I think that there is a need for an expansion of Torpen’s comments.
The selection process for an imperfection shape must consider the specific shell theory being used. Most shell theories include the effects of an imperfection through the derivative of the imperfection shape. The strain-displacement relations include the square of the rotations and the equilibrium equations are modified by the coupling terms of the membrane components and the rotations.
If Sander’s non-linear shell theory is considered, the twist of the normal is included as possible coupling. The derivative of the imperfection in each orthogonal direction is included as an added part of these rotations (including twist) and not the imperfection itself.
The question that requires consideration is: How is the shell to be analyzed? There are essentially two possibilities: Fourier decomposition or three-dimensional shell modeling.
Fourier decomposition considers all components to be expressed in terms of both sine and cosine components. When these conditions are considered, terms of responses include cos(m±n)sin(m±n), etc. Since the main imposed forces (accelerations) are generally from the Fourier functions n = 0 and ± 1, the responding components would be the direct coupling of n = 0 with the critical shape function (k) and the adjacent modes k+1, k-1 and i+j = k and/or i-j = k , where i and j are adjacent secondary critical modes.
If real life imperfection geometries are included, then the Fourier representation of the imperfection geometry should be added to the analysis set. The most critical shapes for cylinders are diamond shapes and for spheres are hexagon shapes.
Three-dimensional shell analysis requires more ingenuity. Since the observed critical buckling behavior follows the Fourier representations, one again has to consider imperfection shapes described as a Fourier series in both orthogonal directions. If a non-linear state is considered at a specific time, then repeated eigenvalue analysis procedures will provide an estimate of the critical load condition from that state (don’t forget to include body forces). If direct load (time dependent) analysis is considered, one has to define what suggests buckling.
Increasing incremental load factors can be considered for either the direct load or Fourier decomposition method. A growth of 1000 fold of the initial imperfection is considered to be an indication that buckling has occurred. For the Fourier method, one can observe the time dependent response. A continuing growth in the critical buckling load will be observed. The energy is transferred from one or multiple modes to the single critical buckling mode over a finite time (usually associated with multiple times of the inverse of the fundamental dynamic natural frequency).
Although Torpen’s comments are essentially correct, a great deal more of engineering thought should be considered.
