Benedikt Kriegesmann

Probabilistic Design of Axially Compressed Composite Cylinders with Geometric and Loading Imperfections

The discrepancy between the analytically determined buckling load of perfect cylindrical shells and experimental test results is traced back to imperfections. The most frequently used guideline for design of cylindrical shells, NASA SP-8007, proposes a deterministic calculation of a knockdown factor with respect to the ratio of radius and wall thickness, which turned out to be very conservative in numerous cases and which is not intended for composite shells. In order to determine a lower bound for the buckling load of an arbitrary type of shell, probabilistic design methods have been developed. Measured imperfection patterns are described using double Fourier series, whereas the Fourier coefficients characterize the scattering of geometry. In this paper, probabilistic analyses of buckling loads are performed regarding Fourier coefficients as random variables. A nonlinear finite element model is used to determine buckling loads, and Monte Carlo simulations are executed. The probabilistic approach is used for a set of six similarly manufactured composite shells. The results indicate that not only geometric but also nontraditional imperfections like loading imperfections have to be considered for obtaining a reliable lower limit of the buckling load. Finally, further Monte Carlo simulations are executed including traditional as well as loading imperfections, and lower bounds of buckling loads are obtained, which are less conservative than NASA SP-8007.

Validation of Lower-Bound Estimates for Compression-Loaded Cylindrical Shells

The traditional approach used in the design of stability critical thin-walled circular cylinders, is to reduce unconservative buckling load predictions with an empirical knockdown factor. An alternative analysis-based approach to determine a lower bound buckling load for cylinders subjected to an axial compressive load is to use a lateral perturbation load to create an initial geometric imperfection and determine the buckling load while that perturbation load is applied. This paper describes a preliminary eort to develop a test capability to verify this analysis-based lower bound approach. Results from tests of three aluminum alloy cylinders are described and compared to nite element predictions.

The Effects of Geometric and Loading Imperfections on the Response and Lower-Bound Buckling Load of a Compression-Loaded Cylindrical Shell

Results from a numerical study of the buckling response of a thin-walled compression- loaded isotropic circular cylindrical shell with initial geometric and loading imperfections are used to determine a lower bound buckling load estimate suitable for preliminary design. The lower bound prediction techniques presented herein include an imperfection caused by a lateral perturbation load, an imperfection in the shape of a single stress-free dimple (similar to the lateral pertubation imperfection), and a distributed load imperfection that induces a nonuniform load in the shell. The ABAQUS finite element code is used for the analyses. Responses of the cylinders for selected imperfection amplitudes and imperfection types are considered, and the effect of each imperfection is compared to the response of a geometrically perfect cylinder. The results indicate that compression-loaded shells subjected to a lateral perturbation load or a single dimple imperfection, and a nonuniform load imperfection, exhibit similar buckling behavior and lower bound trends and the predicted lower bounds are much less conservative than the corresponding design recommendation NASA SP-8007 for the design of buckling-critical shells. In addition, the lateral perturbation technique and the distributed load imperfection produce response characteristics that are physically meaningful and can be validated via laboratory testing.

Closed form probabilistic analysis of lamination parameters for composite structures

Probabilistic analyses allow predicting the stochastic distribution of an output variable (e.g., the buckling load of a structure) based on the stochastic distribution of input parameters (e.g., material and geometric properties). In the probabilistic analysis of composite structures, one important quantity that can be subject to scatter is the ply or fiber orientation of the layers. Laminates with a large number of plies lead to a large number of random variables, which makes the probabilistic analysis very time consuming. Lamination parameters allow describing any ply layup by a maximum of 12 parameters. They have therefore been used for the design optimization of thick laminates. In the current paper, a two-step procedure is considered for using lamination parameters for probabilistic analyses of composite structures with many plies. In the first step, the stochastic distribution of lamination parameters is determined. In the second step, the actual probabilistic analysis is performed. A closed-form so...

