 15.0
3.0 2.5 2.0 3.0 2.5 2.0 2.5 2.0 1.6 2.0 1.6
6.00 5.00 4.00 6.00 5.00 4.00 5.00 4.00 3.20 4.00 3.20
3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
n
Refer Fig. 1(b) for notation.
resultant of manufacturing processes, much of the existing research for common cold-formed sections is not likely to be directly applicable to the LSB. Many studies have been conducted into the structural behaviour and design of LSBs. Both laboratory testing and finite element analysis have been completed to evaluate the behaviour and design of LSBs subject to pure bending [3–5] and shear [6,7], respectively. The above mentioned research has significantly improved the understanding of the structural behaviour of LSBs, particularly in flexural applications. To date, however, no investigation has been conducted into the strength of LSB sections under combined bending and shear actions. Combined bending and shear is especially prevalent at the supports of continuous spans and cantilever beams, where the interaction of high shear force and bending moment can reduce the capacity of a section to well below that for the same section subject only to pure shear or moment. The behaviour of steel beams in combined bending and shear has been investigated by numerous researchers, including LaBoube and Yu [8], Bleich [9], Evans [10], and Shahabian and Roberts [11], with provisions for the design of members subject to such loading included in both AS/NZS 4600 [12] and AS 4100 [13]. Design of LSBs is governed by the Australian cold-formed steel structures code, AS/NZS 4600 [12]. Due to the geometry of the LSB, as well as its unique residual stress characteristics and initial geometric imperfections, it is uncertain whether the design rules for combined bending and
shear outlined in AS/NZS 4600 are appropriate for use with the LSB. First, AS/NZS 4600 [12] suggests a circular interaction equation for the design of flexural members subject to combined bending and shear actions. LaBoube and Yu [8] performed an experimental investigation of the combined bending and shear behaviour of cold-formed channel sections without transverse stiffeners, and identified that a circular relationship for bending and shear interaction is quite conservative. The circular equation was originally derived for individual disjointed plates in combined bending and shear, and is not accurate when applied to webs that are restrained by flanges. This is likely to be especially true for LSB given the significant rigidity of its flanges. For the channel sections used in the study by LaBoube and Yu [8], it was shown that when the applied bending moment was less than approximately 50% of the moment capacity, the full shear capacity was developed. Likewise, little or no reduction in the moment capacity resulted until the applied shear was approximately 65% of the shear capacity [8]. It is expected that similar limits may also apply to LSBs, which would allow more economical design than a circular interaction equation. Second, most studies concerned with combined bending and shear have focused on hot-rolled plate girders or cold-formed members with “open” flanges rather than hollow. Whether the results of such research apply equally to cold-formed LSB sections with hollow flanges is not known. For example, Keerthan and Mahendran [6,7] have shown that the hollow flanges of the LSB provide nearly fixed restraint to the web, which has a significant effect on the shear buckling coefficient and shear buckling strength of LSB. AS/NZS 4600, however, currently considers only simply supported conditions at the edges of the web panel. Keerthan and Mahendran [6,7] have also shown that significant post-buckling strength is available for slender LSBs. Although such post-buckling capacity is ignored in AS/NZS 4600 provisions, it may affect the combined bending and shear behaviour of LSBs. An evaluation of LSBs subject to combined bending and shear is therefore considered important so that refinements to design rules can be made to more adequately reflect the behaviour of the innovative LSB sections. This paper presents the details of an experimental study on the behavior of LSBs subject to combined bending and shear actions and its results.
2. Review of the behaviour of steel beams subject to shear, bending and combined bending and shear actions 2.1. Shear Timoshenko and Gere [14] developed Eq. (1) for the critical elastic buckling stress (fcr) of a rectangular plate in compression, bending or shear, which is a function of plate width (b), thickness (t), Poisson0 s ratio (ν), Young0 s modulus of elasticity (E), and plate buckling coefficient (k). 2 π2 E t f cr ¼ k ð1Þ 12ð1 ν2 Þ b Eq. (1) can be rewritten as an expression for the critical shear buckling stress of a rectangular web plate, replacing fcr with τcr, plate thickness (t) with web thickness (tw), plate width (b) with web panel depth (d1), and substituting the shear buckling coefficient (kv). 2 π2 E tw τcr ¼ kv ð2Þ 12ð1 ν2 Þ d1 The shear buckling coefficient itself is a function of the web panel aspect ratio (a/d1, where a is the shear panel length) and the
P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
degree of fixity at the plate edges. Lee and Yoo [15] showed that for plate girders the support condition is closer to fixed than simple. Similar comments have been made regarding the flangeweb support condition for LiteSteel beams [6,7]. In past research and design provisions, however, a simple flange-web support has often been conservatively assumed, which has led to an underestimation of shear buckling strength. Keerthan and Mahendran [6] developed suitable design equations for the shear capacity of LSBs (Vv) by including the available post-buckling strength and increased shear buckling coefficient (kv). Suitable shear design rules were also developed under the direct strength method (DSM) format. They are given in Appendix A of this paper.
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In the case of cold-formed steel sections, Hancock [18] notes that the ultimate moment capacity is typically less than the plastic moment capacity based on a fully effective cross-section since cold-formed sections are not usually compact. However, compact cold-formed sections generally have inelastic reserve capacity in bending. For example, Anapayan et al. [3] showed the presence of inelastic reserve capacity for LSB sections in bending using both experiments and numerical studies. However, AS/NZS 4600 permits members to be designed based on inelastic reserve capacity only if they satisfy a number of strict criteria, including limits on the shear force taken by the web. The section moment capacities of LSBs could be calculated based on AS/NZS 4600 [12]. The section moment capacity (Ms) is defined in AS/NZS 4600 as follows: M s ¼ f yf Z e
ð3Þ
2.2. Bending Members subject to pure bending generally fail due to lateral– torsional buckling, inward buckling of the compression flange, or flange yielding [16]. In the case of hollow flange beams, failure can also result from lateral–distortional buckling, which is characterised by simultaneous lateral deflection, twist and crosssectional change due to web distortion [3–5]. Discussion herein will focus only on members with full lateral restraint. Slender flange elements subject to uniform compression are prone to local buckling effects, with the critical buckling stress given by Eq. (1), using k¼ 4.0 for plate elements with both longitudinal edges simply supported, and k ¼0.425 for elements with one longitudinal edge simply supported and the other free. Following local buckling, plate elements in compression can develop significant post-buckling capacity, with failure occurring when the highly stressed areas reach yield. AS/NZS 4600 allows for local buckling of compression elements using an effective width concept. If the web of a flexural member is relatively slender, the portion of the web which is in compression is also prone to local buckling. The elastic critical buckling stress of a web in bending may be determined from Eq. (1), substituting tw for t, and d1 for b. The buckling stress (fcr) is then the compressive stress at the extreme plate fibres in pure bending to cause initial elastic buckling. For a web panel with a simply supported flangeweb junction, k¼ 23.8 whereas k ¼39.6 for a fixed flange-web junction [17].
