Buckling and Postbuckling Structures

The following sections are included:

• Preface

• Contents

Experimental results on the behavior of nine single panels and of four torsion boxes, each comprising of two stringer-stiffened cylindrical graphite-epoxy composite panels are presented, these were tested under axial compression, torsion and combined loading. The buckling and postbuckling behavior of these single panels and torsion boxes demonstrated consistent results. Prior to performing the buckling tests, the initial geometric imperfections of the panels and boxes were scanned and recorded. The tests were complemented by finite element calculations, which were performed for each panel and box. These detailed calculations have also assisted in identifying critical regions of the boxes and the boxes were reinforced accordingly to avoid premature failure. The investigation on the single panels revealed a good correlation between the predictions of the finite element codes and the experimental results. The tests indicated that the torsion carrying capacity of the boxes is laminate lay-up dependent; axial compression results were in very good agreement with previous tests performed with single identical panels; and that both the panels and the boxes have a very high postbuckling carrying capacity.

First, the test equipment used for various types of buckling experiments in the Department of Aerospace Engineering at the Politecnico di Milano is presented. It can apply axial and torsion loading, separately and in combination, using displacement control. Then, the results of an experimental investigation on stiffened shells and curved panels, made from graphite-epoxy, are presented. The experimental data acquired during the first non-destructive buckling tests and during the final destructive failure tests demonstrate clearly the strength capacity of these structures to work in the postbuckling range allowing further weight savings. The results show that the structures are able to sustain load in the postbuckling range without any damage. On the negative side, the collapse is sudden and destructive. The measured data are also useful for the development and validation of analytical and numerical high-fidelity methods. These validated analysis tools can provide design criteria that are less conservative than the existing ones.

The phenomenon of secondary instabilities in stiffened composite panels, loaded in uniaxial compression, is studied through a number of experimental investigations. This behaviour is shown to manifest itself as a ‘mode-jump’ – a sudden change in the out-of-plane deformation of a buckled panel. Use of high-speed digital speckle photogrammetry is made to gain further insight into an I-stiffened panel's response during the transient phase associated with mode-jumping. At high loading, this sudden dissipation of energy will be shown to be able to cause failure in vulnerable structures. Predicting this structural response reliably, using the finite element method, poses considerable numerical challenges and the shortcomings of current non-linear solution schemes are discussed. A robust and efficient strategy, which utilizes an automated quasi-static/pseudo-transient hybrid scheme, is presented and validated using a number of experimental tests. This approach is shown to be able to predict mode-jumping with good accuracy.

An analysis-based approach for developing shell-buckling design factors for cylindrical shells that accurately accounts for the effects of initial geometric imperfections is presented. To develop this approach, measured initial geometric imperfection data from six laboratory-scale graphite-epoxy shells are used to determine a manufacturing-process-specific imperfection signature for these shells. This imperfection signature is then used as input into nonlinear finite-element analyses. The imperfection signature represents a “first-approximation” mean imperfection shape that is suitable for developing preliminary-design data. Comparisons of test data and analytical results obtained by using several different imperfection shapes are presented for selected shells. These imperfection shapes include the actual measured imperfection shape of the test specimens, a “first-approximation” mean imperfection shape with plus or minus one standard deviation in the shape, and a linear-bifurcation-mode imperfection shape. In addition, buckling interaction curves for composite shells subjected to combined axial compression and torsion loading are presented that were obtained by using the various imperfection shapes in the analyses. A discussion on the development of experimentally validated nonlinear finite-element analyses is also presented. Overall, the results indicate that the proposed analysis-based approach presented herein, for developing reliable preliminary-design criteria, has the potential to provide improved, less conservative buckling-load estimates, and to reduce the weight and cost of developing buckling-resistant shell structures.

Allowable compressive loads of thin-walled stiffened panels, made of carbon fibre reinforced plastic (CFRP), are currently limited by the first occurrence of buckling, in most cases a skin-based local instability, which drive the design. Therefore, the remaining structural reserves in the form of load-carrying capacity in the postbuckling regime up to the onset of global stringer-based buckling, are not fully exploited. The efficient design of these structural components requires a novel stability design scenario, to permit postbuckling below ultimate load and at the same time to assure that there is no incidence of structural degradation. This requires accurate, reliable and efficient numerical analysis, commonly accomplished by nonlinear finite element calculations.

After introducing the main aspects of the stability design scenario, the design process of the test structures is detailed, followed by an overview of experimental investigations of two nominally identical panels. Finally the nonlinear finite element (FE) analysis and the validation process at different levels of detail is described.

Significant improvements in structural efficiency are possible using specific, inhomogeneous laminates. These are laminates whose stacking sequence has been optimized against constraints such as buckling, damage tolerance and natural frequencies. Furthermore, there are many design scenarios in which further gains can be made by making use of anisotropy, particularly when buckling or postbuckling is concerned. After a brief introduction to current design practice, buckling and postbuckling formulae are derived for anisotropic plates and shells, subject to different loadings. From these expressions, concepts for anisotropic elastic tailoring are discussed and their implications for design considered.

