Multiscale computational homogenisation of shear-flexible beam elements: a Direct FE

Structural beam elements are computationally efficient in modelling slender structures. However, they require homogenised material models and computational homogenisation for such elements is complex because it involves higher order kinematics. A Direct FE 2 homogenisation model for shear-flexible beam elements that is based on the more versatile Timoshenko-Ehrenfest beam theory is presented. Beyond conventional Euler–Bernoulli beam kinematics, an independent shear angle needs to be imposed onto the heterogeneous microscale RVE to govern the microscale shear deformation. It is shown that this can be achieved using an integral constraint involving the moments of axial displacement. With the proposed model, the multiscale analysis can be implemented on commercial FE codes completely as a pre-processing step. Examples presented to demonstrate the performance of the proposed model include 2D and 3D models of fibre reinforced composite beams with material nonlinearity and coupled stretch-twist response. When compared with direct numerical simulations, the proposed model gave closely matching predictions while requiring only a fraction of the computational time. The examples also highlighted the inadequacy of the Euler–Bernoulli beam theory in some cases to further motivate this work.