The effective section modulus (Ze) is based on the initiation of yielding in the extreme compression fibre and therefore does not allow for the inelastic reserve capacity of the section [12]. The effects of local buckling are accounted for by using reduced widths (be) of slender elements in compression for the calculation of the effective section modulus (see Eq. (4)). be ¼
1 0:22=λ brb λ
where 1:052 b λ ¼ pffiffiffi k t
ð4Þ
sffiffiffiffiffi n f E
where k ¼local buckling coefficient, f n ¼applied stress. The section moment capacities of all the LSB sections could be calculated using the AS/NZS 4600 method described above, with local buckling coefficients (k) equal to 4 and 24 for the compression flange and web, respectively. The direct strength method (DSM) is an alternative to the traditional effective width method (EWM) and has been adopted as an alternative design procedure in AISI S100 [19] and AS/NZS 4600 [12]. The nominal section moment capacity at local buckling (Msl) can be determined from Section 7.2.2.3 of AS/NZS 4600 [12]. For, λl r 0:776
M sl ¼ M y
ð5Þ
Table 2 Section moment capacities of LSBs from Anapayan et al.0 s tests [3]. Test no.
LSBs
tfo (mm)
tfi (mm)
tw (mm)
df (mm)
bf (mm)
fyo (MPa)
My (kN m)
Test Mu (kN m)
Mol (kN m)
λ
Mu/My
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
300 75 3.0 300 75 2.5 300 60 2.0 250 75 3.0 250 75 2.5 250 60 2.0 200 60 2.5 200 60 2.0 200 45 1.6 150 45 2.0 150 45 1.6 125 45 2.0 125 45 1.6 300 75 3.0 300 60 2.0 250 75 2.5 250 60 2.0 200 45 1.6 150 45 2.0 150 45 1.6
3.18 2.87 2.15 3.08 2.79 2.09 2.58 2.03 1.56 2.11 1.60 1.98 1.62 3.22 2.22 2.90 2.18 1.79 2.22 1.77
3.18 2.87 2.15 3.08 2.79 2.09 2.58 2.03 1.56 2.11 1.60 1.98 1.62 3.13 2.02 2.60 2.02 1.66 2.02 1.63
2.84 2.51 1.98 2.77 2.48 1.96 2.34 1.85 1.48 1.89 1.60 1.98 1.62 3.09 1.98 2.54 1.95 1.61 1.97 1.58
75.31 75.24 60.28 76.35 75.98 60.47 60.23 60.15 45.05 44.95 45.12 45.10 45.07 74.60 60.00 75.00 60.40 45.50 45.40 45.20
25.17 25.05 19.97 25.22 24.92 20.12 19.95 20.31 14.98 14.73 14.89 14.93 14.95 24.80 19.80 25.50 20.40 15.20 14.80 14.80
528 511 568 506 525 580 496 473 478 498 540 503 549 498 558 552 523 537 538 558
94.27 82.10 59.94 68.25 64.05 46.45 35.92 26.96 17.54 16.51 13.75 12.14 10.96 89.77 57.89 66.35 41.88 21.63 17.93 14.92
103.90 85.80 52.40 77.89 71.49 47.33 52.47 31.80 17.36 19.63 14.94 14.38 12.95 93.00 53.36 70.68 42.12 20.88 20.20 16.18
179.20 126.70 46.75 203.50 148.30 53.63 121.40 59.87 21.22 64.38 34.41 53.77 29.44 209.40 45.51 149.20 52.84 27.44 68.72 35.69
0.73 0.80 1.13 0.58 0.66 0.93 0.54 0.67 0.91 0.51 0.63 0.48 0.61 0.65 1.13 0.67 0.89 0.89 0.51 0.65
1.10 1.05 0.87 1.14 1.12 1.02 1.46 1.18 0.99 1.19 1.09 1.18 1.18 1.04 0.92 1.07 1.01 0.97 1.13 1.08
Note: Refer Fig. 1(b) for bf and df while tw, tfo and tfi are the measured web, outside and inside flange thicknesses.
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1.4 1.2
1.0
S
S1
Mu/My
1.0 0.8
DSM (Equations 5 and 6) Tests
0.6
Vult/Vyw
0.4
Shear Type Mechanism
0.2 Flange Criterion Controls
0.0 0.0
0.2
0.4
0.6
λl =
0.8
1.0
1.2
1.4
1.0
My Mol
M/Mp Fig. 3. Bending and shear interaction diagram [10].
Fig. 2. Comparison of experimental section moment capacities of LSBs from [3] with DSM based design equations.
"
λl 40:776
M ol M sl ¼ 1 0:15 My
0:4 # M ol 0:4 My My
Eq. 9
ð6Þ
Eq. 11 Eq. 10
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi My λl ¼ M ol My ¼ Z f f y where Mol is elastic local buckling moment of the section, M ol ¼ Z f f ol , Zf is elastic section modulus, fol is elastic local buckling stress of the section in bending. Anapayan et al. [3] conducted 20 section moment capacity tests to investigate the behaviour and strength of LSB flexural members. The ultimate section moment capacities from their tests (Mu) are presented in Table 2 along with the measured dimensions and outside flange yield stress (fyo) of tested LSBs used in calculating the first yield moments (My). Elastic local buckling moments of LSBs (Mol) were determined using the finite strip analysis program THIN-WALL, and are also given in Table 2. Fig. 2 compares the test ultimate moment capacities with the section moment capacities predicted by the direct strength method (DSM) using the nondimensional format of Mu/My versus (My/Mol)0.5. It can be seen that compact LSB sections have greater moment capacities than their first yield moments (My). Fig. 2 shows that the direct strength method based equations are able to predict the section moment capacities of LSB sections with reasonably good accuracy (conservative predictions). Anapayan et al. [3] also developed suitable finite element models of LSBs to predict their section moment capacities of LSBs, which agreed well with test results given in Table 2. Hence these LSB finite element models can also be used to accurately predict their section moment capacities. 2.3. Bending and shear interaction The interaction between shear force and bending moment can have a significant effect on the strength of a section, and is generally presented in the form of an interaction diagram (see Figs. 3 and 4). Bleich [9] originally proposed a circular interaction Eq. (7) is for the design of disjointed flat rectangular plates subject to combined bending and shear. 2 2 fb τ þ r 1:0 ð7Þ τcr f cr
Fig. 4. Bending and shear interaction diagram [8].
