Buckling analysis

Adjustment of general parameters may control the calculation.

Calculation for selected stability combinations

If general optionAdvanced solver option is ON, the user may specify which stability combinations will be calculated. Otherwise, all non-calculated are always calculated.

Assumptions of linear buckling calculation

The word ‘linear’ suggests that the following phenomena cannot be taken into account:

• non-linear elastic supports and foundation,

• beams acting in compression or in tension only.

Since the calculation is linear, both types of non-linearity listed above are considered in a way that corresponds to equilibrium equations assembled on a non-deformed structure. In other words, foundation stiffness is equal to stiffness for zero deflection and ‘one-way-action’ beams are taken as usual beams.

Let’s start with the equilibrium equation

(KE + KG) u = R

where KG is a geometric stiffness matrix reflecting the effect of axial forces in beams and slabs.

The basic assumption is that the elements of the matrix KG are linear functions of axial forces in beams. That means, matrix KG corresponding to a K-th multiple of axial forces in the structure is the K-th multiple of original matrix KG. The aim of the buckling calculation is to work out such a multiple K for which the structure loses stability. Such a state happens when the equation below has a non-zero solution.

(KE + K . KG) u = 0

In other words, such a value K should be found for which the determinant of the total stiffness matrix (the term in the brackets) is equal to zero. For information about the methods for solving eigenvalue problems in SCIA Engineer, see the chapter about natural vibration analysis. Similarly to the dynamic calculation, the result is a series of eigenmodes and corresponding K -multiples. The first eigenmode is usually the most important and corresponds to the lowest K -multiple. A possible collapse of the structure usually happens for this first mode.

Note: There is a difference in behaviour of beam and shell elements. For shell elements the axial force is not considered in one direction only. The shell element can be in compression in one direction and simultaneously in tension in the perpendicular direction. Consequently, the element tends to buckle in one direction but is being ‘stiffened’ in the other direction. This is the reason for significant post-critical bearing capacity of such structures.

The calculated buckling eigenmodes can help the structural engineer to get an idea about behaviour of a structure and about possible mechanisms of buckling failure. The resultant critical multiple suggests how far the structure is from possible buckling.

Note: The buckling calculation can be carried out for stability combinations only.