This Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left Blank BEHAVIOUR AND DESIGN OF COLD-FORMED CHANNEL COLUMNS B. Young ~ and K.J.R. Rasmussen 2 i School of Civil and Structural Engineering, Nanyang Technological University, Singapore 639798 (Formerly, Department of Civil Engineering, University of Sydney, Sydney, NSW 2006, Australia) 2 Department of Civil Engineering, University of Sydney, Sydney, NSW 2006, Australia ABSTRACT The paper summarises recent research on cold-formed channel columns compressed between fixed ends and pinned ends. The research program formed the basis of the PhD thesis of the first author. The behaviour of channel columns was investigated experimentally and theoretically. Tests were performed over a range of lengths that involved pure local buckling, distortional buckling as well as overall flexural buckling and flexural-torsional buckling. Elastic and inelastic bifurcation analyses of locally buckled channel columns were used for the theoretical investigation. It is shown experimentally and theoretically that local buckling does not induce overall bending of fixed-ended singly symmetric columns, as it does of pin-ended singly symmetric columns. The test strengths are compared with the design strengths predicted using the Australian/New Zealand, American and European specifications for cold-formed steel structures. It is shown that a fixed-ended channel column can be designed using an effective length of half of the column length and assuming the applied force acts at the centroid of the effective cross-section. Design recommendations for fixed- ended singly symmetric columns are proposed. KEYWORDS Bifurcation analysis, Buckling, Channel column, Cold-formed steel, Effective length, Finite strip method, Fixed-ended, Instability, Pin-ended, Steel structures, Structural design, Tests. INTRODUCTION The use of cold-formed steel structural members has increased rapidly in recent years. Cold-formed members can be used economically in domestic and small industrial building construction and other light structures. As compared to thicker hot-rolled members, cold-formed members provide enhanced strength to weight ratios and ease of construction. Cold-formed sections are normally thinner than hot- 341 342 B. Young and K.J.R. Rasmussen rolled sections and have a different forming process. Consequently, the buckling and material behaviour can be quite different. The boundary conditions for singly symmetric columns are important in the design of thin-walled columns. Local buckling of singly symmetric columns, such as channel sections, may cause overall bending of the column depending on whether the section is compressed between pinned or fixed ends. A uniformly compressed channel section undergoes a shift in the line of action of the intemal force when the section locally buckles. Rhodes and Harvey (1977) explained that the shift results from the asymmetric redistribution of longitudinal stress following the development of local buckling deformations, and leads to an eccentricity of the applied load in pin-ended channels. Hence, local buckling of pin-ended channel columns induces overall bending. Rasmussen and Hancock (1993) performed analytical studies and concluded that the phenomenon does not occur in fixed-ended channel columns, because the shift in the line of action of the internal force is balanced by a shift in the line of action of the external force. Consequently, local buckling does not induce overall bending. It follows that pin-ended and fixed-ended singly symmetric columns behave fundamentally different. The different effects of local buckling on the behaviour of fixed-ended and pin-ended singly symmetric columns lead to inconsistencies in traditional design approaches. In the major codes of practice for cold-formed steel structures, full or partial rotational end restraint is accounted for solely by using effective lengths. Furthermore, the design strength of singly symmetric columns is reduced irrespective of the end support conditions to account for bending induced in pin-ended columns by the shift of the line of action of the internal force. This procedure is not rational for fixed-ended singly symmetric columns, which may remain straight after local buckling. The purpose of this paper is to briefly summarise the tests, theoretical analyses and design analyses (Young 1997) performed at the University of Sydney on cold-formed steel channel columns compressed