Nonlinear Vibration Correlation and Buckling Analysis of Flat Plates and Shells

The employment of nondestructive techniques in aerospace industries is rising thanks to advances in technologies and analysis. This part of the aerospace testing industry is essential to design and validate the new structures’ methodology and safety. Therefore, robust and reliable nondestructive methods have been extensively studied for decades in order to reduce safety problems and maintenance cost. One of the most important and employed nondestructive methods to compute large-scale aerospace structures’ critical buckling load is the Vibration Correlation Technique (VCT). This methodology allows to obtain the buckling load and equivalent boundary conditions by interpolating the natural frequencies of the structures for progressively increasing loadings without considering instabilities. VCT has been successfully investigated and employed for many structures, but it is still under development for composite shell structures. The present work provides a numerical model for carrying out virtual VCT to predict the buckling load, to characterize the natural frequencies variation with progressive higher loadings, and to provide an efficient means for verifying the experimental VCT results. The proposed nonlinear methodology is based on the well-established Carrera Unified Formulation (CUF). CUF represents a hierarchical formulation in which the structural model’s order is considered the analysis’s input. According to CUF, any theory is degenerated into generalized kinematics and is compactly handled. By adopting this formulation, the nonlinear governing equations and the relative FE arrays of the two-dimensional (2D) theories are written in terms of Fundamental Nuclei (FNs). FNs represent the basic building blocks of the proposed formulation. In order to investigate far nonlinear regimes, the full Green-Lagrange strain tensor is considered. Furthermore, the geometrical nonlinear equations are written in a total Lagrangian framework and solved with an opportune Newton-Raphson method. For an assessment of the robustness of the virtual VCT, several flat plate and shell structures are studied and compared with the solutions found in the available VCT literature. The results prove that the proposed approach provides results with an excellent correlation with the experimental ones, allowing to investigate the buckling load and the natural frequencies variation in the nonlinear regime with high reliability

Time Response Stress Analysis of Solid and Reinforced Thin-Walled Structures by Component-Wise Models

This paper deals with the evaluation of time response analyses of typical aerospace metallic structures. Attention is focussed on detailed stress state distributions over time by using the Carrera Unified Formulation (CUF) for modeling thin-walled reinforced shell structures. In detail, the already established component-wise (CW) approach is extended to dynamic time response by mode superposition and Newmark direct integration scheme. CW is a CUF-based modeling technique which allows to model multi-component structures by using the same refined finite element for each structural component, e.g. stringers, panels, ribs. Component coupling is realized by imposing displacement continuity without the need of mathematical artifices in the CW approach, so the stress state is consistent in the entire structural domain. The numerical results discussed include thin-walled open and closed section beams, wing boxes and a benchmark wing subjected to gust loading. They show that the proposed modeling technique is effective. In particular, as CW provides reach modal bases, mode superposition can be significantly efficient, even in the case of complex stress states

Vibration analysis of thermally loaded isotropic and composite beam and plate structures

This work proposed the use of the Carrera Unified Formulation (CUF) for the vibration and buckling analysis of structures subjected to thermal loads. In detail, the variation of natural frequencies for progressively large thermal loads is investigated. Here, particular attention is focused on the study of buckling thermal loads as degenerate cases of the vibration analysis and on the mode aberration caused by thermal stresses. From this standpoint, the use of CUF for the development of high-order beam and plate models is fundamental. Indeed, Lagrange-like (LE) polynomials are considered for developing the kinematic expansion and Layerwise (LW) theories are employed to characterize the complex phenomena that may appear in composite structures. A linearized formulation to study the natural frequencies variation as a function of the progressive increasing thermal loadings is adopted. Different isotropic and laminated composite structures have been analyzed and compared with the Abaqus solution to validate the presented methodology and provide some benchmark solutions. In addition, a parametric study was conducted to evaluate the stacking sequence and thickness effect in the vibration modes and thermal buckling loads. The results document the excellent accuracy and reliability of the presented methodology and show the potentialities of this numerical tool able to analyze cases that are difficult to study experimentally

Nonlinear vibration of composite beams, plates and shells subjected to compression and shear loadings by unified finite elements and comparison with VCT experiments

This work discusses some advances in the nonlinear vibration analysis of beam, plate and shell elements. Based on the Carrera Unified Formulation (CUF), trivial linearized and full nonlinear governing equations are developed in the framework of a hierarchical finite element formulation to study the effect of pre-stress states on the vibration of composite structures. Thanks to CUF and by making use eventually of full Green-Lagrange strain tensor, the proposed methodology is able to characterize simple to complex nonlinear phenomena, including those related to deep post-buckling regimes. Particular attention is given to the characterization of the natural frequencies of thin to thick laminated structure subjected to progressively increasing compression and shear loads. It is demonstrated that whenever stable pre-buckling exists, trivial linearized equations and low kinematics models can be adopted with no loss of generality. In contrast, in the case of unstable pre-buckling regimes and whenever the interlaminar stress state is complex three-dimensional, high order kinematics and full displacement-strain relations must be used for describing vibration. Several problem are discussed, including cylindrical laminated shell under compression and plates in shear. It is demonstrated that the proposed formulation is effective for all the problem considered and, among the others, can be used to validate non-destructive buckling tests such as Vibration Correlation Technique (VCT)

Effect of different geometrically nonlinear strain measures on the static nonlinear response of isotropic and composite shells with constant curvature

The structural analysis of ultra-lightweight flexible shells and membranes may require the adoption of complex nonlinear strain-displacement relations. These may be approximated and simplified in some circumstances, e.g., in the case of moderately large displacements and rotations, in some others may be not. In this paper, the effectiveness of various geometrically nonlinear strain approximations such as the von Krmn strains is investigated by making use of refined shell formulations based on the Carrera Unified Formulation (CUF). Furthermore, geometrical nonlinear equations are written in a total Lagrangian framework and solved with an opportune Newton-Raphson method. Test cases include the study of shells subjected to pinched loadings, combined flexure and compression, and post-buckling including snap-through problems. It is demonstrated that full geometrically nonlinear analysis accounting for full Green-Lagrange strains shall be performed whenever displacements are higher than the order of magnitude of the thickness and if compressive loads are applied

Accurate stress analysis of variable angle tow shells by high-order equivalent-single-layer and layer-wise finite element models

New concepts of lightweight components are conceived nowadays thanks to the advances in the manufacture of composite structures. For instance, mature technologies such as Automatic Fibre Placement (AFP) are employed in the fabrication of structural parts where fibres are steered along curvilinear paths, namely variable angle tow (VAT), which can enhance the mechanical performance and alleviate the structural weight. This is of utmost importance in the aerospace field, where weight savings are one of the main goals. For that reason, shell structures are commonly found in the aerospace industry because of their capabilities of supporting external loadings. Straight-fibre composite shell structures have been studied in recent decades and, now, spatially varying composite shells are attracting the attention of manufacturers. This work analyses the mechanical behaviour of VAT composite shells subjected to different external loadings and boundary conditions. The Carrera Unified Formulation (CUF) is employed to obtain the different structural models in a systematic and hierarchic manner. The outcomes of such numerical models are discussed and compared with commercial software Abaqus