20 Z Y. Shen and Table 4, respectively. ~ and 6 denote the sectional reduction and elongation of the material, respectively. The loading series of the shaking table test are listed in Table 5. The input ground movement in the x- direction and y-direction of the shaking table are El-Centro N-S and El-Centro E-W, respectively, taking the time ratio t//~. /~/.~ TABLE 5 LOADING SERIES OF THE TEST Series No. 1 2 3 4 5 Amplitude of the acceleration X 0.30g 0.30g 0.50g 0.50g 0.70g Direction (0.309g) (0.311 g) (0.4976) (0.499g) (0.7046) Y 0.15g 0.15g 0.25g 0.25g 0.25g direction (0.163g) (0.1556) (0.258g) (0.2546) (0.2526) Series No. 6 7 Amplitude X 0.70g 0.60g of the Direction (0.703g) (0.6036) acceleration Y 0.25g 0.30g Direction (0.2546) (0.3036) ( 9 ) is the actual amplitude of the shaking table. 8 9 10 0.60g 0.80g 0.80g (0.605g) (0.790g) (0.792g) 0.30g 0.35g 0.35g (0.305g) (0.358g) (0.359~;) The natural frequencies of the free vibration of the model measured from the experiment are listed in Table 6. There is no different between the initial frequencies and the frequencies after 4th loading, that indicates the seismic action did not damage the steel frame model and the model was still in the elastic range. After the 4th loading, the frequencies decreased successively after subsequent loading due to the damage of the model. Using the simplified hysteresis model, the natural frequencies were calculated and also listed in Table 6 by parentheses. In Table 6, the errors 6 between tested and calculated frequencies are given as well. From Table 6 it can be seen that the frequencies of a damaged structure can be calculated by using the proposed simplified hysteresis model with sufficient accuracy. TABLE 6 NATURAL FREQUENCIES TESTED AND CALCULATED OF THE FRAME MODEL 1st f (Hz) 2nd f2 (Hz) 3rd f3 (Hz) 4th f4 (Hz) Initial After 4th Loading After 6th Loading After 8th Loading After 10th Loading 3.174(3.181) (g =0.22%) 3.174(3.162) (6 =0.38%) 3.163(3.146) (6=0.54%) 3.135(3.111) (6 =0.77%) 3.092(3.089) (6=0.10%) 4.883(5.264) (6=7.23%) 4.883(5.242) (6 =6.85%) 4.863(5.221) (6=6.85%) 4.833(5.179) (6 =6.68%) 4.800(5.155) (6=6.87%) 9.115(9.243) (6=1.38%) 9.115(9.220) (6=1.14%) 9.081(9.202) (6=1.31%) 9.046(9.164) (6=1.28%) . 8.870(9.097) (6=2.49%) 15.462(17.730) (6=12.79%) 15.462(17.668) (8=12.49%) 15.339(17.618) (6=12.93%) 15.304(17.509) (6=12.59%) 15.200(17.390) (6=12.60%) Cumulative Damage Model for the Analysis of Steel 21 The dynamic displacements Aland A3 of the model during the 10th loading are shown in Figure 6. The calculated results considering and not considering the cumulative damage effects are illustrated by Figure 7 and Figure 8, respectively. Figure 6 The tested displacement curves for points A1 and A3 during 10th loading 22 Z Y. Shen Figure 7 The calculated displacement curves for points A1 and A3 during 10th loading considering the cumulative damage Figure 8 The calculated displacement curves for points A1 and A3 during 10th loading assuming no damage(D=-0) Cumulative Damage Model for the Analysis of Steel 23 Table 7 are the maximum and minimum displacements of points A1 and A3 during different loadings. There are three results corresponding to the measured values, the calculated values taking the cumulative damage into account and the calculated values not considering the damage (/9=-0). TABLE 7 THE MAXIMUM AND MINIMUM DISPLACEMENTS OF POINTS A1 AND A3 DURING DIFFERENT LOADING ~ u.e ~ng~ No. No. 5 No. 6 No. 7 No. 8 No. 9 No. 10 Displacement A1 Max Tested 2.42 2.45 3.30 3.39 3.61 3.82 (cm) Cal. with D 2.20 2.38 2.96 3.16 3.35 3.59 Cal. D=-0 2.19 2.24 2.79 2.87 3.08 3.11 . Min Tested -2.44 -2.49 -3.01 -3.26 -3.57 -3.65 (cm) Cal. with D -2.37 -2.38 -2.63 -2.84 -3.09 -3.29 Cal. D=0 -2.36 -2.36 -2.50 -2.67 -2.97 -3.09 A3 Max Tested 1.29 1.31 1.08 1.10 1.39 1.43 (cm) Cal. with D 1.26 1.28 1.02 1.05 1.30 1.34 Cal. D=-0 1'.24 1.25 0.99 1.00 1.23 1.25 ,, Min Tested -1.30 -1.31 -1.09 -1.09 -1.37 -1.42 (cm) Cal. withD -1.13 -1.14 -1.03 -1.05 -1.31 -1.34 Cal. D=-0 -1.10 -1.11 -1.02 -1.03 -1.28 -1.30 The cumulative damages of the columns of the steel frame model after each loading of the loading series are shown in Table 8. In the Table the two digits of the end member indicate the column (Figure 5) and the first digit means the end where damage occurs. TABLE 8 CUMULATIVE DAMAGES OF THE COLUMNS OF THE STEEL FRAME MODEL Loading End Number No. 1-5 5-1 2-6 6-2 3-7 7-3 4-8 8-4 4 0.083 0.020 0.080 0.024 0.085 0.020 0.081 0.025 5 0.089 0.023 0.153 0.029 0.091 0.024 0.155 0.029 6 0.112 0.037 0.171 0.032 0.165 0.034 0.172 0.033 7 0.155 0.051 0.196 0.065 0.201 0.063 0.198 0.066 8 0.219 0.073 0.348 0.068 0.265 0.088 0.364 0.068 9 0.242 0.081 0.371 0.080 0.288 0.096 0.387 0.080 10 0.287 0.095 0.395 0.097 0.334 0.111 0.432 0.109 Loading End Number No. 5-9 9-5 6-10 10-6 7-11 11-7 8-12 12-8 0 0.021 0.005 0.057 0.017 0.021 0.005 0.042 0.013 0.026 0.128 0.042 0.013 0.084 0.007 From Figures 6 to 8 and Tables 7, 8 the following points can be drawn. First, a severe seismic action 24 Z Y. Shen will cause structures damaged. Second, the damage in a structure will cumulate during successive seismic actions. Third, the cumulative damage will deduce the resistance capacity of structures to the seismic action. Fourth, the dynamic behavior of steel framed structures can be analyzed with acceptable accuracy by using the proposed cumulative damage hysteresis model. CONCLUSION Based on a series of experiments and theoretical analysis mentioned in the previous sections, the following main conclusions can be drawn: (1) The cumulative damage mechanics model of steel under cyclic loading suggested by the author is easy to be used in structural analysis with satisfactory accuracy. (2) Hysteresis curves of steel planar and spatial members can be precisely imitated by using Shen & Lu's integration method with cumulative damage mechanics model as the input of the steel hysteresis characters. (3) The simplified hysteresis model of steel members with damage cumulation effects and the elasto- plastic tangent stiffness matrix of the spatial members derived by the author can put the analysis of steel framed structures subjected to more than one time's earthquakes into practice. (4) Using the method proposed in the paper, the analysis of initially damaged structures becomes practical and the damage of structures due to loading can be calculated in a practical way. REFERENCES Chen W. F. and Ausuta T. (1976). Theory of Beam-Columns, vol. 2, MeGraw-Hill, New York. Duan L. and Chen W. F. (1990). A Yield Surface Equation for Doubly Symmetrical Sections. Engineering Structures 12:4, 114-118. Li G. Q. et al. (1999). Spatial Hysteretic model and Elasto-plastic Stiffness of Steel Columns. Journal of Constructional Steel Research 50:, 283-303. Kachanov L. M.(1986). Introduction to Continuum Damage Mechanics, Martinus Nijhoff Publishers, Dordrecht. Kitipomchai S. et al. (1991). Single-equation Yield Surfaces for Monosymmetric and Asymmetric Sections, Engineering Structures 13:10, 366-370. Shen Z. Y. and Dong B.(1997). An Experiment-based Cumulative Damage Mechanics Model of Steel under Cyclic Loading. Advances in structural Engineering 1:1, 39-46. Shen Z. Y., Dong B. and Cao W. w. (1998). A Hysteresis Model for Plane Steel Members with Damage Cumulation Effects. Journal of Constructional Steel Research 48:2/3, 79-87. Shen Z. Y. and Lu L.W. (1983). Analysis of Initially Crooked, End Restrained Steel Columns, Journal of Constructional Steel Research, 3:1, 40-48. RECENT RESEARCH AND DESIGN DEVELOPMENTS IN COLD- FORMED OPEN SECTION AND TUBULAR MEMBERS Gregory J. Hancock Department of Civil Engineering, University of Sydney NSW, 2006, Australia ABSTRACT A major research program has been performed for 20 years at the University of Sydney on cold- formed open section and tubular structural members. This