Buckling of an imperfection-sensitive cylindrical shell

The Lanczos eigensolver is used to extract the linear buckling modes. This solver is chosen because of its superior accuracy and convergence rate relative to wavefront solvers for problems with closely spaced eigenvalues. Table 1 lists the first 19 eigenvalues of the cylindrical shell. The eigenvalues are closely spaced with a maximum percentage difference of 1.3%.

The geometry, loading, and material properties of the cylindrical shell analyzed in this example are characterized by their axisymmetry. As a consequence of this axisymmetry the eigenmodes associated with the linear buckling problem will be either (1) axisymmetric modes associated with a single eigenvalue, including the possibility of eigenmodes that are axially symmetric but are twisted about the symmetry axis or (2) nonaxisymmetric modes associated with repeated eigenvalues (Wohlever, 1999). The nonaxisymmetric modes are characterized by sinusoidal variations (n-fold symmetry) about the circumference of the cylinder. For most practical engineering problems and as illustrated in Table 1, it is usually found that a majority of the buckling modes of the cylindrical shell are nonaxisymmetric.

The two orthogonal eigenmodes associated with each repeated eigenvalue span a two-dimensional space, and as a result any linear combination of these eigenmodes is also an eigenmode; i.e., there is no preferred direction. Therefore, while the shapes of the orthogonal eigenmodes extracted by the eigensolver will always be the same and span the same two-dimensional space, the phase of the modes is not fixed and might vary from one analysis to another. The lack of preferred directions has consequences with regard to any imperfection study based upon a linear combination of nonaxisymmetric eigenmodes from two or more distinct eigenvalues. As the relative phases of eigenmodes change, the shape of the resulting imperfection and, therefore, the postbuckling response, also changes. To avoid this situation, postprocessing is performed after the linear buckling analysis on each of the nonaxisymmetric eigenmode pairs to fix the phase of the eigenmodes before the imperfection studies are performed. The basic procedure involves calculating a scaling factor for each of the eigenvectors corresponding to a repeated eigenvalue so that their linear combination generates a maximum displacement of 1.0 along the global X-axis. This procedure is completely arbitrary but ensures that the postbuckling response calculations are repeatable.

For the sake of consistency the maximum radial displacement associated with a unique eigenmode is also scaled to 1.0. These factors are further scaled to satisfy the out-of-roundness criterion mentioned earlier.

The modes used to seed the imperfection are taken from the first 19 eigenmodes obtained in the linear eigenvalue buckling analysis. Different combinations are considered: all modes, unique eigenmodes, and pairs of repeated eigenmodes. An imperfection size (i.e., out-of-roundness) of 0.5 times the shell thickness is used in all cases. The first limit-point on the load-displacement curve is recorded as the buckling load. The results indicate that the cylinder buckles at a much lower load than the value predicted by the linear analysis (i.e., the value predicted using only the lowest eigenmode of the system). An imperfection based on mode 1 (a unique eigenmode) results in a buckling load of about 90% of the predicted value. When the imperfection was seeded with a combination of all modes (1–19), a buckling load of 33% of the predicted value was obtained. Table 2 lists the buckling loads predicted by Abaqus (as a fraction of linear eigenvalue buckling load) when different modes are used to seed the imperfection.

The smallest predicted buckling load in this study occurs when using modes 12 and 13 to seed the imperfection, yet the results obtained when the imperfection is seeded using all 19 modes indicate that a larger buckling load can be sustained. One possible explanation for this is that the solution strategy used in this study (discussed earlier) involves using a fixed value for the out-of-roundness of the cylinder as a measure of the imperfection size. Thus, when multiple modes are used to seed the imperfection, the overall effect of any given mode is less than it would be if only that mode were used to seed the imperfection. The large number of closely spaced eigenvalues and innumerable combinations of eigenmodes clearly demonstrates the difficulty of determining the collapse load of structures such as the cylindrical shell. In practice, designing imperfection-sensitive structures against catastrophic failure usually requires a combination of numerical and experimental results as well as practical building experience.

The final deformed configuration shown in Figure 2 uses a displacement magnification factor of 5 and corresponds to using all the modes to seed the imperfection. Even though the cylinder appears to be very short, it can in fact be classified as a moderately long cylinder using the parameters presented in Chajes (1985). The cylinder exhibits thin wall wrinkling; the initial buckling shape can be characterized as dimples appearing on the side of the cylinder. The compression of the cylinder causes a radial expansion due to Poisson's effect; the radial constraint at the ends of the cylinder causes localized bending to occur at the ends. This would cause the shell to fold into an accordion shape. (Presumably this would be seen if self-contact was specified and the analysis was allowed to run further. This is not a trivial task, however, and modifications to the solution controls would probably be required. Such a simulation would be easier to perform with Abaqus/Explicit.) This deformed configuration is in accordance with the perturbed reference geometry, shown in Figure 3. To visualize the imperfect geometry, an imperfection size of 5.0 times the shell thickness (i.e., 10 times the value actually used in the analysis) was used to generate the perturbed mesh shown in this figure. The deformed configuration in the postbuckling analysis depends on the shape of the imperfection introduced into the structure. Seeding the structure with different combinations of modes and imperfection sizes produces different deformed configurations and buckling loads. As the results vary with the size and shape of the imperfection introduced into the structure, there is no solution to which the results from Abaqus can be compared.

The load-displacement curve for the case when the first 19 modes are used to seed the imperfection is shown in Figure 4. The figure shows the variation of the applied load (normalized with respect to the linear eigenvalue buckling load) versus the axial displacement of an end node. The peak load that the cylinder can sustain is clearly visible.