James H. Starnes

FUTURE DIRECTIONS AND CHALLENGES IN SHELL STABILITY ANALYSIS

Recent advances in structural analysis and design technology for buckling-critical shell structures are discussed. These advances include a hierarchical analysis strategy that includes analyses that range from classical analysis methods to high-fidelity nonlinear finite element analysis methods, reliability based design methods, the development of imperfection data bases, and the identification of traditional and nontraditional initial imperfections for composite shell structures. When used judiciously, these advances provide the basis for a viable alternative to the traditional and conservative lower-bound design philosophy of the 1960s. These advances also help answer the question of why, after so many years of concentrated research effort to understand the behavior of buckling-critical thin-walled shells, one has not been able to improve on this conservative lower- bound design philosophy in the past.

Construction of Response Surface Approximations for Design Optimization

Using response surface approximations for design constraints in design optimization provides the designer with an overall perspective of the system response within the design space. Response surface approximations also reduce the numerical noise inherent in many numerical models and simplify the process of integrating several design codes, as is typically required in multidisciplinary optimization. Procedures are discussed for constructing accurate response surface approximations to represent design constraints in design optimization. Response surface approximations are constructed for the stresses and buckling loads of an isotropic plate with an abrupt change of thickness. These response surface approximations are constructed from numerical experiments conducted with a finite element analysis procedure and are used for minimum-weight optimum design of the plate. Nondimensional variahles and stepwise regression are used to reduce the complexity and increase the accuracy of the response surface approximations. Additionally, higher-order polynomials (cubic and quartic instead of the more traditional quadratic) are used as response surface approximations, and a detailed error analysis, using an independent data set, is performed. Finally, it is shown that, by making use of response surface approximations, the optimum weight of the plate may be presented in the form of a design chart for a wide range of geometric, loading, and material constants.

Effects of Imperfections on the Buckling Response of Compression-Loaded Composite Shells

The results of an experimental and analytical study of the effects of initial imperfections on the buckling and postbuckling response of three unstiffened thin-walled compression-loaded graphite-epoxy cylindrical shells with different orthotropic and quasi-isotropic shell-wall laminates are presented. The results identify the effects of traditional and non-traditional initial imperfections on the non-linear response and buckling loads of the shells. The traditional imperfections include the geometric shell-wall mid-surface imperfections that are commonly discussed in the literature on thin shell buckling. The non-traditional imperfections include shell-wall thickness variations, local shell-wall ply-gaps associated with the fabrication process, shell-end geometric imperfections, non-uniform applied end loads, and variations in the boundary conditions including the effects of elastic boundary conditions. A high-fidelity non-linear shell analysis procedure that accurately accounts for the effects of these traditional and non-traditional imperfections on the non-linear responses and buckling loads of the shells is described. The analysis procedure includes a non-linear static analysis that predicts stable response characteristics of the shells and a non-linear transient analysis that predicts unstable response characteristics.

Effect of dropped plies on the strength of graphite-epoxy laminates

The reduction in the compressive and tensile strength of graphite-epox y laminates with thickness discontinuities due to dropped plies was studied by experiment and analysis. The specimens were fabricated with all the dropped plies lumped together in the center of a 16-ply quasi-isotropic layup, such that one surface was flat and the slope of the opposite surface changed abruptly at the dropped ply location to accommodate the thickness change. Even though the thick and thin sections are symmetrically laminated, there exists bending-extensioii coupling due to the geometric eccentricity of the middle planes of the thick and thin sections. Results from tests on 37 specimens are reported that differed in the configuration of the dropped plies only. The axial strength of a laminate with dropped plies is less than the strength of its thin section, and the compressive strength of a laminate with dropped plies is less than its tensile strength. The reduction in axial strength is directly related to the axial stiffness change between the thick and thin sections. To examine the mechanism of failure, the three-dimensional state of stress in the dropped ply region was evaluated by the finite element method. A tensile interlaminar criterion predicted the correct location of failure, but underestimated the failure load, and therefore is a conservative analysis procedure for design.

