An hp-finite element for vibration analysis of laminates reinforced with curvilinear fibres

In this paper, an approach to model thin composite plates reinforced with curvilinear fibres is presented and applied to analyse modes of vibration. Particular attention is given to plates with non-standard geometries, which are less commonly addressed in studies on this topic. Aiming to achieve accuracy with a small number of degrees-of-freedom, the model is based on Kirchhoff’s plate theory, combined with an hp -version finite element method. Assembling p -version Kirchhoff plate elements, while ensuring continuity, presents a significant challenge. Elastic connections are introduced to address this issue. Additionally, elastic boundaries are also considered to impose the boundary conditions. Regarding the reinforcing fibres, cubic polynomial splines are employed to represent the path of the fibres, which also adds to the proposed model generality. To discretise the displacement field of the plate, three sets of interpolation functions are investigated. The convergence properties of the model, and the effects of the intervening features, are analysed based on hp -refinement. The proposed approach is shown to require fewer degrees-of-freedom to effectively analyse irregular-shaped plates, when compared to the more commonly used h -version finite elements. Moreover, the capability of cubic polynomial splines to represent fibre paths is validated. The paper concludes with modal analysis of a composite plate with a complex shape to verify tailoring abilities of reinforcing curvilinear fibres. • A versatile hp -version finite element formulation for variable stiffness composite laminated plates is presented. • Kirchhoff’s hypotheses are adequate for the analysis of thin plates and require less degrees of freedom than Reissner–Mindlin formulations; however, they complicate finite element assemblage, particularly in higher-order elements. This is solved introducing fictitious elastic connections between finite elements. • Cubic polynomial splines are used to define the reinforcement fibre paths, allowing to describe a wide variety of fibre path curves, including, but not limited to, simpler fibre path functions that can be found in the literature. • Three judiciously chosen sets of displacement shape functions are compared and evaluated. • The accuracy and the computational efficiency of the proposed hp -FE approach are demonstrated by analysing modes of vibration in a number of examples, including complex shaped plates. • It is shown that the hp -FE proposed for variable stiffness composite laminated plates benefits both from the flexibility of the h -version finite element method and the computational efficiency of the p -version finite element method.