11.3. Buckling of Pinned-End Columns

Consideration is now given to a relatively slender straight bar subject to axial compression. This member, a column, is similar to the element shown in Fig. 11.1a, in that it too can experience unstable behavior. In the case of a column, the restoring force or moment is provided by elastic forces established within the member rather than by springs external to it.

Refer to Fig. 11.2a, in which is shown a straight, homogeneous, pin-ended column. For such a column, the end moments are zero. This is regarded as the fundamental or most basic case. It is reasonable to suppose that the column can be held in a deformed configuration by a load P while remaining in the elastic range (Fig. 11.2b). Note that the requisite axial motion is permitted by the movable end support. In Fig. 11.2c, the postulated deflection is shown, having been caused by collinear forces P acting at the centroid of the cross section. The bending moment at any section, M = –Pv, when inserted into the equation for the elastic behavior of a beam, EIv″ = M, yields