Keywords: stability, pallet racks, cold-formed steel, splices, sway, non-linear analysis.

Slender beam and column structures are used in the storage industries for palletised goods. Such structures often have a large number of bays and beam levels. All connections between uprights and beams, at upright splices, and between uprights and base-plates, are usually semi-rigid. Conventional methods of accurate analysis of pallet rack structures require the use of non-linear finite element programs or programs involving stability functions which analyse the whole frame.

In order to produce a fast, simple analysis of regular pallet rack structures the authors have developed a single-column model which incorporates the effects of semi-rigid connections and the flexural rigidity of the upright [1,2]. This paper describes the enhancements to the model to include splices in uprights.

Each section of upright between beam levels is represented by a single element in the model, the non-linear effects being included by stability functions incorporating the effects with only translational and rotational degrees of freedom. To avoid increasing the size of the element matrices for sections involving splices a computer algebra package was used to derive a modified set of slope-deflection equations incorporating the splice conditions. Tests were made to ensure that the resulting system of equations was still symmetric.

Splices may appear at any point within a column element between the two beams which bound it. The splices may be very close to beam-column intersection points and may also be treated as pinned connections if there is no experimental data to establish the semi-rigid characteristics. A symbolic algebra package was used to produce limiting equations in these cases. These cases have to be incorporated into the analyses as the modified slope-deflection equations have singularities at the limiting values.

The resulting set of equations are transcendental functions in terms of axial forces but because of the statical determinancy of the simplified column model they can be solved very efficiently using the Gauss Elimination technique. For a rack containing beam levels the system only involves solving simultaneous equations with a maximum of five elements per row.

Three examples of the model are given. Firstly, a single column with three beam levels and a splice in the middle level. The results are compared with a finite element solution of the same problem. The splice model described in the paper required only four elements, whereas the finite element model of the same system required forty-five elements. Results are compared with a range of splice stiffnesses ranging from 0Nmm/rad to Nmm/rad covering the full range of splice conditions from simply-supported to fully fixed. The maximum error was less than 0.3%.

The effects of a splice near to a beam level were investigated by moving the splice in the first example to be either just above, or just below, the first beam level. In this case the stiffness of the splice was kept constant at limiting slope-deflection equations were used. Agreement between a finite element solution and the proposed model was again excellent. However, it is to be noted that the results showed significant variations in bending moments and displacements for the splice in the different positions. The position of a splice must therefore be carefully considered in design.

The third example considered was a pallet-rack containing four bays and five beam levels. The paper details the method of modelling a frame by the single column model by adjusting section properties. The maximum error in deflections and rotations between a finite element model using 550 elements and the 5 element column model was 0.9% and 2.3% respectively. For design purposes the maximum moments must also be considered. The paper shows that the difference in maximum moments was 0.5%. Errors are greater for parts of the frame where moments are lower. When combined with axial forces in design calculations using interaction formulae from Eurocode 3 it can be shown that the errors in using the program are not more than 3-4% with the program generally giving higher values. These results lead to conservative designs.

Overall the paper presents an efficient method of analysing pallet rack frames containing splices. The use of computer algebra procedures has been shown to be an efficient procedure in the derivation of element stiffness matrices and especially dealing with limiting cases of zero stiffness splices and splices near to beam-column intersections.

1

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

2

Feng, X., Godley, M.H.R. and Beale, R.G., "Elastic buckling analysis of pallet rack structures", in "Developments in Structural Engineering Computing", Civil-Comp Press, Edinburgh, 297-305, 1993. doi:10.4203/ccp.48.7.2

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