Advances in Steel Structures - part 29 pptx

260 A.T. Wheeler et al. The failure criteria for the SHS (Figure 5) demonstrates that for the given end plate dimensions, an end plate thicker than 16 mm will result in plastic section failure, while an end plate thinner than 12 mm forms a mechanism (thin plate behaviour). Punching shear failure never governs for this configuration of SHS. In the case of the RHS (Figure 6), the depth-to-width aspect ratio results in punching shear failure becoming the dominant failure mode for end plate thicknesses in the range of 9 mm to 17 mm. Connections comprising end plates thicker than 17 mm will attain full plastic section capacity, while end plates thinner than 9 mm will fail as a result of a plastic mechanism forming in the end plate. The theoretical results depicted in Figures 4 and 5 are consistent with the experimental findings. CONCLUSIONS The analytical models presented in this paper constitute simple methods of predicting the ultimate strength of eight-bolt moment end plate connections joining square and rectangular hollow sections subjected to pure flexure. The model considers three failure modes which are end plate/bolt failure, plastic failure of the connecting beam section, and punching shear (tear out) failure. Plastic mechanism analysis comprising complex two-dimensional patterns of yield lines is employed for the investigation of end plate failure modes, and a modified version of stub tee analysis provides the means through which the effects of prying forces are incorporated in the model. The stub tee analysis is termed the "cumulative modified stub tee model" since it considers prying effects independently in the "in-plane" and "out-of- plane" bending directions for the end plate. The experimental and analytical results indicate that for the SHS connections, plastic section capacity failure dominates, with end plate failure occurring only for the most flexible end plates. For the RHS connections, the failure mode is predominantly that of punching shear, with plastic section capacity limiting the strength for the thicker end plates. The model demonstrates excellent correlation with the test results and is effective in its consideration of all relevant failure modes that can occur. REFERENCES Kato, B. and Mukai, A. (1991). High Strength Bolted Flanges Joints of SHS Stainless Steel Columns. Proceedings International Conference on Steel and Aluminium Structures, Singapore, May 1991. Kennedy, N. A., Vinnakota, S. and Sherbourne A. N. (1981). The Split-Tee Analogy in Bolted Splices and Beam-Column Connections. Joints in Structural Steelwork, John Wiley & Sons, London-Toronto, 1981. Nair, R. S., Birkemoe, P. C. and Munse, W. H. (1974). High Strength Bolts Subject to Tension and Prying. Journal of the Structural Division, ASCE, 100:2, 351-372. Murray, T. M. (1990). Design Guide for Extended End Plate Moment Connections, Steel Design Guide 4, American Institute of Steel Construction. Packer, J. A., Bruno, L. and Birkemoe, P. C. (1989). Limit Analysis of Bolted RHS Flange Plate Joints. Journal of Structural Engineering, ASCE, 115:9, 2226-2241. Wheeler, A. T., Clarke, M. J. and Hancock, G. J. (1995). Tests of Bolted Flange Plate Connections Joining Square and Rectangular Hollow Sections. Proceedings, Fourth Pacific Structural Steel Conference, Singapore, 97-104. Wheeler A. T., Clarke M. J., Hancock G. J. and Murray, T. M. (1998). Design Model for Bolted Moment End Plate Connections Joining Rectangular Hollow Sections. Journal of Structural Engineering, ASCE, 124:2, 164-173. Wheeler, A. T. (1998). The Behaviour of Bolted Moment End Plate Connections in Rectangular Hollow Sections Subjected to Flexure, PhD Thesis, Department of Civil Engineering, The University of Sydney. PREDICTIONS OF ROTATION CAPACITY OF RHS BEAMS USING FINITE ELEMENT ANALYSIS Tim Wilkinson and Gregory J. Hancock Department of Civil Engineering, The University of Sydney, Sydney, NSW, 2006, Australia. ABSTRACT This paper describes finite element analysis of cold-formed RHS beams, to simulate a set of bending tests, and predict the rotation capacity of Class 1 and Class 2 beams. Introducing geometric imperfections into the model was essential to obtain rotation capacities that were close to the experimental results. A perfect specimen without imperfections achieved rotation capacities much higher than those observed. Introducing a bow-out imperfection, constant along the length of the beam, as was (approximately) measured experimentally, did not affect the numerical results significantly. To simulate the effect of the imperfections induced by welding the loading plates to the beams in the experiments, the amplitude of the bow-out imperfection was varied sinusoidally along the length of the beam. The magnitude of the imperfections had an unexpectedly large influence on the rotation capacity of the specimens. Larger imperfections were required on the more slender sections to simulate the experimental results. KEYWORDS Finite element analysis, beams, RHS, cold-formed steel, rotation capacity, local buckling. INTRODUCTION Wilkinson and