A geometrically nonlinear Hellinger–Reissner shell element for the postbuckling analysis of variable stiffness composite laminate structures

Abstract Variable stiffness (VS) composite laminates provide larger freedom to design thin-walled structures than constant stiffness (CS) composite laminates. They showed to allow the redistributing of stresses, improving buckling and post-buckling performance and, therefore, reducing material weight and costs. This work extends a recently developed mixed shell element, MISS-4C, to the post-buckling analysis of VS composite laminate structures. MISS-4C has a linear elastic closed-form solution for the stress interpolation of symmetric composite materials. Its stress feld interpolation is obtained by the minimum number of parameters, making it an isostatic element. Moreover, its kinematic is only assumed along its contour, leading to an efficient evaluation of all operators obtained through analytical integration along the element contour. MISS-4C uses a corotational approach within a fast multi-modal Koiter algorithm to efficiently obtain the initial post-buckling response of VS composite laminate structures.</p

Variable angle tow composites in fibre-reinforced polymer bridges

Fibre reinforced polymers are increasingly used in bridge structures for reducing their environmental impact, extending the service life and saving costs. In this work, we propose the use of composite laminates called Variable Angle Tow (VAT) to realise bridge structures. In VAT composite laminates, the fibre orientation changes pointwise over the structure, enhancing stiffness tailoring capabilities with respect to traditional straight fibre (SF) laminates. In aerospace engineering, VAT composites have been successfully used to tailor structures’ stiffness to enhance linear elastic response, reduce buckling phenomena, and optimise the postbuckling behaviour. However, VAT laminates have never been applied in civil structures. This work shows how using VAT composite laminates in a bridge girder improves its buckling and postbuckling behaviours and increases its overall stiffness. A numerical investigation is conducted to assess the benefits given by VAT composite laminates to the structural performance of bridge girders. For different spans, results of a multi-objective optimisation considering both traditional SF and VAT laminates are presented. It is found that VAT laminates exhibit a buckling load up to 80% over SF laminates, with equal or lower deflections and without adding extra material, thereby opening new design scenarios for bridge structures.</p

A hybrid hex ahedral solid-shell element with self-equilibrated stresses for the geometrically nonlinear static analysis of composite laminated structures

Hybrid finite elements with self-equilibrated assumed stresses have proven to provide several advantages for analysing shell structures. They guarantee high performance when using coarse meshes and accurately represent the stress field. Additionally, they do not require assumptions about the displacement field within the element domain, and the integration is efficiently performed only along their contours. This work exploits those advantages to develop a solid-shell finite element for the geometrically nonlinear static analysis of composite laminated structures. In particular, an eight-node finite element, which has 24 displacement variables and 18 stress parameters, is developed. The displacement field is described only by translations, eliminating the need for complex finite rotation treatments in large displacement problems. A Total Lagrangian formulation is used with the Green-Lagrange strain tensor and the second Piola-Kirchhoff stress tensor. Thickness locking is cured using an assumed natural strain formulation for the transversal normal stress, and the assumed stress field eliminates shear locking. Then, for the analysis of linear-elastic problems, no domain integration is needed, and all the element operators are obtained by line integrals. The resulting formulation is efficient and allows for easy implementation. Computed numerical results show the accuracy and robustness of the presented element when used for both the linear elastic static and geometrically nonlinear elastic static analysis of composite laminated shell structures.</p

Recent advances in the postbuckling analysis of composite laminated structures

The current demand for lightweight structures in a wide range of engineering applications leads to using thin-walled composite laminated structures whose behaviour is governed by buckling and postbuckling phenomena. Such a demand is pushing the borders of computational mechanics to enhance methods and algorithms for studying those structures' geometrically nonlinear responses. This work presents some of the authors' developments in analysing lightweight composite laminated structures. The literature survey introduces a family of finite elements known as MISS elements, where MISS stands for mixed isostatic self-equilibrated stresses. The description of those elements, which are derived from the Hellinger-Reissner functional, is followed by a discussion on their advantages concerning displacement-based elements when studying composite laminated thin-walled structures. Subsequently, a framework for the postbuckling analysis of composite structures with the MISS-4C element that uses the Koiter multi-modal approach is presented.</p

A mixed hexahedral solid-shell finite element with self-equilibrated isostatic assumed stresses for geometrically nonlinear problems

Mixed Finite Elements (FEs) with assumed stresses and displacements provide many advantages in analysing shell structures. They ensure good results for coarse meshes and provide an accurate representation of the stress field. The shell FEs within the family designated by the acronym Mixed Isostatic Self-equilibrated Stresses (MISS) have demonstrated high performance in linear and nonlinear problems thanks to a self-equilibrated stress assumption. This article extends the MISS family by introducing an eight nodes solid-shell FE for the analysis of geometrically nonlinear structures. The element, named MISS-4S, features 24 displacement variables and an isostatic stress representation ruled by 18 parameters. The displacement field is described only by translations, eliminating the need for complex finite rotation treatments in large displacements problems. A total Lagrangian formulation is adopted with the Green–Lagrange strain tensor and the second Piola–Kirchhoff stress tensor. The numerical results concerning popular shell obstacle courses prove the accuracy and robustness of the proposed formulation when using regular or distorted meshes and demonstrate the absence of any locking phenomena. Finally, convergences for pointwise and energy quantities show the superior performance of MISS-4S compared to other elements in the literature, highlighting that an isostatic and self-equilibrated stress representation, already used in shell models, also gives advantages for solid-shell FEs.</p

Postbuckling optimisation of a variable angle tow composite wingbox using a multi-modal Koiter approach

The stiffness-tailoring capability of Variable Angle Tow (VAT) laminates gives enhanced freedom to design thin-walled structures. One key advantage of tow steering is the ability to redistribute stresses improving buckling performance, leading to reduction in material weight and costs. The aim of this work is to optimise the initial postbuckling behaviour of a recently proposed VAT composite wingbox. The optimisation process is based on a fibre path parameterisation. It involves seeking the stacking sequence that minimises the displacements occurring in the postbuckling regime. This problem is solved by coupling the multi-modal Koiter asymptotic approach implemented with a solid-shell Finite Element environment through stochastic optimisation strategies. Results obtained regarding different optimisation scenarios show a much improved performance for the buckling and postbuckling response of the wingbox with respect to the initial VAT design. Additionally, manufacturing constraints are readily included in the optimisation program. The possibility of performing an efficient and robust optimisation process of a complex structure with a multi-modal Koiter asymptotic approach is demonstrated, showing its viability as a design tool for buckling dominated structures. A parametric study regarding the influence of steering radii shows that overcoming the current manufacturing constraint on minimum radius is worthy of investigation