Homogenisation of Slender Periodic Composite Structures

Abstract

The increasing efficiency and performance requirements on current aircraft requires the use of novel configurations and analysis methodologies which capture their behaviour both accurately and efficiently. In spite of the great improvements in computational power, simulating the full 3D structure is too costly and unviable for conceptual design stages. Hence, reduced order models that accurately describe the behaviour of these engineering structures are demanded. This work focuses on those that have a dominant dimension and, because the characteristic length of the sought response so allows it, can be assimilated into a 1-D beam model. A homogenisation technique is introduced to obtain the full 6×6, i.e. including transverse shear effects, equivalent 1-D stiffness properties of complex slender composite structures. The classical 4×4 stiffness matrix is obtained for periodic structures, that is, without the usual assumption of constant cross sections. The problem is posed on a unit cell with periodic boundary conditions such that the small-scale strain state averages to the large-scale one and the deformation energy is conserved between scales. The method is devised such that its implementation can be carried out using a standard finite-element package whose advantages can be exploited. This technique is readily applicable to engineering models. It provides a new level of modelling flexibility by employing tie constraints between different parts so that parametric analyses or optimisation can be performed without re-meshing. The proposed methodology allows local stress recovery and local (periodic) buckling strength predictions; nonlinear effects such as skin wrinkling can therefore be propagated to the large scale. Numerical examples are used to obtain the homogenised properties for several isotropic and composite beams, with and without transverse reinforcements, taper or thickness variation, and for both linear and geometrically-nonlinear deformations. The periodicity in the local post-buckling response disappears in the presence of localisation in the solution and this is also illustrated by a numerical example. Finally, the code originated from this work, SHARP.cells, is coupled with a nonlinear beam solver.