Probabilistic Second-Order Third-Moment Approach for Design of Axially Compressed Composite Shells

The load carrying capability of axially compressed cylindrical shells is dependent on imperfections like geometric deviations from the perfect shell or loading imperfections. The scattering of imperfections induces that the buckling load is randomly distributed. Knowledge about the distribution of buckling load allows an efficient and save design of cylindrical shells. The stochastic distribution can be predicted with purely numerical methods like Monte Carlo simulation, which require a multitude of buckling load calculations. In the present paper a fast semi-analytic procedure is presented, that predicts the distribution of buckling load with the same accuracy as a Monte Carlo simulation, but requires much less computational effort.

Adaptive Strategies for Fail-Safe Topology Optimization

Considering fail-safe requirements in a topology optimization, where the location of the damage is unknown in advance, leads to a high number of potential damage scenarios to be calculated. Since this is driving the overall calculation time of the optimization, a reduction of the considered damage cases is desirable. In this paper, two strategies to achieve a significant reduction of damage cases are shown: An active-set strategy as an extension to the simplified local damage model first introduced by Jansen et al. as well as a newly developed load-path based algorithm for the placement of damage zones.

Optimization of Fail-Safe Lattice Structures

In the current work, a fail-safe optimization of lattice structures is carried out. For the optimization, unit cells are not homogenized, but their members are modeled as beam elements. This allows applying a commonly used engineering approach for obtaining a fail-safe design. It consist of removing one beam element at a time and optimizing the remaining structure. At the end, the maximum beam radii are used for the final design. This approach is computationally extremely expensive for lattice structures, as it requires one optimization per removed beam. In our contribution, we show that the design obtained from this approach actually does not fulfill the desired fail-safe behavior. We therefore apply an alternative approach in which the fail-safe requirement is an optimization constraint. This is still computationally demanding and therefore, criteria are discussion for reducing the number of beam elements to be considered for the fail-safe requirement within the optimization.

Novel trends in failure analysis of composite structures

Thin-walled cylindrical composite shells buckle in the elastic region and thus, there is no need to consider material non-linearities or material failure. Never the less, a robust design guide for composite shells does not exist until today. The most frequently guideline is NASA SP-8007, which gives an empirical based knockdown factor. Because this guideline is not intended for composite shells and it turned out to be very conservative in numerous cases several approaches have been developed to define a lower bound for composite shells. One branch of research is the determination of probabilistic motivated lower bounds, as it is proposed e.g. by Arbocz. Measured shell surfaces are described by double Fourier series whereas the Fourier coefficients characterize the scattering of the imperfection patterns. Additionally, non-traditional imperfections are regarded as random parameters. Then, the stochastic distribution of the buckling load with respect to the imperfections is obtained from a probabilistic analysis. The lower bound is defined as the buckling load that is associated to a chosen level of reliability. Another approach is the deterministic identification of a lower bound as it is proposed by Huhne. A perturbation load is applied, which causes a single buckle and decreases the buckling load. For a certain perturbation load, the buckling load does not decrease by increasing the perturbation load. The buckling load at this ultimate state is defined as lower bound. In case of buckling of stiffened structures like stringer stiffened panels, large deformations occur and material non-linearities have to be considered. Consequently, global failure of structures appears as a combination of stability and material failure. For textile composites, appropriate material and failure models are not available at present, regarding the complex three-dimensional structure. A further challenge is the determination of strength parameters. Especially through-thickness parameters are hardly to obtain. Therefore, in addition to real material testings, virtual material testings are performed by use of an information-passing multiscale approach. The multiscale approach consists of three scales and is based on computation of representative volume elements (RVE’s) on micro-, mesoand macroscale. On microscale, epoxy resin and single fibers are modeled, regarding statistical distribution of fibers. This yields stiffness and strength parameters of unidirectional fiber bundle material. The homogenized material parameters of the microscale