research has included both members and connections and has been performed predominantly for high strength steel sections. The open section members include mainly angles, channels (with and without lips) and zeds, and the tubular members include mainly rectangular (RHS) and square (SHS) hollow sections. The research has been mainly incorporated in the Australian Steel Structures Standard AS 4100-1998 and the Australian/New Zealand cold-formed steel structures standard AS/NZS 4600. The paper summarises the recent developments in the research and points to on-going and future research needs. KEYWORDS Cold-formed, Steel Structures, Structural Design, Open Sections, Tubular Sections, Standards INTRODUCTION Cold-formed structural members are being used more widely in routine structural design as the world steel industry moves from the production of hot-rolled section and plate to coil and strip, often with galvanised and/or painted coatings. Steel in this form is more easily delivered from the steel mill to the manufacturing plant where it is usually cold-rolled into open and closed section members. In Australia, of the approximately one million tonnes of structural steel used each year, 125,000 tonnes is used for cold-formed open sections such as purlins and girts and 400,000 tonnes is used for tubular members. Tubular members are normally produced by cold-forming with an electric resistance weld (ERW) to form the tube. In most applications of open sections, the coil is coated by zinc or aluminium/zinc as part of the steel supply process. In some applications of tubular members, the sections are in-line galvanised with a subsequent enhancement of the tensile properties. The resulting product is called DuraGal (BHP (1996)). In Australia, the total quantity of cold-formed products now exceeds the total quantity of hot-rolled products. The open section members are normally produced from steel manufactured to AS 1397 (Standards Australia, 1993). This steel is cold-reduced and galvanised and typically has yield stress values of 25 26 G.J. Hancock 450 MPa for steel greater than 1.2 mm (called G450), 500 MPa for steel in the range 1.0 to 1.2 mm (called G500) and 550 MPa for steel less than 1.0 mm (called G550). Hence the majority of the sections are constructed from high strength cold-reduced steel. Structural steel hollow sections are normally produced to the Australian Standard AS 1163 (1991). They are all cold-formed and usually have stress grades of 250 MPa (called C250), 350 MPa (called C350) and 450 MPa (called C450). The most common grade is C350 which has the yield strength enhanced from 300 MPa to 350 MPa during the forming process. The C450 grade is often achieved by in-line galvanising (BHP, 1996). The Australian Standard for the design of steel structures AS 4100 was first published in limit states format in 1990 and permitted the use of cold-formed tubular members to AS 1163. Cold-formed tubular members had been permitted to be designed to the permissible stress steel structures design standard AS 1250 (Standards Australia 1981) since an amendment in 1982. However, the research on cold-formed tubular members was limited in many areas, particularly flexural members and connections, and so a significant research program was undertaken. Much of this research which was incorporated in the most recent edition of AS 4100 (Standards Australia, 1998) is described in Zhao, Hancock and Sully (1996). The Australian/New Zealand Standard AS/NZS 4600 (Standards Australia 1996) for the limit states design of cold-formed open section members was published in 1996 and was based mainly on the American Iron and Steel Institute Specification (AISI, 1997). However, the Australian/New Zealand Standard permitted the use of high strength steel to AS 1397 and so research data was incorporated for this purpose. This paper summarises the most recent research in the following areas: 9 High