Construction of response surfaces for design optimization applications

Using response surface approximations in design optimization provides the designer with an overall view of the response. Response surface approximations also reduce the numerical noise inherent in many numerical models and simplify the process of integrating several design codes, as is typically required in the multidisciplanary optimization process. The present paper discusses procedures for constructing accurate response surface approximations to be used in design optimization, by tailoring the response surface to the specific design problem. A homogeneous, isotropic plate with a change in thickness is the design problem considered in the present and response surface approximations are constructed for the stress concentration factor at the thickness discontinuity and for the buckling load of the plate. These response surfaces are constructed from the results of numerical experiments conducted with a finite element analysis. It is shown that by using the proposed procedures, it is possible to obtain response surfaces with a high degree of accuracy. Graduate Research Assistant 1 Professor, Associate Fellow AIAA * Head, Structural Mechanics Branch. Fellow, AIAA Copyright

Shell Buckling Design Criteria Based on Manufacturing Imperfection Signatures

An analysis-based approach for developing shell-buckling design criteria for laminated-composite cylindrical shells that accurately account for the effects of initial geometric imperfections is presented. With this approach, measured initial geometric imperfection data from six 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 shapes include the actual measured imperfection shape of the test specimens, a first-approximation mean imperfection shape, with and without plus or minus one standard deviation, and the 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 of the nonlinear finite element analyses is also presented. Overall, the results indicate that the analysis-based approach presented 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.

Effect of impact damage and holes on the compressive strength of a graphite/epoxy laminate

The effect of low-velocity impact damage and circular holes on the compressive strength of a 48-ply orthotropic graphite/epoxy laminate has been studied experimentally. Unidirectional tapes made of Thornel 300 graphite fibers preimpregnated with Narmco 5208 epoxy resin were used to fabricate the [′45/0 2 /′45/0 2 /′45/0/90] 2 s laminate. Aluminum spheres 1.27 cm in diameter were used as projectiles and propelled at speeds between 52 and 101 m/s. The extent of interior laminate damage caused by impact was determined by ultrasonic inspection and, in some cases, by cross-sectioning specimens for visual inspection. Some specimens were impacted and then loaded to failure in compression to determine their residual strength. Other specimens were loaded to a prescribed compressive strain and impacted at that applied load. Some of these loaded specimens failed catastrophically on impact. Specimens that did not fail catastrophically were subsequently loaded to failure in compression to determine their residual static strength. It was found that low-velocity impact damage can seriously degrade the laminate static compressive strength. Several impact-damaged specimens were subjected to low-strain compression-compression cyclic loading, which was found to degrade further the laminate compressive strength. Specimens with circular holes with diameters up to a third of the specimen width were loaded in static compression to failure, and it was found that holes can also degrade the compressive strength of the laminate. Impact at the higher speeds reduced the compressive strength of the laminate more than the largest holes studied.

Non-linear buckling of a column with initial imperfection via stochastic and non-stochastic convex models

Abstract Buckling of initial imperfection sensitive structure — column on a non-linear elastic foundation — is investigated. A criterion based on the concept of “modal buckling load” is proposed to determine which modes should be included in the analysis when the weighted residuals method is utilized to calculate the limit load — maximum load the structure can support — for a given initial deflection. For stochastic analysis, a random field model is suggested for the uncertain initial imperfection, and Monte Carlo simulations are performed to obtain the probability density of the buckling load and the reliability of the column. Finally, a non-stochastic convex model of uncertainty is employed to describe a situation when only limited information is available on uncertain initial deflection, and the minimum buckling load is obtained for this model. The results from both the stochastic and the non-stochastic approaches are derived and critically contrasted.

Thermomechanical buckling and postbuckling of multilayered composite panels

Abstract A study is made of the thermomechanical buckling and postbuckling responses of flat unstiffened composite panels. The panels are subjected to combined temperature change and applied edge displacement. The analysis is based on a first-order shear deformation, von-Karman type nonlinear plate theory. A mixed formulation is used with the fundamental unknowns consisting of the generalized displacements and the stress resultants of the plate. An efficient multiple-parameter reduction method is used in conjunction with mixed finite element models, for determining the stability boundary and postbuckling response. The reduction method is also used for evaluating the sensitivity coefficients which measure the sensitivity of the buckling and postbuckling responses to variations in the different lamination and material parameters of the panel. Numerical results are presented showing the effects of variations in the laminate stacking sequence, fiber orientation, number of layers and aspect ratio of the panels on the thermomechanical buckling and postbuckling responses and their sensitivity.

Buckling of an axially compressed cylindrical shell of variable thickness

Abstract This paper focuses on the buckling of cylindrical shells with small thickness variations. Two important cases of thickness variation pattern are considered. Asymptotic formulas up to the second order of the thickness variation parameter e are derived by the combination of the perturbation and weighted residual methods. The expressions obtained in this study reduce to Koiters formulas, when only the first-order term of the thickness variation parameter is retained in the analysis. Results from the asymptotic formulas are compared with those obtained through the purely numerical techniques of the finite difference method and the shooting method.