Buckling and Postbuckling Structures II

The following sections are included:

• Dedication
• Preface
• About the Editors
• Contents

Because they so often exhibit extreme forms of nonlinear mode coupling in their imperfection-sensitive response, the buckling design of shells is often thought to require either massive computational power or vast accumulations of test data, or both. Nothing could be further from the truth. Virtually, all the information required to predict lower bounds to the imperfection-sensitive buckling loads, for the vast majority of shells, is conveniently contained within a reanalysis of the stiffness (or energy) components characterizing the linearized, classical critical load (eigenvalue) analysis. This chapter briefly outlines the philosophical basis for the “reduced stiffness method,” refers to its capacity to predict lower bounds to the scatter of past test results, and illustrates its more recent validation through carefully controlled numerical experiments. It also demonstrates how the understanding of the relative importance of the various energy components within each of the prospective buckling modes provides a powerful guide as to the best choice of rib stiffening or fibre reinforcement to enhance the buckling performance. Although attention is focused on the elastic buckling behaviour, it will be indicated how a simple extension of the reduced stiffness method also allows prediction of lower bounds to elastic–plastic buckling loads of shells.

This chapter investigates the so-called “plastic buckling paradox,” which states that the deformation theory of plasticity tends to provide values of the buckling loads which are in better agreement with experimental results than those given by the flow theory of plasticity. The paradox lies in the fact that the latter is physically more sound because it assumes that the current stress depends not only on the current value of the total strain but also on how this strain value has been attained. Instead, the deformation theory of plasticity is based on the assumption that, at any point on the loading path, the stress is uniquely determined by the current state of strain only. The plastic buckling paradox has been observed in a wide range of shell structures, such as plate assemblies, torispherical domes and circular cylinders, under different loading conditions and using different boundary conditions. Among these, the problem of relatively thick cylindrical shell structures subject to either axial compression or combined axial tension and external pressure has attracted substantial interest due to its importance in engineering applications, which has led to a vast literature with many benchmark tests and results. The outcomes from recent studies by the authors are discussed in detail, and it is shown that, when an accurate and consistent FE model is set up, both the flow and deformation theories can predict buckling loads with acceptable plastic strains. In fact, contrary to the common belief, buckling pressures calculated numerically by means of the flow theory are found to be in better agreement with the experimental data. Analytical and semi-analytical formulations are also employed to shed light on the underlying causes of this apparent paradox.

Additional design features have been proposed to increase the structural efficiency of stiffened metal panels, e.g., buckling containment features or crack containment features. Such features typically target a specific design weak point and introduce additional design variables. The introduction of additional design variables, in turn, creates greater design freedom and the potential to produce designs which are better tailored to suit in-service loading and manufacturing constraints. However, changing the traditional stiffened panel configuration introduces new design and analysis challenges. This chapter illustrates the behaviour of buckling containment features within representative wing and fuselage panel design. Also discussed is the potential impact on panel damage tolerance and imperfection sensitivity. Finally, modified analysis and design methods are proposed, quantitatively demonstrating the benefits of additional design variables for a generic large aerospace-stiffened panel component design.

Availability of block-type syntactic foam buoyancy units has facilitated a range of underwater activities otherwise deemed as impossible, for example, dive to the bottom of the oceans in 2012. But buoyancy units which require a vessel-type configuration still remain a subject of research. High specific properties, resistance to degradation in water, and flexible fabrication methods become valuable factors when underwater applications are considered. Light weight is critical for submersibles as this leads to increased payload and to enlarged operational envelope. All of that points towards the use of composite materials.

In this context, this chapter deals with the strength, buckling and structural optimization of externally pressurized bowed-out cylinders, domed end closures onto cylinders, and toroidal shells of closed cross-section subjected to hydrostatic external pressure. It is assumed that shells are from carbon fibre-reinforced plastics (CFRPs). The work is primarily numerical, but some details about the manufacture, testing, and analyses of filamentwound and draped CFRP heads carried out at Liverpool University are provided. The issues regarding the sensitivity of buckling pressures to the initial geometric imperfections, first-ply failure and the last-ply failure are also discussed.

The capabilities of a reduced-order finite element modelling technique for fast nonlinear buckling analysis of structures under static and dynamic loading are illustrated. The reduction method makes use of a well-established analytical perturbation procedure for static and dynamic buckling analysis. This perturbation approach has in past years been used as the foundation for the development and implementation of reduced-order nonlinear finite element buckling analysis. The essentials of the procedure and two characteristic analysis cases are presented. The application to a basic example of a composite plate under dynamic load illustrates the possibility to extend the approach to dynamic response analysis, and the application to static buckling analysis of a composite cylindrical shell shows the potential of this reduced-order modelling framework to analyze and explain modal interaction effects in the buckling behaviour of imperfection-sensitive shells.

A semi-analytical method for the buckling and postbuckling analysis of composite-stiffened panels is presented. The method is formulated in the context of a variational framework and is based on the method of Ritz. Two levels of approximation are introduced, consisting of a plate assembly representation for the buckling analysis and an elastically restrained panel for the postbuckling assessment. Two aeronautical panels with open-section and closed-section stringer profiles are analyzed, illustrating the comparison with finite element predictions and with experimental data taken from the literature. The results indicate that accurate predictions can be obtained in terms of buckling loads, postbuckling reduction in stiffness, and out-of-plane deflections with an analysis time of the order of a few seconds.

This chapter presents a novel multilevel strategy that includes progressive failure analysis and robust design optimization for composite stiffened panels. A multiobjective approach is adopted for the structural sizing of aerospace components at different design stages or levels. The approach is integrated at two design levels, labelled as preliminary design and detailed design. The robust multilevel design methodology integrates the structural sizing to minimize the variance of the structural response. This method improves the product quality by minimizing variability of the output performance function. This innovative approach mirrors the steps taken during design and structural sizing in industry where the manufacture of the final product follows an industrial plan that goes from the material characterization up to trade constraints, through preliminary analysis and detailed design. A composite stiffened panel is optimized to validate the developed methodology. The initial architecture is defined at the preliminary design level by generating a Pareto front, for competing objectives, that is used to choose a design with a required weight. The ultimate load, that the postbuckled panel can bear, is maximized for the chosen weight. Then a robust solution is sought in the neighbourhood of this solution to finally find the layup for the panel capable of bearing the highest load for the given geometry, boundary conditions and optimization constraints.

Experimental and numerical evidence has shown that the occurrence of secondary instabilities, or mode-jumping, can have a detrimental effect on the structural integrity of thin-skinned stiffened composite panels. The sudden release of energy has been shown to initiate skin-stiffener debonding, leading to catastrophic fracture. As a consequence, it is common practice to increase the skin thickness in vulnerable areas, leading to non-optimal, i.e., heavier, structures. In this chapter, an optimization framework is presented which couples nonlinear high-fidelity finite element modelling, which is capable of capturing mode-jumping and the evolution and propagation of damage, with a genetic algorithm for use in the design of efficient and robust postbuckling composite structures.

The following section is included:

• Index