No abstract

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.

A robust topology optimization approach is presented which uses the probabilistic first-order second-moment method for the estimation of mean value and variance of the compliance. The considered sources of uncertainty are the applied load, the spatially varying Young's modulus and the geometry with focus on the latter two. In difference to similar existing approaches for robust topology optimization, the presented approach requires only one solution of an adjoint system to determine the derivatives of the variance, which keeps the computation time close to the deterministic optimization. For validation, also the second-order fourth-moment method and Monte Carlo simulations are embedded into the optimization. For all approaches, the applicability and impact on the resulting design are demonstrated by application to benchmark examples. For random load, the firstorder second-moment approach provides unsatisfying results. For random Young's modulus and geometry however, the robust topology optimization using first-order second-moment approach provides robust designs at very little computational cost.

No abstract

This paper presents a two-stage procedure for density-based optimization towards a fail-safe design. Existing approaches either are computationally extremely expensive or do not explicitly consider fail-safe requirements in the optimization. The current approach trades off both aspects by employing two sequential optimizations to deliver redundant designs that offer robustness to partial failure. In the first stage, a common topology optimization or a topology optimization with local volume constraints is performed. The second stage is referred to as “density-based shape optimization” since it only alters the outline of the structure while still acting on a fixed voxel-type finite element mesh with pseudo-densities assigned to each element. The performance gain and computational efficiency of the current approach are demonstrated by application to various 2D and 3D examples. The results show that, in contrast to explicitly enforcing fail-safety in topology optimization, the current approach can be carried out with reasonable computational cost. Compared to the local volume constraint approach, the suggested procedure further increases the fail-safe performance by 47% for the example considered.