Homogenisation of slender periodic composite structures

A homogenisation technique is introduced to obtain the equivalent 1-D stiffness properties of complex slender periodic composite 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 conditions and the deformation energy is conserved between scales. The method can be implemented in standard finite-element packages and allows for local stress recovery and also for local (periodic) nonlinear effects such as skin wrinkling to 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 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.