3.5.5: Buckling Modes 🔷

Fig. 3.5.5.1: Beam and shell model of a cantilever: shape and load-factors of the first buckling mode

Axial forces in beams and trusses as well as in-plane forces in shells change the element response under transverse load. Tension stiffens, compression has a softening effect.

Slender columns or thin shells may fail due to buckling before the stresses in the cross section reach the material strength. Stability analysis therefore plays an important role in structural design.

When doing cross section optimization with the “Optimize Cross Section”-component, the design formulas applied take account of buckling, based on the buckling length of the members. By default local buckling of individual elements is assumed. So-called global buckling occurs if a structural sub-system consisting of several elements (like e.g. a truss) loses stability. Global buckling can be checked with the “Buckling Modes”-component (see fig. 3.5.5.1).

The “Buckling Modes”-component expects these input parameters:

"Model"	Structure with second order normal forces $N^{II}$ defined. These forces can either be taken from a second order theory calculation (like in fig. 3.5.5.1) or specified via a “Modify Element”-component.
"FromInd"	Index of the first buckling mode to be determined. The default is 1. This is also normally the only buckling shape of interest, since it corresponds to the mode of failure.
"NModes"	Number of buckling modes to be calculated. The default is 1.
"MaxIter"	The determination of the buckling modes is an iterative procedure. “MaxIter” sets the maximum number of iterations.
"Eps"	Represents the convergence criteria. For convergence the iterative change of the norm of the displacements needs to fall below that value.

The model which comes out on the right side lists the computed buckling-modes as result-cases. The buckling shapes get scaled, so that their largest displacement component has the value 1. “BLFacs” returns the buckling load factors which are assumed to be non-negative. When multiplied with those factors the current normal forces $N^{II}$ would lead to an unstable structure. The buckling load factors are listed in ascending order. The calculation of buckling factors assumes small deflections up to the point of instability. This may not always be the case.