where fb is theapplied compressive bending stress, fcr is the theoretical buckling stress in pure bending, τ is the applied shear stress, and τcr is the theoretical buckling stress in pure shear. Whilst Eq. (7) remains the basis of AS/NZS 4600 design provisions for bending and shear interaction in unstiffened flexural members, numerous studies have been completed which show a circular interaction equation to be overly conservative. The circular interaction equation was originally derived for individual disjointed plates in combined bending and shear, and is not likely to be accurate when applied to webs restrained by flanges [8]. Evans [10] presented the interaction diagram shown in Fig. 3 for stiffened plate girders and proposed a somewhat complicated method to determine the relevant points on the diagram. From point S to S1, it was assumed that little reduction in shear capacity resulted due to applied moment, and the capacity of the section was that for pure shear. Gradual reductions in shear capacity due to bending moment were considered as shown in Fig. 3. Shahabian and Roberts [11] developed a simpler approach to bending and shear interaction in response to the complexity of Evans0 method. Based on investigations of plate girders with web panel aspect ratios between 1 and 2, and web slenderness ratios between 150 and 300, the following empirical equation was recommended. 4 4 M V þ r 1:0 ð8Þ Mu Vu where Vu and Mu are the pure shear and moment capacities, respectively, V and M are the applied shear and moment, respectively.
P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
LaBoube and Yu [8] performed an experimental investigation of the combined bending and shear behaviour of cold-formed steel channel sections without transverse stiffeners. For the channels used in their study, it was shown that when the applied bending moment was less than approximately 50% of the moment capacity, the full shear capacity was developed (Fig. 4). Likewise, little or no reduction in the moment capacity resulted until the applied shear was approximately 65% of the shear capacity [8]. Based on the results of the study, LaBoube and Yu [8] presented the following interaction equation. ( ) M 2 M V 1:077 0:732 ð9Þ þ 1:100 r 1:0 Mu Mu Vu
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Clause 5.12.3 of AS 4100 [13] presents suitable design rules for members subject to bending and shear interaction. It is assumed that no reduction in the shear capacity of a section will occur until the applied moment (M) is greater than 75% of the design section moment capacity. Conversely, reductions in the section moment capacity are assumed to occur only when the applied shear force is greater than 60% of the design shear capacity. This approach is similar to that adopted by AS/NZS 4600 [12] for beams with transverse stiffeners. The following equations give the shear capacity (Vvm) of hot-rolled sections in the presence of bending. V vm ¼ V v
for
M r 0:75M u
1:6M V vm ¼ V v 2:2 Mu
ð13Þ
for
0:75M u r M rM u
ð14Þ
Eq. (9) was simplified as: M V þ r 1:6 Mu V u
ð10Þ
Fig. 4 shows the experimental results from LaBoube and Yu0 s study, plotted against Eqs. (9) and (10). Despite these successful research outcomes for cold-formed steel channel sections, bending and shear interaction for beams with unstiffened webs is allowed for in Clause 3.3.5 of AS/NZS 4600 [12] by the circular interaction equation only. 2 2 M V þ r 1:0 ð11Þ Mu Vv where Mn is the design bending moment, Vn is the design shear force, Mu is the nominal section moment capacity, Vv is the nominal shear capacity of the web. For beams with transverse stiffeners, bending and shear interaction is only considered to take effect if the design moment (Mn) is greater than 50% of the design section moment capacity and the design shear force (Vn) is greater than 70% of the design shear capacity. Bending and shear interaction is then governed by the following equation. M V M V 0:6 þ r 1:3 where 40:5 and 40:7 ð12Þ Mu Vv Mu Vv
As discussed in this section, there are a number of bending and shear interaction equations that can be used for steel beams. However, their applicability to LiteSteel beams subject to combined bending and shear is not known. The following section presents the details of the experimental study undertaken for this purpose.
3. Experimental study of LiteSteel beams subject to combined bending and shear 3.1. General An experimental study was conducted to evaluate the behavior and strength of LSB sections subject to combined bending and shear actions. The results obtained from the study were used for the purposes of developing suitable design capacity equations for combined bending and shear, and validation of subsequent finite element analyses. Eighteen full-scale tests were performed on five different LSB sections with web panel aspect ratios of 2.0, 2.5, 3.0, 3.5, 3.9 and 4.0 (see Table 3). They were based on testing of simply supported back to back LSBs subject to a mid-span load (see Figs. 5 and 6). This section gives the details of the mechanical and geometric properties of LSB test specimens, and discusses the test set-up and
Table 3 Measured geometric and mechanical properties of LSB test specimens. Test no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
LSB section
150 45 1.6 150 45 1.6 150 45 1.6 150 45 1.6 150 45 1.6 150 45 2.0 150 45 2.0 150 45 2.0 150 45 2.0 150 45 2.0 200 45 1.6 200 45 1.6 200 45 1.6 200 45 1.6 200 60 2.0 200 60 2.0 250 60 2.0 250 60 2.0
Aspect ratio a/d1
2.0 2.5 3.0 3.5 3.9 2.0 2.5 3.0 3.5 3.9 2.0 2.5 3.0 4.0 2.0 3.5 2.0 3.0
d1 (mm)
121.3 120.0 120.1 120.0 120.0 121.2 120.0 120.0 120.9 120.0 170.0 169.6 170.7 170.8 161.5 160.0 212.2 211.8
Yield stress (MPa)
Thickness (mm)
fyo
fyi
fyw
tfo
tfi
tw
558 558 558 558 558 538 538 538 538 538 537 537 537 537 521 521 523 523
488 488 488 488 488 492 492 492 492 492 491 491 491 491 471 471 473 473
454 454 454 454 454 423 423 423 423 423 452 452 452 452 440 440 452 452
1.77 1.76 1.78 1.77 1.77 2.25 2.25 2.24 2.26 2.25 1.66 1.66 1.68 1.67 2.12 2.10 2.19 2.20
1.66 1.66 1.65 1.66 1.67 2.08 2.09 2.08 2.07 2.08 1.61 1.60 1.61 1.60 2.02 2.01 2.04 2.03
1.60 1.58 1.58 1.58 1.58 1.99 1.97 1.98 1.97 1.97 1.58 1.58 1.58 1.59 1.96 1.97 1.97 1.99
bf (mm)
df (mm)
45.4 45.6 45.4 45.5 45.4 45.5 45.6 45.5 45.4 45.5 45.6 45.4 45.5 45.4 59.9 59.9 60.2 60.1
14.8 14.7 14.8 14.9 14.8 14.8 14.9 14.8 14.7 14.8 15.3 15.0 15.1 14.9 20.4 20.5 20.7 20.7
Note: Refer Fig. 1(b) for bf, df and d1 while tw, tfo and tfi are the measured web, outside and inside flange thicknesses. fyw, fyo and fyi are the measured web, outside and inside flange yield stresses. All thicknesses represent base metal thickness (BMT) exclusive of coating thickness (0.03 mm).