strength angle sections in compression 9 Lipped and unlipped channel sections in compression 9 Unlipped channel sections in bearing 9 Lateral buckling of channel sections 9 Bolted and screwed connections in G550 steel 9 Tubular beam-columns 9 Bolted moment end-plate connections 9 Plastic design of cold-formed square and rectangular hollow sections OPEN-SECTION MEMBERS Axial Compression of Cold-Formed Angles A major research program was performed on cold-formed angles formed by cold-rolling and in-line galvanising so that the final product had a yield stress of 450 MPa (BHP (1996)). Sections ranging from slender (EA 50*50*2.4 mm) to non-slender (EA 50"50"4.7mm) were tested in pin-ended concentric compression such that flexural buckling could occur about the minor principal axis. Detailed measurements of the stress-strain characteristics of the material forming the sections, the residual stresses and overall geometric imperfections were taken. The results are reported in Popovic, Hancock and Rasmussen (1999). The results of the tests are compared with the design rules of AS 4100 (Standards Australia 1998) and AS/NZS 4600 (Standards Australia 1996) in Figs 1 and 2. Comparison of the angle tests is shown with AS 4100 in Fig. 1 and AS/NZS 4600 in Fig. 2 which only includes the slender sections. The slender sections failed in a combination of flexural and flexural-torsional buckling. For the sections Recent Developments in Cold-Formed Open Section and Tubular Members 27 tested, it can be concluded that the design procedure in AS 4100 is not satisfactory if the design yield stress is taken from stub column strengths as shown in Fig. 1 but it is satisfactory if it is based on coupons taken from the flats. The design procedure does not include specific rules for flexural- torsional buckling. Higher design curves than recommended by AS 4100 can be used for the non- slender sections which did not include torsional deformations in the failure mode. As demonstrated in Fig. 2, the design procedure in AS/NZS 4600 is conservative for short length sections where torsional buckling is included twice by virtue of an effective section for local buckling and torsional buckling stresses in the column design. For longer length columns, the additional required moment equal to a load eccentricity of L/1000 need only be applied for slender sections as shown in Fig. 2. Non-slender (fully effective) sections do not need this additional eccentricity as demonstrated in Popovic, Hancock and Rasmussen (1999). 1.4 Long Column Tests - Pinned Ends AS 4100 Column Curves (Ns = Stub Column Strength) 1.2 1.0 (Xb =" 1.0 SSRC 1 (Xb = - 0.5 AISC-LRFD (Xb= 0.0 SSRC2 (Xt)= 0.5 (Xb= 1.0 SSRC3 9 L50x50x2.5 2~ 0.6 / 9 L50x50x4.0 9 L50x50x5.0 ! 0.4 ,~ = r-~.~ .~Kf-~ _:~ o.~ k: ~ 0.0 I I I I I I 0 20 40 60 80 100 120 140 160 180 200 ~n Fig. 1 Comparison of angle section test strengths with hot-rolled design standards 1.4 1.2 1.0 0.8 Z 3 0.6 z 0.4 0.2 Long Column Tests L50x50x2.5 AS 4600 and AISI Column Curves (fy = 396 MPa) l - Pin-Ended f-t buckling controls 9 Fixed-Ended " / ff-t X A e NC / " 9 /. ,L flexure controls [ ~000 . 9 I 0 20 40 60 80 1 O0 120 140 160 180 200 Le/r 0.0 Fig. 2 Comparison of angle section test strengths with cold-formed design standards 28 G.J. Hancock Lipped and Unlipped Channels in Compression A test program on unlipped (plain) and lipped channels in compression was performed where the channels were compressed between fixed ends and pinned ends (Young and Rasmussen, 1998a, 1998b). Whereas it is well-known that local buckling of pin-ended channel columns induces overall bending, this phenomenon does not occur in fixed-ended channel columns that remain straight after local buckling and only bend when overall buckling occurs. These fundamentally different effects of local buckling on the behaviour of pin-ended and fixed-ended channel sections lead to inconsistencies in traditional design