# Thin-walled cylinders are sensitive to imperfections

Decreased buckling loads and geometrical imperfections are closely related for thin-walled cylinders. By definition, a geometrical imperfection is a deviation from the perfect cylindrical geometry. An example is shown in Figure 1. Geometrical imperfections have been demonstrated to have clearly the highest influence on the buckling load when compared to, for example, the effect of thickness variations, stress variations, and boundary stiffness. The real imperfections are often unknown, but in the design process the influence of the imperfections has to be taken into account properly to achieve a safe structure. Design guidelines, such as NASA SP-8007, do not require knowledge about the pattern or even amplitude of the imperfections. Instead, very conservative knock-down factors are used. [1]

Figure 1. A closed skewed imperfection. The end cross-sections of the cylinder have a perfect round shape. There are three axial half-waves and five circumferential waves in the structure. The skewedness parameter has been set to 0.5. The amplitude of the imperfection has been extended for visualization.

Measuring the real imperfections and introducing them into the numerical analysis has been traditionally very costly, but results in less conservative design loads. Realistic geometrical imperfections in a numerical analysis are such that they describe the existing pre-buckling shape and size as good as possible. This approach assumes that specific manufacturing processes statistically result in basically the same imperfections. It has been assumed that the eigenform affine patters are the worst imperfections. Later, axisymmetric imperfections were believed to show the worst behavior. Stimulating imperfections is another approach where a local imperfection is generated to the geometry. The Single Perturbation Load Approach (SPLA) belongs to this class and it has been widely studied in the ongoing DESICOS project (http://www.desicos.eu/images/desicos/307152_DLR_Plakat.pdf).

Hühne showed by numerical and experimental investigations that the collapse of cylindrical shells always starts with a single buckle. [1]

Figure 2. a) The ESAComp specification dialog for cylinder analysis; b) an axisymmetric imperfection with seven half-waves; c) the first buckling mode shape according to the linear buckling analysis; d)

The ESAComp software provides a very user-friendly approach to introduce geometrical imperfections into the geometry. The shape of the imperfection can be generated with a linear buckling analysis and scaled by a user-defined amplitude. For cylindrical shells the distorted shape can be based on analytical formulas as well. Various possibilities are illustrated in Figure 2. ESAComp acts as an efficient tool for studying the imperfection sensitivity through the alternation of the shape and amplitude of the imperfection, and thus helps in finding a robust design.

[1] ECSS‐E‐HB‐32‐24A, Space Engineering – Buckling of Structures, ESA Requirements and Standards Division, 24 March 2010