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Table 4 Predicted failure modes of tested LSB specimens. Test no.
Fig. 5. Details of combined bending and shear test arrangement. (a) Schematic diagram of test set-up. (b) Shear force and bending moment diagrams.
Applied Load
30 mm Gap Back to Back LSBs
Web Side Plates
T-Shaped Stiffener LVDTs
Roller Support
Fig. 6. Experimental set-up used for combined bending and shear tests.
procedure used in the combined bending and shear tests. Experimental results and observations are also described in this section. 3.2. Selection of test specimens In selecting LSB sections and spans for inclusion in the experimental study, it was noted that specimens should ideally represent the full range of shear failure modes (yielding, inelastic buckling and elastic buckling) and section slenderness classifications relevant to bending failure (compact, non-compact and slender section based on AS 4100). Moreover, test spans should give an indication of LSB behaviour over a wide range of applied shear force and bending moment combinations. Keerthan and Mahendran [6] studied the primarily shear behaviour of LSBs with web panel aspect ratios up to 1.5, which was regarded as the lower limit in this study on combined bending and shear. It is considered that LSB specimens with an aspect ratio greater than 1.5 will exhibit the combined effects of bending and shear. In deciding on an upper limit for the web panel aspect ratio,
LSB section
Aspect ratio (a/d1)
d1/tw Predicted shear failure mode
1
150 45 1.6 2.0
75.8 Inelastic buckling
2
150 45 1.6 2.5
75.9 Inelastic buckling
3
150 45 1.6 3.0
76.0 Inelastic buckling
4
150 45 1.6 3.5
75.9 Inelastic buckling
5
150 45 1.6 3.9
75.9 Inelastic buckling
6 7 8 9 10 11 12 13 14 15
150 45 2.0 150 45 2.0 150 45 2.0 150 45 2.0 150 45 2.0 200 45 1.6 200 45 1.6 200 45 1.6 200 45 1.6 200 60 2.0
16
200 60 2.0 3.5
17 18
250 60 2.0 2.0 250 60 2.0 3.0
2.0 2.5 3.0 3.5 3.9 2.0 2.5 3.0 4.0 2.0
60.9 60.9 60.6 61.4 60.9 107.6 107.3 108.0 107.4 82.4
Yielding Yielding Yielding Yielding Yielding Elastic buckling Elastic buckling Elastic buckling Elastic buckling Inelastic buckling
81.2 Inelastic buckling 107.7 Elastic buckling 106.4 Elastic buckling
Section slenderness Noncompact Noncompact Noncompact Noncompact Noncompact Compact Compact Compact Compact Compact Slender Slender Slender Slender Noncompact Noncompact Slender Slender
attention was given to the investigation by Anapayan et al. [3] into the experimental section moment capacities of LSBs using four point loading conditions. Anapayan et al. [3] identified that a 150 45 1.6 LSB displayed shear failure characteristics at spans of up to 1000 mm. However, bending failure was the governing mode when the span was increased to 1500 mm. The web panel aspect ratio for the latter arrangement was approximately equal to 4, which was therefore taken as the upper limit in this study. Table 3 lists the 18 LSB sections that were tested in this experimental study. For the purposes of this study, the length of a web panel (a) was taken to be the distance between the centrelines of the inside pairs of bolts (see Fig. 5a). For LSBs d1 is defined as the clear height of web instead of the depth of the flat portion of web measured along the plane of the web as defined in AS/NZS 4600 [12] for cold-formed channel sections. The reasons for this are given in Keerthan and Mahendran [6]. Table 4 shows the predicted failure modes of LiteSteel beam specimens. The section moment classification of LSBs in Table 4 was determined based on the Australian steel structures code AS 4100 [13]. It was based on the plate slenderness calculations for LSBs using the measured dimensions and yield stresses given in Table 3. Although AS 4100 design rules are not applicable to the cold-formed LSBs, the AS 4100 [13] section classification method was used for LSBs since AS/NZS 4600 [12] design rules do not allow any inelastic reserve bending strength for cold-formed steel beams in general, and limit their section moment capacities to their first yield moments. Shear failure mode of LiteSteel beams was predicted based on Keerthan and Mahendran [6]. 3.3. Geometric and mechanical properties of test specimens Table 3 presents the measured cross-sectional dimensions, thicknesses and yield strengths of LSB test specimens. All the LSB specimens tested as part of this study and those used in the experimental shear investigation by Keerthan and Mahendran [6] were from the same production run. The web and flange yield strengths of such specimens (fyw, fyo and fyi) were determined previously through tensile tests conducted by Keerthan and
P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
Mahendran [6], respectively (see Table 3 for results). The yield strengths in Table 3 were adopted for specimens tested in this study. The thicknesses and cross-sectional dimensions of all LSB specimens were determined prior to testing, based on averages of measurements recorded at a minimum of three locations. Element thicknesses were measured to an accuracy of onehundredth of a millimetre using a micrometer screw gauge, whilst all cross-sectional dimensions were typically measured to the nearest five-hundredths of a millimetre using vernier calipers. LSBs are protected by an aluminium–zinc alloy coating which is not considered to provide any strength [1]. Keerthan and Mahendran [6] measured the thicknesses of coated LSB coupons before and after being immersed in a hydrochloric acid solution, which effectively removes the aluminium–zinc coating. On average, the coating thickness was found to be 