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Lecture

Concepts of stable and unstable elastic equilibrium

Read LectureTo describe the elementary elastic instability modes and to derive the principal critical loads for columns, beams, and plates.

Worked example 6.1: Energy methods I

Worked example 6.2: Energy methods II

This lecture explains how critical buckling loads are determined by solution of the differential equilibrium equations for the structure. The critical loads, assuming simple loading and boundary conditions, are then calculated for the principal cases, namely:

- flexural buckling of columns,
- lateral buckling of beams, and
- buckling of plates.

Instability can occur in all systems or members where compression stresses exist. The simplest type of buckling is that of an initially straight strut compressed by equal and opposite axial forces (see Figure 1).

Other buckling modes also of great practical interest in steel constructions are:

- lateral buckling of beams (see Figure 2)
- plate buckling (see Figure 3)
- shell buckling (see Figure 4)

The fundamental differences in the behaviour of columns, plates, and shells are shown in Figure 5. For behaviour in the elastic range, the critical load and the maximum load carried by an actual (imperfect) column are in reasonable agreement. For the plate, if the post-critical strength is achieved with acceptably small lateral deflections, a greater load than the critical load might be acceptable. For thin-walled cylinders, however, the maximum load in the real (imperfect) situation is much less than the theoretical critical load.

For compressed struts, the flexural buckling illustrated in Figure 1 is not the only possible buckling mode. In some cases, for example, a torsional buckling (see Figure 6) or a combination of torsional and flexural buckling can be seen; if a member is thin-walled, one can also observe a plate buckling of the elements of the cross-section (see Figure 7) which can interact with the overall buckling of the member.

Determination of the critical load using bifurcation theory takes advantage of the fact that the critical situation is associated with a neutral equilibrium condition; equilibrium in a slightly deflected shape can, therefore, be established leading to differential equations which are simple to manage - at least for certain classes of structures. The critical load gives information on the level of stability of a system or member. It is also used as a basic value (bound) for the calculation of the ultimate load for structures in danger of instability, as shown in later lectures. In this lecture, the critical loads are calculated by solving the differential equilibrium equations describing the phenomenon. These solutions are available only for the simplest cases of loading and boundary conditions.

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