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Why the buckling Eigenvalue is not a trustful way to estimate buckling.

In many construction calculations the buckling Eigenvalue is used to check if buckling is a possible failure mechanism. But this method underestimates some influences, which can make buckling more critical than calculated. A general 'rule of thumb' assumption is a minimum Eigenvalue of 2.0. But what are you assuming then?

The engineer should be aware that the buckling Eigenvalue is based on a linear model while buckling is a non-linear phenomenon probably initiated by offsets (due to tolerances) in the construction. The Eigenvalue number is similar to the Euler buckling strength and is valid for slender structures only. The figure below, basically taken from DNVGL-CG-0128, illustrates where the Euler analysis is applicable for (stiffened) plate structures. Compare the Euler buckling or Eigenvalue analysis with the green curve.

DNVGL_CG_0128 figure 3

Note that:

Lambda definition

As illustrated when λ>1.4; Euler is applicable. With non slender designs, the Eigenvalue approach is too optimistic.

But as shown by the 'Ultimate capacity for plate' curve, the capacity of the structure is in many cases higher then predicted by Euler. Elastic buckling in the slender range does not lead immediately to collapse of the structure.  The higher the slenderness factor (λ) the higher the factor between elastic (Euler) buckling and ultimate buckling. After plate buckling the residual load is taken by stiffeners and girders until also these elements will buckle. Therefore initial plate buckling will be not that critical (for slender structures).

The graph in the figure illustrates that the Eigenvalue calculation is far off reality and needs to be treated with caution. Note that buckling is initiated by initial offsets in the structure, the Eigenvalue method is therefore always too optimistic since:

  1. For non-slender designs the elastic buckling is not applicable. Using the Eigenvalue calculation gives wrong answers. A minimum factor of 2 in designs (e.g. with λ<1.2) is no cure and no guarentee for a safe buckling design;
  2. For slender designs the elastic buckling of the panel is initiated by offsets in the structure, which results in lower buckling allowables than calculated by the Eigenvalue. To cover this by a factor of 2 on the Eigenvalue is just a quick 'dirty' solution which will lead to an unnecessary heavy design.

The buckling Eigenvalue is only applicable for slender structures predicting the elastic buckling behavior of the structure and its modes. Although the actual elastic buckling allowable is lower, the (unfactored) Eigenvalue is probably acceptable since it is lower than the collapse stress. This is illustrated by the space between the green line (Elastic buckling) and the 'Ultimate capacity for plate' curve.

It can be concluded that the buckling Eigenvalue is a very rough indication of the elastic buckling strength of the structure. The actual strength capacity can be way off what is calculated. (Probably the mode shapes are more valuable than the actual elastic buckling stress.)  This method is therefore not very suitable for certification of structures. Maybe it is handy for some concept estimations, but why not using the correct method from the beginning?

 Rather use decent methods according to acceptable standards.

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