0.015 mm on each face. In Table 3, base metal thicknesses (BMT) are given, which were determined by subtracting a coating thickness of 0.03 mm from the measured total coated thickness (TCT). Combined bending and shear tests were conducted on pairs of LSBs connected back-toback and hence the dimensions given in Table 3 are the average of those recorded for each specimen in the assembled pair. 3.4. Experimental test set-up and procedure All LSB specimens were tested using the Tinius Olsen machine in the Structures Laboratory. The experimental set-up was based on the arrangement shown in Figs. 5 and 6. LSB sections were tested in pairs bolted back-to-back using T-shaped stiffeners at the supports and loading point, with a 30 mm gap between them to ensure independent behaviour. A concentrated load was applied through the T-shaped stiffener at mid-span. As loads were applied directly to the beam webs and close to the shear centre, bearing failure (including web crippling) and torsional loading were avoided. High strength steel bolts (M16 8.8/S) were used to avoid any bolt failure during testing. Specimens were simply supported at each end on machine ground and lubricated half-rounds placed upon ball bearings, which ensured free rotation at the supports (see Fig. 6). Ten millimeter thick web side plates were used at the supports and loading point to prevent lateral flange movement, with the web side plate height being approximately equal to the clear height of the web. Assembled pairs of LSB sections were positioned in the testing rig, with care taken to align the T-shaped loading stiffener with the
135
centre of the testing machine cross-head. Longer specimen spans were accommodated by placing the half-round supports on a hotrolled universal column section which was set across the Tinius Olsen machine and propped at one end. The hot-rolled section was deemed to be sufficiently stiff so as not to introduce any significant error in deflection measurements. To measure mid-span vertical deflections in each test, 25 mm linear variable displacement transducers (LVDT) were positioned beneath the bottom flange of LSB specimens. An EDCAR data acquisition system was employed to record all load and deflection data from the commencement of testing until failure. At the commencement of testing, a small load was applied to allow the loading and support systems to settle evenly on the bearings. The measuring system was subsequently zeroed and loading initiated. Loading was applied at a constant rate of 1 mm/min until specimen failure. Testing was terminated shortly after reaching the ultimate load to avoid uncontrolled specimen collapse and potential damage to test equipment. 3.5. Experimental results As identified earlier, the experimental study was designed for the purposes of collecting section strength data and observing failure behavior of LSBs subject to combined bending and shear actions. Such results are critical for the validation of finite element analyses and form an important part of the development of suitable design equations. Table 5 summarizes the ultimate loads (Pu) applied in each of the combined bending and shear tests. Also included are the applied maximum shear forces (V) and bending moments (M) in each section at the ultimate load. The applied shear force in each LSB was taken to be Pu/4, assuming half of the load transferred through the T-shaped stiffener to be resisted by a single LSB section. The moment applied at mid-span to each section was calculated as the shear force multiplied by the shear span, a (see Fig. 5a). Specimens were typically observed to fail in a combined bending and shear mode, which was characterised by a combination of shear web buckling/yielding and some degree of flange buckling/yielding. In all members subject to combined bending and shear failures, local buckling/yielding of the compression flange typically occurred near the mid-span loading point (that is, the location of maximum bending moment). It should be noted that no signs of lateral buckling were observed during
Table 5 Summary of experimental results. Test no.
LSB section
Aspect ratio
Ultimate load (Pu) Applied shear force at (kN) failure V (kN)
Applied moment at failure M Shear capacity Vu Moment capacity Mu (kN m) (kN) (kN m)
V/ Vu
M/ Mu
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
150 45 1.6 150 45 1.6 150 45 1.6 150 45 1.6 150 45 1.6 150 45 2.0 150 45 2.0 150 45 2.0 150 45 2.0 150 45 2.0 200 45 1.6 200 45 1.6 200 45 1.6 200 45 1.6 200 60 2.0 200 60 2.0 250 60 2.0 250 60 2.0
2.0 2.5 3.0 3.5 3.9 2.0 2.5 3.0 3.5 3.9 2.0 2.5 3.0 4.0 2.0 3.5 2.0 3.0
178.6 163.5 151.6 133.5 124.4 233.9 207.9 196.4 175.2 157.5 185.9 170.2 155.0 129.6 267.1 208.9 277.3 230.4
10.83 12.26 13.66 14.02 14.55 14.18 15.59 17.68 18.53 18.43 15.80 18.04 19.84 22.14 21.57 29.24 29.42 36.59
0.94 0.90 0.84 0.74 0.70 0.96 0.87 0.81 0.73 0.66 0.96 0.90 0.83 0.70 0.94 0.75 0.92 0.78
0.72 0.81 0.90 0.93 0.96 0.75 0.83 0.94 0.98 0.98 0.77 0.88 0.97 1.09 0.69 0.94 0.72 0.90
44.65 40.88 37.90 33.38 31.10 58.48 51.98 49.10 43.80 39.38 46.47 42.56 38.75 32.40 66.77 52.21 69.32 57.59
47.30 45.49 45.11 44.86 44.72 61.16 59.94 60.25 60.39 59.94 48.58 47.41 46.65 46.59 71.28 69.74 75.49 74.23
15.10 15.10 15.10 15.10 15.10 18.90 18.90 18.90 18.90 18.90 20.40 20.40 20.40 20.40 31.10 31.10 40.80 40.80
136
P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
Bending
Shear
Fig. 7. Failure mode of 250 60 2.0 LSB (aspect ratio¼2.0). (a) Shear buckling in web. (b) Local flange buckling.
Local Flange Buckling
Shear Yielding Fig. 8. Shear yielding of 150 45 2.0 LSB (aspect ratio¼2.0).
