A thin-walled axially compressed CFRP cylinder was studied using the large deformation analysis. Different numbers of axial, circumferential and combined imperfection waves were used to represent the initial shape of the structure. Three examples are shown at the bottom of Figure 1. The horizontal axis of the chart corresponds to the number of axial half-waves and the vertical axis to the number of circumferential waves. The amplitude of the imperfection was 0.2mm, which has been magnified by a factor of 100 for the visualization.
The reference application is summarized in http://www.desicos.eu/images/desicos/307152_DLR_Plakat.pdf
The nominal load used in this sensitivity study corresponded to the SPLA (Single Perturbation Load Approach) Design Buckling Load of 17.99kN. The results of the sensitivity study are presented in the bubble chart of Figure 1. Color codes from red to blue and bubble sizes from small to large indicate the increasing load factor with respect to the SPLA.
Figure 1. Results of the sensitivity study.
In SPLA a geometrical imperfection is created with a radial perturbation load P. For the reference application the perturbation load was as small as 3N. Using the ESAComp panel analysis, which is applicable to semi-cylinders (Figure 2), the deformation level with P=3N was solved. The maximum deformation was slightly over 0.2mm and therefore, the selected amplitude in the sensitivity study was 0.2mm. The radial perturbation force generates a local axial half-wave and oscillating circumferential waves. If we assume that there are multiple equally spaced perturbation loads, the initial imperfect shape can be generated with ESAComp, for example, by assuming that there is a single axial half-wave and eight circumferential waves. Generally the study was made with the imperfection amplitude of 0.2mm. However, a single result was generated with the imperfection amplitude of 2mm. This is presented in the chart of Figure 1 by the red bubble with a black border.
Figure 2. Deformed shape of the semi-cylinder loaded with a single perturbation force P.
In ESAComp the nonlinear equations are solved iteratively by Riks method with the load-controlled incrementation scheme. The number of sub-steps is defined through the analysis options. The nonlinear analysis of ESAComp provides on-line monitoring. The result tracker indicates how the maximum deflection of the structure develops as a function of the load increment and thus gives a direct indication at which load level the structure starts to behave in a nonlinear manner. The graph in Figure 3 is related to the cylinder for which the initial shape has been obtained from the linear buckling analysis. The presented solution has just converged with the design load. The deformation has been scaled by a factor of 100 for the visualization. A load factor of 1.02 was obtained for this configuration and this is indicated in Figure 1 as "Buckl. imp."
Figure 3. The result tracker (left) and the deformed cylinder (right) for which the initial shape was generated using the linear buckling analysis.
To learn more about buckling of shell structures please join the 3rd Int. Conference on Buckling and Postbuckling Behaviour of Composite Laminated Shell Structures in Braunschweig, Germany on March 25-27, 2015 . Keynote speakers come from major research centers and the industry. ESAComp will be presented in the Workshop. Further information: http://www.desicos.eu/images/desicos/draft_program.pdf
© 2015 Componeering Inc, by Harri Katajisto