The objectives for this example include the study of buckling under the action of axial and transverse loads. Such studies are usually classified as follows:

A column may buckle in any one of these modes. Only the lowest value is of practical interest in design calculations. In general cases, buckling failure may occur by a combination of torsion and bending, which is best addressed by a load-displacement study.

We consider slender, elastic straight beams, orientated along the -axis, all with the I-section shown in Figure 1.2.1–1. The section dimensions are suitable for the study of flexural, lateral, and torsional instability problems. The beam is assumed to be made up of an isotropic material with Young's modulus 211 GPa and Poisson's ratio of 0.3125. The mesh consists of 20 B31OS or 10 B32OS beam elements spanning the 12 m length of the beam. This discretization should give good accuracy for the first several modes of buckling. Mesh convergence studies are not reported here.

A cantilever beam is considered for the Euler buckling problem. All degrees of freedom are restrained at the clamped end of the beam. The input data are shown in beambuckle_b31os_isec_flex.inp. An interesting extension of this buckling problem is to examine the response of the column far into the postbuckling range. This is the simplest of the classical “elastica” problems, an elastica being an elastic curve bent by some load (see Timoshenko and Gere, 1961). For this study an initial imperfection in the shape of the lowest buckling mode, with a peak magnitude of 10% of the beam thickness, is introduced. The Riks technique is used. An axial force, equal in magnitude to the critical load, is applied, and the analysis is stopped when the axial force becomes six times the applied load.

All components of displacement, and the rotation about the -axis, are restrained at one of the support nodes for the lateral/torsional buckling problems. Displacements in the - and -directions, and rotation about the -axis, are restrained at the other support node. The *BEAM SECTION option is tested with section types I and ARBITRARY. The *BEAM GENERAL SECTION option is tested with section types I, ARBITRARY and GENERAL. (The use of *BEAM GENERAL SECTION, SECTION=GENERAL in combination with the open section beam elements requires that the warping constants be specified.)

beambuckle_b31os_isec_lat.inp shows the input data used for the eigenvalue buckling analysis. The distributed load is applied as load type PZ, with a magnitude of 1 N/m. A load-displacement analysis is then performed, with collapse being defined by large motion occurring under very small load increments. The model used must provide for switching to the buckling mode. A slight initial imperfection is used for this purpose. The first mode from the eigenvalue buckling analysis is scaled to have a maximum rotation equal to 1% of the flange width. The translational displacements are equally scaled and added to the nodal coordinates to define the perturbed or imperfect geometric data. The normal at each node is defined under *NODE based upon the scaled rotations from the eigenvalue analysis. Since instabilities are expected, the Riks method is used. The analysis is terminated when the lateral displacement () of the middle node is greater than the flange width of the beam. The input for this load-displacement analysis is shown in beambuckle_b31os_arbsec_lat.inp.

The model used for the eigenvalue torsional buckling analysis is the same as that used for the lateral buckling analysis. Here, a concentrated axial load of 10 N is applied to one end of the beam. beambuckle_b31os_tors_gsec.inp shows the input used for this analysis.