Semi-analytical modelling of VAT laminates with cut-outs

Abstract

Using Variable Angle Tow (VAT) laminates, the stiffness and strength properties of laminates can be continuously varied to arrive at designs that outperform traditional straight fibre composite laminates in cases where the loads are distributed in a non-uniform manner. Cases of particular interest are laminates under buckling conditions or laminates containing cut-outs, for which VAT laminates have the ability to decrease concentration factors and increase buckling loads, thus having the ability to create more efficient designs. With the aim of finding solutions with reduced computational cost, a semi-analytical analysis tool is developed which is able to determine the behaviour of VAT laminates with cut-outs. The method is based on finding the position of minimum energy using the Rayleigh-Ritz method. Due to the discontinuous nature of laminates with a cut-out, the approximation functions are enriched with additional functions to capture the behaviour near this discontinuity. In order to determine the energy functional derivatives across the domain of the laminate, Gauss-Legendre Quadrature numerical integration rules are applied to both rectangular and circular sections and the resultant energies are obtained by subtracting the integration on the cut-out domain from the rectangular domain. The approach mixes a stress-based approach to determine the in-plane pre-buckling stresses, and a displacement-based formulation to determine the buckling modes and eigenvalues. Minimising the energies yields the coefficients for the enriched stress-state and enriched displacement field. The model is set-up for balanced and symmetric laminates, thus decoupling the out-of-plane and in-plane behaviours. These components are verified with existing models in the literature and FE solutions, determining the behaviour of pristine VAT laminates and isotropic plates with cut-outs. Upon successful verification, the components are combined to give the buckling behaviour. Furthermore, results from literature for a laminate including a square stiffening insert are used to verify the capabilities of the proposed model. These results have been extended further to circular stiffening insert to verify the capabilities of the proposed enriching trial functions. Finally, a VAT laminate design is analysed where the capabilities of the proposed model, the proposed trial functions and VAT laminate designs are illustrated. For this case, the pre-buckling stresses and subsequent buckling behaviour are analysed for a VAT laminate with a cut-out. The results are shown to be accurate in determining the axial load distribution and showed a decrease in stress concentrations surrounding the cut-out. This pre-buckling stress-state is used to determine the buckling behaviour and compared to the VAT laminate without cut-out.