Keywords: corrugated paperboard, packaging, composite structure, post-buckling, mathematical modelling, finite element analysis.

Computer modelling of buckling is a useful application for predicting the performance of corrugated paperboard packaging. However, discrepancies between modelling results and buckling experiments exist. This paper aims to improve agreement between analytical and experimental results for the buckling problem of a simply supported uniaxially compressed corrugated paperboard panel. The influence of changes to the in-plane boundary conditions on the buckling load and post-buckling behaviour was studied. The three variations of in-plane boundary conditions were: (1) uniform load intensity with free normal in-plane movement and shear free on all edges; (2) uniform compression with constant normal in-plane movement and shear free on all edges; and (3) uniform compression, free normal in-plane movement on unloaded edges, and shear free on all edges.

Analytical models for cases (1) and (2) have been created in the MATLAB software using the Galerkin method with a single-term double sine out-of-plane displacement function. Numerical finite models of corrugated paperboard panel buckling were created for cases (1) and (3), using the ABAQUS software and linear perturbation buckling and general static procedures. The analytical model of case (1) included a novel application to corrugated paperboard of analogous out-of-plane clamped-clamped beam functions for the Airy stress function to satisfy the in-plane boundary condition approximately.

The corrugated paperboard material model and panel dimensions were the same as in [1]. The material is treated as an elastic orthotropic lamina with properties equivalent to the composite structured corrugated paperboard, based on first-order shear deformation laminated plate theory. The elastic lamina material definition and quadratic shell elements were used in the finite element model of a quarter of the panel.

Comparison of buckling load from the analytical results cases (1) and (2), to those published in [1], showed very close agreement and a difference of 0.3% for the numerical finite element result of case (1). The experimental buckling load referred to is 15% lower than the reference analytical buckling load. The post-buckling load vs. deflection amplitude plots for the analytical cases (1) and (2) and the numerical case (1) were compared to the analytical plot in [1]. At more than twice the buckling load, a marked difference of 10% for the analytical plots, 7.3% difference for the numerical plot and 63% difference for the reference experimental plot was found. The results have therefore shown the large difference between the experimental and analytical results should not be attributed to a slight variation of in-plane boundary conditions.

Further work could be done to consider the difference a multi-term analytical solution and improvements of the material modelling of the corrugated paperboard would make in reducing the discrepancy.

1

T. Nordstrand, "Analysis and testing of corrugated board panels into the post-buckling regime", Composite Structures, 63(2), 189-199, 2004. doi:10.1016/S0263-8223(03)00155-7

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