Perturbation-based imperfection analysis for composite cylindrical shells buckling in compression
Introduction
Prediction of the buckling loads of thin-wall cylinders in compression is a classical structural analysis problem, which is relevant to many industries and applications such as the design of space launch vehicles. It is well known that experimental buckling loads are much lower than predicted by classical analytical methods. This has been attributed to the presence of imperfections in real cylinders, such as imperfections in geometry, load application and material.
Current industry approaches for analysis and design of cylindrical launch vehicles and other structures rely on empirical guidelines from NASA in 1968 (SP-8007) [1], which propose a knockdown factor for theoretical buckling load predictions. However, these are generally very conservative, and have been largely determined from experimental data for metallic cylinders. This factor can be inappropriate for buckling of cylinders manufactured using composite materials, where for example two identical cylinders with different stacking sequences can have very different buckling behaviour [2]. As such, there is a need for an analysis approach suitable for industrial application that takes the properties of composite materials into account.
With regards to sandwich materials, NASA also published guidelines for buckling of sandwich structures, which was based on less experimental data, and all involved isotropic facesheets [3]. No other guidelines have been published since regarding knockdown factors for sandwich structures, particularly with facesheets made using composite material. There have been numerous published studies that have highlighted the differences between behaviour of monolithic and sandwich materials, and between isotropic and composite sandwich materials (see, for example, recent review articles [4], [5]). In the meantime, sandwich materials are commonly used on lightweight structures such as space launcher vehicles, which commonly have compression as a critical design case. As such, there is a need for any new buckling analysis technique to incorporate and analyse the effect of using sandwich laminates.
Various other approaches have been adopted for predicting realistic buckling loads, though these have limitations, particularly within the context of a design process. The incorporation of geometric imperfections using the structural eigenmodes is a commonly applied technique [6]. Alternatively, the use of a database of known imperfection patterns has been proposed as a means of assessing the sensitivity of a design using numerical analysis [7]. Other approaches involve taking into account non-traditional imperfections such as those involving the loading or material [8], the use of statistics-based techniques [9], [10], or progressive damage modelling [11], [12]. However, for these approaches, the appropriate parameters to apply are often not known a priori for a given structure and manufacturing process, so that experimental testing would be required for validation. This is expensive, and can become impractical in a design context, particularly where designs are being investigated that are outside of the range of those previously considered. This highlights the need for an analysis methodology to be developed that considers the practicalities of industrial context, where rapid analysis is required for initial design studies without the need for experimental validation.
In response, an approach has been developed that is known as the Single Perturbation Load Analysis (SPLA) [13]. In this approach, a lateral load is first applied to the cylinder, and this load is maintained throughout the application of axial compression leading to buckling. This analysis assumes that a single lateral buckle is a worst-case, realistic and stimulating imperfection pattern for a cylinder in compression. The lateral load causes a knockdown in the buckling load of the cylinder. It was found that small perturbation loads cause a roughly linear reduction in buckling load up to a certain transition point, after which increasing perturbation loads cause only a very small reduction relative to the transition point. This transition point was then proposed as a design point, and shown to provide buckling predictions that were less conservative and more appropriate than those of the benchmark NASA guidelines [13].
In addition to the benefits in conservatism, the use of a technique that is based on a finite element (FE) model has the advantage of being able to account for the exact behaviour of the material and structure. This allows for the incorporation of the effect of composite materials and sandwich laminates with any combination of lay-up and core materials, as part of the analysis of structure with any level of complexity. Further, the use of a technique that is able to identify a design point from only a series of numerical analysis results is highly advantageous within an industrial design context. In this way, the SPLA does not require measuring real imperfections, or information with regards to the manufacturing technique or loading imperfections, or any other information that would require manufacture or experimental testing. Such a technique has the potential to allow for the development of guidelines for imperfection-sensitive composite and sandwich structures, which is the topic of ongoing research [14], [15].