approaches. The research program investigated these phenomena and compared the results with the design approach in AS/NZS 4600. Results for plain channels compressed between fixed ends and pinned ends are shown in Figs. 3 and 4. The fixed ended tests (Fig. 3) clearly show that the formulae for column strength alone accurately predict the test results and the sections carry loads well in excess of the local buckling load. However, the pin-ended tests (Fig. 4) show that the loads carried are not significantly greater than the local buckling load for the pin-ended shorter length test specimens. By comparison, the design predictions accounting for the shift in effective centroid are very conservative. Similar results are achieved for lipped channels but the differences are not so marked as shown in Young and Rasmussen (1998b). Fig. 3 Comparison of fixed-ended plain channel test strengths with design strengths Recent Developments & Cold-Formed Open Section and Tubular Members 29 Fig. 4 Comparison of pin-ended plain channel test strengths with design strengths Unlipped Channel Sections in Bearing An experimental investigation of cold-formed unlipped channels subject to web crippling has been described in Young and Hancock (1998). The concentrated loading forces were applied by means of bearing plates which acted across the full flange widths of the channels. The web crippling results were compared with AS/NZS 4600. The design web crippling strength predictions given by the standard were found to be very unconservative for the unlipped channel sections tested which had web slenderness values ranging from 16.9 to 38.3. These slenderness values are fairly stocky compared with those for the test data base used for the American Iron and Steel Institute Specification (AISI, 1996) on which AS/NZS 4600 was based. A simple plastic mechanism model for the web crippling strength of unlipped channels was proposed. The plastic mechanism model is most appropriate for the stocky web sections which fail as a mechanism due to the load eccentricity resulting from the rounded comers. Lateral Buckling of Channel Sections A research program on the lateral buckling capacities of cold-formed lipped channel-section beams (CFCs) was undertaken and published in Put, Pi and Trahair (1999a). It has been argued that the design approximations based on hot-rolled beams may be inappropriate for CFCs, because of the very different cross-sectional shape and method of manufacture. The paper describes lateral buckling tests on simply supported unbraced CFCs of two different cross-sections which were undertaken to resolve the issue. However, the lateral buckling tests showed that the CFCs failed catastrophically by local and distortional buckling of the compressed element of the cross-section after quite large deformations. The failure moments were lower when the beam lateral deflection increased the compression in the compression lip, and higher when they increased the compression in the flange- web junction. The results in Fig. 5, which are taken from Put, Pi and Trahair (1999a), show some interesting features when compared with the predictions of AS 4100 and AS/NZS 4600. The stockier section C10019 is fairly accurately predicted by AS/NZS 4600 although it is slightly conservative at longer lengths. . 3. 35 3 .59 Cal. D =-0 2.19 2.24 2.79 2.87 3.08 3.11 . Min Tested -2 .44 -2 .49 -3 .01 -3 .26 -3 .57 -3 . 65 (cm) Cal. with D -2 .37 -2 .38 -2 .63 -2 .84 -3 .09 -3 .29 Cal. D=0 -2 .36 -2 .36 -2 .50 -2 .67 -2 .97. COLUMNS OF THE STEEL FRAME MODEL Loading End Number No. 1 -5 5- 1 2-6 6-2 3-7 7-3 4-8 8-4 4 0.083 0.020 0.080 0.024 0.0 85 0.020 0.081 0.0 25 5 0.089 0.023 0. 153 0.029 0.091 0.024 0. 155 0.029 6. -1 .37 -1 .42 (cm) Cal. withD -1 .13 -1 .14 -1 .03 -1 . 05 -1 .31 -1 .34 Cal. D =-0 -1 .10 -1 .11 -1 .02 -1 .03 -1 .28 -1 .30 The cumulative damages of the columns of the steel frame model after each loading