between fixed ends and pinned ends. The main emphasis of the research program was to obtain experimental evidence to demonstrate the differences in the behaviour and strength of fixed- ended and pin-ended singly symmetric columns resulting from local buckling. The research has been published recently in international journals and research reports; and reference is made to these publications for further details. EXPERIMENTAL INVESTIGATION The test program described in Young and Rasmussen (1998a and 1998b) provided experimental ultimate loads and failure modes for cold-formed plain and lipped channel columns compressed between fixed ends and pinned ends. All test specimens were brake-pressed from high strength zinc- coated Grade G450 steel sheets having a nominal yield stress of 450 MPa (Australian Standard, 1993). The test program comprised four different cross-section geometries, two series of plain channels and two series of lipped channels. The four test series were labelled P36, P48, L36 and L48 where "P" and "L" refer to "plain" and "lipped" channel respectively. The average values of measured cross-section dimensions of the test specimens are shown in Table 1 using the nomenclature defined in Fig. 1. The measured cross-section dimensions of each specimen are detailed in Young and Rasmussen (1998a and 1998b). The specimens were tested at various column lengths. The pin-ended specimens were tested using the same effective lengths as those for the fixed-ended specimens. The base metal properties determined from coupon tests are also summarised in Table 1. The table contains the measured Young's modulus (E) and the measured static 0.2% tensile proof stress (0"0.2). The coupon dimensions conformed to the Australian Standard AS1391 (1991) for the tensile testing of metals using 12.5 mm wide coupons and a gauge length of 50 mm. The stress-strain curves obtained Behaviour and Design of Cold-Formed Channel Columns 343 from the coupon tests are detailed in Young and Rasmussen (1998a and 1998b). Residual stress measurements of the lipped channel specimens from Series L48 were obtained by Young and Rasmussen (1995b). The membrane and the flexural residual stresses were found to be less than 3% and 7% of the measured 0.2% tensile proof stress respectively. Hence, the residual stresses were deemed negligible compared with the 0.2% tensile proof stress. Local and overall geometric imperfections were measured prior to testing, and the imperfection profiles are detailed in Young and Rasmussen (1995a and 1995b). Figure 1" Definition of symbols TABLE 1 AVERAGE MEASURED SPECIMEN DIMENSIONS & MATERIAL PROPERTIES P36 P48 L36 Specimen Dimensions Bl Bf Bw t t* (mm) (mm) (mm) (mm) (mm) N/A 36.8 96.9 1.51 1.47 N/A 49.6 95.1 1.52 1.47 12.5 37.0 97.3 1.52 1.48 ri (mm) 0.85 0.85 0.85 A (mm 2) 247 282 281 Material Properties E or02 (GPa) (MPa) 210 550 210 510 210 515 * Base metal thickness Note: 1 in. = 25.4 mm; 1 ksi = 6.89 MPa A test rig shown in Fig. 2 was specifically designed and built for this test program. It consisted of a loading frame and two measurement frames. The measurement frames used spring systems and roller bearings to enable the frames to move longitudinally along the test specimen such that the deformation profiles could be obtained during testing. The pin-ended bearings were designed to allow rotations about the minor axis while restraining major axis rotations as well as twist rotations and warping. The fixed-ended bearings were designed to restrain both minor and major axis rotations as well as twist rotations and warping. Details of the test rig are given in Young and Rasmussen (1999a). The load-deflection curves shown in Fig. 3 demonstrate that the shift in the line of action of the internal force caused by local buckling induces overall bending in a pin-ended channel but not in a fixed-ended channel. As a result of the different effects of local buckling, the strength of the fixed- ended specimen is higher than the strength of the pin-ended specimen, despite the fact that the specimens had the same effective length (ley) of 750 mm for Series L48. The effective length (ley) is assumed equal to half of the column length for the fixed-ended columns (ley= LF / 2) and equal to the column length for the pin-ended columns (ley = Lp), which includes the dimension of the pin-ended bearings. The fixed-ended column remained straight in both principal directions and no twisting of the cross-section