Shear Buckling
Local Flange Buckling
Shear Buckling Fig. 10. Failure mode of 250 60 2.0 LSB (aspect ratio¼ 3.0). (a) Local flange buckling. (b) Shear buckling in web.
Fig. 9. Failure mode of 150 45 1.6 LSB (aspect ratio¼ 3.0).
experimental tests. All specimens displayed significant ductility at failure. Figs. 7–10 show the failure modes of LSBs while Figs. 11–14 show the plots of applied load versus vertical deflection for tested LSBs.
3.5.1. Test specimens failed primarily due to shear (aspect ratio 2.0) Specimens with aspect ratios of 2.0 failed primarily due to shear, with only minor evidence of flange buckling. The failure mode observed for the 250 60 2.0 LSB with an aspect ratio of 2.0 is typical of that for all sections which failed primarily due to shear (Fig. 7). The load–deflection plot from this test is given in
Fig. 12, which shows near linear load–deflection behaviour right up to the point of ultimate load. This was followed by unloading upon failure of the web, with the ultimate load coinciding with relatively low values of mid-span deflection. Diagonal shear web buckling was seen to occur at loads close to the ultimate load. However, it was not possible to accurately determine the web buckling load and hence post-buckling shear capacity during testing. It shows that some post-buckling shear capacity was clearly developed through tension field action. Local compression flange buckling was also initiated close to the ultimate load. Evidence of web shear buckling is shown in Fig. 7a. Shear buckling was characterised by the appearance of web buckles substantially before the ultimate load. Fig. 7b shows local flange buckling in the 250 60 2.0 LSB with an aspect ratio of 2.0.
P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
250
300
Applied Load (kN)
250 Applied load (kN)
137
200 150
LSB 1
100
LSB 2
200 150 LSB 1 LSB 2
100 50
50 0 0
2
4
6
8
0
10
0
2
Vertical Deflection (mm)
6
8
10
Fig. 12. Applied mid-span load versus vertical deflection for 150 45 1.6 LSB (aspect ratio¼ 3.0).
250
250
200
200
Applied load (kN)
Applied Load (kN)
Fig. 11. Applied mid-span load versus vertical deflection for 250 60 2.0 LSB (aspect ratio¼ 2.0).
150 LSB 1 LSB 2
100
4
Vertical Deflection (mm)
150 LSB 1 LSB 2
100 50
50
0
0 0
2
4
6
8
10
Vertical Deflection (mm)
0
2
4
6
8
10
Vertical Deflection (mm)
Fig. 13. Applied mid-span load versus vertical deflection for 150 45 2.0 LSB (aspect ratio¼ 3.0).
Fig. 14. Applied mid-span load versus vertical deflection for 250 60 2.0 LSB (aspect ratio¼ 3.0).
Failure patterns similar to those observed for the 250 60 2.0 LSB with aspect ratio of 2.0 were also observed in the tests on the 200 60 2.0 LSB and the 200 45 1.6 LSB (also with aspect ratios of 2.0). The exception was only extremely slight evidence of flange buckling in the 200 60 2.0 LSB. The failure mode for the 150 45 2.0 LSB with an aspect ratio of 2.0 is depicted in Fig. 8. The specimen displayed strong diagonal shear yielding patterns in the web, but did not exhibit any signs of buckling or yielding in the flanges. The 150 45 1.6 LSB with an aspect ratio of 2.0 displayed a failure mode, which closely matched that of the 150 45 2.0 LSB, with the exception of shear buckling as opposed to yielding. Again, signs of flange buckling were absent.
4. Combined bending and shear interaction diagram and proposed equations
3.5.2. Test specimens failed primarily due to bending (aspect ratios 3.0, 3.5 and 4.0) Specimens with web panel aspect ratios of 3.0, 3.5 and 4.0 were observed to fail primarily due to bending, and generally exhibited linear load–deflection behaviour up to approximately 80% of the ultimate load. Load–deflection plots typically became non-linear beyond this point and rose gradually to the peak ultimate load. This is evident in the load–deflection plot for the 250 60 2.0 LSB with an aspect ratio of 3.0 (see Fig. 14). Web yielding occurred locally at mid-span in all members subject to primary bending failure, which confirms that inelastic reserve capacity was mobilised at the ultimate load. Local buckling/yielding of the compression flange was much more pronounced in the tests on specimens with web panel aspect ratios of 3.0, 3.5 and 4.0, which tended to fail primarily due to bending. Photographs from tests on the 150 45 1.6 LSB and the 250 60 2.0 LSB (both with aspect ratios of 3.0) are given in Figs. 9 and 10. Well defined local flange yielding, which was the primary cause of failure, was seen to occur in both tests at the centre of the span. Shear buckling was also observed in both tests, which is evidence that shear was a contributing factor in the failure mechanism.
Keerthan and Mahendran [6] developed suitable design equations for the shear capacity of LSBs (Vv) by including the available post-buckling strength and the additional level of fixity at the web-flange juncture. Shear capacities of LSBs reported in Table 5 were calculated based on Keerthan and Mahendran0 s improved design rules [6] (see Appendix A). Similarly, the section moment capacities (Mu) of LSBs tested in this study were based on the developed finite element models in [3–5] that were validated against section moment capacity test results [3]. It was found that these finite element models predicted the ultimate section moment capacities of LSBs with good accuracy. Table 5 shows the ratios of V/Vu and M/Mu for each test calculated using the applied shear force (V) and moments (M) at failure and the corresponding shear and section moment capacities (Vu and Mu). Figs. 15 and 16 show the interaction diagram of V/Vu versus M/Mu for LSBs subject to combined bending and shear actions. Ultimate capacity results for members with aspect ratios of 2.0, 2.5, 3.0, 3.5, and 4.0 are included in these figures. Shear capacity results from the experimental study by Keerthan and Mahendran [6] on LSBs with aspect ratios of 1.0 and 1.5 are also included in these figures. Fig. 15 includes the circular interaction diagram based on AS/NZS 4600 to allow a comparison with test results. This figure shows that the current Australian/New Zealand design code provisions based on Eq. (11) are inadequate for predicting the strengths of LSBs subject to combined bending and shear. The overconservative nature of the circular interaction equation in AS/NZS 4600 [12] is clearly evident from Fig. 15. Test results are also compared with current AS 4100 [13] design provisions in Fig. 15. AS 4100 [13] predictions based on Eqs. (13) and (14) compared reasonably well with test results, which reinforces that the behaviour of the LSB is more in line with that of hot-rolled sections. This is likely to be the resultant of the stiff hollow flanges
P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
1.4
1.4
1.2
1.2
1.0
1.0
0.8
0.8
V/Vu
V/Vu
138
0.6
Equation 15
Test_Shear
Equation 17
Equation 12
Equation 11
0.6 (V/V
0.4
0.4
Test_Combined
0.2
0.2
0.0
0.0 0
0.2
0.4
0.6
0.8
1
1.2
0.0
Fig. 15. Interaction diagrams for LSBs subject to combined bending and shear actions. (Mu based on FEA and Vu based on shear design rules with post-buckling from [6]).
