Semi-analytical modelling of variable stiffness laminates with discontinuities

Designs taking advantage of fibre-steered laminated manufacturing can optimally vary the stiffness and strength properties of high-performance structural components according to the geometry, loads and boundary conditions. For the stability behaviour of laminates with discontinuities such as local reinforcements and cut-outs, variable stiffness laminates have the additional ability to decrease stress concentration factors, increase buckling loads and decrease the negative effects of a cut-out; outperforming traditional straight-fibre designs. With the aim of finding closed-form analysis methods or methods with a reduced computational cost, the present study proposes a semi-analytical framework to analyze the stability behaviour of variable stiffness laminates with local reinforcements and cut-outs. Due to the discontinuous nature of the displacement field in these structures, the approximation functions are enriched to capture the behaviour near the discontinuity. In order to determine the energy functional derivatives across the laminate domain, Gauss-Legendre Quadrature numerical integration rules are applied to both rectangular and circular domains and the resultant energies are obtained by subtracting the integration of the cut-out domain from the full domain. A displacement-based formulation is used for the out-of-plane field variable, whereas a stress-based approach is used for the in-plane pre-buckling stress state. The model is set-up for balanced and symmetric laminates, thus decoupling the out-of-plane and the in-plane behaviours. A thorough verification is performed against existing models in the literature and against finite element results. The results for various plates and laminates with varying discontinuities and variable stiffness properties show a good agreement for both in-plane and out-of-plane field variables, ultimately leading to an accurate prediction of the stability behavior of structures with discontinuities.