was observed after local buckling until overall buckling occurred at ultimate. This 344 B. Young and K.J.R. Rasmussen specimen exhibited nearly perfect bifurcation behaviour. In Fig. 3, the minor (u) and major (v) axis deflections and twist rotation (0) about the shear centre were measured at mid-length of the specimens. Further details of the different effects of local buckling on the behaviour of fixed-ended and pin-ended channels have been investigated by comparing strengths, load-shortening and load-deflection curves, as well as longitudinal profiles of buckling deformations, are given in Young and Rasmussen (1995a,b and 1999a) for all test specimens. The experimental and theoretical local buckling loads of the fixed- ended tests were also determined, as detailed in Young and Rasmussen (1998a and 1995b). Figure 2: Failure of a fixed-ended Series L48 column at an effective length of 750mm Figure 3: Load-Deflection curves for Series L48 at an effective length of 750mm THEORETICAL ANALYSES A technique for determining the overall flexural and flexural-torsional bifurcation loads of a locally buckled singly symmetric column is described in Young and Rasmussen (1997a). The overall bifurcation loads of locally buckled fixed-ended channel columns are determined using the theory presented by Rasmussen (1997). The theory is applicable to members of arbitrary cross-section shapes subjected to arbitrary types of loading. The members are assumed to be geometrically perfect in the overall sense but can include imperfections in the local mode. The overall bifurcation analysis uses elastic and inelastic geometric non-linear finite strip local buckling analyses to determine the flexural and torsional tangent rigidities of the locally buckled section. These tangent rigidities are substituted into the flexural and flexural-torsional bifurcation equations to calculate the elastic and inelastic overall buckling loads. The elastic and inelastic tangent rigidities are obtained using the non-linear finite strip buckling analysis programs developed by Hancock (1985) and, Key and Hancock (1993) respectively. Behaviour and Design of Cold-Formed Channel Columns 345 The elastic and inelastic bifurcation loads obtained by Young and Rasmussen (1997b and 1998c) for flexural and flexural-torsional buckling are compared with tests on fixed-ended plain channel columns (Young and Rasmussen, 1998a) in Fig. 4 for Series P36. The bifurcation loads are shown as Ncr on the vertical axis, non-dimensionalised with respect to the elastic local buckling load (Nz = A o'z), where crz is the elastic local buckling stress obtained using a finite strip buckling analysis (Hancock, 1978) and A is the full cross-sectional area. As shown algebraically in Young and Rasmussen (1997a), local buckling induces bending of a pin-ended singly symmetric column in the fundamental state but not of a fixed- ended singly symmetric column. Consequently, only fixed-ended singly symmetric columns exhibit bifurcation behaviour and only tests of fixed-ended columns are compared with bifurcation curves in Fig. 4. Figure 4: Non-dimensionalised load (Ncr/Nz) vs column length (L) for fixed-ended P36 channel column (Wo/t = 0.02) Figure 4 includes the flexural (F) and flexural-torsional (FT) bifurcation curves of both the distorted (locally buckled) and undistorted (non-locally buckled) cross-sections. The curves were obtained using a magnitude (Wo) of the local geometric imperfection (in the shape of the local buckling mode) of 2 % of the thickness (t). The test specimens failed in combined local (L) and flexural (F) buckling modes at short and intermediate lengths (L < 1500 mm), and in a pure flexural buckling mode at long lengths, as shown in Fig. 4. For the inelastic bifurcation analysis, the measured stress-strain curve was modelled using the Ramberg-Osgood expression (Ramberg and Osgood, 1943). In the expression, the parameter n = 8 was obtained for Series P36 using the measured stress-strain curve (Young and Rasmussen, 1998a), where the parameter n describes the shape of the stress-strain curve. The variation of the inelastic bifurcation curves shown in Fig. 4 follows closely the test strengths, except at short lengths where the strength was govemed more by local buckling rather than overall instability. The test strengths are lower than the bifurcation curves, probably because of overall imperfections. The flexural buckling mode observed in the tests was accurately predicted by the elastic and inelastic bifurcation analysis for all column lengths. Further details of the elastic and inelastic bifurcation analyses are given in Young and Rasmussen (1997b and 1998c). It can be concluded from Fig. 4 that overall bifurcation analyses are useful for determining the critical buckling mode and the variation of the buckling load with column length. 