which contribute to a large proportion of the bending capacity much the same as in hot-rolled beams. In this study, Eq. (15) was proposed as a means of more accurately predicting the capacities of LSBs subject to combined bending and shear actions. Fig. 16 shows the proposed combined bending and shear interaction diagram for LSBs. 4 4 M V þ r1 ð15Þ Mu Vu Eqs. (16) and (17) were also proposed as a linear alternative: for
M o 0:65M u
V ¼ V V ½1:65 ðM=M u Þ
for
ð16Þ 0:65M u r M r M u
ð17Þ
With reference to Fig. 16, Eqs. (15) and (17) form lower bound solutions to the experimental results. M and V can be replaced by Mn and Vn, and likewise Mu and VV can be replaced by ϕbMs and ϕvVv to produce equations which are consistent with AS/NZS 4600 provisions. By adopting a lower bound solution, Mn rϕbMs and Vn rϕvVv holds true for all points on the interaction diagram. Note that capacity reduction factors are already included in the calculation of design bending and shear capacities. Hence, there is no need to include any further capacity reduction factor in the lower bound solutions given above. The interaction diagrams shown in Figs. 15 and 16 assume that section moment capacities (Mu) are based on those from finite element analyses (FEA), which include inelastic reserve capacity. Whilst the section moment capacities determined from finite element analysis are greater than those which would be calculated using the current AS/NZS 4600 provisions, the use of such values is justifiable given that they lead to more conservative capacity predictions. Moreover, this approach ensures that Eqs. (15) and (17) are compatible with any future revisions to AS/NZS 4600 which allow for inelastic reserve capacity in LSBs. It is interesting to note that Eq. (15) is identical to that proposed by Shahabian and Roberts [11] for plate girders. Again, this tends to indicate that LSB behaviour is more consistent with that of hot-rolled sections rather than typical cold-formed sections. Fig. 16 also includes the interaction diagram based on Eq. (12) although AS/NZS 4600 states that this equation is for beams with transverse stiffeners. Comparison with test results shows that Eq. (12) is unable to predict the strengths of LSBs subject to combined
0.2
0.4
0.6
0.8
1.0
1.2
M/Mu
M/Mu
V ¼ VV
+ (M/M ) = 1
Fig. 16. Proposed interaction diagrams based on Eqs. (15) and (17) for LSBs subject to combined bending and shear actions. (Mu based on FEA and Vu based on shear design rules with post-buckling from [6]).
bending and shear compared to the similar linear alternative given by Eq. (17). Hence Eq. (12) is not considered suitable for LSBs. Fig. 17(a) shows the interaction diagram of M/Mu and V/Vu where Mu is calculated based on the direct strength method (DSM) and Vu is calculated based on Keerthan and Mahendran0 s [6] design rules with post-buckling strength. This figure shows that the section moment capacities predicted by the DSM are also reasonably accurate. Hence direct strength method (DSM) equations can also be used to determine the section moment capacities of LSBs instead of using finite element analyses of Anapayan and Mahendran [5]. Fig. 17(b) and (c) shows the interaction diagram of M/Mu and V/Vu where Mu is calculated based on finite element analyses and direct strength method (DSM), respectively and Vu is calculated based on Keerthan and Mahendran0 s [6] design rules without post-buckling strength. Since these shear design rules did not include the available post-buckling strength in LSBs, the ratios of V/Vu exceed 1 for test results. It shows that the shear capacities predicted by Keerthan and Mahendran0 s [6] design rules without post-buckling strength are conservative. Fig. 17(d) and (e) shows the interaction diagram of M/Mu and V/Vu where Mu is calculated based on finite element analyses and direct strength method (DSM), respectively and Vu is calculated based on AS/NZS 4600 [12] design rules. AS/NZS 4600 design rules assume that the web panel of lipped channel beam is simply supported at the web-flange juncture and hence the shear buckling coefficient of coldformed steel beams is only 5.34. Since AS/NZS 4600 design rules neither include the available post-buckling strength in LSBs nor the additional fixity at the web-flange juncture, the ratios of V/Vu vary from 0.8 to 2.07 for test results. It shows that the shear capacities predicted by AS/NZS 4600 design rules are very conservative.
5. Conclusions This paper has presented the details of an experimental investigation into the structural behaviour and design of LSBs subject to combined bending and shear actions. Eighteen tests were undertaken using a three point loading arrangement. It was found that noticeable reductions in shear capacity were observed when applied bending moments exceeded about 65% of the section moment capacity. Likewise, noticeable reductions in bending capacity were observed when applied shear forces exceeded about 65% of the shear capacity. LSBs which failed in a primary shear mode displayed well defined web yield zones and a diagonal post-buckling tension field mechanism. Minor local buckling of the
P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
1.4
1.4
1.2
1.2
1.0
1.0
0.8
V/Vu
V/Vu
139
0.6
0.8 0.6
0.4
0.4
0.2
0.2 0.0
0.0 0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
M/Mu
0.6
1
0.8
1.2
M/Mu
2.5
1.4
2.0
1.2 1.0 V/Vu
V/Vu
1.5 0.8
1.0
0.6 0.4
0.5 0.2 0.0
0.0 0
0.2
0.4
0.6
0.8
1
0
1.2
0.2
0.4
M/Mu
0.6
0.8
1
1.2
M/Mu
2.5 2.0
V/Vu
1.5 1.0 0.5 0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
M/Mu
Fig. 17. Comparison of test results with proposed interaction diagrams. (a) Mu based on DSM and Vu based on shear design rules with post-buckling from [6]. (b) Mu based on FEA and Vu based on shear design rules without post-buckling from [6]. (c) Mu based on DSM and Vu based on shear design rules without post-buckling from [6]. (d) Mu based on FEA and Vu based on AS/NZS 4600 design rules [12]. (e) Mu based on DSM and Vu based on AS/NZS 4600 design rules [12].
