A slenderness based method for web crippling design of aluminium tubular sections

123 Existing web crippling design provisions available in European, American and Australian/New 124 Zealand standards are based on empiric equations that differ from the approach adopted for the 125 treatment of other instabilities such as local or overall buckling which employ χ-λ curves. 126 Assessment of those empiric web crippling provisions based on test data available in the 127 literature and reported herein for Interior-One-Flange (IOF) and Interior-Two-Flange (ITF) 128 loading conditions has demonstrated that they provide unsafe and inconsistent predictions 129 thereby highlighting the need to develop an alternative approach. Focusing on aluminium 130 tubular sections subjected to IOF and ITF loading conditions, this article reports experimental 131 and numerical results that were used to develop χ-λ curves for web crippling design. The 132 tubular sections were made of 6060 and 6063-T6 aluminium alloys and were manufactured by 133 extrusion. A total number of twelve tests were carried out and subsequently used to develop 134 and calibrate a numerical model. The measured dimensions, material properties and web

There is extensive research available on web crippling performance of cross-sections made of 156 cold-formed steel and stainless steel, however, the amount of web crippling studies on their structural performance needs to be further investigated to ultimately explore further 160 applications. Particular attention must be placed upon the fact that its Young's modulus is one 161 third of steel's which makes aluminium structures more susceptible to deflections and 162 instabilities including web crippling. experimental data were used to find strength-to-yield stress ratios for normal and high strength 171 alloys and assess the impact of the bearing length on the web crippling strength. It was found 172 that high strength alloys can achieve web crippling resistances 20% higher than normal alloys 173 and that the influence of the bearing length (i.e. the length over which the transverse load is 174 applied) on the web crippling resistance is higher for ETF loading than for ITF. In the 175 companion paper, Young and Zhou (2008) Table 2. 219 This table also reports the flat portion of the web hw (i.e. H-2t) to thickness ratio hw/t. All tubes 220 were delivered in lengths of 500 mm approximately and were cut at the laboratories of the 221 University of Wolverhampton to the lengths L shown in Table 2 Table 3 where E is the Young's 225 modulus, f0 is the 0.2% proof strength and fu is the ultimate stress.  Wolverhampton. The loads were applied by means of 50 mm thick steel bearing plates with a 242 bearing length ss of 50 mm. All the bearing plates were manufactured to load the full width of 243 the tube and were allowed to rotate as shown by the test set ups presented in Fig. 4. The flanges 244 of the tubes were not fastened to the bearing plates. In the IOF loading test, the load was applied 245 at mid-span over the bearing length and blocks made of engineered wood (Bock and Real 2014) 246 were inserted in the tubes at the supports, which were 50 mm long, to ensure that web crippling 247 failure occurs under the bearing length only. In the ITF loading tests, two identical bearings 248 with a length ss of 50 mm were positioned at mid lengths of each specimen. The transverse 249 load was applied by a 100 kN Zwick Roell testing frame incorporating load cells under the 250 hydraulic jack. Displacement control was used to drive the machine at a rate of 0.5mm/min.

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The load readings and machine's crosshead vertical displacement were logged at 1 s intervals.

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In with high scatter highlighting the need for an alternative approach to web crippling design.

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Statistical results are presented in Table 5, with regards to mean and coefficient of variation The geometry of the tubes was modelled by using 3D deformable shell elements and discretised

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The boundary conditions implemented in the numerical model were the same as those used in 330 the tests. All supporting plates and bearing lengths were modelled by using 3D analytical rigid 331 shells. Reference points (RPs) located in the centre were assigned to all rigid shells to which 332 boundary conditions were applied as shown in Figures 8 and 9 for IOF and ITF, respectively.

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In these figures U1, U2 and U3 are displacements corresponding to the x, y and z directions 334 respectively while UR1, UR2 and UR3 are rotations about the x-x, the y-y and z-z axes.

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For IOF loading models the supports were allowed to rotate about the cross-sectional horizontal 336 axis only and the rest degrees of freedom were restrained. At one of the ends, the longitudinal displacement was also allowed. All degrees of freedom were restrained at the reference point 338 of the bearing length except vertical displacement.

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For ITF loading models, all degrees of freedom of the bottom bearing plate were restrained 341 while only vertical displacement was allowed at the top bearing plate.

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In both IOF and ITF loading, surface-to-surface contact interaction was applied to the regions 343 of the tube in contact with the bearing lengths and supports as shown in Figures 8 and 9. The

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3D rigid plates were defined as "Master surface" while the regions of the tube in contact were 345 defined as "Slave surface". Contact was modelled by using hard normal behaviour and penalty 346 isotropic tangential behaviour with a friction coefficient of 0.6. The element thickness of the 347 tube was excluded and adjustment of the slave surface was allowed only to remove overclosure.

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To apply the load in the FE models, a vertical displacement was defined at the RPs. To solve 349 the models, the general static method in two steps allowing for geometric nonlinearities was     Linear elastic analyses were performed to obtain buckling shapes and associated eigenvalues.

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The load taken as the critical buckling load Rw,cr was the lowest elastic buckling load pertinent 397 to web buckling. Because contact interaction is incompatible with buckling analysis, for IOF 398 models the interaction between the specimen and supporting plates was simulated by using tie 399 type constraint and the bearing plate was simulated with a pressure load. For ITF models, the 400 interaction between the bearing plates and specimen was simulated by using tie type constraint

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Having obtained the ultimate web crippling load Rw,u, the buckling load Rw,cr and plastic load obtained and first slenderness χ(λ) based equations were derived by using regression analysis.  The χ(λ) curves shown in Figure 17 are the best fit trend lines that could be used to predict  Figure 19. Figure 18 shows that after the incorporation of the derived predictive models, the  values of the ratios were determined for various data sets which are presented in Table 8.

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Compared to the statistics from the assessment of existing design standards presented in Table   501 5, it is observed that the mean and COV values of the predictions by the proposed method 502 significantly improve. A graphical comparison between numerical/FE and predicted values is 503 presented in Figure 20 showing that both experimental and FE data sets fall closer to the 45 504 degree line.  The research presented in this article has yet again proved that slenderness based approaches 529 have potential to be adopted for web crippling design. Further research on additional cross-530 section shapes and materials is underway to extend this design approach.

Data Availability Statement
Some or all data, models, or code that support the findings of this study are available from the 534 corresponding author upon reasonable request.             (a) (b) Figure 18. χ(λ) curves incorporating predictive models Rw,cr,pred and Rw,pl,pred for (a) IOF and (b) ITF loading