346 DESIGN ANALYSES B. Young and K.J.R. Rasmussen The fixed-ended test strengths obtained by Young and Rasmussen (1998b) for Series L48 channels are compared in Fig. 5 with the unfactored design strengths predicted using the Australian/New Zealand Standard (AS/NZS 4600, 1996), European (1996) and American Iron and Steel Institute (AISI, 1996) specifications for cold-formed steel structures. Details of the calculation of the design strengths are given in Young and Rasmussen (1997c). The design strengths of the fixed-ended columns were calculated by assuming concentric loading (loading through the centroid of the effective cross-section). The ultimate loads of the tests are plotted against the effective length for minor axis flexural buckling (ley). The effective length is assumed equal to half of the column length for the fixed-ended columns (ley= LF / 2). The theoretical minor axis flexural buckling loads and flexural-torsional buckling loads of the undistorted cross-section as well as the experimental local buckling load are also shown in Fig. 5. These loads were determined in Young and Rasmussen (1995b). Figure 5: Comparison of fixed-ended test strengths with design strengths for Series L48 The design strength predictions by the three specifications are conservative, as shown in Fig. 5. The failure modes observed in the Series L48 tests were combined local and distortional buckling modes at short lengths, combinations of these modes with the flexural-torsional buckling mode at intermediate lengths, and combined flexural and flexural-torsional buckling modes at long lengths. Flexural- torsional buckling failure modes were predicted by the three specifications for all column lengths which was in agreement with the failure modes observed in the tests, except at short lengths where neither flexural nor flexural-torsional buckling was found. The fact that the test strengths were conservatively predicted confirms the assumption that fixed-ended channel columns shall be designed by assuming loading through the effective centroid and an effective length of half of the column length. It also follows that the strength of a fixed-ended column is not reduced by the shift of the effective centroid. A comparison between the test strengths and the design strengths using the three specifications for both fixed-ended and pin-ended channel columns are given in Young and Rasmussen (1997c and 1998a,b) for all four test series. Young and Rasmussen (1999b) presented a comparison between the experimentally measured shift of the effective centroid and the shift of the effective centroid predicted by AS/NZS 4600 and the AISI Behaviour and Design of Cold-Formed Channel Columns 347 Specification. It was concluded that effective width rules of the specifications accurately predict the direction and magnitude of the shift of the effective centroid for plain channels but not for lipped channels with slender flanges. Young and Rasmussen (1999b) proposed simple modifications to the current effective width rules that provide agreement between the measured and predicted shifts of the effective centroid for lipped channels. The modifications were shown to produce more accurate design strengths for lipped channel columns. CONCLUSIONS The research program of the PhD thesis of the first author has been summarised. The program was undertaken at the University of Sydney on cold-formed steel channel columns that involved tests, theoretical analyses and design analyses. It was demonstrated experimentally and theoretically that the shift in the line of action of the internal force caused by local buckling deformations does not induce overall bending of fixed-ended singly symmetric columns as it does of pin-ended singly symmetric columns. For fixed-ended singly symmetric columns, the applied load always passes through the effective centroid of the cross-section. Hence, the effect of the shift in the line of action of the internal force due to local buckling should be ignored in the determination of the member strength of fixed-ended singly symmetric