3.6.5: Buckling Modes

Fig. 3.6.5.1: 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, alter the element response under transverse load. Tension increases stiffness, while compression reduces it. Slender columns or thin shells may fail due to buckling before the stresses in the cross-section reach the material strength, making stability analysis crucial in structural design.

When performing cross-section optimization with the “Optimize Cross Section” component, the design formulas applied account for buckling, based on the buckling length of the members. By default, local buckling of individual elements is assumed. Global buckling, which occurs when a structural sub-system consisting of several elements (such as a truss) loses stability, can be checked with the “Buckling Modes” component (see Fig. 3.6.5.1).

For calculating buckling modes there need to be second order normal forces $N^{II}$ present in the system. These can be defined directly via a "Modify Element"-component or calculated through a secon order theory analysis.

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 for the shell elements) or specified via a “Modify Element”-component (like in fig. 3.5.5.1 for the beam elements).
"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.
"LCasesNII"	Names of the load-cases from which the largest compressive force $N^{II}$ is chosen for each element. By default all calculated load-cases are considered. In case no calculated results are present the user-defined NII-values are used (see "ModifyElement"-component in section 3.1.8).
"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 inputs "MaxIter" and "Eps" control the accuracy of the Eigen Modes calculation. The default values work in most cases.

The model which comes out on the right side lists the computed buckling-modes as result-cases of the load-case-combination "BucklingModes". 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.

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Buckling_Cantilever.gh