Hancock (1997, 1998) describe tests on cold-formed RHS beams to examine the Class 1 flange and web slenderness limits. The sections represented a broad range of web and flange slenderness values, but it would have been desirable to test a much larger selection of specimens. A more extensive test program would have been expensive and time consuming. Finite element analysis provides a relatively inexpensive, and time efficient alternative to physical experiments In order to model the plastic bending tests, the finite element program should include the effects of material and geometric non-linearity, residual stresses, imperfections, and local buckling. The program ABAQUS (Version 5.7-1) (Hibbit, Karlsson and Sorensen 1997), installed on Digital Alpha WorkStations in the Department of Civil Engineering, The University of Sydney, performed the numerical analysis. 261 262 T. Wilkinson and G.J. Hancock PHYSICAL MODEL AND FINITE ELEMENT MESH A typical RHS has dimensions d, b, t, r e, referring to the depth, width, thickness and external corner radius. The depth refers to the larger of the dimensions of the rectangular shape. The rotation capacity, R, of a beam is defined only when the section can sustain its plastic moment, Mp. R is defined as R = K1/K p -1, where r,p=Mp/El is the plastic curvature, and K1 is the curvature (K) at which the moment drops back below the plastic moment. Figure 1 shows the simplified testing arrangement for the RHS beams. The RHS were supplied by BHP Steel Structural and Pipeline Products, in either Grade C350 or Grade C450 (DuraGal). All beams were bent about the major axis, and most reached the plastic moment, Mp, and continued to deform plastically until a local buckle formed adjacent to the loading plate. A typical finite element mesh, replicating the test arrangement, is shown in Figure 2. The two relevant symmetry planes, at the mid-length of the beam, and through the minor principal axis of the RHS, have been used to reduce the size of the model. Figure 1: Physical Model Figure 2: Typical Finite Element Mesh ELEMENT TYPE The most appropriate element type to model the local buckling of the RHS was the shell element. The $4R5 element, defined as "4-node doubly curved general purpose shell, reduced integration with hourglass control, using five degrees of freedom per node" (Hibbit, Karlson, and Sorensen 1997), was used. The loading plates attached to the RHS beam were modelled as 3-dimensional brick elements, type C3D8 (8 node linear brick). The weld between the RHS and the loading plate was element type C3D6 (6 node linear triangular prism). The RHS was joined to the loading plates only by the weld elements. Details on the mesh refinement process have been omitted for brevity. MATERIAL PROPERTIES The cold-formed RHS have stress-strain curves that include gradual yielding, no distinct yield plateau, and strain hardening. There is variation of yield stress around the section, due to different amounts of work on the flat faces and corners during the production process, with higher yield stresses in the corners. Details of the material properties can be found in Wilkinson and Hancock (1997). The finite element model used three sets of material properties, as shown in Figure 3. Figure 4 compares the responses of a 150 x 50 x 3.0 C450 RHS from the experiment and for a geometrically perfect finite element. The post yielding moment in the ABAQUS was lower than in the experiment by approximately 3 %. The numerical model assumed the same material properties across the whole flange, web or corner, with discontinuity Rotation Capacity of RHS Beams Us&g Finite Element Analys& 263 of properties at the junctions of the regions. In reality, there is a smooth increase of yield stress from the centre of a flat face, to the comer. The numerical model assigned the measured properties from the coupon cut from the centre of the face (which were the lowest across the face) to the entire face, resulting in a small underestimation of the moment. The slight error in predicted moment was not considered important, as the main aim of the analysis was to predict the rotation capacity. Figure 4 also shows that buckling occurred at much higher curvatures in the geometrically perfect model, compared to the experiment. Figure 3: Different material properties around RHS Figure 4: Comparison of results from experiment, and perfect mesh GEOMETRIC IMPERFECTIONS The initial numerical analyses were performed on geometrically perfect specimens. It is known that imperfections must be included in a finite element model to simulate the true shape of the specimen and introduce some inherent instability into the model, in order to induce buckling. Bow-out Imperfection Measurement of the imperfections indicated that most RHS had an approximately constant "bow-out" along the length of each beam. For most cases, the web bulged outwards and the flange inwards. The magnitude of the bow was approximately d/500 (for the web), and -b/500 for the flange. Imperfection profiles are graphed in Wilkinson and Hancock (1997). However, the nature of the imperfection immediately