Despite the success of the previous work with the SPLA approach, the applicability of the technique in different contexts and for different structures has not been demonstrated. In their work, Hühne et al. only reported results for cylinders of a fixed geometry, with the same radius and thickness, although different layups were studied [13]. The results for the SPLA were also presented in terms of buckling load against perturbation load. However, it is unclear if the SPLA is applicable to cylinders of different geometry, how the SPLA applies to cylinders made from sandwich laminates and how best to apply the SPLA in a design context for cylinders of different thickness. It is also unclear how the SPLA compares to analysis of imperfections using eigenmodes, and how the SPLA applies to cylinders with small and large cutouts.
In this work, FE analysis is used to investigate the SPLA technique for different cylinder configurations. The SPLA is applied to cylinders of monolithic composite laminate and sandwich laminates of varying thickness. This requires a modification to the way in which the SPLA results are considered, which considerably improves the generality of the technique. The results from the SPLA investigation are compared to an eigenmode-based analysis, in terms of the imperfection sensitivity and practicality in a design context. Comparison of the SPLA approach with eigenmode-based analysis has not previously been presented in literature. Cylinders with small and large cutouts are then investigated with the SPLA approach and the interaction between the perturbation and the cutout assessed. The results provide the first application of the SPLA technique to sandwich laminates and cylinders with cutouts. This extends the understanding of both the imperfection sensitivity of monolithic and sandwich cylinders with and without cutouts, and of the applicability and usefulness of the SPLA technique.
Section snippets
Cylinder definition
A nominal cylinder geometry was defined, which was based on previous experimental and numerical investigations by one of the authors [16], [17], [18]. The cylinder geometry is illustrated in Fig. 1, and is based on using a free length (L) of 540 mm and a radius (R) of 350 mm. Monolithic laminates were defined that were based on a carbon/epoxy uni-directional tape IM7/8552, with material properties taken from literature [19], [20] and given in Table 1, where E, G, ν are the stiffness modulus,
Perturbation analysis
A finite element model was generated of the cylinder for analysis in Abaqus/Explicit 6.8-1. The model used 4-node reduced integration thick shell elements (S4R) [22]. The mesh used was based on element dimensions of 10 mm × 10 mm, which involved 220 elements around the cylinder circumference and 54 elements in the cylinder axial direction. Analyses were run on a 32-bit Intel Core 2 Duo 2.25 GHz CPU processor.
The modelling and analysis strategy applied in this work was based on the analysis reported
Eigenmode analysis
The cylinder of the previous section was analysed using eigenmode-based analysis, for comparison with the SPLA. All monolithic and sandwich laminate configurations previously described were considered. The lowest buckling mode was first found, and this was used to impose an imperfection pattern on the nominal cylinder geometry. The imperfection pattern was measured based on the ratio of the maximum displacement magnitude to the shell thickness, a/t, and a range of a/t values were investigated.
Perturbation analysis with panel cutouts
An investigation was conducted into the application of the SPLA to cylinders with cutouts. A “large” cutout of 180 × 180 mm and a “small” cutout of 25 × 25 mm were considered. The large cutout was selected as it represented a significant portion of the cylinder length, and corresponded to a significant structural feature such as windows or access panels found on aerospace structures. The small cutout was selected as it represented a minor structural feature, with a size similar to the damage
Conclusion
An investigation into the application of the SPLA has been conducted across a range of configurations. The SPLA has been applied to cylinders of different shell thickness and for sandwich laminates. It was found that across configurations the SPLA results are best presented in terms of buckling knockdown factor and perturbation displacement as a ratio of shell thickness. Different orthotropic laminates showed varying sensitivity to perturbations, and sandwich laminates were less sensitive than
Acknowledgments
The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under Grant agreement 282522. The first author acknowledges the financial support of a Research Leave Award from the School of Aerospace, Mechanical and Manufacturing Engineering in RMIT University, and an RMIT University Early Career Researcher International Travel Award.
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