Buckling Eigenmodes

The following compression beam is fixed at one end and subjected to load according to Euler case 1. The profile and system dimensions are illustrated below. The beam is made of construction steel St 37. The load is applied at the center of mass.

According to Peterson [1], the ideal buckling load is PKi = 56.4 kN
According to Peterson [1], the ideal buckling load is PKi = 56.4 kN

For the system detailed above, the FEM calculation delivers the following results for the
first 4 buckling eigenmodes. Influences such as shear strain and warping are taken into account.

Buckling eigenmode 1, load factor = 54.14 (torsional buckling)
Buckling eigenmode 1, load factor = 54.14 (torsional buckling)
Buckling eigenmode 2, load factor = 190.52 (buckling around the z axis of the section)
Buckling eigenmode 2, load factor = 190.52 (buckling around the z axis of the section)
Buckling eigenmode 3, load factor = 345.88 (buckling around the y axis of the section)
Buckling eigenmode 3, load factor = 345.88 (buckling around the y axis of the section)
Buckling eigenmode 4, load factor = 422.61 (buckling around the z axis of the second eigenmode)
Buckling eigenmode 4, load factor = 422.61 (buckling around the z axis of the second eigenmode)

Buckling Eigenmodes of a Column

System and loads
System and loads

The illustrated structure was analyzed by Rubin and Aminbagai [2].

The stability analysis delivers, in complete agreement with the above
publication, the following buckling eigenmodes and results:

Buckling lengths and load factors
Buckling lengths and load factors

Buckling Eigenmode of a Shell Structure

This example addresses slab buckling. The illustrated slab is supported flexibly all around. The slab thickness is 0.005 m, the material is construction steel St 37. Two load cases will be analyzed.

System and loads
System and loads

Uniformly Distributed Load

The theoretical buckling stress according to Petersen is sKi = 1.18625 MN/m².
Given the above system setup, this is approximated by 1.186 < sKi < 1.188 MN/m².

Triangular Load

Calculated buckling eigenmode
Calculated buckling eigenmode

The theoretical buckling stress according to Petersen is sKi = 2.3132 MN/m².
Given the above system setup, this is approximated by 2.322 < sKi < 2.324 MN/m².

References

[1] Petersen, C.
Statik und Stabilität der Baukonstruktionen (Statics and Stability of Constuctions). Vieweg Verlag, Braunschweig 1980.

[2] Rubin, H.; Aminbaghai, M.
Ein Stabilitätsproblem mit Überraschungen (A Stability Problem with Surprises).
Stahlbau 66, Issue 5, pp. 271 - 276. Verlag Ernst & Sohn, Berlin 1997.