columns. It follows that for singly symmetric columns of the same effective length, the fixed-ended column strength is higher than the pin-ended column strength when the ultimate load exceeds the local buckling load. A technique for determining overall flexural and flexural-torsional bifurcation loads of locally buckled singly symmetric columns has been presented and applied to fixed-ended channel sections. The analysis uses elastic or inelastic non-linear finite strip buckling analyses to determine the tangent rigidities of the locally buckled section. The tangent rigidities are substituted into the overall flexural and flexural-torsional bifurcation equations to produce the overall buckling loads. The comparison between elastic and inelastic bifurcation loads of plain channel section columns with tests indicated generally good agreement. It was concluded that overall bifurcation analyses are useful for determining the critical buckling mode and the variation of the buckling load with column length. The test strengths were compared with design strengths obtained using the Australian/New Zealand (1996), European (1996) and American (1996) specifications for cold-formed steel structures. For fixed-ended columns, the design strength predictions by the three specifications were conservative for test Series L48. The design strengths were calculated assuming concentric loading through the effective centroid and an effective length of half of the column length. The overall failure modes predicted by the three specifications were in agreement with the failure modes observed in the tests at intermediate and long effective lengths but not at short effective lengths. On the basis of the comparison between test strengths and design strengths, it was recommended that fixed-ended columns failing by local and overall buckling shall be designed by assuming loading through the effective centroid (centroid of the effective cross-section) and using an effective length of half of the column length. ACKNOWLEDGEMENTS The first author wish to express his most sincere gratitude and appreciation to his PhD supervisor Assoc. Prof. Kim J. R. Rasmussen, for his invaluable guidance and support throughout the entire course of the candidature. The first author will always be greatly indebted to Assoc. Prof. Rasmussen for his helps. 348 REFERENCES B. Young and K.J.R. Rasmussen American Iron and Steel Institute (1996). Specification for the Design of Cold-Formed Steel Structural Members, AISI, Washington, DC. Australian Standard (1991). Methods for Tensile Testing of Metals, AS 1391, Standards Association of Australia, Sydney, Australia. Australian Standard (1993). Steel Sheet and Strip Hot-dipped zinc-coated or aluminium/zinc-coated, AS 1397, Standards Association of Australia, Sydney, Australia. Australian/New Zealand Standard (1996). Cold-Formed Steel Structures, AS/NZS 4600:1996, Standards Australia, Sydney, Australia. Eurocode 3, (1996). ENV 1993-1-3, Design of Steel Structures, Part 1.3: Supplementary Rules for Cold-Formed Thin Gauge Members and Sheeting. Draft February 1996, CEN, Brussels. Hancock G.J. (1978). Local, Distortional and Lateral Buckling of I-Beams. Journal of the Structural Division, ASCE 104:11, 1787-1798. Hancock G.J. (1985). Non-linear Analysis of Thin-walled I-Sections in Bending. Aspects of Analysis of Plate Structures, eds D.J. Dawe, R.W. Horsington, A.G. Kamtekar & G.H. Little, 251-268. Key P.W. and Hancock G.J. (1993). A Finite Strip Method for the Elastic-Plastic Large Displacement Analysis of Thin-Walled and Cold-Formed Steel Sections. Thin-walled Structures 16, 3-29. Ramberg W. and Osgood W.R. (1943). Description of Stress-Strain Curves by Three Parameters. Technical Note No. 902, National Advisory Committee for Aeronautics, Washington, D.C. Rasmussen K.J.R. (1997). Bifurcation of Locally Buckled Members. Thin-Walled Structures 28:2, 117-154. Rasmussen K.J.R. and Hancock G.J. (1993). The Flexural Behaviour of Fixed-ended Channel Section Columns. Thin-Walled Structures 17:1, 45-63. Rhodes J. and Harvey J.M. (1977). Interaction Behaviour of Plain Channel Columns under Concentric or Eccentric Loading. Proceedings of the 2nd International Colloquium on the Stability of Steel Structures, ECCS, Liege, 439-444. Young B. (1997). The Behaviour and Design of Cold-Formed Channel Columns, PhD Thesis Vol. 1 & 2, Department of Civil Engineering, University of Sydney, Australia. Young B. and Rasmussen