Keywords: pallet racks, connection, moment-rotation, non-linear, semi-rigid.

Slender beam and column structures, pallet racks, are used in the storage industries for palletised goods. The racks are commonly made from cold-formed steel and the sections are in shapes such as channels and hat-sections [1]. Investigations into rack structures have been made by various authors [2,3]. Previous work by the authors [3] used a single model to analyse rack structures. Connections between uprights and beams, at upright splices and at the bases of uprights are usually semi-rigid. The joints between columns and beams are often made using boltless connections [4]. Experimental moment-rotation curves for these joints are highly non-linear. All the reported analyses into rack structures have assumed a bi-linear model with the same properties assumed for clockwise and anti- clockwise rotations. However, this assumption is often not valid. No results have been reported into the analysis of rack structures which include moment reversal at the beam-to-column joints.

This paper is concerned with the analysis of a pinned-base portal frame under a combination of point loads normal to the ground and with proportionally increasing side loads. In the early stages of loading the bending moments at both beam-upright joints have the same sign. However, as the portal frame begins to sway the non- linear P- effect will cause the bending moments in one of the joints to reverse. Stability functions were used to model the beam and the uprights. Semi-rigid joints at the end of the beam were incorporated into the beam stiffness matrices. The experimental moment-rotation curve, including the elastic unloading line, was approximated by a various curves. The program was first validated against non- linear one and two storey frames analysed by previous researchers [5] who used polynomial and exponential curves to model loading curves. The experimental moment-rotation curves for pallet racks show that, due to plasticity at low loads, the elastic unloading curves are distinct from the loading curves and hence can not be modelled by continuous polynomials or exponential functions.

Using a mirror arrangement of portal frames tied together by a tension jack an experimental investigation was undertaken at Oxford Brookes University [6]. Single storey portal frames, free to sway, were tested with different percentages of side load. To investigate the behaviour of the portal frames numerically, three different models of joint behaviour were used in the analysis. The most accurate model was a tri-linear loading model together with a bi-linear unloading model. The second and third models used bi-linear loading and unloading lines with the same slopes used for loading and unloading. The values of the elastic stiffnesses were chosen to be the stiffness at half the ultimate failure moment (as used in the SEMA code [7]) and the value of stiffness as calculated by the FEM code [8]. The tri-linear loading model agreed closely with the experimental results for both moments and the deflection due to sway. The FEM model gave results for moments which gave the same accuracy as the tri-linear analysis but gave sway deflections which were typically 10% higher than the experimental values. Hence this model can be conservatively used to predict the sway behaviour of pallet rack frames. The value of stiffness used in the SEMA code is higher than the values used in the other analyses and produced deflections which were much lower than experiment. Having validated the program against experiment the program was run with the typical design code values of 0.5% and 1% side load. In these cases the differences between the FEM code and the more accurate model were small and hence the FEM code can be used to predict the behaviour of pallet rack portal frames, even when unloading takes place. The SEMA code consistently gave values of deflection significantly smaller than experiment.

The multi-linear model was then applied to two storey, single bay and two bay, single storey frames to illustrate the application of the analysis procedure to more complex structures.

1

W.-W. Yu, "Cold Formed Structures", McGraw-Hill, 1973

2

J.M. Davies, "Down-Aisle Stability of Rack Structures", Eleventh Int. Speciality Conf. on Cold-Formed Steel Struct., 417-435, St. Louis, USA, 1992.

3

M.H.R. Godley, R.G. Beale and X. Feng. "Analysis and design of down-aisle pallet rack structures", Computers and Structures, 77, 391-401, 2000. doi:10.1016/S0045-7949(00)00031-6

4

F.D. Markazi, M.H.R. Godley and R.G. Beale, "Experimental Analysis of Semi-Rigid Boltless Connections" Thin-Walled Structures, 28(1), 57-87, 1997. doi:10.1016/S0263-8231(97)00003-7

5

C. Bernuzzi and C.A. Castiglioni, "Experimental analysis on the cyclic behaviour of beam-to-column joints in steel storage racks", Thin-Walled Structures, 39, 841-859, 2001. doi:10.1016/S0263-8231(01)00034-9

6

B. Dhillon and J.W. O'malley, "Interactive Design of Semi-Rigid Steel Frames", Journal of Structural Engineering, 125, 556-564, 1999. doi:10.1061/(ASCE)0733-9445(1999)125:5(556)

7

M. Abdel-Jaber, "The Influence of Semi-Rigid Connections on the Behaviour of Beam and Column Structural Systems", PhD Thesis, Oxford Brookes University, 2002

8

SEMA, Code of Practice for the Design of Static Racking, The Storage Equipment Manufacturers' Association, UK, 1985

9

FEM, The Design of Steel Static Pallet Racking and Shelving, The Federation Europeene de la Manutention,1998

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