compression flange was also observed for such specimens. LSBs which failed in a primary bending mode displayed local buckling of the compression flange and yielding. AS 4100 [13] predictions compared reasonably well with the experimental results, which indicates that the behaviour of LSB is more in line with that of hot-rolled steel sections. Experimental results were compared with existing AS/NZS 4600 [12] design provisions based on a circular interaction equation for combined bending and shear actions. Such provisions were found to be overly conservative and to inadequately reflect the behaviour of the LSB. Two lower bound equations were proposed as a means of more accurately predicting the capacities of LSBs subject to combined bending and shear actions. It was found that current direct strength method (DSM) based equations can be used to determine the section moment capacities of LSBs.
Acknowledgements The authors would like to thank Australian Research Council for their financial support and the Queensland University of Technology
for providing the necessary facilities and support to conduct this research project.
Appendix A. Proposed DSM design equations for the shear capacity of LSBs (a) Shear capacity of LiteSteel beams without post-buckling Vv ¼1 V yw
λ r 0:815
Vv 0:815 ¼ λ V yw Vv 1 ¼ V yw λ2
0:815 o λ r 1:23
λ 4 1:23
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s f yw V yw d1 ¼ 0:815 λ¼ V cr tw EkLSB
ðA1Þ
ðA2Þ
ðA3Þ
ðA4Þ
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P. Keerthan et al. / Thin-Walled Structures 77 (2014) 129–140
V yw ¼ 0:6f yw d1 t w
ðA5Þ
kLSB π 2 Et 3w V cr ¼ 12ð1 ν2 Þd1 For LSBs kss ¼ 4 þ
kLSB ¼ kss þ 0:87ðksf kss Þ 5:34
for
ða=d1 Þ2
kss ¼ 5:34 þ ksf ¼
ðA6Þ
4
5:34 ða=d1 Þ2
ksf ¼ 8:98 þ
þ
a o1 d1
For
ða=d1 Þ2
ðA8Þ
a Z1 d1
ðA9Þ
2:31 3:44 þ 8:39ða=d1 Þ ða=d1 Þ
5:61 1
ða=d Þ2
ðA7Þ
1:99 1
ða=d Þ3
For
For
a o1 d1
a Z1 d1
ðA10Þ
ðA11Þ
(b) Shear capacity of LiteSteel beams with post-buckling Option 1 Vv ¼1 V yw
λ r 0:815
ðA12Þ
Vv 0:815 0:815 þ 0:25 1 ¼ λ λ V yw Vv 1 1 ¼ 2 þ 0:25 1 2 V yw λ λ
0:815 oλ r 1:23
λ 4 1:23
ðA13Þ ðA14Þ
Option 2 Vv ¼1 V yw
λ r 0:815
Vv 1 0:15 ¼ 2 V yw λ λ
λ 40:815
ðA15Þ ðA16Þ
Eq. (A16) can also be written as follows. " # Vv V cr 0:5 V cr 0:5 ¼ 1 0:15 λ 40:815 V yw V yw V yw
ðA17Þ
References [1] OneSteel Australian Tube Mills, (OATM) (2008), Design of LiteSteel Beams, Brisbane, Australia. [2] Dempsey, RI. (1990), Structural behaviour and design of hollow flange beams. In: Proceedings of the second national structural engineering conference, Adelaide, Australia. [3] Anapayan T, Mahendran M, Mahaarachchi D. Section moment capacity tests of LiteSteel beams. Thin-Walled Struct 2011;49:502–12. [4] Anapayan T, Mahendran M. Improved design rules for hollow flange sections subject to lateral distortional buckling. Thin-Walled Struct 2012;50:128–40. [5] Anapayan T, Mahendran M. Numerical modelling and design of LiteSteel beams subject to lateral buckling. J Construct Steel Res 2012;70:51–64. [6] Keerthan P, Mahendran M. New design rules for the shear strength of LiteSteel beams. J Construct Steel Res 2011;67:1050–63. [7] Keerthan P, Mahendran M. Experimental studies on the shear behaviour and strength of LiteSteel beams. J Eng Struct 2010;32:3235–47. [8] LaBoube, RA, Yu, WW. (1978), Cold-formed steel web elements under combined bending and shear. In: Proceedings of the fourth international specialty conference on cold-formed steel structures, University of MissouriRolla, St Louis, Missouri, USA. [9] Bleich F. Buckling strength of metal structures. New York, USA: McGraw Hill; 1952. [10] Evans HR. Longitudinally and transversely reinforced plate girders. In: Plated structures, stability and strength. London, UK: Applied Science Publishers; 1983. [11] Shahabian F, Roberts TM. Behaviour of plate girders subjected to combined bending and shear loading. Sci Iran 2008;15:16–20. [12] Standards Australia/Standards New Zealand (SA) (2005), Australia/New Zealand Standard AS/NZS 4600 Cold-Formed Steel Structures, Sydney, Australia. [13] Standards Australia (SA) (1998). Australian Standard AS 4100 Steel Structures, Sydney, Australia. [14] Timoshenko SP, Gere JM. Theory of elastic stability. New York, USA: McGrawHill Book Co. Inc; 1961. [15] Lee SC, Yoo CH. Strength of plate girder web panels under pure shear. J Struct Eng, ASCE 1998;124:184–94. [16] Galambos TV. Guide to stability design criteria for metal structures. 5th ed.. New York, USA: John Wiley & Sons; 1998. [17] Bulson PS. The stability of flat plates. London, UK: Chatto and Windus; 1970. [18] Hancock GJ. Design of cold-formed steel structures to AS/NZS 4600:1996. 3rd ed.. Sydney, Australia: Australian Institute of Steel Construction; 1998. [19] American Iron and Steel Institute (AISI) (2007), North American specification for the design of cold-formed steel structural members, AISI, Washington, DC, USA.