## 14. Post-Buckling of Cylindrical Panel

This test case is a variant of the panel described by Verolme [1] and Remmers [2]. The panel illustrates the post-buckling behaviour of an initially undeformed panel, which 'jumps' from its pre-buckling state to a stable post-buckling state. Remmers [2] simulates the behaviour by preforming a dynamic (transient) analysis at a load just after the instability point, following the mode-jumping technique according to Riks and Rankin [3, 4].

The term mode jumping is often used to describe a sudden dynamic change in a static process. In computational mechanics, the abrupt change in wave numbers (such as from pre- to post-buckling state) is indicated as a mode-jump. This phenomenon was first mentioned by Stein [5], describing a buckling experiment with a flat panel. Mode-jumping is an important phenomenon for stiffened panels, where a mode-jump may occur from local skin buckling (between the stiffeners) to a global buckling pattern.

In order to simulate such jumps with FEA, one can use a transient solving routine including some form of damping, such as Rayleigh damping, in order to find a stable static path [2]. The artificial damping scheme of B2000++ allows for finding a stable path with 'static' analysis. This test case uses this artificial damping method to find a stable post-buckling solution (an undamped simulation will fail).

The panel is made out of 2024-T3 aluminum and simply supported at the straight edges and clamped at the curved edges, as shown in the figure below. The panel is compressed at one of the curved edges, creating an edge shortening of 3 mm.

Table 2 lists the geometry and material parameter values.

Table 2. Verolme panel geometry and material

 Geometry length (L) 0.5 m width (W) 0.356 m radius (R) 0.38 m thickness (t) 0.001 m Material data Young's modulus (E) 73.1 GPa Poisson ratio (P) 0.33

The numerical model for the test case uses a mesh of 20x20 elements. This will execute quickly, but is too coarse for an accurate result. An 80x80 mesh will produce an accurate result. Higher mesh densities do not show any significant improvement.

The following options are defined in the cases command:

cases
case 1
residue_function_type      artificial_damping
dissipated_energy_fraction 2.e-6
step_size_init             0.01
step_size_max              0.005
step_size_min              1e-13
max_newton_iterations      30
max_divergences            1
end
end

The figure below shows the deformed panel and the load-shortening curve of the analysis for a 160x160 mesh and the load-shortening curves for other mesh densities.

[1] K. Verolme. The development of a design tool for fiber metal laminate compression panels. PhD thesis, Delft University of Technology, 1995.

[2] J. Remmers. Mode jumping with B2000. Master's thesis, Delft University of Technology, 1998.

[3] E. Riks, C. Rankin and F. Brogan. On the solution of mode-jumping phenomena in thin-walled shell structures. Computer Methods in Applied Mechanics and Engineering 136, pp 59-92, 1996.

[4] E. Riks and C. Rankin. Computer simulation of the buckling behaviour of thin shells under quasi-static loads. Archive of Computational Mechanics in Engineering 4, pp 325-351, 1997.

[5] M. Stein. Loads and deformation of buckled rectangular plates. NASA Technical Report, National Aeronautics and Space Administration, 1959.