adjacent to the loading plate was unknown, as it was not possible to measure the imperfections extremely close to the loading plate. The process of welding a flat plate to a web with a slight bow-out imperfection is certain to induce local imperfections close to the plate. Figure 5 shows a typical mesh with the bow-out imperfection included. Figure 6 shows the moment curvature relationships obtained for a series of analyses on 150 x 50 • 3.0 C450 RHS with bow-out imperfections. The magnitude of the imperfection was either d/500 and -b/500 (approximately the magnitude of the measured imperfections), or d/75 and -b/75 (very much larger than the observed imperfections). Compared to a specimen with no imperfection, the magnitude of the bow-out imperfection had a minor effect on the rotation capacity. In fact, the rotation capacity increased slightly as the imperfection increased. Even when the bow-out imperfections were included, the numerical results exceeded the observed rotation capacity by a significant amount. The conclusion is that the bow-out imperfection was not a suitable type of imperfection to include in the model. 264 T. Wilkinson and G.J. Hancock Figure 5: Mesh incorporating "bow-out" Figure 6: Results for "bow-out" imperfection Sinusoidal Varying Imperfection It is more common to include imperfections that follow the buckled shape of a "perfect" specimen, such as by linear superposition of various eigenmodes. The approach taken was to vary the magnitude of the bow-out imperfection sinusoidally along the length of the specimen. The half wavelength of the imperfection is defined as L w. A typical specimen with the sinusoidally varying bow-out along the length is shown in Figure 7. Figure 8 shows the results of selected analyses for a variety of imperfection wavelengths. The code "cont" in the legend to Figure 8 indicates the continuously varying imperfection. The specimen analysed was 150 x 50 x 3 RHS. A half wavelength of approximately d/2 (d is the depth of the RHS web) tended to yield the lowest rotation capacity and most closely matched the experimental behaviour (refer to the specimen with Lw = 70 mm). A half wavelength of d/2 was approximately equal to the half wavelength of the local buckle observed experimentally and in the ABAQUS simulations. In Figure 8, the specimen with L w = 70 mm experienced a rapid drop in load after buckling, and had a buckled shape as shown in Figure 9 which matched the location of the local buckle in the experiments. A specimen with a slightly different imperfection profile, Lw = 60 mm, had a much flatter post buckling response, and the buckled shape include two local buckles, as shown in Figure 10. Both specimens buckle at approximately the same curvature. To force one local buckle to form, and in the desired location, the imperfections were imposed only near the loading plate, as shown in Figure 11. Figure 8 includes the response of an additional specimen, with L w = 60 mm, but only the single imperfection. The curvature at which buckling initiated was barely unchanged, but the buckled shape changed, producing the desired shape of one buckle (Figure 9). Imperfection Size A variety of imperfection magnitudes was considered. The magnitude of the imperfections was varied from 6w = d/2000 to 6w = d/250, and 6f = -b/2000 to 6f = -b/250. Figure 12 shows the moment curvature graphs for a section with varying magnitudes of imperfection. It can be seen that increasing the imperfection size decreases the rotation capacity. For this example of a 150 x 50 x 3 RHS, applying an imperfection of 1/500 most closely matches the experimental response. Rotation Capacity of RHS Beams Using Finite Element Analysis 265 Figure 7: Mesh with sinusoidal imperfection Figure 8: Results for sinusoidal imperfection Figure 9: Specimen with one local buckle Figure 10: Specimen with two local buckles Figure 11: Single imperfection Figure 12: Effect of imperfection magnitude 266 T. Wilkinson and G.J. Hancock PREDICTIONS OF ROTATION CAPACITY A large range of sections was then analysed. The sizes considered were either 150 • 150 (d/b = 1.0), 150 x 90 (d/b = 1.66), 150 x 75 (d/b = 2.0), 150 x 50 (d/b = 3.0), and 150 x 37.5 (d/b = 4.0), with a variety of thicknesses, and different imperfection sizes: d/250, d/500, d/lO00, d/1500, or d/2000 (for the web), and b/250, b/500, b/lO00, b/1500, or b/2000 (for the flange). The material properties assumed were those for specimen BS02 (see Wilkinson and Hancock 1997). Figures 13 to 16 plot the relationship between web slenderness and rotation capacity for each aspect ratio considered and each imperfection size. The results are compared with the tests of Wilkinson and Hancock (1997, 1998), Hasan and Hancock (1988), and Zhao and Hancock (1991). It needs to be reinforced that the ABAQUS analyses were all performed on RHS with web depth d = 150 mm and material properties for specimen BS02 (Grade C450). The experimental results shown in comparison were from a variety of RHS with varying dimensions and material properties. Figure 17 compares the effect of aspect ratio with a given imperfection