K.J.R. (1995a). Compression Tests of Fixed-ended and Pin-ended Cold-Formed Plain Channels. Research Report R714, School of Civil and Mining Engineering, University of Sydney, Australia. Young B. and Rasmussen K.J.R. (1995b). Compression Tests of Fixed-ended and Pin-ended Cold-Formed Lipped Channels. Research Report R715, School of Civil and Mining Engineering, University of Sydney, Australia. Young B. and Rasmussen K.J.R. (1997a). Bifurcation of Locally Buckled Channel Columns. Research Report R760, Department of Civil Engineering, University of Sydney, Australia. Young B. and Rasmussen K.J.R. (1997b). Bifurcation of Singly Symmetric Columns. Thin-Walled Structures 28:2, 155-177. Young B. and Rasmussen K.J.R. (1997c). Design of Cold-Formed Singly Symmetric Compression Members. Research Report R759, Department of Civil Engineering, University of Sydney, Australia. Young B. and Rasmussen K.J.R. (1998a). Tests of Fixed-ended Plain Channel Columns. Journal of Structural Engineering, ASCE 124:2, 131-139. Young B. and Rasmussen K.J.R. (1998b). Design of Lipped Channel Columns. Journal of Structural Engineering, ASCE 124:2, 140-148. Young B. and Rasmussen K.J.R. (1998c). Inelastic Bifurcation Analysis of Locally Buckled Channel Columns. Proceedings of the 2nd International Conference on Thin-Walled Structures, Singapore, Elsevier Science, 409-416. Young B. and Rasmussen K.J.R. (1999a). Behaviour of Cold-formed Singly Symmetric Columns. Thin-walled Structures 33:2, 83-102. Young B. and Rasmussen K.J.R. (1999b). Shift of the Effective Centroid of Channel Columns. Journal of Structural Engineering, ASCE 125:5, 524-531. SECTION MOMENT CAPACITY OF COLD-FORMED UNLIPPED CHANNELS B. Young ~ and G.J. Hancock 2 1 School of Civil and Structural Engineering, Nanyang Technological University, Singapore 639798 (Formerly, Department of Civil Engineering, University of Sydney, Sydney, NSW 2006, Australia) 2 Department of Civil Engineering, University of Sydney, Sydney, NSW 2006, Australia ABSTRACT This paper describes a series of tests performed on cold-formed unlipped channels subjected to major axis bending. Traditionally, cold-formed steel members are thinner than hot-rolled steel members due to the limitations of the cold-formed technology. In the past, the typical thickness of cold-formed members is less than 3 mm. However, with the recent advancements made in the cold-forming technology, members of 12 mm and greater thickness can now be produced. Therefore, there is a possibility that thicker cold-formed members can be designed using hot-rolled steel structures standards. The objective of this test program is to determine the possibility of using the Australian Standard for hot-rolled steel structures in the design of thicker cold-formed unlipped channels subjected to major axis bending. In addition, the appropriateness of the section moment capacity design equations specified in the current Australian/New Zealand Standard and the American Specification for cold-formed steel structures for thicker cold-formed members is also investigated in this paper. The test strengths are compared with the design strengths obtained using the Australian/New Zealand Standard and the American Specification for both the hot-rolled and cold-formed steel structures. The comparisons showed that the design strengths predicted by the hot-rolled and the cold-formed steel structures standards and specifications are conservative for thicker cold-formed channels. KEYWORDS Beam, Bending, Channel members, Cold-formed steel, Design strength, Experimental investigation, Hot-rolled steel, Moment capacity, Steel structures, Structural design, Test strength. INTRODUCTION Cold-formed steel structural members can be used very efficiently in many applications where conventional hot-rolled members proved to be uneconomic (Hancock, 1998). Hence, the use of cold- 349 . analysis, Buckling, Channel column, Cold-formed steel, Effective length, Finite strip method, Fixed-ended, Instability, Pin-ended, Steel structures, Structural design, Tests. INTRODUCTION The. buckling on the behaviour of fixed-ended and pin-ended singly symmetric columns lead to inconsistencies in traditional design approaches. In the major codes of practice for cold-formed steel structures, . Buckling of I-Beams. Journal of the Structural Division, ASCE 104:11, 178 7-1 798. Hancock G.J. (1985). Non-linear Analysis of Thin-walled I-Sections in Bending. Aspects of Analysis of Plate Structures,