In certain types of problems, and especially those involving compressive stresses, we find that a structural member may develop relatively large distortions
under certain critical loading conditions. Such structural members are said to buckle, or become unstable, at these critical loads. As an example of elastic buckling, we consider firstly the buckling of a slender column under an axial compressive load.
FLEXURAL BUCKLING OF A PIN-ENDED STRUT:
A perfectly straight bar of uniform cross-section has two axes of symmetry Cx and Cy in the crosssection on the right of Fig1. We suppose the bar to be a flat sirip of material, Cx being the weakest axis of the cross-section. End thrusts P are applied along the centroidai axis Cz of the bar, and EI its uniform flexural stiffness for bending about Cx.
Fig1 :Flexural buckling of a pin-ended strut under axial thrust.
Now Cx is the weakest axis of bending of the bar, and if bowing of the compressed bar occurs we should expect bending to take place in the yz-plane. Consider the possibility that at some value of P, the end thrust, the strut can buckle laterally in the yz-plane. There can be no lateral deflections at the ends of the strut; suppose v is the displacement of the centre line of the bar parallel to Cy at any point. There can be no forces at the hinges parallel to Cy, as these would imply bending moments at the ends of the bar. The only two external forces are the end thrusts P, which are assumed to maintain their original line of action after the onset ofbuckling. The bending moment at any section of the bar is then
which is a sagging moment in relation to the axes Cz and Cy. But the moment-curvature relationship for the beam at any section is
provided the deflection v is small. Thus
Solving above dfferential equation