size. Some analyses were repeated using the material properties of a Grade C350 (Specimen BS 11) specimen, and the comparison between steel grades is shown in Figure 17. Note that the following figures use the AS 4100 definition of web slenderness (~.w), where Xw = (d- 2t)/rd'(fy/250). Figure 13: Results for d/b = 1.0 Figure 14: Results for d/b = 1.66 Figure 15: Results for d/b = 2.0 Figure 16: Results for d/b = 3.0 Figure 17: Comparison of aspect ratio Figure 18: Comparison of steel grade Rotation Capacity of RHS Beams Using Finite Element Analysis Several observations can be made from the results: 267 Imperfection size had a lesser effect on the rotation capacity of the more slender sections (R < 1), and had a greater effect for stockier sections. For a given aspect ratio, the band of results encompassing the varying imperfection sizes widens as the slenderness decreases. This is an unexpected result of the study. There is a clear non-linear trend between the web slenderness and rotation capacity for a given aspect ratio and imperfection size. The shape of the trend is similar regardless of aspect ratio and imperfection size (eg Figure 17). It may be possible to simplify the trend by a bi-linear relationship: a steep line for lower slenderness, and a line of less gradient at higher slenderness values. Sully (1996) found a similar bi-linear trend when comparing the critical local buckling strain of SHS (under pure compression and pure bending) to the plate slenderness. No single line is a very good match for the experimental results. For example, the ABAQUS results for d/b = 3.0 and imperfection of 1/250 match the experimental results well when ~.w = 48, while in the range 58 < Xw < 65, the results for an imperfection of 1/500 provide a reasonable estimation of the experimental results, and for 75 < ~.w < 85 an imperfection of 1/2000 gives results closest to the experimental values. For d/b = 1.0, the ABAQUS results for an imperfection of 1/250 are close to the experimental results in the range 37 < Xw < 48, while in the range 25 < ~.w < 35, an imperfection of 1/2000 most accurately simulates the test results. This suggests considerable variability in the imperfections with changing aspect ratios and slenderness, and that as the slenderness increases, larger imperfections are required to simulate the experimental behaviour. There is no reason why the same magnitude of imperfections should be applicable to sections with a range of slenderness values. A possible explanation is that the true imperfections in the specimen were caused by the welding of the loading plate to the RHS. A thinner section was deformed more by a similar heat input, hence larger imperfections were induced. The sinusoidally varying bow-out imperfections simulated the effect of the imperfections caused by the weld, and hence greater imperfections were required as the slenderness increased. Using the material properties of either Grade C350, or Grade C450 steel does not make a significant difference to the relationship between rotation capacity and web slenderness (Figure 18). However, both the Grade C350 and Grade C450 steel comes from the same virgin strip steel, and extra strength in the C450 specimens is obtained via the proprietary in-line galvanising process referred to as DuraGal. There is no reason to assume that the steel from a different supplier with different properties (eg a hot-formed steel) would produce the same relationship between rotation capacity and web slenderness. For sections with d/b = 1.0, the values of R predicted by ABAQUS are consistently below the observed experimental values of Hasan and Hancock (1988) and Zhao and Hancock (1991), even when small imperfections are imposed. The ABAQUS simulations were performed with material properties taken from the specimens of Wilkinson and Hancock (1997), since the exact material properties from Hasan and Hancock, and Zhao and Hancock were unknown. Preliminary parametric studies showed that increasing strain hardening modulus increased the rotation capacity. If the strain hardening portion of the material properties assumed was different from tile "true" response of the sections of Hasan and Hancock, and Zhao and Hancock, the numerical simulations are likely to produce inaccurate results. In particular, Zhao and Hancock used Grade C450 specimens from a different supplier, Palmer Tube Mills Australia Pty Ltd, which were not in-line galvanised, so it is reasonable to assume that the material properties were different to those used in the ABAQUS simulations. The significance of material properties is a notable finding of the finite element study. 