The key features of the finite strip buckling analysis program VICONOPT are described. A bi-level strategy, for the efficient optimum design of composite stiffened panels is also presented. Panel level optimization finds a minimum weight cross-sectional geometry based on a substitution of equivalent orthotropic plates for laminated plates. Optimization at the laminate level finds stacking sequences satisfying laminate design rules using either manual selection or a Genetic Algorithm. VICONOPT models are validated with ABAQUS finite element models, and with experimental compressive testing of two blade-stiffened panels. The buckling and postbuckling behaviour of the two panels, with initial buckling in the stiffeners and skin, respectively, is investigated in a high load and high strain range. The bi-level strategy is evaluated by design of a Z-stiffened example, which has been manufactured and tested, and also by design of a wing cover panel which is subject to compression and lateral pressure. The optimum wing panel, considering a variety of stiffener shapes, is 17% lighter than an existing datum design.

An overview of buckling and postbuckling behavior of pipes and tubes is presented, with particular emphasis on nonlinear finite element modeling. Classical bifurcation solutions for isotropic elastic cylinders under external pressure and bending are presented first. Subsequently, the stability of metal tubes and pipes under bending and external pressure is examined, and comparison with available experimental data is conducted. Finally, the behavior of composite cylinders under external pressure is presented, and the important issue of delamination is thoroughly discussed.

Partly due to the major discrepancies between observed buckling loads and the prediction from classical theory, the buckling of pressurised spherical caps, as well as axially compressed cylinders, has excited extraordinary interest over the past half century. This chapter reviews the historical change in the understanding of the buckling mechanism of the spherical shell caps. Then fifty-two selected experiments, previously conducted by Yamada are compared with the numerical simulation results recently obtained using an advanced mixed finite element technique. It is shown that since measured geometric initial imperfection distributions are carefully introduced, good agreement between the numerical simulations and the experimental measurements are obtained in not only buckling loads but also postbuckling large deflection behaviour.

Recent developments in the variational modelling of nonlinear interactive buckling phenomena, found in compression sandwich panels, are presented. Strong interactions between local and global modes are found to lead to localized buckling, in the more compressed face-plate, after the global instability. Important features such as the effect of using orthotropic core materials and face-plates of differing thicknesses, the potential for face–core delamination, panels under combined bending and compression, and the sensitivity to imperfections are discussed. The combination of modal interactions and the introduction of initial geometric imperfections in the struts shows that the critical buckling loads from linear analysis can significantly overestimate their true elastic strength.

This chapter describes recent developments on the buckling and post-buckling analysis of thin walled structures by the Boundary Elements Method (BEM). Boundary integral equations are obtained by coupling two dimensional plane stress elasticity with boundary element formulation of Reissner plate bending. The linear buckling problem is formulated as a standard eigenvalue problem, in order to obtain directly critical loads and buckling modes as part of the solution. A multi-region BEM formulation is also introduced for the analysis of local buckling in thin walled assemblies. Nonlinear equations for large deflection and post-buckling analysis are also investigated. Nonlinear system of equations is solved using an incremental solution strategy, where displacements and tractions are applied in small but finite increments. Several examples of flat and curved plates (shallow shells) with different dimensions and boundary conditions are analyzed, including plate assemblies. Results are compared with other numerical and analytical solutions, and good agreement is obtained.

A mechanism-based progressive failure analyses approach is presented for fiber reinforced composite laminates that are loaded predominantly in compression. Each ply of the laminate is modeled as a nonlinear elastic degrading lamina (characterized through laboratory scale experiments, in shear and in transverse directions) in a state of plane stress. In the fiber direction, elastic behavior prevails, however, in the present work, the phenomenon of fiber microbuckling, is explicitly accounted for by allowing the in situ fiber rotation to be a variable in the problem. The present model disbands the notion of a fixed compressive strength, and instead uses the mechanics of the failure process to provide the in situ compression strength in a lamina, dictated by the current local stress state, the current state of the degrading lamina and the local fiber rotation. These features are built into a user defined material subroutine that is implemented through the ABAQUS commercial code in conjunction with Classical Lamination Theory. The inputs to the present work are data for lamina transverse property degradation, the elastic lamina orthotropic properties, the ultimate tensile strength of the lamina in the fiber direction, the laminate stacking sequence and the structural panel geometry. The validity of the approach is demonstrated through numerical simulations of the response of a double notched flat unstiffened panel subjected to compression. The predictions of the simulations are compared against experimental data and good agreement is reported.

The reason for the onset of undesirable nonlinear mechanical behaviour of structures can often be found in instability phenomena on length scales that are much smaller than those of the component. Micro-buckling in cellular materials, e.g., may lead to layer-wise collapse of the cell structure, corresponding to a characteristic plateau in the overall compressive force-displacement relationship. Correspondingly, local buckling of the face sheets or the cores of sandwich plates may mark the onset of buckling of the whole sandwich plate. To account for the importance of understanding and predicting of buckling on small length scales this paper discusses buckling phenomena in open and closed-cell cellular materials as well as in sandwich structures with unstructured and structured cores. Analytical and numerical methods for predicting corresponding critical loads are presented. Most of these methods rely on the assumption of an actual, or representative, periodic micro-structure for facilitating the analysis. The influence of boundary conditions and some localisation phenomena are also discussed.

The following sections are included:

• Index