268 SUMMARY T. Wilkinson and G.J. Hancock This paper has described the finite element analysis of RHS beams. The finite element program ABAQUS was used for the analysis. The maximum loads predicted were slightly lower than those observed experimentally, since the numerical model assumed the same material properties across the whole flange, web or comer of the RHS In reality, the variation of material properties is gradual, with a smooth increase of yield stress from the centre of a flat face, to a maximum in the comer. A perfect specimen without imperfections achieved rotation capacities much higher than those observed experimentally. Introducing a bow-out imperfection, constant along the length of the beam, as was (approximately) measured experimentally, did not affect the numerical results significantly. In order to simulate the effect of the imperfections induced by welding the loading plates to the beams in the experiments, the amplitude of the bow-out imperfection was varied sinusoidally along the length of the beam, and limited to be just near the loading plates. The size of the imperfections had an unexpectedly large influence on the rotation capacity of the specimens. It is likely that the imperfection caused by welding the loading plates to the RHS was a major factor affecting the experientially observed behaviour. The sinusoidally varying imperfections in the ABAQUS model simulated the effects of the localised imperfections in the physical situation. Larger imperfections were required on the more slender sections to simulate the experimental results, since for the same type of welding, larger imperfections are induced in more slender sections. REFERENCES Hasan, S. W., and Hancock, G. J., (1988), "Plastic Bending Tests of Cold-Formed Rectangular Hollow Sections", Research Report, No R586, School of Civil and Mining Engineering, The University of Sydney, Sydney, Australia. (also published in Steel Construction, Journal of the Australian Institute of Steel Construction, Vol 23, No 4, November 1989, pp 2-19.) Hibbit, Karlsson and Sorensen, (1997), "ABAQUS", Version 5.7, Users Manual, Pawtucket, RI, USA. Sully, R. M., (1996) "The Behaviour of Cold-Formed RHS and SHS Beam-Columns", PhD Thesis, School of Civil and Mining Engineering, The University of Sydney, Sydney, Australia. Wilkinson, T. and Hancock, G. J., (1997), "Tests for the Compact Web Slenderness of Cold-Formed Rectangular Hollow Sections", Research Report, No R744, Department of Civil Engineering, University of Sydney, Sydney, Australia. Wilkinson T. and Hancock G. J., (1998), "Tests to examine the compact web slenderness of cold-formed RHS", Journal of Structural Engineering, American Society of Civil Engineers, Vol 124, No 10, October 1998, pp 1166-1174. Zhao, X. L. & Hancock, G. J., (1991), "Tests to Determine Plate Slenderness Limits for Cold-Formed Rectangular Hollow Sections of Grade C450", Steel Construction, Journal of Australian Institute of Steel Construction, Vol 25, No 4, November 1991, pp 2-16. ACKNOWLEDGEMENTS This paper describes part of a research project is funded by CIDECT. The first author is funded by an Australian Postgraduate Award from the Commonwealth of Australia, supplemented by the Centre for Advanced Structural Engineering at The University of Sydney. FAILURE MODES OF BOLTED COLD-FORMED STEEL CONNECTIONS UNDER STATIC SHEAR LOADING K. H. Ip 1 and K. F. Chung 2 1Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hum, Hong Kong 2Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung, Hum, Hong Kong ABSTRACT Three failure modes of bolted cold-formed steel (CFS) connections were predicted using a three-dimensional finite element (FE) model with material geometric and contact nonlinearity. The connections were under static shear loading up to 3 mm end extension, which is appropriate for the design of moment connections. The model can predict the propagation of yielding in the CFS strips, which characterizes the connection failure mode. Three distinct failure modes were observed from the simulation results, namely, (i) the bearing failure, (ii) the shear-out failure and (iii) the net-section failure. Through parametric runs, the effects of geometry and material properties on the failure modes were studied The results were also compared with the bearing resistances based on design rules in BS5950: Part 5. KEYWORDS Bolted connections, cold-formed steel, failure modes INTRODUCTION Galvanized cold-formed steel (CFS) sections can be found in various building applications, ranging from purlins and steel framing, to roof sheeting and floor decking. The advantages of using CFS sections are derived from their long-term durability together with high yield strengths and high buildability. In building construction, CFS sections are usually bolted to hot rolled steel (HRS) members to form shear and moment connections. With the development of material technology, high strength CFS sections are available for building applications. The established 269 . be found in various building applications, ranging from purlins and steel framing, to roof sheeting and floor decking. The advantages of using CFS sections are derived from their long-term durability. International Conference on Steel and Aluminium Structures, Singapore, May 1991. Kennedy, N. A., Vinnakota, S. and Sherbourne A. N. (1981). The Split-Tee Analogy in Bolted Splices and Beam-Column. shape of the specimen and introduce some inherent instability into the model, in order to induce buckling. Bow-out Imperfection Measurement of the imperfections indicated that most RHS had